From c26f3b7e84f0bb75c4bf30bf32c8ece9760ee36b Mon Sep 17 00:00:00 2001 From: ID Bot Date: Wed, 27 Mar 2024 19:48:43 +0000 Subject: [PATCH] Script updating gh-pages from b5c55ce. [ci skip] --- .../draft-irtf-cfrg-cpace.html | 5197 +++++++++++++++++ .../draft-irtf-cfrg-cpace.txt | 3275 +++++++++++ draft-irtf-cfrg-cpace-11/index.html | 45 + index.html | 8 + 4 files changed, 8525 insertions(+) create mode 100644 draft-irtf-cfrg-cpace-11/draft-irtf-cfrg-cpace.html create mode 100644 draft-irtf-cfrg-cpace-11/draft-irtf-cfrg-cpace.txt create mode 100644 draft-irtf-cfrg-cpace-11/index.html diff --git a/draft-irtf-cfrg-cpace-11/draft-irtf-cfrg-cpace.html b/draft-irtf-cfrg-cpace-11/draft-irtf-cfrg-cpace.html new file mode 100644 index 0000000..7fd22f2 --- /dev/null +++ b/draft-irtf-cfrg-cpace-11/draft-irtf-cfrg-cpace.html @@ -0,0 +1,5197 @@ + + + + + + +CPace, a balanced composable PAKE + + + + + + + + + + + + + + + + + + + + + + + + + +
Internet-DraftCPaceMarch 2024
Abdalla, et al.Expires 28 September 2024[Page]
+
+
+
+
Workgroup:
+
Network Working Group
+
Internet-Draft:
+
draft-irtf-cfrg-cpace-latest
+
Published:
+
+ +
+
Intended Status:
+
Informational
+
Expires:
+
+
Authors:
+
+
+
M. Abdalla
+
DFINITY - Zurich
+
+
+
B. Haase
+
Endress + Hauser Liquid Analysis - Gerlingen
+
+
+
J. Hesse
+
IBM Research Europe - Zurich
+
+
+
+
+

CPace, a balanced composable PAKE

+
+

Abstract

+

This document describes CPace which is a protocol that allows two +parties that share a low-entropy secret (password) to derive a strong shared key without +disclosing the secret to offline dictionary attacks. +The CPace protocol was tailored for constrained devices and +can be used on groups of prime- and non-prime order.

+
+
+

+Discussion Venues +

+

This note is to be removed before publishing as an RFC.

+

Discussion of this document takes place on the + Crypto Forum Research Group mailing list (cfrg@ietf.org), + which is archived at https://mailarchive.ietf.org/arch/search/?email_list=cfrg.

+

Source for this draft and an issue tracker can be found at + https://github.com/cfrg/draft-irtf-cfrg-cpace.

+
+
+
+

+Status of This Memo +

+

+ This Internet-Draft is submitted in full conformance with the + provisions of BCP 78 and BCP 79.

+

+ Internet-Drafts are working documents of the Internet Engineering Task + Force (IETF). Note that other groups may also distribute working + documents as Internet-Drafts. The list of current Internet-Drafts is + at https://datatracker.ietf.org/drafts/current/.

+

+ Internet-Drafts are draft documents valid for a maximum of six months + and may be updated, replaced, or obsoleted by other documents at any + time. It is inappropriate to use Internet-Drafts as reference + material or to cite them other than as "work in progress."

+

+ This Internet-Draft will expire on 28 September 2024.

+
+
+ +
+
+ ▲

+Table of Contents +

+ +
+
+
+
+

+1. Introduction +

+

This document describes CPace which is a balanced Password-Authenticated-Key-Establishment (PAKE) +protocol for two parties where both parties derive a cryptographic key +of high entropy from a shared secret of low-entropy. +CPace protects the passwords against offline dictionary attacks by requiring +adversaries to actively interact with a protocol party and by allowing +for at most one single password guess per active interaction.

+

The CPace design was tailored considering the following main objectives:

+
    +
  • +

    Efficiency: Deployment of CPace is feasible on resource-constrained devices.

    +
    • +
  • +

    Versatility: CPace supports different application scenarios via versatile input formats, and by supporting applications with and without clear initiator and responder roles.

    +
    • +
  • +

    Implementation error resistance: CPace aims at avoiding common implementation pitfalls already by-design, such as avoiding incentives for insecure execution-time speed optimizations. For smooth integration into different cryptographic library ecosystems, this document provides a variety of cipher suites.

    +
    • +
  • +

    Post-quantum annoyance: CPace comes with mitigations with respect to adversaries that become capable of breaking the discrete logarithm problem on elliptic curves.

    +
    • +
+
+
+

+1.1. Outline of this document +

+
    +
  • +

    Section 3 describes the expected properties of an application using CPace, and discusses in particular which application-level aspects are relevant for CPace's security.

    +
    • +
  • +

    Section 4 gives an overview of the recommended +cipher suites for CPace which were optimized for different types of cryptographic +library ecosystems.

    +
    • +
  • +

    Section 5 introduces the notation used throughout this document.

    +
    • +
  • +

    Section 6 specifies the CPace protocol.

    +
    • +
  • +

    The final section provides explicit reference implementations and test vectors of all of the +functions defined for CPace in the appendix.

    +
    • +
+

As this document is primarily written for implementers and application designers, we would like to refer the theory-inclined reader to the scientific paper [AHH21] which covers the detailed security analysis of the different CPace instantiations as defined in this document via the cipher suites.

+
+
+
+
+
+
+

+2. Requirements Notation +

+

The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL +NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", +"MAY", and "OPTIONAL" in this document are to be interpreted as +described in BCP 14 [RFC2119] [RFC8174] when, and only when, they +appear in all capitals, as shown here.

+
+
+
+
+

+3. High-level application perspective +

+

CPace enables balanced password-authenticated key establishment. CPace requires a shared secret octet string, the password-related string (PRS), is available for both parties A and B. PRS can be a low-entropy secret itself, for instance a clear-text password encoded according to [RFC8265], or any string derived from a common secret, for instance by use of a password-based key derivation function.

+

Applications with clients and servers where the server side is storing account and password information in its persistent memory are recommended to use augmented PAKE protocols such as OPAQUE [I-D.irtf-cfrg-opaque].

+

In the course of the CPace protocol, A sends one message MSGa to B and B sends one message MSGb to A. CPace does not mandate any ordering of these two messages. We use the term "initiator-responder" for CPace where A always speaks first, and the term "symmetric" setting where anyone can speak first.

+

CPace's output is an intermediate session key (ISK), but any party might abort in case of an invalid received message. A and B will produce the same ISK value only if both sides did initiate the protocol using the same protocol inputs, specifically the same PRS string and the same value for the optional input parameters CI, ADa, ADb and sid that will be specified in the upcoming sections.

+

The naming of ISK key as "intermediate" session key highlights the fact that it is RECOMMENDED that applications process ISK by use of a suitable strong key derivation function KDF (such as defined in [RFC5869]) before using the key in a higher-level protocol.

+
+
+

+3.1. Optional CPace inputs +

+

For accomodating different application settings, CPace offers the following OPTIONAL inputs, i.e. inputs which MAY also be the empty string:

+
    +
  • +

    Channel identifier (CI). CI can be used to bind a session key exchanged with CPace to a specific networking channel which interconnects the protocol parties. Both parties are required to have the same view of CI. CI will not be publicly sent on the wire and may also include confidential information.

    +
    • +
  • +

    Associated data fields (ADa and ADb). +These fields can be used to authenticate public associated data alongside the CPace protocol. The values ADa (and ADb, respectively) are guaranteed to be authenticated in case both parties agree on a key.

    +

    +ADa and ADb can for instance include party identities or protocol +version information of an application protocol (e.g. to avoid downgrade attacks).

    +

    +If party identities are not encoded as part of CI, party identities SHOULD be included in ADa and ADb + (see Section 9.1). +In a setting with clear initiator and responder roles, identity information in ADa +sent by the initiator can be used by the responder for choosing the right PRS string (respectively password) for this identity.

    +
    • +
  • +

    Session identifier (sid). +CPace comes with a security analysis [AHH21] in the framework of universal composability. +This framework allows for modular analysis of a larger application protocol which uses CPace as a building block. For such analysis +the CPace protocol is bound to a specific session of the larger protocol by use of a sid string that is globally unique. As a result, when used with a unique sid, CPace instances remain secure when running concurrently with other CPace instances, and even arbitrary other protocols.

    +

    +For this reason, it is RECOMMENDED that applications establish a unique session identifier sid +prior to running the CPace protocol. This can be implemented by concatenating random bytes produced by A +with random bytes produced by B. If such preceding round is not an option but +parties are assigned clear initiator-responder roles, it is RECOMMENDED to let the initiator A choose a fresh +random sid and send it to B together with the first message. +If a sid string is used it SHOULD HAVE a length of at least 8 bytes.

    +
    • +
+
+
+
+
+

+3.2. Responsibilities of the application layer +

+

The following tasks are out of the scope of this document and left to the application layer

+
    +
  • +

    Setup phase:

    +
      +
    • +

      The application layer is responsible for the handshake that makes parties agree on a common CPace cipher suite.

      +
      • +
    • +

      The application layer needs to specify how to encode the CPace byte strings Ya / Yb and ADa / ADb defined in section + Section 6 +for transfer over the network. +For CPace it is RECOMMENDED to encode network messages by using MSGa = lv_cat(Ya,ADa) and MSGb = lv_cat(Yb,ADb) +using the length-value concatenation function lv_cat +speficied in Section 5.3. +This document provides test vectors for lv_cat-encoded messages. +Alternative network encodings, e.g., the encoding method +used for the client hello and server hello messages of the TLS protocol, MAY be used +when considering the guidance given in Section 9.

      +
      • +
    +
    • +
  • +

    This document does not specify which encodings applications use for the mandatory PRS input and the optional inputs +CI, sid, ADa and ADb. If PRS is a clear-text password or an octet string derived from a clear-text password, +e.g. by use of a key-derivation function, the clear-text password SHOULD BE encoded according to [RFC8265].

    +
    • +
  • +

    The application needs to settle whether CPace is used in the initiator-responder or the symmetric setting, as in the symmetric +setting transcripts must be generated using ordered string concatenation. In this document we will provide test vectors +for both, initiator-responder and symmetric settings.

    +
    • +
+
+
+
+
+
+
+

+4. CPace cipher suites +

+

In the setup phase of CPace, both communication partners need to agree on a common cipher suite. +Cipher suites consist of a combination of a hash function H and an elliptic curve environment G.

+

For naming cipher suites we use the convention "CPACE-G-H". We RECOMMEND the following cipher suites:

+
    +
  • +

    CPACE-X25519-SHA512. This suite uses the group environment G_X25519 defined in Section 7.2 and SHA-512 as hash function. +This cipher suite comes with the smallest messages on the wire and a low computational cost.

    +
    • +
  • +

    CPACE-P256_XMD:SHA-256_SSWU_NU_-SHA256. +This suite instantiates the group environment G as specified in Section 7.4 using the encode_to_curve function P256_XMD:SHA-256_SSWU_NU_ +from [RFC9380] on curve NIST-P256, and hash function SHA-256.

    +
    • +
+

The following RECOMMENDED cipher suites provide higher security margins.

+
    +
  • +

    CPACE-X448-SHAKE256. This suite uses the group environment G_X448 defined in Section 7.2 and SHAKE-256 as hash function.

    +
    • +
  • +

    CPACE-P384_XMD:SHA-384_SSWU_NU_-SHA384. +This suite instantiates G as specified in Section 7.4 using the encode_to_curve function P384_XMD:SHA-384_SSWU_NU_ +from [RFC9380] on curve NIST-P384 with H = SHA-384.

    +
    • +
  • +

    CPACE-P521_XMD:SHA-512_SSWU_NU_-SHA512. +This suite instantiates G as specified in Section 7.4 using the encode_to_curve function P521_XMD:SHA-512_SSWU_NU_ +from [RFC9380] on curve NIST-P521 with H = SHA-512.

    +
    • +
+

CPace can also securely be implemented using the cipher suites CPACE-RISTR255-SHA512 and CPACE-DECAF448-SHAKE256 defined in +Section 7.3. Section 9 gives guidance on how to implement CPace on further elliptic curves.

+
+
+
+
+

+5. Definitions and notation +

+
+
+

+5.1. Hash function H +

+

Common choices for H are SHA-512 [RFC6234] or SHAKE-256 [FIPS202]. (I.e. the hash function +outputs octet strings, and not group elements.) +For considering both variable-output-length hashes and fixed-output-length hashes, we use the following convention. +In case that the hash function is specified for a fixed-size output, we define H.hash(m,l) such +that it returns the first l octets of the output.

+

We use the following notation for referring to the specific properties of a hash function H:

+
    +
  • +

    H.hash(m,l) is a function that operates on an input octet string m and returns a hashing result of l octets.

    +
    • +
  • +

    H.b_in_bytes denotes the minimum output size in bytes for collision resistance for the +security level target of the hash function. E.g. H.b_in_bytes = 64 for SHA-512 and SHAKE-256 and H.b_in_bytes = 32 for +SHA-256 and SHAKE-128. We use the notation H.hash(m) = H.hash(m, H.b_in_bytes) and let the hash operation +output the default length if no explicit length parameter is given.

    +
    • +
  • +

    H.bmax_in_bytes denotes the maximum output size in octets supported by the hash function. In case of fixed-size +hashes such as SHA-256, this is the same as H.b_in_bytes, while there is no such limit for hash functions such as SHAKE-256.

    +
    • +
  • +

    H.s_in_bytes denotes the input block size used by H. This number denotes the maximum number of bytes that can be processed +in a single block before applying the compression function or permutation becomes necessary. (See also [RFC2104] for the corresponding block size concepts). +For instance, for SHA-512 the input block size s_in_bytes is 128 as the compression function can process up to 128 bytes, +while for SHAKE-256 the input block size amounts to 136 bytes before the permutation of the sponge state needs to be applied.

    +
    • +
+
+
+
+
+

+5.2. Group environment G +

+

The group environment G specifies an elliptic curve group (also denoted G for convenience) and associated constants +and functions as detailed below. In this document we use additive notation for the group operation.

+
    +
  • +

    G.calculate_generator(H,PRS,CI,sid) denotes a function that outputs a representation of a generator (referred to as "generator" from now on) of the group +which is derived from input octet strings PRS, CI, and sid and with the help of the hash function H.

    +
    • +
  • +

    G.sample_scalar() is a function returning a representation of an integer (referred to as "scalar" from now on) appropriate as a +private Diffie-Hellman key for the group.

    +
    • +
  • +

    G.scalar_mult(y,g) is a function operating on a scalar +y and a group element g. It returns an octet string representation of the group element Y = g*y.

    +
    • +
  • +

    G.I denotes a unique octet string representation of the neutral element of the group. G.I is used for detecting and signaling certain error conditions.

    +
    • +
  • +

    G.scalar_mult_vfy(y,g) is a function operating on +a scalar y and a group element g. It returns an octet string +representation of the group element g*y. Additionally, scalar_mult_vfy specifies validity conditions for y,g and g*y and outputs G.I in case they are not met.

    +
    • +
  • +

    G.DSI denotes a domain-separation identifier octet string which SHALL be uniquely identifying the group environment G.

    +
    • +
+
+
+
+
+

+5.3. Notation for string operations +

+
    +
  • +

    bytes1 || bytes2 and denotes concatenation of octet strings.

    +
    • +
  • +

    len(S) denotes the number of octets in an octet string S.

    +
    • +
  • +

    nil denotes an empty octet string, i.e., len(nil) = 0.

    +
    • +
  • +

    This document uses quotation marks "" both for general language (e.g. for citation of notation used in other documents) and +as syntax for specifying octet strings as in b"CPace25519".

    +

    +We use a preceeding lower-case letter b"" in front of the quotation marks if a character sequence is representing an octet string sequence. +I.e. we use the notation for byte string representations with single-byte ASCII character encodings from the python programming language.

    +
    • +
  • +

    prepend_len(octet_string) denotes the octet sequence that is obtained from prepending +the length of the octet string to the string itself. The length shall be prepended by using an LEB128 encoding of the length. +This will result in a single-byte encoding for values below 128. (Test vectors and reference implementations +for prepend_len and the LEB128 encodings are given in the appendix.)

    +
    • +
  • +

    lv_cat(a0,a1, ...) is the "length-value" encoding function which returns the concatenation of the input strings with an encoding of +their respective length prepended. E.g. lv_cat(a0,a1) returns +prepend_len(a0) || prepend_len(a1). The detailed specification of lv_cat and a reference implementations are given in the appendix.

    +
    • +
  • +

    network_encode(Y,AD) denotes the function specified by the application layer that outputs an octet string encoding +of the input octet strings Y and AD +for transfer on the network. The implementation of MSG = network_encode(Y,AD) SHALL allow the receiver party to parse MSG for the +individual subcomponents Y and AD. +For CPace we RECOMMEND to implement network_encode(Y,AD) as network_encode(Y,AD) = lv_cat(Y,AD).

    +

    +Other encodings, such as the network encoding used for the client-hello and server-hello messages in TLS MAY also be used when +following the guidance given in the security consideration section.

    +
    • +
  • +

    sample_random_bytes(n) denotes a function that returns n octets, each of which is to be independently sampled from an uniform distribution between 0 and 255.

    +
    • +
  • +

    zero_bytes(n) denotes a function that returns n octets with value 0.

    +
    • +
  • +

    o_cat(bytes1,bytes2) denotes a function for ordered concatenation of octet strings. It places the lexiographically larger octet +string first and prepends the two bytes from the octet string b"oc" to the result. (Explicit reference code for this function is given in the appendix.)

    +
    • +
  • +

    transcript(MSGa,MSGb) denotes function outputing a string for the protocol transcript with messages MSGa and MSGb. +In applications where CPace is used without clear initiator and responder roles, i.e. where the ordering of messages is +not enforced by the protocol flow, transcript(MSGa,MSGb) = o_cat(MSGa,MSGb) SHALL be used. +In the initiator-responder setting transcript(MSGa,MSGb) SHALL BE implemented such that the later message is appended to the +earlier message, i.e., transcript(MSGa,MSGb) = MSGa||MSGb if MSGa is sent first.

    +
    • +
+
+
+
+
+

+5.4. Notation for group operations +

+

We use additive notation for the group, i.e., X*2 denotes the element that is obtained by computing X+X, for group element X and group operation +.

+
+
+
+
+
+
+

+6. The CPace protocol +

+

CPace is a one round protocol between two parties, A and B. At invocation, A and B are provisioned with PRS,G,H and OPTIONAL CI,sid,ADa (for A) and CI,sid,ADb (for B). +A sends a message MSGa to B. MSGa contains the public share Ya +and OPTIONAL associated data ADa (i.e. an ADa field that MAY have a length of 0 bytes). +Likewise, B sends a message MSGb to A. MSGb contains the public share Yb +and OPTIONAL associated data ADb (i.e. an ADb field that MAY have a length of 0 bytes). +Both A and B use the received messages for deriving a shared intermediate session key, ISK.

+
+
+

+6.1. Protocol flow +

+

Optional parameters and messages are denoted with [].

+
+
+                        public: G, H
+
+  A: PRS,[ADa],[CI],[sid]         B: PRS,[ADb],[CI],[sid]
+    ---------------------------------------
+ compute Ya    |     Ya,[ADa]     |  compute Yb
+               |----------------->|
+               |     Yb,[ADb]     |
+ verify inputs |<-----------------|  verify inputs
+ derive ISK    |                  |  derive ISK
+    ---------------------------------------
+ output ISK                          output ISK
+
+
+
+
+
+
+
+

+6.2. CPace protocol instructions +

+

A computes a generator g = G.calculate_generator(H,PRS,CI,sid), scalar ya = G.sample_scalar() and group element Ya = G.scalar_mult (ya,g). A then transmits MSGa = network_encode(Ya, ADa) with +optional associated data ADa to B.

+

B computes a generator g = G.calculate_generator(H,PRS,CI,sid), scalar yb = G.sample_scalar() and group element Yb = G.scalar_mult(yb,g). B sends MSGb = network_encode(Yb, ADb) with optional associated data ADb to A.

+

Upon reception of MSGa, B checks that MSGa was properly generated in conformity with the chosen encoding of network messages (notably correct length fields). +If this parsing fails, then B MUST abort. (Testvectors of examples for invalid messages when using lv_cat() as network_encode function for +CPace are given in the appendix.) +B then computes K = G.scalar_mult_vfy(yb,Ya). B MUST abort if K=G.I. +Otherwise B calculates +ISK = H.hash(lv_cat(G.DSI || b"_ISK", sid, K)||transcript(MSGa, MSGb)). B returns ISK and terminates.

+

Likewise upon reception of MSGb, A parses MSGb for Yb and ADb and checks for a valid encoding. +If this parsing fails, then A MUST abort. A then computes K = G.scalar_mult_vfy(ya,Yb). A MUST abort if K=G.I. +Otherwise A calculates +ISK = H.hash(lv_cat(G.DSI || b"_ISK", sid, K) || transcript(MSGa, MSGb)). A returns ISK and terminates.

+

The session key ISK returned by A and B is identical if and only if the supplied input parameters PRS, CI and sid match on both sides and transcript view (containing of MSGa and MSGb) of both parties match.

+

(Note that in case of a symmetric protocol execution without clear initiator/responder roles, transcript(MSGa, MSGb) needs to be implemented using ordered concatenation for generating a matching view by both parties.)

+
+
+
+
+ +
+
+

+8. Implementation verification +

+

Any CPace implementation MUST be tested against invalid or weak point attacks. +Implementation MUST be verified to abort upon conditions where G.scalar_mult_vfy functions outputs G.I. +For testing an implementation it is RECOMMENDED to include weak or invalid point encodings within MSGa and MSGb and introduce this +in a protocol run. It SHALL be verified that the abort condition is properly handled.

+

Moreover regarding the network format any implementation MUST be tested with respect to invalid encodings of MSGa and MSGb. +E.g. when lv_cat is used as network format for encoding MSGa and MSGb, +the sum of the prepended lengths of the fields must be verified to match the actual length of the message. +Tests SHALL verify that a party aborts in case that incorrectly encoded messages are received.

+

Corresponding test vectors are given in the appendix for all recommended cipher suites.

+
+
+
+
+

+9. Security Considerations +

+

A security proof of CPace is found in [AHH21]. This proof covers all recommended cipher suites included in this document. +In the following sections we describe how to protect CPace against several attack families, such as relay-, length extension- or side channel attacks. We also describe aspects to consider when deviating from recommended cipher suites.

+
+
+

+9.1. Party identifiers and relay attacks +

+

If unique strings identifying the protocol partners are included either as part of the channel identifier CI, the session id sid or the associated data fields ADa, ADb, the ISK will provide implicit authentication also regarding the party identities. Incorporating party identifier strings +is important for fending off relay attacks. +Such attacks become relevant in a setting where several parties, say, A, B and C, share the same password PRS. An adversary might relay messages from a honest user A, who aims at interacting with user B, to a party C instead. If no party identifier strings are used, and B and C use the same PRS value, A might be establishing a common ISK key with C while assuming to interact with party B. +Including and checking party identifiers can fend off such relay attacks.

+
+
+
+
+

+9.2. Network message encoding and hashing protocol transcripts +

+

It is RECOMMENDED to encode the (Ya,ADa) and (Yb,ADb) fields on the network by using network_encode(Y,AD) = lv_cat(Y,AD). I.e. we RECOMMEND +to prepend an encoding of the length of the subfields. Prepending the length of +all variable-size input strings results in a so-called prefix-free encoding of transcript strings, using terminology introduced in [CDMP05]. This property allows for disregarding length-extension imperfections that come with the commonly used Merkle-Damgard hash function constructions such as SHA256 and SHA512.

+

Other alternative network encoding formats which prepend an encoding of the length of variable-size data fields in the protocol +messages are equally suitable. +This includes, e.g., the type-length-value format specified in the DER encoding standard (X.690) or the protocol message encoding used in the TLS protocol family for the TLS client-hello or server-hello messages.

+

In case that an application uses another form of network message encoding which is not prefix-free, +the guidance given in [CDMP05] SHOULD BE considered (e.g. by replacing hash functions with the HMAC constructions from[RFC2104]).

+
+
+
+
+

+9.3. Key derivation +

+

Although already K is a shared value, it MUST NOT itself be used as an application key. Instead, ISK MUST BE used. Leakage of K to an adversary can lead to offline dictionary attacks.

+

As noted already in Section 6 it is RECOMMENDED to process ISK +by use of a suitable strong key derivation function KDF (such as defined in [RFC5869]) first, +before using the key in a higher-level protocol.

+
+
+
+
+

+9.4. Key confirmation +

+

In many applications it is advisable to add an explicit key confirmation round after the CPace protocol flow. However, as some applications +might only require implicit authentication and as explicit authentication messages are already a built-in feature in many higher-level protocols (e.g. TLS 1.3) the CPace protocol described here does not mandate +use of a key confirmation on the level of the CPace sub-protocol.

+

Already without explicit key confirmation, CPace enjoys weak forward security under the sCDH and sSDH assumptions [AHH21]. +With added explicit confirmation, CPace enjoys perfect forward security also under the strong sCDH and sSDH assumptions [AHH21].

+

Note that in [ABKLX21] it was shown that an idealized variant of CPace +also enjoys perfect forward security without explicit key confirmation. However this proof does not explicitly cover +the recommended cipher suites +in this document and requires the stronger assumption of an algebraic adversary model. For this reason, we recommend adding +explicit key confirmation if perfect forward security is required.

+

When implementing explicit key confirmation, it is recommended to use an appropriate message-authentication code (MAC) +such as HMAC [RFC2104] or +CMAC [RFC4493] using a key mac_key derived from ISK.

+

One suitable option that works also in the parallel setting without message ordering is to proceed as follows.

+
    +
  • +

    First calculate mac_key as mac_key = H.hash(b"CPaceMac" || ISK).

    +
    • +
  • +

    Then let each party send an authenticator tag Ta, Tb that is calculated over the protocol message that it has sent previously. I.e. +let party A calculate its transmitted authentication code Ta as Ta = MAC(mac_key, MSGa) and let party B calculate its transmitted +authentication code Tb as Tb = MAC(mac_key, MSGb).

    +
    • +
  • +

    Let the receiving party check the remote authentication tag for the correct value and abort in case that it's incorrect.

    +
    • +
+
+
+
+
+

+9.5. Sampling of scalars +

+

For curves over fields F_q where q is a prime close to a power of two, we recommend sampling scalars as a uniform bit string of length field_size_bits. We do so in order to reduce both, complexity of the implementation and the attack surface +with respect to side-channels for embedded systems in hostile environments. +The effect of non-uniform sampling on security was demonstrated to be begnin in [AHH21] for the case of Curve25519 and Curve448. +This analysis however does not transfer to most curves in Short-Weierstrass form.

+

As a result, we recommend rejection sampling if G is as in Section 7.4. Alternatively an algorithm designed allong the lines of the hash_to_field() function from [RFC9380] can also be +used. There oversampling to an integer significantly larger than the curve order is followed by a modular reduction to the group order.

+
+
+
+
+

+9.6. Preconditions for using the simplified CPace specification from Section 7.2 +

+

The security of the algorithms used for the recommended cipher suites for the Montgomery curves Curve25519 and Curve448 in Section 7.2 rely on the following properties [AHH21]:

+
    +
  • +

    The curve has order (p * c) with p prime and c a small cofactor. Also the curve's quadratic twist must be of order (p' * c') with p' prime and c' a cofactor.

    +
    • +
  • +

    The cofactor c of the curve MUST BE EQUAL to or an integer multiple of the cofactor c' of the curve's quadratic twist. Also, importantly, the +implementation of the scalar_mult and scalar_mult_vfy +functions must ensure that all scalars actually used for the group operation are integer multiples of +c (e.g. such as asserted by the specification of the decodeScalar functions in [RFC7748]).

    +
    • +
  • +

    Both field order q and group order p MUST BE close to a power of two along the lines of [AHH21], Appendix E. Otherwise the simplified scalar sampling specified in Section 7.2 +needs to be changed.

    +
    • +
  • +

    The representation of the neutral element G.I MUST BE the same for both, the curve and its twist.

    +
    • +
  • +

    The implementation of G.scalar_mult_vfy(y,X) MUST map all c low-order points on the curve and all c' low-order points on the twist to G.I.

    +
    • +
+

Algorithms for curves other than the ones recommended here can be based on the principles from Section 7.2 given that the above properties hold.

+
+
+
+
+

+9.7. Nonce values +

+

Secret scalars ya and yb MUST NOT be reused. Values for sid SHOULD NOT be reused since the composability +guarantees established by the simulation-based proof rely on the uniqueness of session ids [AHH21].

+

If CPace is used in a concurrent system, it is RECOMMENDED that a unique sid is generated by the higher-level protocol and passed to CPace. One suitable option is that sid is generated by concatenating ephemeral random strings contributed by both parties.

+
+
+
+
+

+9.8. Side channel attacks +

+

All state-of-the art methods for realizing constant-time execution SHOULD be used. +Special care is RECOMMENDED specifically for elliptic curves in Short-Weierstrass form +as important standard documents including [IEEE1363] describe curve operations with +non-constant-time algorithms.

+

In case that side channel attacks are to be considered practical for a given application, it is RECOMMENDED to pay special +attention on computing the secret generator G.calculate_generator(PRS,CI,sid). +The most critical substep to consider might be the processing of the first block of the hash that includes +the PRS string. +The zero-padding introduced when hashing the sensitive PRS string can be expected to make +the task for a side-channel attack somewhat more complex. Still this feature alone is not sufficient for ruling out power analysis attacks.

+

Even though the calculate_generator operation might be considered to form the primary target for side-channel attacks as information on long-term secrets might be exposed, +also the subsequent operations on ephemeral values, such as scalar +sampling and scalar multiplication should be protected from side-channels.

+
+
+
+
+

+9.9. Quantum computers +

+

CPace is proven secure under the hardness of the strong computational Simultaneous Diffie-Hellmann (sSDH) and strong computational Diffie-Hellmann (sCDH) +assumptions in the group G (as defined in [AHH21]). +These assumptions are not expected to hold any longer when large-scale quantum computers (LSQC) are available. +Still, even in case that LSQC emerge, it is reasonable to assume that discrete-logarithm computations will remain costly. CPace with ephemeral session id values +sid forces the adversary to solve one computational Diffie-Hellman problem per password guess [ES21]. +In this sense, using the wording suggested by Steve Thomas on the CFRG mailing list, CPace is "quantum-annoying".

+
+
+
+
+
+
+

+10. IANA Considerations +

+

No IANA action is required.

+
+
+
+
+

+11. Acknowledgements +

+

We would like to thank the participants on the CFRG list for comments and advice. Any comment and advice is appreciated.

+
+
+
+

+12. References +

+
+
+

+12.1. Normative References +

+
+
[I-D.draft-irtf-cfrg-ristretto255-decaf448]
+
+de Valence, H., Grigg, J., Hamburg, M., Lovecruft, I., Tankersley, G., and F. Valsorda, "The ristretto255 and decaf448 Groups", Work in Progress, Internet-Draft, draft-irtf-cfrg-ristretto255-decaf448-08, , <https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-ristretto255-decaf448-08>.
+
+
[I-D.irtf-cfrg-opaque]
+
+Bourdrez, D., Krawczyk, H., Lewi, K., and C. A. Wood, "The OPAQUE Augmented PAKE Protocol", Work in Progress, Internet-Draft, draft-irtf-cfrg-opaque-14, , <https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-opaque-14>.
+
+
[IEEE1363]
+
+"Standard Specifications for Public Key Cryptography, IEEE 1363", .
+
+
[RFC2119]
+
+Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", BCP 14, RFC 2119, DOI 10.17487/RFC2119, , <https://www.rfc-editor.org/rfc/rfc2119>.
+
+
[RFC7748]
+
+Langley, A., Hamburg, M., and S. Turner, "Elliptic Curves for Security", RFC 7748, DOI 10.17487/RFC7748, , <https://www.rfc-editor.org/rfc/rfc7748>.
+
+
[RFC8174]
+
+Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC 2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174, , <https://www.rfc-editor.org/rfc/rfc8174>.
+
+
[SEC1]
+
+Standards for Efficient Cryptography Group (SECG), "SEC 1: Elliptic Curve Cryptography", , <http://www.secg.org/sec1-v2.pdf>.
+
+
+
+
+
+
+

+12.2. Informative References +

+
+
[ABKLX21]
+
+Abdalla, M., Barbosa, M., Katz, J., Loss, J., and J. Xu, "Algebraic Adversaries in the Universal Composability Framework.", n.d., <https://eprint.iacr.org/2021/1218>.
+
+
[AHH21]
+
+Abdalla, M., Haase, B., and J. Hesse, "Security analysis of CPace", n.d., <https://eprint.iacr.org/2021/114>.
+
+
[CDMP05]
+
+Coron, J.-S., Dodis, Y., Malinaud, C., and P. Puniya, "Merkle-Damgaard Revisited: How to Construct a Hash Function", In Advances in Cryptology - CRYPTO 2005, pages 430-448, DOI 10.1007/11535218_26, , <https://doi.org/10.1007/11535218_26>.
+
+
[ES21]
+
+Eaton, E. and D. Stebila, "The 'quantum annoying' property of password-authenticated key exchange protocols.", n.d., <https://eprint.iacr.org/2021/696>.
+
+
[FIPS202]
+
+National Institute of Standards and Technology (NIST), "SHA-3 Standard: Permutation-Based Hash and Extendable-Output Functions", , <https://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.202.pdf>.
+
+
[RFC2104]
+
+Krawczyk, H., Bellare, M., and R. Canetti, "HMAC: Keyed-Hashing for Message Authentication", RFC 2104, DOI 10.17487/RFC2104, , <https://www.rfc-editor.org/rfc/rfc2104>.
+
+
[RFC4493]
+
+Song, JH., Poovendran, R., Lee, J., and T. Iwata, "The AES-CMAC Algorithm", RFC 4493, DOI 10.17487/RFC4493, , <https://www.rfc-editor.org/rfc/rfc4493>.
+
+
[RFC5246]
+
+Dierks, T. and E. Rescorla, "The Transport Layer Security (TLS) Protocol Version 1.2", RFC 5246, DOI 10.17487/RFC5246, , <https://www.rfc-editor.org/rfc/rfc5246>.
+
+
[RFC5639]
+
+Lochter, M. and J. Merkle, "Elliptic Curve Cryptography (ECC) Brainpool Standard Curves and Curve Generation", RFC 5639, DOI 10.17487/RFC5639, , <https://www.rfc-editor.org/rfc/rfc5639>.
+
+
[RFC5869]
+
+Krawczyk, H. and P. Eronen, "HMAC-based Extract-and-Expand Key Derivation Function (HKDF)", RFC 5869, DOI 10.17487/RFC5869, , <https://www.rfc-editor.org/rfc/rfc5869>.
+
+
[RFC6234]
+
+Eastlake 3rd, D. and T. Hansen, "US Secure Hash Algorithms (SHA and SHA-based HMAC and HKDF)", RFC 6234, DOI 10.17487/RFC6234, , <https://www.rfc-editor.org/rfc/rfc6234>.
+
+
[RFC8265]
+
+Saint-Andre, P. and A. Melnikov, "Preparation, Enforcement, and Comparison of Internationalized Strings Representing Usernames and Passwords", RFC 8265, DOI 10.17487/RFC8265, , <https://www.rfc-editor.org/rfc/rfc8265>.
+
+
[RFC8446]
+
+Rescorla, E., "The Transport Layer Security (TLS) Protocol Version 1.3", RFC 8446, DOI 10.17487/RFC8446, , <https://www.rfc-editor.org/rfc/rfc8446>.
+
+
[RFC9380]
+
+Faz-Hernandez, A., Scott, S., Sullivan, N., Wahby, R. S., and C. A. Wood, "Hashing to Elliptic Curves", RFC 9380, DOI 10.17487/RFC9380, , <https://www.rfc-editor.org/rfc/rfc9380>.
+
+
[SEC2]
+
+Standards for Efficient Cryptography Group (SECG), "SEC 2: Recommended Elliptic Curve Domain Parameters", , <http://www.secg.org/sec2-v2.pdf>.
+
+
+
+
+
+
+
+

+Appendix A. CPace function definitions +

+
+
+

+A.1. Definition and test vectors for string utility functions +

+
+
+

+A.1.1. prepend_len function +

+
+
+def prepend_len(data):
+    "prepend LEB128 encoding of length"
+    length = len(data)
+    length_encoded = b""
+    while True:
+        if length < 128:
+            length_encoded += bytes([length])
+        else:
+            length_encoded += bytes([(length & 0x7f) + 0x80])
+        length = int(length >> 7)
+        if length == 0:
+            break;
+    return length_encoded + data
+
+
+
+
+
+
+

+A.1.2. prepend_len test vectors +

+
+
+  prepend_len(b""): (length: 1 bytes)
+    00
+  prepend_len(b"1234"): (length: 5 bytes)
+    0431323334
+  prepend_len(bytes(range(127))): (length: 128 bytes)
+    7f000102030405060708090a0b0c0d0e0f101112131415161718191a1b
+    1c1d1e1f202122232425262728292a2b2c2d2e2f303132333435363738
+    393a3b3c3d3e3f404142434445464748494a4b4c4d4e4f505152535455
+    565758595a5b5c5d5e5f606162636465666768696a6b6c6d6e6f707172
+    737475767778797a7b7c7d7e
+  prepend_len(bytes(range(128))): (length: 130 bytes)
+    8001000102030405060708090a0b0c0d0e0f101112131415161718191a
+    1b1c1d1e1f202122232425262728292a2b2c2d2e2f3031323334353637
+    38393a3b3c3d3e3f404142434445464748494a4b4c4d4e4f5051525354
+    55565758595a5b5c5d5e5f606162636465666768696a6b6c6d6e6f7071
+    72737475767778797a7b7c7d7e7f
+
+
+
+
+
+
+

+A.1.3. lv_cat function +

+
+
+  def lv_cat(*args):
+      result = b""
+      for arg in args:
+          result += prepend_len(arg)
+      return result
+
+
+
+
+
+
+

+A.1.4. Testvector for lv_cat() +

+
+
+  lv_cat(b"1234",b"5",b"",b"6789"): (length: 13 bytes)
+    04313233340135000436373839
+
+
+
+
+
+
+

+A.1.5. Examples for messages not obtained from a lv_cat-based encoding +

+

The following messages are examples which have invalid encoded length fields. I.e. they are examples +where parsing for the sum of the length of subfields as expected for a message generated using lv_cat(Y,AD) +does not give the correct length of the message. Parties MUST abort upon reception of such invalid messages as MSGa or MSGb.

+
+
+  Inv_MSG1 not encoded by lv_cat: (length: 3 bytes)
+    ffffff
+  Inv_MSG2 not encoded by lv_cat: (length: 3 bytes)
+    ffff03
+  Inv_MSG3 not encoded by lv_cat: (length: 4 bytes)
+    00ffff03
+  Inv_MSG4 not encoded by lv_cat: (length: 4 bytes)
+    00ffffff
+
+
+
+
+
+
+
+
+

+A.2. Definition of generator_string function. +

+
+
+def generator_string(DSI,PRS,CI,sid,s_in_bytes):
+    # Concat all input fields with prepended length information.
+    # Add zero padding in the first hash block after DSI and PRS.
+    len_zpad = max(0,s_in_bytes - 1 - len(prepend_len(PRS))
+                     - len(prepend_len(DSI)))
+    return lv_cat(DSI, PRS, zero_bytes(len_zpad),
+                           CI, sid)
+
+
+
+
+
+
+

+A.3. Definitions and test vector ordered concatenation +

+
+
+

+A.3.1. Definitions for lexiographical ordering +

+

For ordered concatenation lexiographical ordering of byte sequences is used:

+
+
+   def lexiographically_larger(bytes1,bytes2):
+      "Returns True if bytes1 > bytes2 using lexiographical ordering."
+      min_len = min (len(bytes1), len(bytes2))
+      for m in range(min_len):
+          if bytes1[m] > bytes2[m]:
+              return True;
+          elif bytes1[m] < bytes2[m]:
+              return False;
+      return len(bytes1) > len(bytes2)
+
+
+
+
+
+
+

+A.3.2. Definitions for ordered concatenation +

+

With the above definition of lexiographical ordering ordered concatenation is specified as follows.

+
+
+  def o_cat(bytes1,bytes2):
+      if lexiographically_larger(bytes1,bytes2):
+          return b"oc" + bytes1 + bytes2
+      else:
+          return b"oc" + bytes2 + bytes1
+
+
+
+
+
+
+

+A.3.3. Test vectors ordered concatenation +

+
+
+  string comparison for o_cat:
+    lexiographically_larger(b"\0", b"\0\0") == False
+    lexiographically_larger(b"\1", b"\0\0") == True
+    lexiographically_larger(b"\0\0", b"\0") == True
+    lexiographically_larger(b"\0\0", b"\1") == False
+    lexiographically_larger(b"\0\1", b"\1") == False
+    lexiographically_larger(b"ABCD", b"BCD") == False
+
+  o_cat(b"ABCD",b"BCD"): (length: 9 bytes)
+    6f6342434441424344
+  o_cat(b"BCD",b"ABCDE"): (length: 10 bytes)
+    6f634243444142434445
+
+
+
+
+
+
+
+
+

+A.4. Decoding and Encoding functions according to RFC7748 +

+
+
+   def decodeLittleEndian(b, bits):
+       return sum([b[i] << 8*i for i in range((bits+7)/8)])
+
+   def decodeUCoordinate(u, bits):
+       u_list = [ord(b) for b in u]
+       # Ignore any unused bits.
+       if bits % 8:
+           u_list[-1] &= (1<<(bits%8))-1
+       return decodeLittleEndian(u_list, bits)
+
+   def encodeUCoordinate(u, bits):
+       return ''.join([chr((u >> 8*i) & 0xff)
+                       for i in range((bits+7)/8)])
+
+
+
+
+
+
+

+A.5. Elligator 2 reference implementation +

+

The Elligator 2 map requires a non-square field element Z which shall be calculated +as follows.

+
+
+    def find_z_ell2(F):
+        # Find nonsquare for Elligator2
+        # Argument: F, a field object, e.g., F = GF(2^255 - 19)
+        ctr = F.gen()
+        while True:
+            for Z_cand in (F(ctr), F(-ctr)):
+                # Z must be a non-square in F.
+                if is_square(Z_cand):
+                    continue
+                return Z_cand
+            ctr += 1
+
+
+

The values of the non-square Z only depend on the curve. The algorithm above +results in a value of Z = 2 for Curve25519 and Z=-1 for Ed448.

+

The following code maps a field element r to an encoded field element which +is a valid u-coordinate of a Montgomery curve with curve parameter A.

+
+
+    def elligator2(r, q, A, field_size_bits):
+        # Inputs: field element r, field order q,
+        #         curve parameter A and field size in bits
+        Fq = GF(q); A = Fq(A); B = Fq(1);
+
+        # get non-square z as specified in the hash2curve draft.
+        z = Fq(find_z_ell2(Fq))
+        powerForLegendreSymbol = floor((q-1)/2)
+
+        v = - A / (1 + z * r^2)
+        epsilon = (v^3 + A * v^2 + B * v)^powerForLegendreSymbol
+        x = epsilon * v - (1 - epsilon) * A/2
+        return encodeUCoordinate(Integer(x), field_size_bits)
+
+
+
+
+
+
+
+
+

+Appendix B. Test vectors +

+
+
+

+B.1. Test vector for CPace using group X25519 and hash SHA-512 +

+
+
+

+B.1.1. Test vectors for calculate_generator with group X25519 +

+
+
+  Inputs
+    H   = SHA-512 with input block size 128 bytes.
+    PRS = b'Password' ; ZPAD length: 109 ; DSI = b'CPace255'
+    CI = b'\nAinitiator\nBresponder'
+    CI = 0a41696e69746961746f720a42726573706f6e646572
+    sid = 7e4b4791d6a8ef019b936c79fb7f2c57
+  Outputs
+    generator_string(G.DSI,PRS,CI,sid,H.s_in_bytes):
+    (length: 168 bytes)
+      0843506163653235350850617373776f72646d000000000000000000
+      00000000000000000000000000000000000000000000000000000000
+      00000000000000000000000000000000000000000000000000000000
+      00000000000000000000000000000000000000000000000000000000
+      00000000000000000000000000000000160a41696e69746961746f72
+      0a42726573706f6e646572107e4b4791d6a8ef019b936c79fb7f2c57
+    hash generator string: (length: 32 bytes)
+      10047198e8c4cacf0ab8a6d0ac337b8ae497209d042f7f3a50945863
+      94e821fc
+    decoded field element of 255 bits: (length: 32 bytes)
+      10047198e8c4cacf0ab8a6d0ac337b8ae497209d042f7f3a50945863
+      94e8217c
+    generator g: (length: 32 bytes)
+      4e6098733061c0e8486611a904fe5edb049804d26130a44131a6229e
+      55c5c321
+
+
+
+
+
+
+

+B.1.2. Test vector for MSGa +

+
+
+  Inputs
+    ADa = b'ADa'
+    ya (little endian): (length: 32 bytes)
+      21b4f4bd9e64ed355c3eb676a28ebedaf6d8f17bdc365995b3190971
+      53044080
+  Outputs
+    Ya: (length: 32 bytes)
+      f970e36f37cfcd9a39e37dd2d1fbc9156d6d2f9ae422f4722cbd9d32
+      e9b1e704
+    MSGa = lv_cat(Ya,ADa): (length: 37 bytes)
+      20f970e36f37cfcd9a39e37dd2d1fbc9156d6d2f9ae422f4722cbd9d
+      32e9b1e70403414461
+
+
+
+
+
+
+

+B.1.3. Test vector for MSGb +

+
+
+  Inputs
+    ADb = b'ADb'
+    yb (little endian): (length: 32 bytes)
+      848b0779ff415f0af4ea14df9dd1d3c29ac41d836c7808896c4eba19
+      c51ac40a
+  Outputs
+    Yb: (length: 32 bytes)
+      0178bbbab0804a4455b8f02e5d6e7d80997c6470bfb3618d7e74c396
+      47af5a29
+    MSGb = lv_cat(Yb,ADb): (length: 37 bytes)
+      200178bbbab0804a4455b8f02e5d6e7d80997c6470bfb3618d7e74c3
+      9647af5a2903414462
+
+
+
+
+
+
+

+B.1.4. Test vector for secret points K +

+
+
+    scalar_mult_vfy(ya,Yb): (length: 32 bytes)
+      42ba4c6dc4c184a1cf405d4503f64bf7f015e2a0107450e38b9efff3
+      bee52412
+    scalar_mult_vfy(yb,Ya): (length: 32 bytes)
+      42ba4c6dc4c184a1cf405d4503f64bf7f015e2a0107450e38b9efff3
+      bee52412
+
+
+
+
+
+
+

+B.1.5. Test vector for ISK calculation initiator/responder +

+
+
+    unordered cat of transcript : (length: 74 bytes)
+      20f970e36f37cfcd9a39e37dd2d1fbc9156d6d2f9ae422f4722cbd9d
+      32e9b1e70403414461200178bbbab0804a4455b8f02e5d6e7d80997c
+      6470bfb3618d7e74c39647af5a2903414462
+    DSI = G.DSI_ISK, b'CPace255_ISK': (length: 12 bytes)
+      43506163653235355f49534b
+    lv_cat(DSI,sid,K)||MSGa||MSGb: (length: 137 bytes)
+      0c43506163653235355f49534b107e4b4791d6a8ef019b936c79fb7f
+      2c572042ba4c6dc4c184a1cf405d4503f64bf7f015e2a0107450e38b
+      9efff3bee5241220f970e36f37cfcd9a39e37dd2d1fbc9156d6d2f9a
+      e422f4722cbd9d32e9b1e70403414461200178bbbab0804a4455b8f0
+      2e5d6e7d80997c6470bfb3618d7e74c39647af5a2903414462
+    ISK result: (length: 64 bytes)
+      f5ef3c13fdb9dfe839bdbf8a9256e8cee7db8a8f1dfa74958a925450
+      cf8089cd560d9a4e7956b7334b6f625c8559b75ea0764ac2be894b8f
+      3d434b30e87797d5
+
+
+
+
+
+
+

+B.1.6. Test vector for ISK calculation parallel execution +

+
+
+    ordered cat of transcript : (length: 76 bytes)
+      6f6320f970e36f37cfcd9a39e37dd2d1fbc9156d6d2f9ae422f4722c
+      bd9d32e9b1e70403414461200178bbbab0804a4455b8f02e5d6e7d80
+      997c6470bfb3618d7e74c39647af5a2903414462
+    DSI = G.DSI_ISK, b'CPace255_ISK': (length: 12 bytes)
+      43506163653235355f49534b
+    lv_cat(DSI,sid,K)||o_cat(MSGa,MSGb): (length: 139 bytes)
+      0c43506163653235355f49534b107e4b4791d6a8ef019b936c79fb7f
+      2c572042ba4c6dc4c184a1cf405d4503f64bf7f015e2a0107450e38b
+      9efff3bee524126f6320f970e36f37cfcd9a39e37dd2d1fbc9156d6d
+      2f9ae422f4722cbd9d32e9b1e70403414461200178bbbab0804a4455
+      b8f02e5d6e7d80997c6470bfb3618d7e74c39647af5a2903414462
+    ISK result: (length: 64 bytes)
+      f4051edc63b2620e10d5ecf76d9f0c5ccd1447858a98d4bf847fafac
+      737478c1350e14619bc0fcd4f028d10e4102dfca39f91fe9b829a503
+      ab3e0549bd835edf
+
+
+
+
+
+
+

+B.1.7. Corresponding C programming language initializers +

+
+
+const unsigned char tc_PRS[] = {
+ 0x50,0x61,0x73,0x73,0x77,0x6f,0x72,0x64,
+};
+const unsigned char tc_CI[] = {
+ 0x0a,0x41,0x69,0x6e,0x69,0x74,0x69,0x61,0x74,0x6f,0x72,0x0a,
+ 0x42,0x72,0x65,0x73,0x70,0x6f,0x6e,0x64,0x65,0x72,
+};
+const unsigned char tc_sid[] = {
+ 0x7e,0x4b,0x47,0x91,0xd6,0xa8,0xef,0x01,0x9b,0x93,0x6c,0x79,
+ 0xfb,0x7f,0x2c,0x57,
+};
+const unsigned char tc_g[] = {
+ 0x4e,0x60,0x98,0x73,0x30,0x61,0xc0,0xe8,0x48,0x66,0x11,0xa9,
+ 0x04,0xfe,0x5e,0xdb,0x04,0x98,0x04,0xd2,0x61,0x30,0xa4,0x41,
+ 0x31,0xa6,0x22,0x9e,0x55,0xc5,0xc3,0x21,
+};
+const unsigned char tc_ya[] = {
+ 0x21,0xb4,0xf4,0xbd,0x9e,0x64,0xed,0x35,0x5c,0x3e,0xb6,0x76,
+ 0xa2,0x8e,0xbe,0xda,0xf6,0xd8,0xf1,0x7b,0xdc,0x36,0x59,0x95,
+ 0xb3,0x19,0x09,0x71,0x53,0x04,0x40,0x80,
+};
+const unsigned char tc_ADa[] = {
+ 0x41,0x44,0x61,
+};
+const unsigned char tc_Ya[] = {
+ 0xf9,0x70,0xe3,0x6f,0x37,0xcf,0xcd,0x9a,0x39,0xe3,0x7d,0xd2,
+ 0xd1,0xfb,0xc9,0x15,0x6d,0x6d,0x2f,0x9a,0xe4,0x22,0xf4,0x72,
+ 0x2c,0xbd,0x9d,0x32,0xe9,0xb1,0xe7,0x04,
+};
+const unsigned char tc_yb[] = {
+ 0x84,0x8b,0x07,0x79,0xff,0x41,0x5f,0x0a,0xf4,0xea,0x14,0xdf,
+ 0x9d,0xd1,0xd3,0xc2,0x9a,0xc4,0x1d,0x83,0x6c,0x78,0x08,0x89,
+ 0x6c,0x4e,0xba,0x19,0xc5,0x1a,0xc4,0x0a,
+};
+const unsigned char tc_ADb[] = {
+ 0x41,0x44,0x62,
+};
+const unsigned char tc_Yb[] = {
+ 0x01,0x78,0xbb,0xba,0xb0,0x80,0x4a,0x44,0x55,0xb8,0xf0,0x2e,
+ 0x5d,0x6e,0x7d,0x80,0x99,0x7c,0x64,0x70,0xbf,0xb3,0x61,0x8d,
+ 0x7e,0x74,0xc3,0x96,0x47,0xaf,0x5a,0x29,
+};
+const unsigned char tc_K[] = {
+ 0x42,0xba,0x4c,0x6d,0xc4,0xc1,0x84,0xa1,0xcf,0x40,0x5d,0x45,
+ 0x03,0xf6,0x4b,0xf7,0xf0,0x15,0xe2,0xa0,0x10,0x74,0x50,0xe3,
+ 0x8b,0x9e,0xff,0xf3,0xbe,0xe5,0x24,0x12,
+};
+const unsigned char tc_ISK_IR[] = {
+ 0xf5,0xef,0x3c,0x13,0xfd,0xb9,0xdf,0xe8,0x39,0xbd,0xbf,0x8a,
+ 0x92,0x56,0xe8,0xce,0xe7,0xdb,0x8a,0x8f,0x1d,0xfa,0x74,0x95,
+ 0x8a,0x92,0x54,0x50,0xcf,0x80,0x89,0xcd,0x56,0x0d,0x9a,0x4e,
+ 0x79,0x56,0xb7,0x33,0x4b,0x6f,0x62,0x5c,0x85,0x59,0xb7,0x5e,
+ 0xa0,0x76,0x4a,0xc2,0xbe,0x89,0x4b,0x8f,0x3d,0x43,0x4b,0x30,
+ 0xe8,0x77,0x97,0xd5,
+};
+const unsigned char tc_ISK_SY[] = {
+ 0xf4,0x05,0x1e,0xdc,0x63,0xb2,0x62,0x0e,0x10,0xd5,0xec,0xf7,
+ 0x6d,0x9f,0x0c,0x5c,0xcd,0x14,0x47,0x85,0x8a,0x98,0xd4,0xbf,
+ 0x84,0x7f,0xaf,0xac,0x73,0x74,0x78,0xc1,0x35,0x0e,0x14,0x61,
+ 0x9b,0xc0,0xfc,0xd4,0xf0,0x28,0xd1,0x0e,0x41,0x02,0xdf,0xca,
+ 0x39,0xf9,0x1f,0xe9,0xb8,0x29,0xa5,0x03,0xab,0x3e,0x05,0x49,
+ 0xbd,0x83,0x5e,0xdf,
+};
+
+
+
+
+
+
+

+B.1.8. Test vectors for G_X25519.scalar_mult_vfy: low order points +

+

Test vectors for which G_X25519.scalar_mult_vfy(s_in,ux) must return the neutral +element or would return the neutral element if bit #255 of field element +representation was not correctly cleared. (The decodeUCoordinate function from RFC7748 mandates clearing bit #255 for field element representations for use in the X25519 function.).

+
+
+u0: 0000000000000000000000000000000000000000000000000000000000000000
+u1: 0100000000000000000000000000000000000000000000000000000000000000
+u2: ecffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff7f
+u3: e0eb7a7c3b41b8ae1656e3faf19fc46ada098deb9c32b1fd866205165f49b800
+u4: 5f9c95bca3508c24b1d0b1559c83ef5b04445cc4581c8e86d8224eddd09f1157
+u5: edffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff7f
+u6: daffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
+u7: eeffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff7f
+u8: dbffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
+u9: d9ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
+ua: cdeb7a7c3b41b8ae1656e3faf19fc46ada098deb9c32b1fd866205165f49b880
+ub: 4c9c95bca3508c24b1d0b1559c83ef5b04445cc4581c8e86d8224eddd09f11d7
+
+u0 ... ub MUST be verified to produce the correct results q0 ... qb:
+
+Additionally, u0,u1,u2,u3,u4,u5 and u7 MUST trigger the abort case
+when included in MSGa or MSGb.
+
+s = af46e36bf0527c9d3b16154b82465edd62144c0ac1fc5a18506a2244ba449aff
+qN = G_X25519.scalar_mult_vfy(s, uX)
+q0: 0000000000000000000000000000000000000000000000000000000000000000
+q1: 0000000000000000000000000000000000000000000000000000000000000000
+q2: 0000000000000000000000000000000000000000000000000000000000000000
+q3: 0000000000000000000000000000000000000000000000000000000000000000
+q4: 0000000000000000000000000000000000000000000000000000000000000000
+q5: 0000000000000000000000000000000000000000000000000000000000000000
+q6: d8e2c776bbacd510d09fd9278b7edcd25fc5ae9adfba3b6e040e8d3b71b21806
+q7: 0000000000000000000000000000000000000000000000000000000000000000
+q8: c85c655ebe8be44ba9c0ffde69f2fe10194458d137f09bbff725ce58803cdb38
+q9: db64dafa9b8fdd136914e61461935fe92aa372cb056314e1231bc4ec12417456
+qa: e062dcd5376d58297be2618c7498f55baa07d7e03184e8aada20bca28888bf7a
+qb: 993c6ad11c4c29da9a56f7691fd0ff8d732e49de6250b6c2e80003ff4629a175
+
+
+
+
+
+
+
+
+

+B.2. Test vector for CPace using group X448 and hash SHAKE-256 +

+
+
+

+B.2.1. Test vectors for calculate_generator with group X448 +

+
+
+  Inputs
+    H   = SHAKE-256 with input block size 136 bytes.
+    PRS = b'Password' ; ZPAD length: 117 ; DSI = b'CPace448'
+    CI = b'\nAinitiator\nBresponder'
+    CI = 0a41696e69746961746f720a42726573706f6e646572
+    sid = 5223e0cdc45d6575668d64c552004124
+  Outputs
+    generator_string(G.DSI,PRS,CI,sid,H.s_in_bytes):
+    (length: 176 bytes)
+      0843506163653434380850617373776f726475000000000000000000
+      00000000000000000000000000000000000000000000000000000000
+      00000000000000000000000000000000000000000000000000000000
+      00000000000000000000000000000000000000000000000000000000
+      000000000000000000000000000000000000000000000000160a4169
+      6e69746961746f720a42726573706f6e646572105223e0cdc45d6575
+      668d64c552004124
+    hash generator string: (length: 56 bytes)
+      769e06d6c41c8cf1c87aa3df8e687167f6d0a2e41821e856276a0221
+      d88272359d0b43204b546174c9179c83c107b707f296eafaa1c5a293
+    decoded field element of 448 bits: (length: 56 bytes)
+      769e06d6c41c8cf1c87aa3df8e687167f6d0a2e41821e856276a0221
+      d88272359d0b43204b546174c9179c83c107b707f296eafaa1c5a293
+    generator g: (length: 56 bytes)
+      6fdae14718eb7506dd96e3f7797896efdb8db9ec0797485c9c48a192
+      2e44961da097f2908b084a5de33ab671630660d27d79ffd6ee8ec846
+
+
+
+
+
+
+

+B.2.2. Test vector for MSGa +

+
+
+  Inputs
+    ADa = b'ADa'
+    ya (little endian): (length: 56 bytes)
+      21b4f4bd9e64ed355c3eb676a28ebedaf6d8f17bdc365995b3190971
+      53044080516bd083bfcce66121a3072646994c8430cc382b8dc543e8
+  Outputs
+    Ya: (length: 56 bytes)
+      396bd11daf223711e575cac6021e3fa31558012048a1cec7876292b9
+      6c61eda353fe04f33028d2352779668a934084da776c1c51a58ce4b5
+    MSGa = lv_cat(Ya,ADa): (length: 61 bytes)
+      38396bd11daf223711e575cac6021e3fa31558012048a1cec7876292
+      b96c61eda353fe04f33028d2352779668a934084da776c1c51a58ce4
+      b503414461
+
+
+
+
+
+
+

+B.2.3. Test vector for MSGb +

+
+
+  Inputs
+    ADb = b'ADb'
+    yb (little endian): (length: 56 bytes)
+      848b0779ff415f0af4ea14df9dd1d3c29ac41d836c7808896c4eba19
+      c51ac40a439caf5e61ec88c307c7d619195229412eaa73fb2a5ea20d
+  Outputs
+    Yb: (length: 56 bytes)
+      53c519fb490fde5a04bda8c18b327d0fc1a9391d19e0ac00c59df9c6
+      0422284e593d6b092eac94f5aa644ed883f39bd4f04e4beb6af86d58
+    MSGb = lv_cat(Yb,ADb): (length: 61 bytes)
+      3853c519fb490fde5a04bda8c18b327d0fc1a9391d19e0ac00c59df9
+      c60422284e593d6b092eac94f5aa644ed883f39bd4f04e4beb6af86d
+      5803414462
+
+
+
+
+
+
+

+B.2.4. Test vector for secret points K +

+
+
+    scalar_mult_vfy(ya,Yb): (length: 56 bytes)
+      e00af217556a40ccbc9822cc27a43542e45166a653aa4df746d5f8e1
+      e8df483e9baff71c9eb03ee20a688ad4e4d359f70ac9ec3f6a659997
+    scalar_mult_vfy(yb,Ya): (length: 56 bytes)
+      e00af217556a40ccbc9822cc27a43542e45166a653aa4df746d5f8e1
+      e8df483e9baff71c9eb03ee20a688ad4e4d359f70ac9ec3f6a659997
+
+
+
+
+
+
+

+B.2.5. Test vector for ISK calculation initiator/responder +

+
+
+    unordered cat of transcript : (length: 122 bytes)
+      38396bd11daf223711e575cac6021e3fa31558012048a1cec7876292
+      b96c61eda353fe04f33028d2352779668a934084da776c1c51a58ce4
+      b5034144613853c519fb490fde5a04bda8c18b327d0fc1a9391d19e0
+      ac00c59df9c60422284e593d6b092eac94f5aa644ed883f39bd4f04e
+      4beb6af86d5803414462
+    DSI = G.DSI_ISK, b'CPace448_ISK': (length: 12 bytes)
+      43506163653434385f49534b
+    lv_cat(DSI,sid,K)||MSGa||MSGb: (length: 209 bytes)
+      0c43506163653434385f49534b105223e0cdc45d6575668d64c55200
+      412438e00af217556a40ccbc9822cc27a43542e45166a653aa4df746
+      d5f8e1e8df483e9baff71c9eb03ee20a688ad4e4d359f70ac9ec3f6a
+      65999738396bd11daf223711e575cac6021e3fa31558012048a1cec7
+      876292b96c61eda353fe04f33028d2352779668a934084da776c1c51
+      a58ce4b5034144613853c519fb490fde5a04bda8c18b327d0fc1a939
+      1d19e0ac00c59df9c60422284e593d6b092eac94f5aa644ed883f39b
+      d4f04e4beb6af86d5803414462
+    ISK result: (length: 64 bytes)
+      4030297722c1914711da6b2a224a44b53b30c05ab02c2a3d3ccc7272
+      a3333ce3a4564c17031b634e89f65681f52d5c3d1df7baeb88523d2e
+      481b3858aed86315
+
+
+
+
+
+
+

+B.2.6. Test vector for ISK calculation parallel execution +

+
+
+    ordered cat of transcript : (length: 124 bytes)
+      6f633853c519fb490fde5a04bda8c18b327d0fc1a9391d19e0ac00c5
+      9df9c60422284e593d6b092eac94f5aa644ed883f39bd4f04e4beb6a
+      f86d580341446238396bd11daf223711e575cac6021e3fa315580120
+      48a1cec7876292b96c61eda353fe04f33028d2352779668a934084da
+      776c1c51a58ce4b503414461
+    DSI = G.DSI_ISK, b'CPace448_ISK': (length: 12 bytes)
+      43506163653434385f49534b
+    lv_cat(DSI,sid,K)||o_cat(MSGa,MSGb): (length: 211 bytes)
+      0c43506163653434385f49534b105223e0cdc45d6575668d64c55200
+      412438e00af217556a40ccbc9822cc27a43542e45166a653aa4df746
+      d5f8e1e8df483e9baff71c9eb03ee20a688ad4e4d359f70ac9ec3f6a
+      6599976f633853c519fb490fde5a04bda8c18b327d0fc1a9391d19e0
+      ac00c59df9c60422284e593d6b092eac94f5aa644ed883f39bd4f04e
+      4beb6af86d580341446238396bd11daf223711e575cac6021e3fa315
+      58012048a1cec7876292b96c61eda353fe04f33028d2352779668a93
+      4084da776c1c51a58ce4b503414461
+    ISK result: (length: 64 bytes)
+      4cd30768e2f75f0583449614bce823b421c31163c5a3bde4eed1c664
+      284a32995ea3430b5c47fc7dd771b534ad38eaea5d8c8f97bd548966
+      7facfc044615075f
+
+
+
+
+
+
+

+B.2.7. Corresponding C programming language initializers +

+
+
+const unsigned char tc_PRS[] = {
+ 0x50,0x61,0x73,0x73,0x77,0x6f,0x72,0x64,
+};
+const unsigned char tc_CI[] = {
+ 0x0a,0x41,0x69,0x6e,0x69,0x74,0x69,0x61,0x74,0x6f,0x72,0x0a,
+ 0x42,0x72,0x65,0x73,0x70,0x6f,0x6e,0x64,0x65,0x72,
+};
+const unsigned char tc_sid[] = {
+ 0x52,0x23,0xe0,0xcd,0xc4,0x5d,0x65,0x75,0x66,0x8d,0x64,0xc5,
+ 0x52,0x00,0x41,0x24,
+};
+const unsigned char tc_g[] = {
+ 0x6f,0xda,0xe1,0x47,0x18,0xeb,0x75,0x06,0xdd,0x96,0xe3,0xf7,
+ 0x79,0x78,0x96,0xef,0xdb,0x8d,0xb9,0xec,0x07,0x97,0x48,0x5c,
+ 0x9c,0x48,0xa1,0x92,0x2e,0x44,0x96,0x1d,0xa0,0x97,0xf2,0x90,
+ 0x8b,0x08,0x4a,0x5d,0xe3,0x3a,0xb6,0x71,0x63,0x06,0x60,0xd2,
+ 0x7d,0x79,0xff,0xd6,0xee,0x8e,0xc8,0x46,
+};
+const unsigned char tc_ya[] = {
+ 0x21,0xb4,0xf4,0xbd,0x9e,0x64,0xed,0x35,0x5c,0x3e,0xb6,0x76,
+ 0xa2,0x8e,0xbe,0xda,0xf6,0xd8,0xf1,0x7b,0xdc,0x36,0x59,0x95,
+ 0xb3,0x19,0x09,0x71,0x53,0x04,0x40,0x80,0x51,0x6b,0xd0,0x83,
+ 0xbf,0xcc,0xe6,0x61,0x21,0xa3,0x07,0x26,0x46,0x99,0x4c,0x84,
+ 0x30,0xcc,0x38,0x2b,0x8d,0xc5,0x43,0xe8,
+};
+const unsigned char tc_ADa[] = {
+ 0x41,0x44,0x61,
+};
+const unsigned char tc_Ya[] = {
+ 0x39,0x6b,0xd1,0x1d,0xaf,0x22,0x37,0x11,0xe5,0x75,0xca,0xc6,
+ 0x02,0x1e,0x3f,0xa3,0x15,0x58,0x01,0x20,0x48,0xa1,0xce,0xc7,
+ 0x87,0x62,0x92,0xb9,0x6c,0x61,0xed,0xa3,0x53,0xfe,0x04,0xf3,
+ 0x30,0x28,0xd2,0x35,0x27,0x79,0x66,0x8a,0x93,0x40,0x84,0xda,
+ 0x77,0x6c,0x1c,0x51,0xa5,0x8c,0xe4,0xb5,
+};
+const unsigned char tc_yb[] = {
+ 0x84,0x8b,0x07,0x79,0xff,0x41,0x5f,0x0a,0xf4,0xea,0x14,0xdf,
+ 0x9d,0xd1,0xd3,0xc2,0x9a,0xc4,0x1d,0x83,0x6c,0x78,0x08,0x89,
+ 0x6c,0x4e,0xba,0x19,0xc5,0x1a,0xc4,0x0a,0x43,0x9c,0xaf,0x5e,
+ 0x61,0xec,0x88,0xc3,0x07,0xc7,0xd6,0x19,0x19,0x52,0x29,0x41,
+ 0x2e,0xaa,0x73,0xfb,0x2a,0x5e,0xa2,0x0d,
+};
+const unsigned char tc_ADb[] = {
+ 0x41,0x44,0x62,
+};
+const unsigned char tc_Yb[] = {
+ 0x53,0xc5,0x19,0xfb,0x49,0x0f,0xde,0x5a,0x04,0xbd,0xa8,0xc1,
+ 0x8b,0x32,0x7d,0x0f,0xc1,0xa9,0x39,0x1d,0x19,0xe0,0xac,0x00,
+ 0xc5,0x9d,0xf9,0xc6,0x04,0x22,0x28,0x4e,0x59,0x3d,0x6b,0x09,
+ 0x2e,0xac,0x94,0xf5,0xaa,0x64,0x4e,0xd8,0x83,0xf3,0x9b,0xd4,
+ 0xf0,0x4e,0x4b,0xeb,0x6a,0xf8,0x6d,0x58,
+};
+const unsigned char tc_K[] = {
+ 0xe0,0x0a,0xf2,0x17,0x55,0x6a,0x40,0xcc,0xbc,0x98,0x22,0xcc,
+ 0x27,0xa4,0x35,0x42,0xe4,0x51,0x66,0xa6,0x53,0xaa,0x4d,0xf7,
+ 0x46,0xd5,0xf8,0xe1,0xe8,0xdf,0x48,0x3e,0x9b,0xaf,0xf7,0x1c,
+ 0x9e,0xb0,0x3e,0xe2,0x0a,0x68,0x8a,0xd4,0xe4,0xd3,0x59,0xf7,
+ 0x0a,0xc9,0xec,0x3f,0x6a,0x65,0x99,0x97,
+};
+const unsigned char tc_ISK_IR[] = {
+ 0x40,0x30,0x29,0x77,0x22,0xc1,0x91,0x47,0x11,0xda,0x6b,0x2a,
+ 0x22,0x4a,0x44,0xb5,0x3b,0x30,0xc0,0x5a,0xb0,0x2c,0x2a,0x3d,
+ 0x3c,0xcc,0x72,0x72,0xa3,0x33,0x3c,0xe3,0xa4,0x56,0x4c,0x17,
+ 0x03,0x1b,0x63,0x4e,0x89,0xf6,0x56,0x81,0xf5,0x2d,0x5c,0x3d,
+ 0x1d,0xf7,0xba,0xeb,0x88,0x52,0x3d,0x2e,0x48,0x1b,0x38,0x58,
+ 0xae,0xd8,0x63,0x15,
+};
+const unsigned char tc_ISK_SY[] = {
+ 0x4c,0xd3,0x07,0x68,0xe2,0xf7,0x5f,0x05,0x83,0x44,0x96,0x14,
+ 0xbc,0xe8,0x23,0xb4,0x21,0xc3,0x11,0x63,0xc5,0xa3,0xbd,0xe4,
+ 0xee,0xd1,0xc6,0x64,0x28,0x4a,0x32,0x99,0x5e,0xa3,0x43,0x0b,
+ 0x5c,0x47,0xfc,0x7d,0xd7,0x71,0xb5,0x34,0xad,0x38,0xea,0xea,
+ 0x5d,0x8c,0x8f,0x97,0xbd,0x54,0x89,0x66,0x7f,0xac,0xfc,0x04,
+ 0x46,0x15,0x07,0x5f,
+};
+
+
+
+
+
+
+

+B.2.8. Test vectors for G_X448.scalar_mult_vfy: low order points +

+

Test vectors for which G_X448.scalar_mult_vfy(s_in,ux) must return the neutral +element. +This includes points that are non-canonicaly encoded, i.e. have coordinate values +larger +than the field prime.

+

Weak points for X448 smaller than the field prime (canonical)

+
+
+  u0: (length: 56 bytes)
+    0000000000000000000000000000000000000000000000000000000000
+    000000000000000000000000000000000000000000000000000000
+  u1: (length: 56 bytes)
+    0100000000000000000000000000000000000000000000000000000000
+    000000000000000000000000000000000000000000000000000000
+  u2: (length: 56 bytes)
+    fefffffffffffffffffffffffffffffffffffffffffffffffffffffffe
+    ffffffffffffffffffffffffffffffffffffffffffffffffffffff
+
+
+

Weak points for X448 larger or equal to the field prime (non-canonical)

+
+
+  u3: (length: 56 bytes)
+    fffffffffffffffffffffffffffffffffffffffffffffffffffffffffe
+    ffffffffffffffffffffffffffffffffffffffffffffffffffffff
+  u4: (length: 56 bytes)
+    00000000000000000000000000000000000000000000000000000000ff
+    ffffffffffffffffffffffffffffffffffffffffffffffffffffff
+
+All of the above points u0 ... u4 MUST trigger the abort case
+when included in the protocol messages MSGa or MSGb.
+
+
+

Expected results for X448 resp. G_X448.scalar_mult_vfy

+
+
+  scalar s: (length: 56 bytes)
+    af8a14218bf2a2062926d2ea9b8fe4e8b6817349b6ed2feb1e5d64d7a4
+    523f15fceec70fb111e870dc58d191e66a14d3e9d482d04432cadd
+  G_X448.scalar_mult_vfy(s,u0): (length: 56 bytes)
+    0000000000000000000000000000000000000000000000000000000000
+    000000000000000000000000000000000000000000000000000000
+  G_X448.scalar_mult_vfy(s,u1): (length: 56 bytes)
+    0000000000000000000000000000000000000000000000000000000000
+    000000000000000000000000000000000000000000000000000000
+  G_X448.scalar_mult_vfy(s,u2): (length: 56 bytes)
+    0000000000000000000000000000000000000000000000000000000000
+    000000000000000000000000000000000000000000000000000000
+  G_X448.scalar_mult_vfy(s,u3): (length: 56 bytes)
+    0000000000000000000000000000000000000000000000000000000000
+    000000000000000000000000000000000000000000000000000000
+  G_X448.scalar_mult_vfy(s,u4): (length: 56 bytes)
+    0000000000000000000000000000000000000000000000000000000000
+    000000000000000000000000000000000000000000000000000000
+
+
+

Test vectors for scalar_mult with nonzero outputs

+
+
+  scalar s: (length: 56 bytes)
+    af8a14218bf2a2062926d2ea9b8fe4e8b6817349b6ed2feb1e5d64d7a4
+    523f15fceec70fb111e870dc58d191e66a14d3e9d482d04432cadd
+  point coordinate u_curve on the curve: (length: 56 bytes)
+    ab0c68d772ec2eb9de25c49700e46d6325e66d6aa39d7b65eb84a68c55
+    69d47bd71b41f3e0d210f44e146dec8926b174acb3f940a0b82cab
+  G_X448.scalar_mult_vfy(s,u_curve): (length: 56 bytes)
+    3b0fa9bc40a6fdc78c9e06ff7a54c143c5d52f365607053bf0656f5142
+    0496295f910a101b38edc1acd3bd240fd55dcb7a360553b8a7627e
+
+  point coordinate u_twist on the twist: (length: 56 bytes)
+    c981cd1e1f72d9c35c7d7cf6be426757c0dc8206a2fcfa564a8e7618c0
+    3c0e61f9a2eb1c3e0dd97d6e9b1010f5edd03397a83f5a914cb3ff
+  G_X448.scalar_mult_vfy(s,u_twist): (length: 56 bytes)
+    d0a2bb7e9c5c2c627793d8342f23b759fe7d9e3320a85ca4fd61376331
+    50ffd9a9148a9b75c349fac43d64bec49a6e126cc92cbfbf353961
+
+
+
+
+
+
+
+
+

+B.3. Test vector for CPace using group ristretto255 and hash SHA-512 +

+
+
+

+B.3.1. Test vectors for calculate_generator with group ristretto255 +

+
+
+  Inputs
+    H   = SHA-512 with input block size 128 bytes.
+    PRS = b'Password' ; ZPAD length: 100 ;
+    DSI = b'CPaceRistretto255'
+    CI = b'\nAinitiator\nBresponder'
+    CI = 0a41696e69746961746f720a42726573706f6e646572
+    sid = 7e4b4791d6a8ef019b936c79fb7f2c57
+  Outputs
+    generator_string(G.DSI,PRS,CI,sid,H.s_in_bytes):
+    (length: 168 bytes)
+      11435061636552697374726574746f3235350850617373776f726464
+      00000000000000000000000000000000000000000000000000000000
+      00000000000000000000000000000000000000000000000000000000
+      00000000000000000000000000000000000000000000000000000000
+      00000000000000000000000000000000160a41696e69746961746f72
+      0a42726573706f6e646572107e4b4791d6a8ef019b936c79fb7f2c57
+    hash result: (length: 64 bytes)
+      a5ce446f63a1ae6d1fee80fa67d0b4004a4b1283ec5549a462bf33a6
+      c1ae06a0871f9bf48545f49b2a792eed255ac04f52758c9c60448306
+      810b44e986e3dcbb
+    encoded generator g: (length: 32 bytes)
+      5e25411ca1ad7c9debfd0b33ad987a95cefef2d3f15dcc8bd26415a5
+      dfe2e15a
+
+
+
+
+
+
+

+B.3.2. Test vector for MSGa +

+
+
+  Inputs
+    ADa = b'ADa'
+    ya (little endian): (length: 32 bytes)
+      da3d23700a9e5699258aef94dc060dfda5ebb61f02a5ea77fad53f4f
+      f0976d08
+  Outputs
+    Ya: (length: 32 bytes)
+      383a85dd236978f17f8c8545b50dabc52a39fcdab2cf8bc531ce040f
+      f77ca82d
+    MSGa = lv_cat(Ya,ADa): (length: 37 bytes)
+      20383a85dd236978f17f8c8545b50dabc52a39fcdab2cf8bc531ce04
+      0ff77ca82d03414461
+
+
+
+
+
+
+

+B.3.3. Test vector for MSGb +

+
+
+  Inputs
+    ADb = b'ADb'
+    yb (little endian): (length: 32 bytes)
+      d2316b454718c35362d83d69df6320f38578ed5984651435e2949762
+      d900b80d
+  Outputs
+    Yb: (length: 32 bytes)
+      a6206309c0e8e5f579295e35997ac4300ab3fecec3c17f7b604f3e69
+      8fa1383c
+    MSGb = lv_cat(Yb,ADb): (length: 37 bytes)
+      20a6206309c0e8e5f579295e35997ac4300ab3fecec3c17f7b604f3e
+      698fa1383c03414462
+
+
+
+
+
+
+

+B.3.4. Test vector for secret points K +

+
+
+    scalar_mult_vfy(ya,Yb): (length: 32 bytes)
+      fa1d0318864e2cacb26875f1b791c9ae83204fe8359addb53e95a2e9
+      8893853f
+    scalar_mult_vfy(yb,Ya): (length: 32 bytes)
+      fa1d0318864e2cacb26875f1b791c9ae83204fe8359addb53e95a2e9
+      8893853f
+
+
+
+
+
+
+

+B.3.5. Test vector for ISK calculation initiator/responder +

+
+
+    unordered cat of transcript : (length: 74 bytes)
+      20383a85dd236978f17f8c8545b50dabc52a39fcdab2cf8bc531ce04
+      0ff77ca82d0341446120a6206309c0e8e5f579295e35997ac4300ab3
+      fecec3c17f7b604f3e698fa1383c03414462
+    DSI = G.DSI_ISK, b'CPaceRistretto255_ISK':
+    (length: 21 bytes)
+      435061636552697374726574746f3235355f49534b
+    lv_cat(DSI,sid,K)||MSGa||MSGb: (length: 146 bytes)
+      15435061636552697374726574746f3235355f49534b107e4b4791d6
+      a8ef019b936c79fb7f2c5720fa1d0318864e2cacb26875f1b791c9ae
+      83204fe8359addb53e95a2e98893853f20383a85dd236978f17f8c85
+      45b50dabc52a39fcdab2cf8bc531ce040ff77ca82d0341446120a620
+      6309c0e8e5f579295e35997ac4300ab3fecec3c17f7b604f3e698fa1
+      383c03414462
+    ISK result: (length: 64 bytes)
+      e91ccb2c0f5e0d0993a33956e3be59754f3f2b07db57631f5394452e
+      a2e7b4354674eb1f5686c078462bf83bec72e8743df440108e638f35
+      26d9b90e85be096f
+
+
+
+
+
+
+

+B.3.6. Test vector for ISK calculation parallel execution +

+
+
+    ordered cat of transcript : (length: 76 bytes)
+      6f6320a6206309c0e8e5f579295e35997ac4300ab3fecec3c17f7b60
+      4f3e698fa1383c0341446220383a85dd236978f17f8c8545b50dabc5
+      2a39fcdab2cf8bc531ce040ff77ca82d03414461
+    DSI = G.DSI_ISK, b'CPaceRistretto255_ISK':
+    (length: 21 bytes)
+      435061636552697374726574746f3235355f49534b
+    lv_cat(DSI,sid,K)||o_cat(MSGa,MSGb): (length: 148 bytes)
+      15435061636552697374726574746f3235355f49534b107e4b4791d6
+      a8ef019b936c79fb7f2c5720fa1d0318864e2cacb26875f1b791c9ae
+      83204fe8359addb53e95a2e98893853f6f6320a6206309c0e8e5f579
+      295e35997ac4300ab3fecec3c17f7b604f3e698fa1383c0341446220
+      383a85dd236978f17f8c8545b50dabc52a39fcdab2cf8bc531ce040f
+      f77ca82d03414461
+    ISK result: (length: 64 bytes)
+      1638fb6ff564a80a12af07c036870e10c4efb539fa847fdf3e9c4621
+      7bf52cd4df4ca0fe51146492a9ba6dd6a42ac402bc2d60adb4084c81
+      758d754d1d81482a
+
+
+
+
+
+
+

+B.3.7. Corresponding C programming language initializers +

+
+
+const unsigned char tc_PRS[] = {
+ 0x50,0x61,0x73,0x73,0x77,0x6f,0x72,0x64,
+};
+const unsigned char tc_CI[] = {
+ 0x0a,0x41,0x69,0x6e,0x69,0x74,0x69,0x61,0x74,0x6f,0x72,0x0a,
+ 0x42,0x72,0x65,0x73,0x70,0x6f,0x6e,0x64,0x65,0x72,
+};
+const unsigned char tc_sid[] = {
+ 0x7e,0x4b,0x47,0x91,0xd6,0xa8,0xef,0x01,0x9b,0x93,0x6c,0x79,
+ 0xfb,0x7f,0x2c,0x57,
+};
+const unsigned char tc_g[] = {
+ 0x5e,0x25,0x41,0x1c,0xa1,0xad,0x7c,0x9d,0xeb,0xfd,0x0b,0x33,
+ 0xad,0x98,0x7a,0x95,0xce,0xfe,0xf2,0xd3,0xf1,0x5d,0xcc,0x8b,
+ 0xd2,0x64,0x15,0xa5,0xdf,0xe2,0xe1,0x5a,
+};
+const unsigned char tc_ya[] = {
+ 0xda,0x3d,0x23,0x70,0x0a,0x9e,0x56,0x99,0x25,0x8a,0xef,0x94,
+ 0xdc,0x06,0x0d,0xfd,0xa5,0xeb,0xb6,0x1f,0x02,0xa5,0xea,0x77,
+ 0xfa,0xd5,0x3f,0x4f,0xf0,0x97,0x6d,0x08,
+};
+const unsigned char tc_ADa[] = {
+ 0x41,0x44,0x61,
+};
+const unsigned char tc_Ya[] = {
+ 0x38,0x3a,0x85,0xdd,0x23,0x69,0x78,0xf1,0x7f,0x8c,0x85,0x45,
+ 0xb5,0x0d,0xab,0xc5,0x2a,0x39,0xfc,0xda,0xb2,0xcf,0x8b,0xc5,
+ 0x31,0xce,0x04,0x0f,0xf7,0x7c,0xa8,0x2d,
+};
+const unsigned char tc_yb[] = {
+ 0xd2,0x31,0x6b,0x45,0x47,0x18,0xc3,0x53,0x62,0xd8,0x3d,0x69,
+ 0xdf,0x63,0x20,0xf3,0x85,0x78,0xed,0x59,0x84,0x65,0x14,0x35,
+ 0xe2,0x94,0x97,0x62,0xd9,0x00,0xb8,0x0d,
+};
+const unsigned char tc_ADb[] = {
+ 0x41,0x44,0x62,
+};
+const unsigned char tc_Yb[] = {
+ 0xa6,0x20,0x63,0x09,0xc0,0xe8,0xe5,0xf5,0x79,0x29,0x5e,0x35,
+ 0x99,0x7a,0xc4,0x30,0x0a,0xb3,0xfe,0xce,0xc3,0xc1,0x7f,0x7b,
+ 0x60,0x4f,0x3e,0x69,0x8f,0xa1,0x38,0x3c,
+};
+const unsigned char tc_K[] = {
+ 0xfa,0x1d,0x03,0x18,0x86,0x4e,0x2c,0xac,0xb2,0x68,0x75,0xf1,
+ 0xb7,0x91,0xc9,0xae,0x83,0x20,0x4f,0xe8,0x35,0x9a,0xdd,0xb5,
+ 0x3e,0x95,0xa2,0xe9,0x88,0x93,0x85,0x3f,
+};
+const unsigned char tc_ISK_IR[] = {
+ 0xe9,0x1c,0xcb,0x2c,0x0f,0x5e,0x0d,0x09,0x93,0xa3,0x39,0x56,
+ 0xe3,0xbe,0x59,0x75,0x4f,0x3f,0x2b,0x07,0xdb,0x57,0x63,0x1f,
+ 0x53,0x94,0x45,0x2e,0xa2,0xe7,0xb4,0x35,0x46,0x74,0xeb,0x1f,
+ 0x56,0x86,0xc0,0x78,0x46,0x2b,0xf8,0x3b,0xec,0x72,0xe8,0x74,
+ 0x3d,0xf4,0x40,0x10,0x8e,0x63,0x8f,0x35,0x26,0xd9,0xb9,0x0e,
+ 0x85,0xbe,0x09,0x6f,
+};
+const unsigned char tc_ISK_SY[] = {
+ 0x16,0x38,0xfb,0x6f,0xf5,0x64,0xa8,0x0a,0x12,0xaf,0x07,0xc0,
+ 0x36,0x87,0x0e,0x10,0xc4,0xef,0xb5,0x39,0xfa,0x84,0x7f,0xdf,
+ 0x3e,0x9c,0x46,0x21,0x7b,0xf5,0x2c,0xd4,0xdf,0x4c,0xa0,0xfe,
+ 0x51,0x14,0x64,0x92,0xa9,0xba,0x6d,0xd6,0xa4,0x2a,0xc4,0x02,
+ 0xbc,0x2d,0x60,0xad,0xb4,0x08,0x4c,0x81,0x75,0x8d,0x75,0x4d,
+ 0x1d,0x81,0x48,0x2a,
+};
+
+
+
+
+
+
+

+B.3.8. Test case for scalar_mult with valid inputs +

+
+
+    s: (length: 32 bytes)
+      7cd0e075fa7955ba52c02759a6c90dbbfc10e6d40aea8d283e407d88
+      cf538a05
+    X: (length: 32 bytes)
+      2c3c6b8c4f3800e7aef6864025b4ed79bd599117e427c41bd47d93d6
+      54b4a51c
+    G.scalar_mult(s,decode(X)): (length: 32 bytes)
+      7c13645fe790a468f62c39beb7388e541d8405d1ade69d1778c5fe3e
+      7f6b600e
+    G.scalar_mult_vfy(s,X): (length: 32 bytes)
+      7c13645fe790a468f62c39beb7388e541d8405d1ade69d1778c5fe3e
+      7f6b600e
+
+
+
+
+
+
+

+B.3.9. Invalid inputs for scalar_mult_vfy +

+

For these test cases scalar_mult_vfy(y,.) MUST return the representation of the neutral element G.I. When points Y_i1 or Y_i2 are included in MSGa or MSGb the protocol MUST abort.

+
+
+    s: (length: 32 bytes)
+      7cd0e075fa7955ba52c02759a6c90dbbfc10e6d40aea8d283e407d88
+      cf538a05
+    Y_i1: (length: 32 bytes)
+      2b3c6b8c4f3800e7aef6864025b4ed79bd599117e427c41bd47d93d6
+      54b4a51c
+    Y_i2 == G.I: (length: 32 bytes)
+      00000000000000000000000000000000000000000000000000000000
+      00000000
+    G.scalar_mult_vfy(s,Y_i1) = G.scalar_mult_vfy(s,Y_i2) = G.I
+
+
+
+
+
+
+
+
+

+B.4. Test vector for CPace using group decaf448 and hash SHAKE-256 +

+
+
+

+B.4.1. Test vectors for calculate_generator with group decaf448 +

+
+
+  Inputs
+    H   = SHAKE-256 with input block size 136 bytes.
+    PRS = b'Password' ; ZPAD length: 112 ;
+    DSI = b'CPaceDecaf448'
+    CI = b'\nAinitiator\nBresponder'
+    CI = 0a41696e69746961746f720a42726573706f6e646572
+    sid = 5223e0cdc45d6575668d64c552004124
+  Outputs
+    generator_string(G.DSI,PRS,CI,sid,H.s_in_bytes):
+    (length: 176 bytes)
+      0d435061636544656361663434380850617373776f72647000000000
+      00000000000000000000000000000000000000000000000000000000
+      00000000000000000000000000000000000000000000000000000000
+      00000000000000000000000000000000000000000000000000000000
+      000000000000000000000000000000000000000000000000160a4169
+      6e69746961746f720a42726573706f6e646572105223e0cdc45d6575
+      668d64c552004124
+    hash result: (length: 112 bytes)
+      8955b426ff1d3a22032d21c013cf94134cee9a4235e93261a4911edb
+      f68f2945f0267c983954262c7f59badb9caf468ebe21b7e9885657af
+      b8f1a3b783c2047ba519e113ecf81b2b580dd481f499beabd401cc77
+      1d28915fb750011209040f5f03b2ceb5e5eb259c96b478382d5a5c57
+    encoded generator g: (length: 56 bytes)
+      682d1a4f49fc2a4834356ae4d7f58636bc9481521c845e66e6fb0b29
+      69341df45fbaeaea9e2221b3f5babc54c5f8ce456988ffc519defaeb
+
+
+
+
+
+
+

+B.4.2. Test vector for MSGa +

+
+
+  Inputs
+    ADa = b'ADa'
+    ya (little endian): (length: 56 bytes)
+      33d561f13cfc0dca279c30e8cde895175dc25483892819eba132d58c
+      13c0462a8eb0d73fda941950594bef5191d8394691f86edffcad6c1e
+  Outputs
+    Ya: (length: 56 bytes)
+      e233867540319ec86eaecc09a85dec233745db729f61c36bde14c034
+      200994fc4b6e8d263008c169585fd1d186d8ac560cb9f7ad0d166965
+    MSGa = lv_cat(Ya,ADa): (length: 61 bytes)
+      38e233867540319ec86eaecc09a85dec233745db729f61c36bde14c0
+      34200994fc4b6e8d263008c169585fd1d186d8ac560cb9f7ad0d1669
+      6503414461
+
+
+
+
+
+
+

+B.4.3. Test vector for MSGb +

+
+
+  Inputs
+    ADb = b'ADb'
+    yb (little endian): (length: 56 bytes)
+      2523c969f68fa2b2aea294c2539ef36eb1e0558abd14712a7828f16a
+      85ed2c7e77e2bdd418994405fb1b57b6bbaadd66849892aac9d81402
+  Outputs
+    Yb: (length: 56 bytes)
+      5062a0f33478914bf162a80dad39b5b266c1dd02f408573b41827e38
+      599b682afbf7a0735adfd68c39ab4994fd1b034846270e38332b4da9
+    MSGb = lv_cat(Yb,ADb): (length: 61 bytes)
+      385062a0f33478914bf162a80dad39b5b266c1dd02f408573b41827e
+      38599b682afbf7a0735adfd68c39ab4994fd1b034846270e38332b4d
+      a903414462
+
+
+
+
+
+
+

+B.4.4. Test vector for secret points K +

+
+
+    scalar_mult_vfy(ya,Yb): (length: 56 bytes)
+      dc9edef7c127e79d32f2584f9fcd3269174fe32226c2082963879a6d
+      eafefb9c14efcee9fc1245917ad3658037d2d62aff2d3f76fa4fca99
+    scalar_mult_vfy(yb,Ya): (length: 56 bytes)
+      dc9edef7c127e79d32f2584f9fcd3269174fe32226c2082963879a6d
+      eafefb9c14efcee9fc1245917ad3658037d2d62aff2d3f76fa4fca99
+
+
+
+
+
+
+

+B.4.5. Test vector for ISK calculation initiator/responder +

+
+
+    unordered cat of transcript : (length: 122 bytes)
+      38e233867540319ec86eaecc09a85dec233745db729f61c36bde14c0
+      34200994fc4b6e8d263008c169585fd1d186d8ac560cb9f7ad0d1669
+      6503414461385062a0f33478914bf162a80dad39b5b266c1dd02f408
+      573b41827e38599b682afbf7a0735adfd68c39ab4994fd1b03484627
+      0e38332b4da903414462
+    DSI = G.DSI_ISK, b'CPaceDecaf448_ISK': (length: 17 bytes)
+      435061636544656361663434385f49534b
+    lv_cat(DSI,sid,K)||MSGa||MSGb: (length: 214 bytes)
+      11435061636544656361663434385f49534b105223e0cdc45d657566
+      8d64c55200412438dc9edef7c127e79d32f2584f9fcd3269174fe322
+      26c2082963879a6deafefb9c14efcee9fc1245917ad3658037d2d62a
+      ff2d3f76fa4fca9938e233867540319ec86eaecc09a85dec233745db
+      729f61c36bde14c034200994fc4b6e8d263008c169585fd1d186d8ac
+      560cb9f7ad0d16696503414461385062a0f33478914bf162a80dad39
+      b5b266c1dd02f408573b41827e38599b682afbf7a0735adfd68c39ab
+      4994fd1b034846270e38332b4da903414462
+    ISK result: (length: 64 bytes)
+      a752612fe6dec542e96629a6eb68ecb9bfe2257224975e916035aee7
+      47c6aba32af2e6fe25eeb96261e6140100edcf95686e0aaa134026b4
+      b5254fd271b7a4da
+
+
+
+
+
+
+

+B.4.6. Test vector for ISK calculation parallel execution +

+
+
+    ordered cat of transcript : (length: 124 bytes)
+      6f6338e233867540319ec86eaecc09a85dec233745db729f61c36bde
+      14c034200994fc4b6e8d263008c169585fd1d186d8ac560cb9f7ad0d
+      16696503414461385062a0f33478914bf162a80dad39b5b266c1dd02
+      f408573b41827e38599b682afbf7a0735adfd68c39ab4994fd1b0348
+      46270e38332b4da903414462
+    DSI = G.DSI_ISK, b'CPaceDecaf448_ISK': (length: 17 bytes)
+      435061636544656361663434385f49534b
+    lv_cat(DSI,sid,K)||o_cat(MSGa,MSGb): (length: 216 bytes)
+      11435061636544656361663434385f49534b105223e0cdc45d657566
+      8d64c55200412438dc9edef7c127e79d32f2584f9fcd3269174fe322
+      26c2082963879a6deafefb9c14efcee9fc1245917ad3658037d2d62a
+      ff2d3f76fa4fca996f6338e233867540319ec86eaecc09a85dec2337
+      45db729f61c36bde14c034200994fc4b6e8d263008c169585fd1d186
+      d8ac560cb9f7ad0d16696503414461385062a0f33478914bf162a80d
+      ad39b5b266c1dd02f408573b41827e38599b682afbf7a0735adfd68c
+      39ab4994fd1b034846270e38332b4da903414462
+    ISK result: (length: 64 bytes)
+      e6c79d30d4381a45bd47b14b769d41354211aff553ece937d4ac134f
+      09844896c72a723b1f1b6da1ab281d759a15624d2bcd0e423b70b8b8
+      50a4d0ed126a3026
+
+
+
+
+
+
+

+B.4.7. Corresponding C programming language initializers +

+
+
+const unsigned char tc_PRS[] = {
+ 0x50,0x61,0x73,0x73,0x77,0x6f,0x72,0x64,
+};
+const unsigned char tc_CI[] = {
+ 0x0a,0x41,0x69,0x6e,0x69,0x74,0x69,0x61,0x74,0x6f,0x72,0x0a,
+ 0x42,0x72,0x65,0x73,0x70,0x6f,0x6e,0x64,0x65,0x72,
+};
+const unsigned char tc_sid[] = {
+ 0x52,0x23,0xe0,0xcd,0xc4,0x5d,0x65,0x75,0x66,0x8d,0x64,0xc5,
+ 0x52,0x00,0x41,0x24,
+};
+const unsigned char tc_g[] = {
+ 0x68,0x2d,0x1a,0x4f,0x49,0xfc,0x2a,0x48,0x34,0x35,0x6a,0xe4,
+ 0xd7,0xf5,0x86,0x36,0xbc,0x94,0x81,0x52,0x1c,0x84,0x5e,0x66,
+ 0xe6,0xfb,0x0b,0x29,0x69,0x34,0x1d,0xf4,0x5f,0xba,0xea,0xea,
+ 0x9e,0x22,0x21,0xb3,0xf5,0xba,0xbc,0x54,0xc5,0xf8,0xce,0x45,
+ 0x69,0x88,0xff,0xc5,0x19,0xde,0xfa,0xeb,
+};
+const unsigned char tc_ya[] = {
+ 0x33,0xd5,0x61,0xf1,0x3c,0xfc,0x0d,0xca,0x27,0x9c,0x30,0xe8,
+ 0xcd,0xe8,0x95,0x17,0x5d,0xc2,0x54,0x83,0x89,0x28,0x19,0xeb,
+ 0xa1,0x32,0xd5,0x8c,0x13,0xc0,0x46,0x2a,0x8e,0xb0,0xd7,0x3f,
+ 0xda,0x94,0x19,0x50,0x59,0x4b,0xef,0x51,0x91,0xd8,0x39,0x46,
+ 0x91,0xf8,0x6e,0xdf,0xfc,0xad,0x6c,0x1e,
+};
+const unsigned char tc_ADa[] = {
+ 0x41,0x44,0x61,
+};
+const unsigned char tc_Ya[] = {
+ 0xe2,0x33,0x86,0x75,0x40,0x31,0x9e,0xc8,0x6e,0xae,0xcc,0x09,
+ 0xa8,0x5d,0xec,0x23,0x37,0x45,0xdb,0x72,0x9f,0x61,0xc3,0x6b,
+ 0xde,0x14,0xc0,0x34,0x20,0x09,0x94,0xfc,0x4b,0x6e,0x8d,0x26,
+ 0x30,0x08,0xc1,0x69,0x58,0x5f,0xd1,0xd1,0x86,0xd8,0xac,0x56,
+ 0x0c,0xb9,0xf7,0xad,0x0d,0x16,0x69,0x65,
+};
+const unsigned char tc_yb[] = {
+ 0x25,0x23,0xc9,0x69,0xf6,0x8f,0xa2,0xb2,0xae,0xa2,0x94,0xc2,
+ 0x53,0x9e,0xf3,0x6e,0xb1,0xe0,0x55,0x8a,0xbd,0x14,0x71,0x2a,
+ 0x78,0x28,0xf1,0x6a,0x85,0xed,0x2c,0x7e,0x77,0xe2,0xbd,0xd4,
+ 0x18,0x99,0x44,0x05,0xfb,0x1b,0x57,0xb6,0xbb,0xaa,0xdd,0x66,
+ 0x84,0x98,0x92,0xaa,0xc9,0xd8,0x14,0x02,
+};
+const unsigned char tc_ADb[] = {
+ 0x41,0x44,0x62,
+};
+const unsigned char tc_Yb[] = {
+ 0x50,0x62,0xa0,0xf3,0x34,0x78,0x91,0x4b,0xf1,0x62,0xa8,0x0d,
+ 0xad,0x39,0xb5,0xb2,0x66,0xc1,0xdd,0x02,0xf4,0x08,0x57,0x3b,
+ 0x41,0x82,0x7e,0x38,0x59,0x9b,0x68,0x2a,0xfb,0xf7,0xa0,0x73,
+ 0x5a,0xdf,0xd6,0x8c,0x39,0xab,0x49,0x94,0xfd,0x1b,0x03,0x48,
+ 0x46,0x27,0x0e,0x38,0x33,0x2b,0x4d,0xa9,
+};
+const unsigned char tc_K[] = {
+ 0xdc,0x9e,0xde,0xf7,0xc1,0x27,0xe7,0x9d,0x32,0xf2,0x58,0x4f,
+ 0x9f,0xcd,0x32,0x69,0x17,0x4f,0xe3,0x22,0x26,0xc2,0x08,0x29,
+ 0x63,0x87,0x9a,0x6d,0xea,0xfe,0xfb,0x9c,0x14,0xef,0xce,0xe9,
+ 0xfc,0x12,0x45,0x91,0x7a,0xd3,0x65,0x80,0x37,0xd2,0xd6,0x2a,
+ 0xff,0x2d,0x3f,0x76,0xfa,0x4f,0xca,0x99,
+};
+const unsigned char tc_ISK_IR[] = {
+ 0xa7,0x52,0x61,0x2f,0xe6,0xde,0xc5,0x42,0xe9,0x66,0x29,0xa6,
+ 0xeb,0x68,0xec,0xb9,0xbf,0xe2,0x25,0x72,0x24,0x97,0x5e,0x91,
+ 0x60,0x35,0xae,0xe7,0x47,0xc6,0xab,0xa3,0x2a,0xf2,0xe6,0xfe,
+ 0x25,0xee,0xb9,0x62,0x61,0xe6,0x14,0x01,0x00,0xed,0xcf,0x95,
+ 0x68,0x6e,0x0a,0xaa,0x13,0x40,0x26,0xb4,0xb5,0x25,0x4f,0xd2,
+ 0x71,0xb7,0xa4,0xda,
+};
+const unsigned char tc_ISK_SY[] = {
+ 0xe6,0xc7,0x9d,0x30,0xd4,0x38,0x1a,0x45,0xbd,0x47,0xb1,0x4b,
+ 0x76,0x9d,0x41,0x35,0x42,0x11,0xaf,0xf5,0x53,0xec,0xe9,0x37,
+ 0xd4,0xac,0x13,0x4f,0x09,0x84,0x48,0x96,0xc7,0x2a,0x72,0x3b,
+ 0x1f,0x1b,0x6d,0xa1,0xab,0x28,0x1d,0x75,0x9a,0x15,0x62,0x4d,
+ 0x2b,0xcd,0x0e,0x42,0x3b,0x70,0xb8,0xb8,0x50,0xa4,0xd0,0xed,
+ 0x12,0x6a,0x30,0x26,
+};
+
+
+
+
+
+
+

+B.4.8. Test case for scalar_mult with valid inputs +

+
+
+    s: (length: 56 bytes)
+      dd1bc7015daabb7672129cc35a3ba815486b139deff9bdeca7a4fc61
+      34323d34658761e90ff079972a7ca8aa5606498f4f4f0ebc0933a819
+    X: (length: 56 bytes)
+      601431d5e51f43d422a92d3fb2373bde28217aab42524c341aa404ea
+      ba5aa5541f7042dbb3253ce4c90f772b038a413dcb3a0f6bf3ae9e21
+    G.scalar_mult(s,decode(X)): (length: 56 bytes)
+      388b35c60eb41b66085a2118316218681d78979d667702de105fdc1f
+      21ffe884a577d795f45691781390a229a3bd7b527e831380f2f585a4
+    G.scalar_mult_vfy(s,X): (length: 56 bytes)
+      388b35c60eb41b66085a2118316218681d78979d667702de105fdc1f
+      21ffe884a577d795f45691781390a229a3bd7b527e831380f2f585a4
+
+
+
+
+
+
+

+B.4.9. Invalid inputs for scalar_mult_vfy +

+

For these test cases scalar_mult_vfy(y,.) MUST return the representation of the neutral element G.I. When points Y_i1 or Y_i2 are included in MSGa or MSGb the protocol MUST abort.

+
+
+    s: (length: 56 bytes)
+      dd1bc7015daabb7672129cc35a3ba815486b139deff9bdeca7a4fc61
+      34323d34658761e90ff079972a7ca8aa5606498f4f4f0ebc0933a819
+    Y_i1: (length: 56 bytes)
+      5f1431d5e51f43d422a92d3fb2373bde28217aab42524c341aa404ea
+      ba5aa5541f7042dbb3253ce4c90f772b038a413dcb3a0f6bf3ae9e21
+    Y_i2 == G.I: (length: 56 bytes)
+      00000000000000000000000000000000000000000000000000000000
+      00000000000000000000000000000000000000000000000000000000
+    G.scalar_mult_vfy(s,Y_i1) = G.scalar_mult_vfy(s,Y_i2) = G.I
+
+
+
+
+
+
+
+
+

+B.5. Test vector for CPace using group NIST P-256 and hash SHA-256 +

+
+
+

+B.5.1. Test vectors for calculate_generator with group NIST P-256 +

+
+
+  Inputs
+    H   = SHA-256 with input block size 64 bytes.
+    PRS = b'Password' ; ZPAD length: 23 ;
+    DSI = b'CPaceP256_XMD:SHA-256_SSWU_NU_'
+    DST = b'CPaceP256_XMD:SHA-256_SSWU_NU__DST'
+    CI = b'\nAinitiator\nBresponder'
+    CI = 0a41696e69746961746f720a42726573706f6e646572
+    sid = 34b36454cab2e7842c389f7d88ecb7df
+  Outputs
+    generator_string(PRS,G.DSI,CI,sid,H.s_in_bytes):
+    (length: 104 bytes)
+      1e4350616365503235365f584d443a5348412d3235365f535357555f
+      4e555f0850617373776f726417000000000000000000000000000000
+      0000000000000000160a41696e69746961746f720a42726573706f6e
+      6465721034b36454cab2e7842c389f7d88ecb7df
+    generator g: (length: 65 bytes)
+      041b51433114e096c9d595f0955f5717a75169afb95557f4a6f51155
+      035dee19c76887bce5c7c054fa1fe48a4a62c7fb96dc75e34259d2f7
+      2b8d41f31b8e586bcd
+
+
+
+
+
+
+

+B.5.2. Test vector for MSGa +

+
+
+  Inputs
+    ADa = b'ADa'
+    ya (big endian): (length: 32 bytes)
+      37574cfbf1b95ff6a8e2d7be462d4d01e6dde2618f34f4de9df869b2
+      4f532c5d
+  Outputs
+    Ya: (length: 65 bytes)
+      04b75c1bcda84a0f324aabb7f25cf853ed7fb327c33f23db6aeb320d
+      81df014649c2ac691925fce0eceac7dbc75eca25e6a1558066a610b4
+      021488279e3b989d52
+    Alternative correct value for Ya: g*(-ya):
+    (length: 65 bytes)
+      04b75c1bcda84a0f324aabb7f25cf853ed7fb327c33f23db6aeb320d
+      81df0146493d5396e5da031f1415382438a135da195eaa7f9a59ef4b
+      fdeb77d861c46762ad
+    MSGa = lv_cat(Ya,ADa): (length: 70 bytes)
+      4104b75c1bcda84a0f324aabb7f25cf853ed7fb327c33f23db6aeb32
+      0d81df014649c2ac691925fce0eceac7dbc75eca25e6a1558066a610
+      b4021488279e3b989d5203414461
+
+
+
+
+
+
+

+B.5.3. Test vector for MSGb +

+
+
+  Inputs
+    ADb = b'ADb'
+    yb (big endian): (length: 32 bytes)
+      e5672fc9eb4e721f41d80181ec4c9fd9886668acc48024d33c82bb10
+      2aecba52
+  Outputs
+    Yb: (length: 65 bytes)
+      04bb2783a57337e74671f76452876b27839c0ea9e044e3aadaad2e64
+      777ed27a90e80a99438e2f1c072462f2895c6dadf1b43867b92ffb65
+      562b78c793947dcada
+    Alternative correct value for Yb: g*(-yb):
+    (length: 65 bytes)
+      04bb2783a57337e74671f76452876b27839c0ea9e044e3aadaad2e64
+      777ed27a9017f566bb71d0e3f9db9d0d76a392520e4bc79847d0049a
+      a9d487386c6b823525
+    MSGb = lv_cat(Yb,ADb): (length: 70 bytes)
+      4104bb2783a57337e74671f76452876b27839c0ea9e044e3aadaad2e
+      64777ed27a90e80a99438e2f1c072462f2895c6dadf1b43867b92ffb
+      65562b78c793947dcada03414462
+
+
+
+
+
+
+

+B.5.4. Test vector for secret points K +

+
+
+    scalar_mult_vfy(ya,Yb): (length: 32 bytes)
+      8fd12b283805750aeee6151bcd4211a6b71019e8fc416293ade24ed2
+      bce12c39
+    scalar_mult_vfy(yb,Ya): (length: 32 bytes)
+      8fd12b283805750aeee6151bcd4211a6b71019e8fc416293ade24ed2
+      bce12c39
+
+
+
+
+
+
+

+B.5.5. Test vector for ISK calculation initiator/responder +

+
+
+    unordered cat of transcript : (length: 140 bytes)
+      4104b75c1bcda84a0f324aabb7f25cf853ed7fb327c33f23db6aeb32
+      0d81df014649c2ac691925fce0eceac7dbc75eca25e6a1558066a610
+      b4021488279e3b989d52034144614104bb2783a57337e74671f76452
+      876b27839c0ea9e044e3aadaad2e64777ed27a90e80a99438e2f1c07
+      2462f2895c6dadf1b43867b92ffb65562b78c793947dcada03414462
+    DSI = G.DSI_ISK, b'CPaceP256_XMD:SHA-256_SSWU_NU__ISK':
+    (length: 34 bytes)
+      4350616365503235365f584d443a5348412d3235365f535357555f4e
+      555f5f49534b
+    lv_cat(DSI,sid,K)||MSGa||MSGb: (length: 225 bytes)
+      224350616365503235365f584d443a5348412d3235365f535357555f
+      4e555f5f49534b1034b36454cab2e7842c389f7d88ecb7df208fd12b
+      283805750aeee6151bcd4211a6b71019e8fc416293ade24ed2bce12c
+      394104b75c1bcda84a0f324aabb7f25cf853ed7fb327c33f23db6aeb
+      320d81df014649c2ac691925fce0eceac7dbc75eca25e6a1558066a6
+      10b4021488279e3b989d52034144614104bb2783a57337e74671f764
+      52876b27839c0ea9e044e3aadaad2e64777ed27a90e80a99438e2f1c
+      072462f2895c6dadf1b43867b92ffb65562b78c793947dcada034144
+      62
+    ISK result: (length: 32 bytes)
+      7ae1e916606e44652e3c0d7231198af6519226339c241e546afd0bbf
+      48e1c96a
+
+
+
+
+
+
+

+B.5.6. Test vector for ISK calculation parallel execution +

+
+
+    ordered cat of transcript : (length: 142 bytes)
+      6f634104bb2783a57337e74671f76452876b27839c0ea9e044e3aada
+      ad2e64777ed27a90e80a99438e2f1c072462f2895c6dadf1b43867b9
+      2ffb65562b78c793947dcada034144624104b75c1bcda84a0f324aab
+      b7f25cf853ed7fb327c33f23db6aeb320d81df014649c2ac691925fc
+      e0eceac7dbc75eca25e6a1558066a610b4021488279e3b989d520341
+      4461
+    DSI = G.DSI_ISK, b'CPaceP256_XMD:SHA-256_SSWU_NU__ISK':
+    (length: 34 bytes)
+      4350616365503235365f584d443a5348412d3235365f535357555f4e
+      555f5f49534b
+    lv_cat(DSI,sid,K)||o_cat(MSGa,MSGb): (length: 227 bytes)
+      224350616365503235365f584d443a5348412d3235365f535357555f
+      4e555f5f49534b1034b36454cab2e7842c389f7d88ecb7df208fd12b
+      283805750aeee6151bcd4211a6b71019e8fc416293ade24ed2bce12c
+      396f634104bb2783a57337e74671f76452876b27839c0ea9e044e3aa
+      daad2e64777ed27a90e80a99438e2f1c072462f2895c6dadf1b43867
+      b92ffb65562b78c793947dcada034144624104b75c1bcda84a0f324a
+      abb7f25cf853ed7fb327c33f23db6aeb320d81df014649c2ac691925
+      fce0eceac7dbc75eca25e6a1558066a610b4021488279e3b989d5203
+      414461
+    ISK result: (length: 32 bytes)
+      5600a5c5bea5e92695dd68bd33d7f7b58326199c27c9b7326d76e4f9
+      cb2fb276
+
+
+
+
+
+
+

+B.5.7. Corresponding C programming language initializers +

+
+
+const unsigned char tc_PRS[] = {
+ 0x50,0x61,0x73,0x73,0x77,0x6f,0x72,0x64,
+};
+const unsigned char tc_CI[] = {
+ 0x0a,0x41,0x69,0x6e,0x69,0x74,0x69,0x61,0x74,0x6f,0x72,0x0a,
+ 0x42,0x72,0x65,0x73,0x70,0x6f,0x6e,0x64,0x65,0x72,
+};
+const unsigned char tc_sid[] = {
+ 0x34,0xb3,0x64,0x54,0xca,0xb2,0xe7,0x84,0x2c,0x38,0x9f,0x7d,
+ 0x88,0xec,0xb7,0xdf,
+};
+const unsigned char tc_g[] = {
+ 0x04,0x1b,0x51,0x43,0x31,0x14,0xe0,0x96,0xc9,0xd5,0x95,0xf0,
+ 0x95,0x5f,0x57,0x17,0xa7,0x51,0x69,0xaf,0xb9,0x55,0x57,0xf4,
+ 0xa6,0xf5,0x11,0x55,0x03,0x5d,0xee,0x19,0xc7,0x68,0x87,0xbc,
+ 0xe5,0xc7,0xc0,0x54,0xfa,0x1f,0xe4,0x8a,0x4a,0x62,0xc7,0xfb,
+ 0x96,0xdc,0x75,0xe3,0x42,0x59,0xd2,0xf7,0x2b,0x8d,0x41,0xf3,
+ 0x1b,0x8e,0x58,0x6b,0xcd,
+};
+const unsigned char tc_ya[] = {
+ 0x37,0x57,0x4c,0xfb,0xf1,0xb9,0x5f,0xf6,0xa8,0xe2,0xd7,0xbe,
+ 0x46,0x2d,0x4d,0x01,0xe6,0xdd,0xe2,0x61,0x8f,0x34,0xf4,0xde,
+ 0x9d,0xf8,0x69,0xb2,0x4f,0x53,0x2c,0x5d,
+};
+const unsigned char tc_ADa[] = {
+ 0x41,0x44,0x61,
+};
+const unsigned char tc_Ya[] = {
+ 0x04,0xb7,0x5c,0x1b,0xcd,0xa8,0x4a,0x0f,0x32,0x4a,0xab,0xb7,
+ 0xf2,0x5c,0xf8,0x53,0xed,0x7f,0xb3,0x27,0xc3,0x3f,0x23,0xdb,
+ 0x6a,0xeb,0x32,0x0d,0x81,0xdf,0x01,0x46,0x49,0xc2,0xac,0x69,
+ 0x19,0x25,0xfc,0xe0,0xec,0xea,0xc7,0xdb,0xc7,0x5e,0xca,0x25,
+ 0xe6,0xa1,0x55,0x80,0x66,0xa6,0x10,0xb4,0x02,0x14,0x88,0x27,
+ 0x9e,0x3b,0x98,0x9d,0x52,
+};
+const unsigned char tc_yb[] = {
+ 0xe5,0x67,0x2f,0xc9,0xeb,0x4e,0x72,0x1f,0x41,0xd8,0x01,0x81,
+ 0xec,0x4c,0x9f,0xd9,0x88,0x66,0x68,0xac,0xc4,0x80,0x24,0xd3,
+ 0x3c,0x82,0xbb,0x10,0x2a,0xec,0xba,0x52,
+};
+const unsigned char tc_ADb[] = {
+ 0x41,0x44,0x62,
+};
+const unsigned char tc_Yb[] = {
+ 0x04,0xbb,0x27,0x83,0xa5,0x73,0x37,0xe7,0x46,0x71,0xf7,0x64,
+ 0x52,0x87,0x6b,0x27,0x83,0x9c,0x0e,0xa9,0xe0,0x44,0xe3,0xaa,
+ 0xda,0xad,0x2e,0x64,0x77,0x7e,0xd2,0x7a,0x90,0xe8,0x0a,0x99,
+ 0x43,0x8e,0x2f,0x1c,0x07,0x24,0x62,0xf2,0x89,0x5c,0x6d,0xad,
+ 0xf1,0xb4,0x38,0x67,0xb9,0x2f,0xfb,0x65,0x56,0x2b,0x78,0xc7,
+ 0x93,0x94,0x7d,0xca,0xda,
+};
+const unsigned char tc_K[] = {
+ 0x8f,0xd1,0x2b,0x28,0x38,0x05,0x75,0x0a,0xee,0xe6,0x15,0x1b,
+ 0xcd,0x42,0x11,0xa6,0xb7,0x10,0x19,0xe8,0xfc,0x41,0x62,0x93,
+ 0xad,0xe2,0x4e,0xd2,0xbc,0xe1,0x2c,0x39,
+};
+const unsigned char tc_ISK_IR[] = {
+ 0x7a,0xe1,0xe9,0x16,0x60,0x6e,0x44,0x65,0x2e,0x3c,0x0d,0x72,
+ 0x31,0x19,0x8a,0xf6,0x51,0x92,0x26,0x33,0x9c,0x24,0x1e,0x54,
+ 0x6a,0xfd,0x0b,0xbf,0x48,0xe1,0xc9,0x6a,
+};
+const unsigned char tc_ISK_SY[] = {
+ 0x56,0x00,0xa5,0xc5,0xbe,0xa5,0xe9,0x26,0x95,0xdd,0x68,0xbd,
+ 0x33,0xd7,0xf7,0xb5,0x83,0x26,0x19,0x9c,0x27,0xc9,0xb7,0x32,
+ 0x6d,0x76,0xe4,0xf9,0xcb,0x2f,0xb2,0x76,
+};
+
+
+
+
+
+
+

+B.5.8. Test case for scalar_mult_vfy with correct inputs +

+
+
+    s: (length: 32 bytes)
+      f012501c091ff9b99a123fffe571d8bc01e8077ee581362e1bd21399
+      0835643b
+    X: (length: 65 bytes)
+      0424648eb986c2be0af636455cef0550671d6bcd8aa26e0d72ffa1b1
+      fd12ba4e0f78da2b6d2184f31af39e566aef127014b6936c9a37346d
+      10a4ab2514faef5831
+    G.scalar_mult(s,X) (full coordinates): (length: 65 bytes)
+      04f5a191f078c87c36633b78c701751159d56c59f3fe9105b5720673
+      470f303ab925b6a7fd1cdd8f649a21cf36b68d9e9c4a11919a951892
+      519786104b27033757
+    G.scalar_mult_vfy(s,X) (only X-coordinate):
+    (length: 32 bytes)
+      f5a191f078c87c36633b78c701751159d56c59f3fe9105b572067347
+      0f303ab9
+
+
+
+
+
+
+

+B.5.9. Invalid inputs for scalar_mult_vfy +

+

For these test cases scalar_mult_vfy(y,.) MUST return the representation of the neutral element G.I. When including Y_i1 or Y_i2 in MSGa or MSGb the protocol MUST abort.

+
+
+    s: (length: 32 bytes)
+      f012501c091ff9b99a123fffe571d8bc01e8077ee581362e1bd21399
+      0835643b
+    Y_i1: (length: 65 bytes)
+      0424648eb986c2be0af636455cef0550671d6bcd8aa26e0d72ffa1b1
+      fd12ba4e0f78da2b6d2184f31af39e566aef127014b6936c9a37346d
+      10a4ab2514faef5857
+    Y_i2: (length: 1 bytes)
+      00
+    G.scalar_mult_vfy(s,Y_i1) = G.scalar_mult_vfy(s,Y_i2) = G.I
+
+
+
+
+
+
+
+
+

+B.6. Test vector for CPace using group NIST P-384 and hash SHA-384 +

+
+
+

+B.6.1. Test vectors for calculate_generator with group NIST P-384 +

+
+
+  Inputs
+    H   = SHA-384 with input block size 128 bytes.
+    PRS = b'Password' ; ZPAD length: 87 ;
+    DSI = b'CPaceP384_XMD:SHA-384_SSWU_NU_'
+    DST = b'CPaceP384_XMD:SHA-384_SSWU_NU__DST'
+    CI = b'\nAinitiator\nBresponder'
+    CI = 0a41696e69746961746f720a42726573706f6e646572
+    sid = 5b3773aa90e8f23c61563a4b645b276c
+  Outputs
+    generator_string(PRS,G.DSI,CI,sid,H.s_in_bytes):
+    (length: 168 bytes)
+      1e4350616365503338345f584d443a5348412d3338345f535357555f
+      4e555f0850617373776f726457000000000000000000000000000000
+      00000000000000000000000000000000000000000000000000000000
+      00000000000000000000000000000000000000000000000000000000
+      00000000000000000000000000000000160a41696e69746961746f72
+      0a42726573706f6e646572105b3773aa90e8f23c61563a4b645b276c
+    generator g: (length: 97 bytes)
+      04f35a925fe82e54350e80b084a8013b1960cb3f73c49b0c2ae9b523
+      997846ddd14c66f24f62223112cf35b866065f91ad86674cce2a2876
+      84904e49f01287b54666bb518df2ea53cec627fa6e1283f14c6ed4bc
+      d11b33fbb962da3e2e4ff1345c
+
+
+
+
+
+
+

+B.6.2. Test vector for MSGa +

+
+
+  Inputs
+    ADa = b'ADa'
+    ya (big endian): (length: 48 bytes)
+      ef433dd5ad142c860e7cb6400dd315d388d5ec5420c550e9d6f0907f
+      375d988bc4d704837e43561c497e7dd93edcdb9d
+  Outputs
+    Ya: (length: 97 bytes)
+      04fd864c1a81f0e657a8a3f8e4ebafa421da712b6fb98f0abfa139ff
+      971718cab474fa74c6a44b80a46468699280dd5d271252f3b9c05acc
+      93dbd8b939152987cd5a8d1fb7b70c45512c993ec5456cc10f1797c9
+      2fac2f1b7e363478a9ecd79e74
+    Alternative correct value for Ya: g*(-ya):
+    (length: 97 bytes)
+      04fd864c1a81f0e657a8a3f8e4ebafa421da712b6fb98f0abfa139ff
+      971718cab474fa74c6a44b80a46468699280dd5d27edad0c463fa533
+      6c242746c6ead67832a572e04848f3baaed366c13aba933eefe86836
+      cf53d0e481c9cb87571328618b
+    MSGa = lv_cat(Ya,ADa): (length: 102 bytes)
+      6104fd864c1a81f0e657a8a3f8e4ebafa421da712b6fb98f0abfa139
+      ff971718cab474fa74c6a44b80a46468699280dd5d271252f3b9c05a
+      cc93dbd8b939152987cd5a8d1fb7b70c45512c993ec5456cc10f1797
+      c92fac2f1b7e363478a9ecd79e7403414461
+
+
+
+
+
+
+

+B.6.3. Test vector for MSGb +

+
+
+  Inputs
+    ADb = b'ADb'
+    yb (big endian): (length: 48 bytes)
+      50b0e36b95a2edfaa8342b843dddc90b175330f2399c1b36586dedda
+      3c255975f30be6a750f9404fccc62a6323b5e471
+  Outputs
+    Yb: (length: 97 bytes)
+      04822b9874755c51adfdf624101eb4dc12a8ae433750be4fd6f4f7eb
+      f6954ddb57837752a4effa4a5b44627a64b62a2db9d3c9c031c4ad37
+      dbe7bf180d6bcba54feb4e84eeb876ebfa64a85d4c5ac2063dc05ba7
+      26810824c41e1893faa9373a84
+    Alternative correct value for Yb: g*(-yb):
+    (length: 97 bytes)
+      04822b9874755c51adfdf624101eb4dc12a8ae433750be4fd6f4f7eb
+      f6954ddb57837752a4effa4a5b44627a64b62a2db92c363fce3b52c8
+      241840e7f294345ab014b17b11478914059b57a2b3a53df9c13fa458
+      d87ef7db3be1e76c0656c8c57b
+    MSGb = lv_cat(Yb,ADb): (length: 102 bytes)
+      6104822b9874755c51adfdf624101eb4dc12a8ae433750be4fd6f4f7
+      ebf6954ddb57837752a4effa4a5b44627a64b62a2db9d3c9c031c4ad
+      37dbe7bf180d6bcba54feb4e84eeb876ebfa64a85d4c5ac2063dc05b
+      a726810824c41e1893faa9373a8403414462
+
+
+
+
+
+
+

+B.6.4. Test vector for secret points K +

+
+
+    scalar_mult_vfy(ya,Yb): (length: 48 bytes)
+      374290a54e07015baad085b311b18fbae1a20652e137c7c4bd13d565
+      7d8b1ace028eb5acfba8c68d6211a79fff0965c9
+    scalar_mult_vfy(yb,Ya): (length: 48 bytes)
+      374290a54e07015baad085b311b18fbae1a20652e137c7c4bd13d565
+      7d8b1ace028eb5acfba8c68d6211a79fff0965c9
+
+
+
+
+
+
+

+B.6.5. Test vector for ISK calculation initiator/responder +

+
+
+    unordered cat of transcript : (length: 204 bytes)
+      6104fd864c1a81f0e657a8a3f8e4ebafa421da712b6fb98f0abfa139
+      ff971718cab474fa74c6a44b80a46468699280dd5d271252f3b9c05a
+      cc93dbd8b939152987cd5a8d1fb7b70c45512c993ec5456cc10f1797
+      c92fac2f1b7e363478a9ecd79e74034144616104822b9874755c51ad
+      fdf624101eb4dc12a8ae433750be4fd6f4f7ebf6954ddb57837752a4
+      effa4a5b44627a64b62a2db9d3c9c031c4ad37dbe7bf180d6bcba54f
+      eb4e84eeb876ebfa64a85d4c5ac2063dc05ba726810824c41e1893fa
+      a9373a8403414462
+    DSI = G.DSI_ISK, b'CPaceP384_XMD:SHA-384_SSWU_NU__ISK':
+    (length: 34 bytes)
+      4350616365503338345f584d443a5348412d3338345f535357555f4e
+      555f5f49534b
+    lv_cat(DSI,sid,K)||MSGa||MSGb: (length: 305 bytes)
+      224350616365503338345f584d443a5348412d3338345f535357555f
+      4e555f5f49534b105b3773aa90e8f23c61563a4b645b276c30374290
+      a54e07015baad085b311b18fbae1a20652e137c7c4bd13d5657d8b1a
+      ce028eb5acfba8c68d6211a79fff0965c96104fd864c1a81f0e657a8
+      a3f8e4ebafa421da712b6fb98f0abfa139ff971718cab474fa74c6a4
+      4b80a46468699280dd5d271252f3b9c05acc93dbd8b939152987cd5a
+      8d1fb7b70c45512c993ec5456cc10f1797c92fac2f1b7e363478a9ec
+      d79e74034144616104822b9874755c51adfdf624101eb4dc12a8ae43
+      3750be4fd6f4f7ebf6954ddb57837752a4effa4a5b44627a64b62a2d
+      b9d3c9c031c4ad37dbe7bf180d6bcba54feb4e84eeb876ebfa64a85d
+      4c5ac2063dc05ba726810824c41e1893faa9373a8403414462
+    ISK result: (length: 48 bytes)
+      a62d337820ce9cc1195a1adfb3c1efc2d844c0d8c6bc44bd060fe3cd
+      d4ee8d2343aca0168c2b58478354a37d8d8856bd
+
+
+
+
+
+
+

+B.6.6. Test vector for ISK calculation parallel execution +

+
+
+    ordered cat of transcript : (length: 206 bytes)
+      6f636104fd864c1a81f0e657a8a3f8e4ebafa421da712b6fb98f0abf
+      a139ff971718cab474fa74c6a44b80a46468699280dd5d271252f3b9
+      c05acc93dbd8b939152987cd5a8d1fb7b70c45512c993ec5456cc10f
+      1797c92fac2f1b7e363478a9ecd79e74034144616104822b9874755c
+      51adfdf624101eb4dc12a8ae433750be4fd6f4f7ebf6954ddb578377
+      52a4effa4a5b44627a64b62a2db9d3c9c031c4ad37dbe7bf180d6bcb
+      a54feb4e84eeb876ebfa64a85d4c5ac2063dc05ba726810824c41e18
+      93faa9373a8403414462
+    DSI = G.DSI_ISK, b'CPaceP384_XMD:SHA-384_SSWU_NU__ISK':
+    (length: 34 bytes)
+      4350616365503338345f584d443a5348412d3338345f535357555f4e
+      555f5f49534b
+    lv_cat(DSI,sid,K)||o_cat(MSGa,MSGb): (length: 307 bytes)
+      224350616365503338345f584d443a5348412d3338345f535357555f
+      4e555f5f49534b105b3773aa90e8f23c61563a4b645b276c30374290
+      a54e07015baad085b311b18fbae1a20652e137c7c4bd13d5657d8b1a
+      ce028eb5acfba8c68d6211a79fff0965c96f636104fd864c1a81f0e6
+      57a8a3f8e4ebafa421da712b6fb98f0abfa139ff971718cab474fa74
+      c6a44b80a46468699280dd5d271252f3b9c05acc93dbd8b939152987
+      cd5a8d1fb7b70c45512c993ec5456cc10f1797c92fac2f1b7e363478
+      a9ecd79e74034144616104822b9874755c51adfdf624101eb4dc12a8
+      ae433750be4fd6f4f7ebf6954ddb57837752a4effa4a5b44627a64b6
+      2a2db9d3c9c031c4ad37dbe7bf180d6bcba54feb4e84eeb876ebfa64
+      a85d4c5ac2063dc05ba726810824c41e1893faa9373a8403414462
+    ISK result: (length: 48 bytes)
+      eebf988a62b5c854f0ba32822ab45d23329bd1c78c84a4a0e1b40704
+      c99c0a6f6c01c29af5fc6943254b883ce8a65ea1
+
+
+
+
+
+
+

+B.6.7. Corresponding C programming language initializers +

+
+
+const unsigned char tc_PRS[] = {
+ 0x50,0x61,0x73,0x73,0x77,0x6f,0x72,0x64,
+};
+const unsigned char tc_CI[] = {
+ 0x0a,0x41,0x69,0x6e,0x69,0x74,0x69,0x61,0x74,0x6f,0x72,0x0a,
+ 0x42,0x72,0x65,0x73,0x70,0x6f,0x6e,0x64,0x65,0x72,
+};
+const unsigned char tc_sid[] = {
+ 0x5b,0x37,0x73,0xaa,0x90,0xe8,0xf2,0x3c,0x61,0x56,0x3a,0x4b,
+ 0x64,0x5b,0x27,0x6c,
+};
+const unsigned char tc_g[] = {
+ 0x04,0xf3,0x5a,0x92,0x5f,0xe8,0x2e,0x54,0x35,0x0e,0x80,0xb0,
+ 0x84,0xa8,0x01,0x3b,0x19,0x60,0xcb,0x3f,0x73,0xc4,0x9b,0x0c,
+ 0x2a,0xe9,0xb5,0x23,0x99,0x78,0x46,0xdd,0xd1,0x4c,0x66,0xf2,
+ 0x4f,0x62,0x22,0x31,0x12,0xcf,0x35,0xb8,0x66,0x06,0x5f,0x91,
+ 0xad,0x86,0x67,0x4c,0xce,0x2a,0x28,0x76,0x84,0x90,0x4e,0x49,
+ 0xf0,0x12,0x87,0xb5,0x46,0x66,0xbb,0x51,0x8d,0xf2,0xea,0x53,
+ 0xce,0xc6,0x27,0xfa,0x6e,0x12,0x83,0xf1,0x4c,0x6e,0xd4,0xbc,
+ 0xd1,0x1b,0x33,0xfb,0xb9,0x62,0xda,0x3e,0x2e,0x4f,0xf1,0x34,
+ 0x5c,
+};
+const unsigned char tc_ya[] = {
+ 0xef,0x43,0x3d,0xd5,0xad,0x14,0x2c,0x86,0x0e,0x7c,0xb6,0x40,
+ 0x0d,0xd3,0x15,0xd3,0x88,0xd5,0xec,0x54,0x20,0xc5,0x50,0xe9,
+ 0xd6,0xf0,0x90,0x7f,0x37,0x5d,0x98,0x8b,0xc4,0xd7,0x04,0x83,
+ 0x7e,0x43,0x56,0x1c,0x49,0x7e,0x7d,0xd9,0x3e,0xdc,0xdb,0x9d,
+};
+const unsigned char tc_ADa[] = {
+ 0x41,0x44,0x61,
+};
+const unsigned char tc_Ya[] = {
+ 0x04,0xfd,0x86,0x4c,0x1a,0x81,0xf0,0xe6,0x57,0xa8,0xa3,0xf8,
+ 0xe4,0xeb,0xaf,0xa4,0x21,0xda,0x71,0x2b,0x6f,0xb9,0x8f,0x0a,
+ 0xbf,0xa1,0x39,0xff,0x97,0x17,0x18,0xca,0xb4,0x74,0xfa,0x74,
+ 0xc6,0xa4,0x4b,0x80,0xa4,0x64,0x68,0x69,0x92,0x80,0xdd,0x5d,
+ 0x27,0x12,0x52,0xf3,0xb9,0xc0,0x5a,0xcc,0x93,0xdb,0xd8,0xb9,
+ 0x39,0x15,0x29,0x87,0xcd,0x5a,0x8d,0x1f,0xb7,0xb7,0x0c,0x45,
+ 0x51,0x2c,0x99,0x3e,0xc5,0x45,0x6c,0xc1,0x0f,0x17,0x97,0xc9,
+ 0x2f,0xac,0x2f,0x1b,0x7e,0x36,0x34,0x78,0xa9,0xec,0xd7,0x9e,
+ 0x74,
+};
+const unsigned char tc_yb[] = {
+ 0x50,0xb0,0xe3,0x6b,0x95,0xa2,0xed,0xfa,0xa8,0x34,0x2b,0x84,
+ 0x3d,0xdd,0xc9,0x0b,0x17,0x53,0x30,0xf2,0x39,0x9c,0x1b,0x36,
+ 0x58,0x6d,0xed,0xda,0x3c,0x25,0x59,0x75,0xf3,0x0b,0xe6,0xa7,
+ 0x50,0xf9,0x40,0x4f,0xcc,0xc6,0x2a,0x63,0x23,0xb5,0xe4,0x71,
+};
+const unsigned char tc_ADb[] = {
+ 0x41,0x44,0x62,
+};
+const unsigned char tc_Yb[] = {
+ 0x04,0x82,0x2b,0x98,0x74,0x75,0x5c,0x51,0xad,0xfd,0xf6,0x24,
+ 0x10,0x1e,0xb4,0xdc,0x12,0xa8,0xae,0x43,0x37,0x50,0xbe,0x4f,
+ 0xd6,0xf4,0xf7,0xeb,0xf6,0x95,0x4d,0xdb,0x57,0x83,0x77,0x52,
+ 0xa4,0xef,0xfa,0x4a,0x5b,0x44,0x62,0x7a,0x64,0xb6,0x2a,0x2d,
+ 0xb9,0xd3,0xc9,0xc0,0x31,0xc4,0xad,0x37,0xdb,0xe7,0xbf,0x18,
+ 0x0d,0x6b,0xcb,0xa5,0x4f,0xeb,0x4e,0x84,0xee,0xb8,0x76,0xeb,
+ 0xfa,0x64,0xa8,0x5d,0x4c,0x5a,0xc2,0x06,0x3d,0xc0,0x5b,0xa7,
+ 0x26,0x81,0x08,0x24,0xc4,0x1e,0x18,0x93,0xfa,0xa9,0x37,0x3a,
+ 0x84,
+};
+const unsigned char tc_K[] = {
+ 0x37,0x42,0x90,0xa5,0x4e,0x07,0x01,0x5b,0xaa,0xd0,0x85,0xb3,
+ 0x11,0xb1,0x8f,0xba,0xe1,0xa2,0x06,0x52,0xe1,0x37,0xc7,0xc4,
+ 0xbd,0x13,0xd5,0x65,0x7d,0x8b,0x1a,0xce,0x02,0x8e,0xb5,0xac,
+ 0xfb,0xa8,0xc6,0x8d,0x62,0x11,0xa7,0x9f,0xff,0x09,0x65,0xc9,
+};
+const unsigned char tc_ISK_IR[] = {
+ 0xa6,0x2d,0x33,0x78,0x20,0xce,0x9c,0xc1,0x19,0x5a,0x1a,0xdf,
+ 0xb3,0xc1,0xef,0xc2,0xd8,0x44,0xc0,0xd8,0xc6,0xbc,0x44,0xbd,
+ 0x06,0x0f,0xe3,0xcd,0xd4,0xee,0x8d,0x23,0x43,0xac,0xa0,0x16,
+ 0x8c,0x2b,0x58,0x47,0x83,0x54,0xa3,0x7d,0x8d,0x88,0x56,0xbd,
+};
+const unsigned char tc_ISK_SY[] = {
+ 0xee,0xbf,0x98,0x8a,0x62,0xb5,0xc8,0x54,0xf0,0xba,0x32,0x82,
+ 0x2a,0xb4,0x5d,0x23,0x32,0x9b,0xd1,0xc7,0x8c,0x84,0xa4,0xa0,
+ 0xe1,0xb4,0x07,0x04,0xc9,0x9c,0x0a,0x6f,0x6c,0x01,0xc2,0x9a,
+ 0xf5,0xfc,0x69,0x43,0x25,0x4b,0x88,0x3c,0xe8,0xa6,0x5e,0xa1,
+};
+
+
+
+
+
+
+

+B.6.8. Test case for scalar_mult_vfy with correct inputs +

+
+
+    s: (length: 48 bytes)
+      6e8a99a5cdd408eae98e1b8aed286e7b12adbbdac7f2c628d9060ce9
+      2ae0d90bd57a564fd3500fbcce3425dc94ba0ade
+    X: (length: 97 bytes)
+      045b4cd53c4506cc04ba4c44f2762d5d32c3e55df25b8baa5571b165
+      7ad9576efea8259f0684de065a470585b4be876748c7797054f3defe
+      f21b77f83d53bac57c89d52aa4d6dd5872bd281989b138359698009f
+      8ac1f301538badcce9d9f4036e
+    G.scalar_mult(s,X) (full coordinates): (length: 97 bytes)
+      0465c28db05fd9f9a93651c5cc31eae49c4e5246b46489b8f6105873
+      3173a033cda76c3e3ea5352b804e67fdbe2e334be8245dad5c8c993e
+      63bacf0456478f29b71b6c859f13676f84ff150d2741f028f560584a
+      0bdba19a63df62c08949c2fd6d
+    G.scalar_mult_vfy(s,X) (only X-coordinate):
+    (length: 48 bytes)
+      65c28db05fd9f9a93651c5cc31eae49c4e5246b46489b8f610587331
+      73a033cda76c3e3ea5352b804e67fdbe2e334be8
+
+
+
+
+
+
+

+B.6.9. Invalid inputs for scalar_mult_vfy +

+

For these test cases scalar_mult_vfy(y,.) MUST return the representation of the neutral element G.I. When including Y_i1 or Y_i2 in MSGa or MSGb the protocol MUST abort.

+
+
+    s: (length: 48 bytes)
+      6e8a99a5cdd408eae98e1b8aed286e7b12adbbdac7f2c628d9060ce9
+      2ae0d90bd57a564fd3500fbcce3425dc94ba0ade
+    Y_i1: (length: 97 bytes)
+      045b4cd53c4506cc04ba4c44f2762d5d32c3e55df25b8baa5571b165
+      7ad9576efea8259f0684de065a470585b4be876748c7797054f3defe
+      f21b77f83d53bac57c89d52aa4d6dd5872bd281989b138359698009f
+      8ac1f301538badcce9d9f40302
+    Y_i2: (length: 1 bytes)
+      00
+    G.scalar_mult_vfy(s,Y_i1) = G.scalar_mult_vfy(s,Y_i2) = G.I
+
+
+
+
+
+
+
+
+

+B.7. Test vector for CPace using group NIST P-521 and hash SHA-512 +

+
+
+

+B.7.1. Test vectors for calculate_generator with group NIST P-521 +

+
+
+  Inputs
+    H   = SHA-512 with input block size 128 bytes.
+    PRS = b'Password' ; ZPAD length: 87 ;
+    DSI = b'CPaceP521_XMD:SHA-512_SSWU_NU_'
+    DST = b'CPaceP521_XMD:SHA-512_SSWU_NU__DST'
+    CI = b'\nAinitiator\nBresponder'
+    CI = 0a41696e69746961746f720a42726573706f6e646572
+    sid = 7e4b4791d6a8ef019b936c79fb7f2c57
+  Outputs
+    generator_string(PRS,G.DSI,CI,sid,H.s_in_bytes):
+    (length: 168 bytes)
+      1e4350616365503532315f584d443a5348412d3531325f535357555f
+      4e555f0850617373776f726457000000000000000000000000000000
+      00000000000000000000000000000000000000000000000000000000
+      00000000000000000000000000000000000000000000000000000000
+      00000000000000000000000000000000160a41696e69746961746f72
+      0a42726573706f6e646572107e4b4791d6a8ef019b936c79fb7f2c57
+    generator g: (length: 133 bytes)
+      0400dc927958f0b69ccad8fb67ef008905354b58c7c9c92ad50060a9
+      e6afb10437d6ca8a26164e8573702b897275a25d05ed4407af2a3849
+      86dca7e243b92c5dd500d40057012121a9c8e34373fa619f918f7d47
+      9c23f85f0485379ef0f05284398de26653b49a155324c9d7b138be84
+      d0b49bb58e232b7bf697798de6ee8afd6b92b6fa2f
+
+
+
+
+
+
+

+B.7.2. Test vector for MSGa +

+
+
+  Inputs
+    ADa = b'ADa'
+    ya (big endian): (length: 66 bytes)
+      006367e9c2aeff9f1db19af600cca73343d47cbe446cebbd1ccd783f
+      82755a872da86fd0707eb3767c6114f1803deb62d63bdd1e613f67e6
+      3e8c141ee5310e3ee819
+  Outputs
+    Ya: (length: 133 bytes)
+      04003701ec35caafa3dd416cad29ba1774551f9d2ed89f7e1065706d
+      ca230b86a11d02e4cee8b3fde64380d4a05983167d8a2414bc594ad5
+      286c068792ab7ca60ff6ea00919c41c00e789dabc2f42fd94178d7bf
+      d8fbe1aff1c1854b3dafb3a0ea13f5a5fc1703860f022bd271740469
+      bb322b07c179c7c225499b31727c0ea3ee65578634
+    Alternative correct value for Ya: g*(-ya):
+    (length: 133 bytes)
+      04003701ec35caafa3dd416cad29ba1774551f9d2ed89f7e1065706d
+      ca230b86a11d02e4cee8b3fde64380d4a05983167d8a2414bc594ad5
+      286c068792ab7ca60ff6ea016e63be3ff18762543d0bd026be872840
+      27041e500e3e7ab4c2504c5f15ec0a5a03e8fc79f0fdd42d8e8bfb96
+      44cdd4f83e86383ddab664ce8d83f15c119aa879cb
+    MSGa = lv_cat(Ya,ADa): (length: 139 bytes)
+      850104003701ec35caafa3dd416cad29ba1774551f9d2ed89f7e1065
+      706dca230b86a11d02e4cee8b3fde64380d4a05983167d8a2414bc59
+      4ad5286c068792ab7ca60ff6ea00919c41c00e789dabc2f42fd94178
+      d7bfd8fbe1aff1c1854b3dafb3a0ea13f5a5fc1703860f022bd27174
+      0469bb322b07c179c7c225499b31727c0ea3ee6557863403414461
+
+
+
+
+
+
+

+B.7.3. Test vector for MSGb +

+
+
+  Inputs
+    ADb = b'ADb'
+    yb (big endian): (length: 66 bytes)
+      009227bf8dc741dacc9422f8bf3c0e96fce9587bc562eaafe0dc5f6f
+      82f28594e4a6f98553560c62b75fa4abb198cecbbb86ebd41b0ea025
+      4cde78ac68d39a240ae7
+  Outputs
+    Yb: (length: 133 bytes)
+      0400f5cb68bf0117bd1a65412a2bc800af92013f9969cf546e1ea6d3
+      bcf08643fdc482130aec1eecc33a2b5f33600be51295047fa3399fa2
+      82cc1a78de91f3a4e30b5d01a085b453f22bf3dc947386b042e5fc4e
+      c691fee47fe3c3ec6408c22a17c26bc0ab73940910614d6fcee32daf
+      bfd2d340d6e382d71b1fc763d7cec502fbcbcf93b4
+    Alternative correct value for Yb: g*(-yb):
+    (length: 133 bytes)
+      0400f5cb68bf0117bd1a65412a2bc800af92013f9969cf546e1ea6d3
+      bcf08643fdc482130aec1eecc33a2b5f33600be51295047fa3399fa2
+      82cc1a78de91f3a4e30b5d005f7a4bac0dd40c236b8c794fbd1a03b1
+      396e011b801c3c139bf73dd5e83d943f548c6bf6ef9eb290311cd250
+      402d2cbf291c7d28e4e0389c28313afd0434306c4b
+    MSGb = lv_cat(Yb,ADb): (length: 139 bytes)
+      85010400f5cb68bf0117bd1a65412a2bc800af92013f9969cf546e1e
+      a6d3bcf08643fdc482130aec1eecc33a2b5f33600be51295047fa339
+      9fa282cc1a78de91f3a4e30b5d01a085b453f22bf3dc947386b042e5
+      fc4ec691fee47fe3c3ec6408c22a17c26bc0ab73940910614d6fcee3
+      2dafbfd2d340d6e382d71b1fc763d7cec502fbcbcf93b403414462
+
+
+
+
+
+
+

+B.7.4. Test vector for secret points K +

+
+
+    scalar_mult_vfy(ya,Yb): (length: 66 bytes)
+      00503e75e38e012a6dc6f3561980e4cf540dbcff3de3a4a6f09d79c3
+      2cc45764d3a6605eb45df1dc63fb7937b7879f2820da1b3266b69fa0
+      99bf8720dd8f6a07e8ed
+    scalar_mult_vfy(yb,Ya): (length: 66 bytes)
+      00503e75e38e012a6dc6f3561980e4cf540dbcff3de3a4a6f09d79c3
+      2cc45764d3a6605eb45df1dc63fb7937b7879f2820da1b3266b69fa0
+      99bf8720dd8f6a07e8ed
+
+
+
+
+
+
+

+B.7.5. Test vector for ISK calculation initiator/responder +

+
+
+    unordered cat of transcript : (length: 278 bytes)
+      850104003701ec35caafa3dd416cad29ba1774551f9d2ed89f7e1065
+      706dca230b86a11d02e4cee8b3fde64380d4a05983167d8a2414bc59
+      4ad5286c068792ab7ca60ff6ea00919c41c00e789dabc2f42fd94178
+      d7bfd8fbe1aff1c1854b3dafb3a0ea13f5a5fc1703860f022bd27174
+      0469bb322b07c179c7c225499b31727c0ea3ee655786340341446185
+      010400f5cb68bf0117bd1a65412a2bc800af92013f9969cf546e1ea6
+      d3bcf08643fdc482130aec1eecc33a2b5f33600be51295047fa3399f
+      a282cc1a78de91f3a4e30b5d01a085b453f22bf3dc947386b042e5fc
+      4ec691fee47fe3c3ec6408c22a17c26bc0ab73940910614d6fcee32d
+      afbfd2d340d6e382d71b1fc763d7cec502fbcbcf93b403414462
+    DSI = G.DSI_ISK, b'CPaceP521_XMD:SHA-512_SSWU_NU__ISK':
+    (length: 34 bytes)
+      4350616365503532315f584d443a5348412d3531325f535357555f4e
+      555f5f49534b
+    lv_cat(DSI,sid,K)||MSGa||MSGb: (length: 397 bytes)
+      224350616365503532315f584d443a5348412d3531325f535357555f
+      4e555f5f49534b107e4b4791d6a8ef019b936c79fb7f2c574200503e
+      75e38e012a6dc6f3561980e4cf540dbcff3de3a4a6f09d79c32cc457
+      64d3a6605eb45df1dc63fb7937b7879f2820da1b3266b69fa099bf87
+      20dd8f6a07e8ed850104003701ec35caafa3dd416cad29ba1774551f
+      9d2ed89f7e1065706dca230b86a11d02e4cee8b3fde64380d4a05983
+      167d8a2414bc594ad5286c068792ab7ca60ff6ea00919c41c00e789d
+      abc2f42fd94178d7bfd8fbe1aff1c1854b3dafb3a0ea13f5a5fc1703
+      860f022bd271740469bb322b07c179c7c225499b31727c0ea3ee6557
+      86340341446185010400f5cb68bf0117bd1a65412a2bc800af92013f
+      9969cf546e1ea6d3bcf08643fdc482130aec1eecc33a2b5f33600be5
+      1295047fa3399fa282cc1a78de91f3a4e30b5d01a085b453f22bf3dc
+      947386b042e5fc4ec691fee47fe3c3ec6408c22a17c26bc0ab739409
+      10614d6fcee32dafbfd2d340d6e382d71b1fc763d7cec502fbcbcf93
+      b403414462
+    ISK result: (length: 64 bytes)
+      ed208a15af3ef8a67a5cac4acb360d03154570e3b1b1c54867f53a72
+      53cb919d13aa47efc647375be2250cb39ad965afa4ddfcb6be47d586
+      d28c7eef6d654525
+
+
+
+
+
+
+

+B.7.6. Test vector for ISK calculation parallel execution +

+
+
+    ordered cat of transcript : (length: 280 bytes)
+      6f6385010400f5cb68bf0117bd1a65412a2bc800af92013f9969cf54
+      6e1ea6d3bcf08643fdc482130aec1eecc33a2b5f33600be51295047f
+      a3399fa282cc1a78de91f3a4e30b5d01a085b453f22bf3dc947386b0
+      42e5fc4ec691fee47fe3c3ec6408c22a17c26bc0ab73940910614d6f
+      cee32dafbfd2d340d6e382d71b1fc763d7cec502fbcbcf93b4034144
+      62850104003701ec35caafa3dd416cad29ba1774551f9d2ed89f7e10
+      65706dca230b86a11d02e4cee8b3fde64380d4a05983167d8a2414bc
+      594ad5286c068792ab7ca60ff6ea00919c41c00e789dabc2f42fd941
+      78d7bfd8fbe1aff1c1854b3dafb3a0ea13f5a5fc1703860f022bd271
+      740469bb322b07c179c7c225499b31727c0ea3ee6557863403414461
+    DSI = G.DSI_ISK, b'CPaceP521_XMD:SHA-512_SSWU_NU__ISK':
+    (length: 34 bytes)
+      4350616365503532315f584d443a5348412d3531325f535357555f4e
+      555f5f49534b
+    lv_cat(DSI,sid,K)||o_cat(MSGa,MSGb): (length: 399 bytes)
+      224350616365503532315f584d443a5348412d3531325f535357555f
+      4e555f5f49534b107e4b4791d6a8ef019b936c79fb7f2c574200503e
+      75e38e012a6dc6f3561980e4cf540dbcff3de3a4a6f09d79c32cc457
+      64d3a6605eb45df1dc63fb7937b7879f2820da1b3266b69fa099bf87
+      20dd8f6a07e8ed6f6385010400f5cb68bf0117bd1a65412a2bc800af
+      92013f9969cf546e1ea6d3bcf08643fdc482130aec1eecc33a2b5f33
+      600be51295047fa3399fa282cc1a78de91f3a4e30b5d01a085b453f2
+      2bf3dc947386b042e5fc4ec691fee47fe3c3ec6408c22a17c26bc0ab
+      73940910614d6fcee32dafbfd2d340d6e382d71b1fc763d7cec502fb
+      cbcf93b403414462850104003701ec35caafa3dd416cad29ba177455
+      1f9d2ed89f7e1065706dca230b86a11d02e4cee8b3fde64380d4a059
+      83167d8a2414bc594ad5286c068792ab7ca60ff6ea00919c41c00e78
+      9dabc2f42fd94178d7bfd8fbe1aff1c1854b3dafb3a0ea13f5a5fc17
+      03860f022bd271740469bb322b07c179c7c225499b31727c0ea3ee65
+      57863403414461
+    ISK result: (length: 64 bytes)
+      e7b10b6da531d9a8fd47fdd08441e8bb803d16c59a93e366d5cd9a10
+      277bbc543d943182889154704d80f2b0756ed62da87e0eb4e6d07920
+      480100d5e800ca85
+
+
+
+
+
+
+

+B.7.7. Corresponding C programming language initializers +

+
+
+const unsigned char tc_PRS[] = {
+ 0x50,0x61,0x73,0x73,0x77,0x6f,0x72,0x64,
+};
+const unsigned char tc_CI[] = {
+ 0x0a,0x41,0x69,0x6e,0x69,0x74,0x69,0x61,0x74,0x6f,0x72,0x0a,
+ 0x42,0x72,0x65,0x73,0x70,0x6f,0x6e,0x64,0x65,0x72,
+};
+const unsigned char tc_sid[] = {
+ 0x7e,0x4b,0x47,0x91,0xd6,0xa8,0xef,0x01,0x9b,0x93,0x6c,0x79,
+ 0xfb,0x7f,0x2c,0x57,
+};
+const unsigned char tc_g[] = {
+ 0x04,0x00,0xdc,0x92,0x79,0x58,0xf0,0xb6,0x9c,0xca,0xd8,0xfb,
+ 0x67,0xef,0x00,0x89,0x05,0x35,0x4b,0x58,0xc7,0xc9,0xc9,0x2a,
+ 0xd5,0x00,0x60,0xa9,0xe6,0xaf,0xb1,0x04,0x37,0xd6,0xca,0x8a,
+ 0x26,0x16,0x4e,0x85,0x73,0x70,0x2b,0x89,0x72,0x75,0xa2,0x5d,
+ 0x05,0xed,0x44,0x07,0xaf,0x2a,0x38,0x49,0x86,0xdc,0xa7,0xe2,
+ 0x43,0xb9,0x2c,0x5d,0xd5,0x00,0xd4,0x00,0x57,0x01,0x21,0x21,
+ 0xa9,0xc8,0xe3,0x43,0x73,0xfa,0x61,0x9f,0x91,0x8f,0x7d,0x47,
+ 0x9c,0x23,0xf8,0x5f,0x04,0x85,0x37,0x9e,0xf0,0xf0,0x52,0x84,
+ 0x39,0x8d,0xe2,0x66,0x53,0xb4,0x9a,0x15,0x53,0x24,0xc9,0xd7,
+ 0xb1,0x38,0xbe,0x84,0xd0,0xb4,0x9b,0xb5,0x8e,0x23,0x2b,0x7b,
+ 0xf6,0x97,0x79,0x8d,0xe6,0xee,0x8a,0xfd,0x6b,0x92,0xb6,0xfa,
+ 0x2f,
+};
+const unsigned char tc_ya[] = {
+ 0x00,0x63,0x67,0xe9,0xc2,0xae,0xff,0x9f,0x1d,0xb1,0x9a,0xf6,
+ 0x00,0xcc,0xa7,0x33,0x43,0xd4,0x7c,0xbe,0x44,0x6c,0xeb,0xbd,
+ 0x1c,0xcd,0x78,0x3f,0x82,0x75,0x5a,0x87,0x2d,0xa8,0x6f,0xd0,
+ 0x70,0x7e,0xb3,0x76,0x7c,0x61,0x14,0xf1,0x80,0x3d,0xeb,0x62,
+ 0xd6,0x3b,0xdd,0x1e,0x61,0x3f,0x67,0xe6,0x3e,0x8c,0x14,0x1e,
+ 0xe5,0x31,0x0e,0x3e,0xe8,0x19,
+};
+const unsigned char tc_ADa[] = {
+ 0x41,0x44,0x61,
+};
+const unsigned char tc_Ya[] = {
+ 0x04,0x00,0x37,0x01,0xec,0x35,0xca,0xaf,0xa3,0xdd,0x41,0x6c,
+ 0xad,0x29,0xba,0x17,0x74,0x55,0x1f,0x9d,0x2e,0xd8,0x9f,0x7e,
+ 0x10,0x65,0x70,0x6d,0xca,0x23,0x0b,0x86,0xa1,0x1d,0x02,0xe4,
+ 0xce,0xe8,0xb3,0xfd,0xe6,0x43,0x80,0xd4,0xa0,0x59,0x83,0x16,
+ 0x7d,0x8a,0x24,0x14,0xbc,0x59,0x4a,0xd5,0x28,0x6c,0x06,0x87,
+ 0x92,0xab,0x7c,0xa6,0x0f,0xf6,0xea,0x00,0x91,0x9c,0x41,0xc0,
+ 0x0e,0x78,0x9d,0xab,0xc2,0xf4,0x2f,0xd9,0x41,0x78,0xd7,0xbf,
+ 0xd8,0xfb,0xe1,0xaf,0xf1,0xc1,0x85,0x4b,0x3d,0xaf,0xb3,0xa0,
+ 0xea,0x13,0xf5,0xa5,0xfc,0x17,0x03,0x86,0x0f,0x02,0x2b,0xd2,
+ 0x71,0x74,0x04,0x69,0xbb,0x32,0x2b,0x07,0xc1,0x79,0xc7,0xc2,
+ 0x25,0x49,0x9b,0x31,0x72,0x7c,0x0e,0xa3,0xee,0x65,0x57,0x86,
+ 0x34,
+};
+const unsigned char tc_yb[] = {
+ 0x00,0x92,0x27,0xbf,0x8d,0xc7,0x41,0xda,0xcc,0x94,0x22,0xf8,
+ 0xbf,0x3c,0x0e,0x96,0xfc,0xe9,0x58,0x7b,0xc5,0x62,0xea,0xaf,
+ 0xe0,0xdc,0x5f,0x6f,0x82,0xf2,0x85,0x94,0xe4,0xa6,0xf9,0x85,
+ 0x53,0x56,0x0c,0x62,0xb7,0x5f,0xa4,0xab,0xb1,0x98,0xce,0xcb,
+ 0xbb,0x86,0xeb,0xd4,0x1b,0x0e,0xa0,0x25,0x4c,0xde,0x78,0xac,
+ 0x68,0xd3,0x9a,0x24,0x0a,0xe7,
+};
+const unsigned char tc_ADb[] = {
+ 0x41,0x44,0x62,
+};
+const unsigned char tc_Yb[] = {
+ 0x04,0x00,0xf5,0xcb,0x68,0xbf,0x01,0x17,0xbd,0x1a,0x65,0x41,
+ 0x2a,0x2b,0xc8,0x00,0xaf,0x92,0x01,0x3f,0x99,0x69,0xcf,0x54,
+ 0x6e,0x1e,0xa6,0xd3,0xbc,0xf0,0x86,0x43,0xfd,0xc4,0x82,0x13,
+ 0x0a,0xec,0x1e,0xec,0xc3,0x3a,0x2b,0x5f,0x33,0x60,0x0b,0xe5,
+ 0x12,0x95,0x04,0x7f,0xa3,0x39,0x9f,0xa2,0x82,0xcc,0x1a,0x78,
+ 0xde,0x91,0xf3,0xa4,0xe3,0x0b,0x5d,0x01,0xa0,0x85,0xb4,0x53,
+ 0xf2,0x2b,0xf3,0xdc,0x94,0x73,0x86,0xb0,0x42,0xe5,0xfc,0x4e,
+ 0xc6,0x91,0xfe,0xe4,0x7f,0xe3,0xc3,0xec,0x64,0x08,0xc2,0x2a,
+ 0x17,0xc2,0x6b,0xc0,0xab,0x73,0x94,0x09,0x10,0x61,0x4d,0x6f,
+ 0xce,0xe3,0x2d,0xaf,0xbf,0xd2,0xd3,0x40,0xd6,0xe3,0x82,0xd7,
+ 0x1b,0x1f,0xc7,0x63,0xd7,0xce,0xc5,0x02,0xfb,0xcb,0xcf,0x93,
+ 0xb4,
+};
+const unsigned char tc_K[] = {
+ 0x00,0x50,0x3e,0x75,0xe3,0x8e,0x01,0x2a,0x6d,0xc6,0xf3,0x56,
+ 0x19,0x80,0xe4,0xcf,0x54,0x0d,0xbc,0xff,0x3d,0xe3,0xa4,0xa6,
+ 0xf0,0x9d,0x79,0xc3,0x2c,0xc4,0x57,0x64,0xd3,0xa6,0x60,0x5e,
+ 0xb4,0x5d,0xf1,0xdc,0x63,0xfb,0x79,0x37,0xb7,0x87,0x9f,0x28,
+ 0x20,0xda,0x1b,0x32,0x66,0xb6,0x9f,0xa0,0x99,0xbf,0x87,0x20,
+ 0xdd,0x8f,0x6a,0x07,0xe8,0xed,
+};
+const unsigned char tc_ISK_IR[] = {
+ 0xed,0x20,0x8a,0x15,0xaf,0x3e,0xf8,0xa6,0x7a,0x5c,0xac,0x4a,
+ 0xcb,0x36,0x0d,0x03,0x15,0x45,0x70,0xe3,0xb1,0xb1,0xc5,0x48,
+ 0x67,0xf5,0x3a,0x72,0x53,0xcb,0x91,0x9d,0x13,0xaa,0x47,0xef,
+ 0xc6,0x47,0x37,0x5b,0xe2,0x25,0x0c,0xb3,0x9a,0xd9,0x65,0xaf,
+ 0xa4,0xdd,0xfc,0xb6,0xbe,0x47,0xd5,0x86,0xd2,0x8c,0x7e,0xef,
+ 0x6d,0x65,0x45,0x25,
+};
+const unsigned char tc_ISK_SY[] = {
+ 0xe7,0xb1,0x0b,0x6d,0xa5,0x31,0xd9,0xa8,0xfd,0x47,0xfd,0xd0,
+ 0x84,0x41,0xe8,0xbb,0x80,0x3d,0x16,0xc5,0x9a,0x93,0xe3,0x66,
+ 0xd5,0xcd,0x9a,0x10,0x27,0x7b,0xbc,0x54,0x3d,0x94,0x31,0x82,
+ 0x88,0x91,0x54,0x70,0x4d,0x80,0xf2,0xb0,0x75,0x6e,0xd6,0x2d,
+ 0xa8,0x7e,0x0e,0xb4,0xe6,0xd0,0x79,0x20,0x48,0x01,0x00,0xd5,
+ 0xe8,0x00,0xca,0x85,
+};
+
+
+
+
+
+
+

+B.7.8. Test case for scalar_mult_vfy with correct inputs +

+
+
+    s: (length: 66 bytes)
+      0182dd7925f1753419e4bf83429763acd37d64000cd5a175edf53a15
+      87dd986bc95acc1506991702b6ba1a9ee2458fee8efc00198cf0088c
+      480965ef65ff2048b856
+    X: (length: 133 bytes)
+      0400dc5078b24c4af1620cc10fbecc6cd8cf1cab0b011efb73c782f2
+      26dc21c7ca7eb406be74a69ecba5b4a87c07cfc6e687b4beca9a6eda
+      c95940a3b4120573b26a80005e697833b0ba285fce7b3f1f25243008
+      860b8f1de710a0dcc05b0d20341efe90eb2bcca26797c2d85ae6ca74
+      c00696cb1b13e40bda15b27964d7670576647bfab9
+    G.scalar_mult(s,X) (full coordinates): (length: 133 bytes)
+      040122f88ce73ec5aa2d1c8c5d04148760c3d97ba87daa10d8cb8bb7
+      c73cf6e951fc922721bf1437995cfb13e132a78beb86389e60d3517c
+      df6d99a8a2d6db19ef27bd0055af9e8ddcf337ce0a7c22a9c8099bc4
+      a44faeded1eb72effd26e4f322217b67d60b944b267b3df5046078fd
+      577f1785728f49b241fd5e8c83223a994a2d219281
+    G.scalar_mult_vfy(s,X) (only X-coordinate):
+    (length: 66 bytes)
+      0122f88ce73ec5aa2d1c8c5d04148760c3d97ba87daa10d8cb8bb7c7
+      3cf6e951fc922721bf1437995cfb13e132a78beb86389e60d3517cdf
+      6d99a8a2d6db19ef27bd
+
+
+
+
+
+
+

+B.7.9. Invalid inputs for scalar_mult_vfy +

+

For these test cases scalar_mult_vfy(y,.) MUST return the representation of the neutral element G.I. When including Y_i1 or Y_i2 in MSGa or MSGb the protocol MUST abort.

+
+
+    s: (length: 66 bytes)
+      0182dd7925f1753419e4bf83429763acd37d64000cd5a175edf53a15
+      87dd986bc95acc1506991702b6ba1a9ee2458fee8efc00198cf0088c
+      480965ef65ff2048b856
+    Y_i1: (length: 133 bytes)
+      0400dc5078b24c4af1620cc10fbecc6cd8cf1cab0b011efb73c782f2
+      26dc21c7ca7eb406be74a69ecba5b4a87c07cfc6e687b4beca9a6eda
+      c95940a3b4120573b26a80005e697833b0ba285fce7b3f1f25243008
+      860b8f1de710a0dcc05b0d20341efe90eb2bcca26797c2d85ae6ca74
+      c00696cb1b13e40bda15b27964d7670576647bfaf9
+    Y_i2: (length: 1 bytes)
+      00
+    G.scalar_mult_vfy(s,Y_i1) = G.scalar_mult_vfy(s,Y_i2) = G.I
+
+
+
+
+
+
+
+
+
+
+

+Authors' Addresses +

+
+
Michel Abdalla
+
DFINITY - Zurich
+ +
+
+
Bjoern Haase
+
Endress + Hauser Liquid Analysis - Gerlingen
+ +
+
+
Julia Hesse
+
IBM Research Europe - Zurich
+ +
+
+
+ + + diff --git a/draft-irtf-cfrg-cpace-11/draft-irtf-cfrg-cpace.txt b/draft-irtf-cfrg-cpace-11/draft-irtf-cfrg-cpace.txt new file mode 100644 index 0000000..6bd8a90 --- /dev/null +++ b/draft-irtf-cfrg-cpace-11/draft-irtf-cfrg-cpace.txt @@ -0,0 +1,3275 @@ + + + + +Network Working Group M. Abdalla +Internet-Draft DFINITY - Zurich +Intended status: Informational B. Haase +Expires: 28 September 2024 Endress + Hauser Liquid Analysis - Gerlingen + J. Hesse + IBM Research Europe - Zurich + 27 March 2024 + + + CPace, a balanced composable PAKE + draft-irtf-cfrg-cpace-latest + +Abstract + + This document describes CPace which is a protocol that allows two + parties that share a low-entropy secret (password) to derive a strong + shared key without disclosing the secret to offline dictionary + attacks. The CPace protocol was tailored for constrained devices and + can be used on groups of prime- and non-prime order. + +Discussion Venues + + This note is to be removed before publishing as an RFC. + + Discussion of this document takes place on the Crypto Forum Research + Group mailing list (cfrg@ietf.org), which is archived at + https://mailarchive.ietf.org/arch/search/?email_list=cfrg. + + Source for this draft and an issue tracker can be found at + https://github.com/cfrg/draft-irtf-cfrg-cpace. + +Status of This Memo + + This Internet-Draft is submitted in full conformance with the + provisions of BCP 78 and BCP 79. + + Internet-Drafts are working documents of the Internet Engineering + Task Force (IETF). Note that other groups may also distribute + working documents as Internet-Drafts. The list of current Internet- + Drafts is at https://datatracker.ietf.org/drafts/current/. + + Internet-Drafts are draft documents valid for a maximum of six months + and may be updated, replaced, or obsoleted by other documents at any + time. It is inappropriate to use Internet-Drafts as reference + material or to cite them other than as "work in progress." + + This Internet-Draft will expire on 28 September 2024. + +Copyright Notice + + Copyright (c) 2024 IETF Trust and the persons identified as the + document authors. All rights reserved. + + This document is subject to BCP 78 and the IETF Trust's Legal + Provisions Relating to IETF Documents (https://trustee.ietf.org/ + license-info) in effect on the date of publication of this document. + Please review these documents carefully, as they describe your rights + and restrictions with respect to this document. Code Components + extracted from this document must include Revised BSD License text as + described in Section 4.e of the Trust Legal Provisions and are + provided without warranty as described in the Revised BSD License. + +Table of Contents + + 1. Introduction + 1.1. Outline of this document + 2. Requirements Notation + 3. High-level application perspective + 3.1. Optional CPace inputs + 3.2. Responsibilities of the application layer + 4. CPace cipher suites + 5. Definitions and notation + 5.1. Hash function H + 5.2. Group environment G + 5.3. Notation for string operations + 5.4. Notation for group operations + 6. The CPace protocol + 6.1. Protocol flow + 6.2. CPace protocol instructions + 7. Implementation of recommended CPace cipher suites + 7.1. Common function for computing generators + 7.2. CPace group objects G_X25519 and G_X448 for + single-coordinate Ladders on Montgomery curves + 7.2.1. Verification tests + 7.3. CPace group objects G_Ristretto255 and G_Decaf448 for + prime-order group abstractions + 7.3.1. Verification tests + 7.4. CPace group objects for curves in Short-Weierstrass + representation + 7.4.1. Curves and associated functions + 7.4.2. Suitable encode_to_curve methods + 7.4.3. Definition of the group environment G for + Short-Weierstrass curves + 7.4.4. Verification tests + 8. Implementation verification + 9. Security Considerations + 9.1. Party identifiers and relay attacks + 9.2. Network message encoding and hashing protocol transcripts + 9.3. Key derivation + 9.4. Key confirmation + 9.5. Sampling of scalars + 9.6. Preconditions for using the simplified CPace specification + from Section 7.2 + 9.7. Nonce values + 9.8. Side channel attacks + 9.9. Quantum computers + 10. IANA Considerations + 11. Acknowledgements + 12. References + 12.1. Normative References + 12.2. Informative References + Appendix A. CPace function definitions + A.1. Definition and test vectors for string utility functions + A.1.1. prepend_len function + A.1.2. prepend_len test vectors + A.1.3. lv_cat function + A.1.4. Testvector for lv_cat() + A.1.5. Examples for messages not obtained from a lv_cat-based + encoding + A.2. Definition of generator_string function. + A.3. Definitions and test vector ordered concatenation + A.3.1. Definitions for lexiographical ordering + A.3.2. Definitions for ordered concatenation + A.3.3. Test vectors ordered concatenation + A.4. Decoding and Encoding functions according to RFC7748 + A.5. Elligator 2 reference implementation + Appendix B. Test vectors + B.1. Test vector for CPace using group X25519 and hash SHA-512 + B.1.1. Test vectors for calculate_generator with group X25519 + B.1.2. Test vector for MSGa + B.1.3. Test vector for MSGb + B.1.4. Test vector for secret points K + B.1.5. Test vector for ISK calculation initiator/responder + B.1.6. Test vector for ISK calculation parallel execution + B.1.7. Corresponding C programming language initializers + B.1.8. Test vectors for G_X25519.scalar_mult_vfy: low order + points + B.2. Test vector for CPace using group X448 and hash SHAKE-256 + B.2.1. Test vectors for calculate_generator with group X448 + B.2.2. Test vector for MSGa + B.2.3. Test vector for MSGb + B.2.4. Test vector for secret points K + B.2.5. Test vector for ISK calculation initiator/responder + B.2.6. Test vector for ISK calculation parallel execution + B.2.7. Corresponding C programming language initializers + B.2.8. Test vectors for G_X448.scalar_mult_vfy: low order + points + B.3. Test vector for CPace using group ristretto255 and hash + SHA-512 + B.3.1. Test vectors for calculate_generator with group + ristretto255 + B.3.2. Test vector for MSGa + B.3.3. Test vector for MSGb + B.3.4. Test vector for secret points K + B.3.5. Test vector for ISK calculation initiator/responder + B.3.6. Test vector for ISK calculation parallel execution + B.3.7. Corresponding C programming language initializers + B.3.8. Test case for scalar_mult with valid inputs + B.3.9. Invalid inputs for scalar_mult_vfy + B.4. Test vector for CPace using group decaf448 and hash + SHAKE-256 + B.4.1. Test vectors for calculate_generator with group + decaf448 + B.4.2. Test vector for MSGa + B.4.3. Test vector for MSGb + B.4.4. Test vector for secret points K + B.4.5. Test vector for ISK calculation initiator/responder + B.4.6. Test vector for ISK calculation parallel execution + B.4.7. Corresponding C programming language initializers + B.4.8. Test case for scalar_mult with valid inputs + B.4.9. Invalid inputs for scalar_mult_vfy + B.5. Test vector for CPace using group NIST P-256 and hash + SHA-256 + B.5.1. Test vectors for calculate_generator with group NIST + P-256 + B.5.2. Test vector for MSGa + B.5.3. Test vector for MSGb + B.5.4. Test vector for secret points K + B.5.5. Test vector for ISK calculation initiator/responder + B.5.6. Test vector for ISK calculation parallel execution + B.5.7. Corresponding C programming language initializers + B.5.8. Test case for scalar_mult_vfy with correct inputs + B.5.9. Invalid inputs for scalar_mult_vfy + B.6. Test vector for CPace using group NIST P-384 and hash + SHA-384 + B.6.1. Test vectors for calculate_generator with group NIST + P-384 + B.6.2. Test vector for MSGa + B.6.3. Test vector for MSGb + B.6.4. Test vector for secret points K + B.6.5. Test vector for ISK calculation initiator/responder + B.6.6. Test vector for ISK calculation parallel execution + B.6.7. Corresponding C programming language initializers + B.6.8. Test case for scalar_mult_vfy with correct inputs + B.6.9. Invalid inputs for scalar_mult_vfy + B.7. Test vector for CPace using group NIST P-521 and hash + SHA-512 + B.7.1. Test vectors for calculate_generator with group NIST + P-521 + B.7.2. Test vector for MSGa + B.7.3. Test vector for MSGb + B.7.4. Test vector for secret points K + B.7.5. Test vector for ISK calculation initiator/responder + B.7.6. Test vector for ISK calculation parallel execution + B.7.7. Corresponding C programming language initializers + B.7.8. Test case for scalar_mult_vfy with correct inputs + B.7.9. Invalid inputs for scalar_mult_vfy + Authors' Addresses + +1. Introduction + + This document describes CPace which is a balanced Password- + Authenticated-Key-Establishment (PAKE) protocol for two parties where + both parties derive a cryptographic key of high entropy from a shared + secret of low-entropy. CPace protects the passwords against offline + dictionary attacks by requiring adversaries to actively interact with + a protocol party and by allowing for at most one single password + guess per active interaction. + + The CPace design was tailored considering the following main + objectives: + + * Efficiency: Deployment of CPace is feasible on resource- + constrained devices. + + * Versatility: CPace supports different application scenarios via + versatile input formats, and by supporting applications with and + without clear initiator and responder roles. + + * Implementation error resistance: CPace aims at avoiding common + implementation pitfalls already by-design, such as avoiding + incentives for insecure execution-time speed optimizations. For + smooth integration into different cryptographic library + ecosystems, this document provides a variety of cipher suites. + + * Post-quantum annoyance: CPace comes with mitigations with respect + to adversaries that become capable of breaking the discrete + logarithm problem on elliptic curves. + +1.1. Outline of this document + + * Section 3 describes the expected properties of an application + using CPace, and discusses in particular which application-level + aspects are relevant for CPace's security. + + * Section 4 gives an overview of the recommended cipher suites for + CPace which were optimized for different types of cryptographic + library ecosystems. + + * Section 5 introduces the notation used throughout this document. + + * Section 6 specifies the CPace protocol. + + * The final section provides explicit reference implementations and + test vectors of all of the functions defined for CPace in the + appendix. + + As this document is primarily written for implementers and + application designers, we would like to refer the theory-inclined + reader to the scientific paper [AHH21] which covers the detailed + security analysis of the different CPace instantiations as defined in + this document via the cipher suites. + +2. Requirements Notation + + The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", + "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and + "OPTIONAL" in this document are to be interpreted as described in + BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all + capitals, as shown here. + +3. High-level application perspective + + CPace enables balanced password-authenticated key establishment. + CPace requires a shared secret octet string, the password-related + string (PRS), is available for both parties A and B. PRS can be a + low-entropy secret itself, for instance a clear-text password encoded + according to [RFC8265], or any string derived from a common secret, + for instance by use of a password-based key derivation function. + + Applications with clients and servers where the server side is + storing account and password information in its persistent memory are + recommended to use augmented PAKE protocols such as OPAQUE + [I-D.irtf-cfrg-opaque]. + + In the course of the CPace protocol, A sends one message MSGa to B + and B sends one message MSGb to A. CPace does not mandate any + ordering of these two messages. We use the term "initiator- + responder" for CPace where A always speaks first, and the term + "symmetric" setting where anyone can speak first. + + CPace's output is an intermediate session key (ISK), but any party + might abort in case of an invalid received message. A and B will + produce the same ISK value only if both sides did initiate the + protocol using the same protocol inputs, specifically the same PRS + string and the same value for the optional input parameters CI, ADa, + ADb and sid that will be specified in the upcoming sections. + + The naming of ISK key as "intermediate" session key highlights the + fact that it is RECOMMENDED that applications process ISK by use of a + suitable strong key derivation function KDF (such as defined in + [RFC5869]) before using the key in a higher-level protocol. + +3.1. Optional CPace inputs + + For accomodating different application settings, CPace offers the + following OPTIONAL inputs, i.e. inputs which MAY also be the empty + string: + + * Channel identifier (CI). CI can be used to bind a session key + exchanged with CPace to a specific networking channel which + interconnects the protocol parties. Both parties are required to + have the same view of CI. CI will not be publicly sent on the + wire and may also include confidential information. + + * Associated data fields (ADa and ADb). These fields can be used to + authenticate public associated data alongside the CPace protocol. + The values ADa (and ADb, respectively) are guaranteed to be + authenticated in case both parties agree on a key. + + ADa and ADb can for instance include party identities or protocol + version information of an application protocol (e.g. to avoid + downgrade attacks). + + If party identities are not encoded as part of CI, party + identities SHOULD be included in ADa and ADb (see Section 9.1). + In a setting with clear initiator and responder roles, identity + information in ADa sent by the initiator can be used by the + responder for choosing the right PRS string (respectively + password) for this identity. + + * Session identifier (sid). CPace comes with a security analysis + [AHH21] in the framework of universal composability. This + framework allows for modular analysis of a larger application + protocol which uses CPace as a building block. For such analysis + the CPace protocol is bound to a specific session of the larger + protocol by use of a sid string that is globally unique. As a + result, when used with a unique sid, CPace instances remain secure + when running concurrently with other CPace instances, and even + arbitrary other protocols. + + For this reason, it is RECOMMENDED that applications establish a + unique session identifier sid prior to running the CPace protocol. + This can be implemented by concatenating random bytes produced by + A with random bytes produced by B. If such preceding round is not + an option but parties are assigned clear initiator-responder + roles, it is RECOMMENDED to let the initiator A choose a fresh + random sid and send it to B together with the first message. If a + sid string is used it SHOULD HAVE a length of at least 8 bytes. + +3.2. Responsibilities of the application layer + + The following tasks are out of the scope of this document and left to + the application layer + + * Setup phase: + + - The application layer is responsible for the handshake that + makes parties agree on a common CPace cipher suite. + + - The application layer needs to specify how to encode the CPace + byte strings Ya / Yb and ADa / ADb defined in section Section 6 + for transfer over the network. For CPace it is RECOMMENDED to + encode network messages by using MSGa = lv_cat(Ya,ADa) and MSGb + = lv_cat(Yb,ADb) using the length-value concatenation function + lv_cat speficied in Section 5.3. This document provides test + vectors for lv_cat-encoded messages. Alternative network + encodings, e.g., the encoding method used for the client hello + and server hello messages of the TLS protocol, MAY be used when + considering the guidance given in Section 9. + + * This document does not specify which encodings applications use + for the mandatory PRS input and the optional inputs CI, sid, ADa + and ADb. If PRS is a clear-text password or an octet string + derived from a clear-text password, e.g. by use of a key- + derivation function, the clear-text password SHOULD BE encoded + according to [RFC8265]. + + * The application needs to settle whether CPace is used in the + initiator-responder or the symmetric setting, as in the symmetric + setting transcripts must be generated using ordered string + concatenation. In this document we will provide test vectors for + both, initiator-responder and symmetric settings. + +4. CPace cipher suites + + In the setup phase of CPace, both communication partners need to + agree on a common cipher suite. Cipher suites consist of a + combination of a hash function H and an elliptic curve environment G. + + For naming cipher suites we use the convention "CPACE-G-H". We + RECOMMEND the following cipher suites: + + * CPACE-X25519-SHA512. This suite uses the group environment + G_X25519 defined in Section 7.2 and SHA-512 as hash function. + This cipher suite comes with the smallest messages on the wire and + a low computational cost. + + * CPACE-P256_XMD:SHA-256_SSWU_NU_-SHA256. This suite instantiates + the group environment G as specified in Section 7.4 using the + encode_to_curve function P256_XMD:SHA-256_SSWU_NU_ from [RFC9380] + on curve NIST-P256, and hash function SHA-256. + + The following RECOMMENDED cipher suites provide higher security + margins. + + * CPACE-X448-SHAKE256. This suite uses the group environment G_X448 + defined in Section 7.2 and SHAKE-256 as hash function. + + * CPACE-P384_XMD:SHA-384_SSWU_NU_-SHA384. This suite instantiates G + as specified in Section 7.4 using the encode_to_curve function + P384_XMD:SHA-384_SSWU_NU_ from [RFC9380] on curve NIST-P384 with H + = SHA-384. + + * CPACE-P521_XMD:SHA-512_SSWU_NU_-SHA512. This suite instantiates G + as specified in Section 7.4 using the encode_to_curve function + P521_XMD:SHA-512_SSWU_NU_ from [RFC9380] on curve NIST-P521 with H + = SHA-512. + + CPace can also securely be implemented using the cipher suites CPACE- + RISTR255-SHA512 and CPACE-DECAF448-SHAKE256 defined in Section 7.3. + Section 9 gives guidance on how to implement CPace on further + elliptic curves. + +5. Definitions and notation + +5.1. Hash function H + + Common choices for H are SHA-512 [RFC6234] or SHAKE-256 [FIPS202]. + (I.e. the hash function outputs octet strings, and not group + elements.) For considering both variable-output-length hashes and + fixed-output-length hashes, we use the following convention. In case + that the hash function is specified for a fixed-size output, we + define H.hash(m,l) such that it returns the first l octets of the + output. + + We use the following notation for referring to the specific + properties of a hash function H: + + * H.hash(m,l) is a function that operates on an input octet string m + and returns a hashing result of l octets. + + * H.b_in_bytes denotes the minimum output size in bytes for + collision resistance for the security level target of the hash + function. E.g. H.b_in_bytes = 64 for SHA-512 and SHAKE-256 and + H.b_in_bytes = 32 for SHA-256 and SHAKE-128. We use the notation + H.hash(m) = H.hash(m, H.b_in_bytes) and let the hash operation + output the default length if no explicit length parameter is + given. + + * H.bmax_in_bytes denotes the _maximum_ output size in octets + supported by the hash function. In case of fixed-size hashes such + as SHA-256, this is the same as H.b_in_bytes, while there is no + such limit for hash functions such as SHAKE-256. + + * H.s_in_bytes denotes the _input block size_ used by H. This + number denotes the maximum number of bytes that can be processed + in a single block before applying the compression function or + permutation becomes necessary. (See also [RFC2104] for the + corresponding block size concepts). For instance, for SHA-512 the + input block size s_in_bytes is 128 as the compression function can + process up to 128 bytes, while for SHAKE-256 the input block size + amounts to 136 bytes before the permutation of the sponge state + needs to be applied. + +5.2. Group environment G + + The group environment G specifies an elliptic curve group (also + denoted G for convenience) and associated constants and functions as + detailed below. In this document we use additive notation for the + group operation. + + * G.calculate_generator(H,PRS,CI,sid) denotes a function that + outputs a representation of a generator (referred to as + "generator" from now on) of the group which is derived from input + octet strings PRS, CI, and sid and with the help of the hash + function H. + + * G.sample_scalar() is a function returning a representation of an + integer (referred to as "scalar" from now on) appropriate as a + private Diffie-Hellman key for the group. + + * G.scalar_mult(y,g) is a function operating on a scalar y and a + group element g. It returns an octet string representation of the + group element Y = g*y. + + * G.I denotes a unique octet string representation of the neutral + element of the group. G.I is used for detecting and signaling + certain error conditions. + + * G.scalar_mult_vfy(y,g) is a function operating on a scalar y and a + group element g. It returns an octet string representation of the + group element g*y. Additionally, scalar_mult_vfy specifies + validity conditions for y,g and g*y and outputs G.I in case they + are not met. + + * G.DSI denotes a domain-separation identifier octet string which + SHALL be uniquely identifying the group environment G. + +5.3. Notation for string operations + + * bytes1 || bytes2 and denotes concatenation of octet strings. + + * len(S) denotes the number of octets in an octet string S. + + * nil denotes an empty octet string, i.e., len(nil) = 0. + + * This document uses quotation marks "" both for general language + (e.g. for citation of notation used in other documents) and as + syntax for specifying octet strings as in b"CPace25519". + + We use a preceeding lower-case letter b"" in front of the + quotation marks if a character sequence is representing an octet + string sequence. I.e. we use the notation for byte string + representations with single-byte ASCII character encodings from + the python programming language. + + * prepend_len(octet_string) denotes the octet sequence that is + obtained from prepending the length of the octet string to the + string itself. The length shall be prepended by using an LEB128 + encoding of the length. This will result in a single-byte + encoding for values below 128. (Test vectors and reference + implementations for prepend_len and the LEB128 encodings are given + in the appendix.) + + * lv_cat(a0,a1, ...) is the "length-value" encoding function which + returns the concatenation of the input strings with an encoding of + their respective length prepended. E.g. lv_cat(a0,a1) returns + prepend_len(a0) || prepend_len(a1). The detailed specification of + lv_cat and a reference implementations are given in the appendix. + + * network_encode(Y,AD) denotes the function specified by the + application layer that outputs an octet string encoding of the + input octet strings Y and AD for transfer on the network. The + implementation of MSG = network_encode(Y,AD) SHALL allow the + receiver party to parse MSG for the individual subcomponents Y and + AD. For CPace we RECOMMEND to implement network_encode(Y,AD) as + network_encode(Y,AD) = lv_cat(Y,AD). + + Other encodings, such as the network encoding used for the client- + hello and server-hello messages in TLS MAY also be used when + following the guidance given in the security consideration + section. + + * sample_random_bytes(n) denotes a function that returns n octets, + each of which is to be independently sampled from an uniform + distribution between 0 and 255. + + * zero_bytes(n) denotes a function that returns n octets with value + 0. + + * o_cat(bytes1,bytes2) denotes a function for ordered concatenation + of octet strings. It places the lexiographically larger octet + string first and prepends the two bytes from the octet string + b"oc" to the result. (Explicit reference code for this function + is given in the appendix.) + + * transcript(MSGa,MSGb) denotes function outputing a string for the + protocol transcript with messages MSGa and MSGb. In applications + where CPace is used without clear initiator and responder roles, + i.e. where the ordering of messages is not enforced by the + protocol flow, transcript(MSGa,MSGb) = o_cat(MSGa,MSGb) SHALL be + used. In the initiator-responder setting transcript(MSGa,MSGb) + SHALL BE implemented such that the later message is appended to + the earlier message, i.e., transcript(MSGa,MSGb) = MSGa||MSGb if + MSGa is sent first. + +5.4. Notation for group operations + + We use additive notation for the group, i.e., X*2 denotes the element + that is obtained by computing X+X, for group element X and group + operation +. + +6. The CPace protocol + + CPace is a one round protocol between two parties, A and B. At + invocation, A and B are provisioned with PRS,G,H and OPTIONAL + CI,sid,ADa (for A) and CI,sid,ADb (for B). A sends a message MSGa to + B. MSGa contains the public share Ya and OPTIONAL associated data + ADa (i.e. an ADa field that MAY have a length of 0 bytes). Likewise, + B sends a message MSGb to A. MSGb contains the public share Yb and + OPTIONAL associated data ADb (i.e. an ADb field that MAY have a + length of 0 bytes). Both A and B use the received messages for + deriving a shared intermediate session key, ISK. + +6.1. Protocol flow + + Optional parameters and messages are denoted with []. + + public: G, H + + A: PRS,[ADa],[CI],[sid] B: PRS,[ADb],[CI],[sid] + --------------------------------------- + compute Ya | Ya,[ADa] | compute Yb + |----------------->| + | Yb,[ADb] | + verify inputs |<-----------------| verify inputs + derive ISK | | derive ISK + --------------------------------------- + output ISK output ISK + +6.2. CPace protocol instructions + + A computes a generator g = G.calculate_generator(H,PRS,CI,sid), + scalar ya = G.sample_scalar() and group element Ya = G.scalar_mult + (ya,g). A then transmits MSGa = network_encode(Ya, ADa) with + optional associated data ADa to B. + + B computes a generator g = G.calculate_generator(H,PRS,CI,sid), + scalar yb = G.sample_scalar() and group element Yb = + G.scalar_mult(yb,g). B sends MSGb = network_encode(Yb, ADb) with + optional associated data ADb to A. + + Upon reception of MSGa, B checks that MSGa was properly generated in + conformity with the chosen encoding of network messages (notably + correct length fields). If this parsing fails, then B MUST abort. + (Testvectors of examples for invalid messages when using lv_cat() as + network_encode function for CPace are given in the appendix.) B then + computes K = G.scalar_mult_vfy(yb,Ya). B MUST abort if K=G.I. + Otherwise B calculates ISK = H.hash(lv_cat(G.DSI || b"_ISK", sid, + K)||transcript(MSGa, MSGb)). B returns ISK and terminates. + + Likewise upon reception of MSGb, A parses MSGb for Yb and ADb and + checks for a valid encoding. If this parsing fails, then A MUST + abort. A then computes K = G.scalar_mult_vfy(ya,Yb). A MUST abort + if K=G.I. Otherwise A calculates ISK = H.hash(lv_cat(G.DSI || + b"_ISK", sid, K) || transcript(MSGa, MSGb)). A returns ISK and + terminates. + + The session key ISK returned by A and B is identical if and only if + the supplied input parameters PRS, CI and sid match on both sides and + transcript view (containing of MSGa and MSGb) of both parties match. + + (Note that in case of a symmetric protocol execution without clear + initiator/responder roles, transcript(MSGa, MSGb) needs to be + implemented using ordered concatenation for generating a matching + view by both parties.) + +7. Implementation of recommended CPace cipher suites + +7.1. Common function for computing generators + + The different cipher suites for CPace defined in the upcoming + sections share the same method for deterministically combining the + individual strings PRS, CI, sid and the domain-separation identifier + DSI to a generator string that we describe here. + + * generator_string(DSI, PRS, CI, sid, s_in_bytes) denotes a function + that returns the string lv_cat(DSI, PRS, zero_bytes(len_zpad), CI, + sid). + + * len_zpad = MAX(0, s_in_bytes - len(prepend_len(PRS)) - + len(prepend_len(G.DSI)) - 1) + + The zero padding of length len_zpad is designed such that the + encoding of DSI and PRS together with the zero padding field + completely fills at least the first input block (of length + s_in_bytes) of the hash. As a result for the common case of short + PRS the number of bytes to hash becomes independent of the actual + length of the password (PRS). (A reference implementation and test + vectors are provided in the appendix.) + + The introduction of a zero-padding within the generator string also + helps mitigating attacks of a side-channel adversary that analyzes + correlations between publicly known variable information with a short + low-entropy PRS string. Note that the hash of the first block is + intentionally made independent of session-specific inputs, such as + sid or CI and that there is no limitation regarding the maximum + length of the PRS string. + +7.2. CPace group objects G_X25519 and G_X448 for single-coordinate + Ladders on Montgomery curves + + In this section we consider the case of CPace when using the X25519 + and X448 Diffie-Hellman functions from [RFC7748] operating on the + Montgomery curves Curve25519 and Curve448 [RFC7748]. CPace + implementations using single-coordinate ladders on further Montgomery + curves SHALL use the definitions in line with the specifications for + X25519 and X448 and review the guidance given in Section 9. + + For the group environment G_X25519 the following definitions apply: + + * G_X25519.field_size_bytes = 32 + + * G_X25519.field_size_bits = 255 + + * G_X25519.sample_scalar() = sample_random_bytes(G.field_size_bytes) + + * G_X25519.scalar_mult(y,g) = G.scalar_mult_vfy(y,g) = X25519(y,g) + + * G_X25519.I = zero_bytes(G.field_size_bytes) + + * G_X25519.DSI = b"CPace255" + + CPace cipher suites using G_X25519 MUST use a hash function producing + at least H.b_max_in_bytes >= 32 bytes of output. It is RECOMMENDED + to use G_X25519 in combination with SHA-512. + + For X448 the following definitions apply: + + * G_X448.field_size_bytes = 56 + + * G_X448.field_size_bits = 448 + + * G_X448.sample_scalar() = sample_random_bytes(G.field_size_bytes) + + * G_X448.scalar_mult(y,g) = G.scalar_mult_vfy(y,g) = X448(y,g) + + * G_X448.I = zero_bytes(G.field_size_bytes) + + * G_X448.DSI = b"CPace448" + + CPace cipher suites using G_X448 MUST use a hash function producing + at least H.b_max_in_bytes >= 56 bytes of output. It is RECOMMENDED + to use G_X448 in combination with SHAKE-256. + + For both G_X448 and G_X25519 the G.calculate_generator(H, PRS,sid,CI) + function shall be implemented as follows. + + * First gen_str = generator_string(G.DSI,PRS,CI,sid, H.s_in_bytes) + SHALL BE calculated using the input block size of the chosen hash + function. + + * This string SHALL then BE hashed to the required length + gen_str_hash = H.hash(gen_str, G.field_size_bytes). Note that + this implies that the permissible output length H.maxb_in_bytes + MUST BE larger or equal to the field size of the group G for + making a hashing function suitable. + + * This result is then considered as a field coordinate using the u = + decodeUCoordinate(gen_str_hash, G.field_size_bits) function from + [RFC7748] which we repeat in the appendix for convenience. + + * The result point g is then calculated as (g,v) = + map_to_curve_elligator2(u) using the function from [RFC9380]. + Note that the v coordinate produced by the map_to_curve_elligator2 + function is not required for CPace and discarded. The appendix + repeats the definitions from [RFC9380] for convenience. + + In the appendix we show sage code that can be used as reference + implementation. + +7.2.1. Verification tests + + For single-coordinate Montgomery ladders on Montgomery curves + verification tests according to Section 8 SHALL check for proper + handling of the abort conditions, when a party is receiving u + coordinate values that encode a low-order point on either the curve + or the quadratic twist. + + In addition to that in case of G_X25519 the tests SHALL also verify + that the implementation of G.scalar_mult_vfy(y,g) produces the + expected results for non-canonical u coordinate values with bit #255 + set, which may also encode low-order points. + + Corresponding test vectors are provided in the appendix. + +7.3. CPace group objects G_Ristretto255 and G_Decaf448 for prime-order + group abstractions + + In this section we consider the case of CPace using the Ristretto255 + and Decaf448 group abstractions + [I-D.draft-irtf-cfrg-ristretto255-decaf448]. These abstractions + define an encode and decode function, group operations using an + internal encoding and an element-derivation function that maps a byte + string to a group element. With the group abstractions there is a + distinction between an internal representation of group elements and + an external encoding of the same group element. In order to + distinguish between these different representations, we prepend an + underscore before values using the internal representation within + this section. + + For Ristretto255 the following definitions apply: + + * G_Ristretto255.DSI = b"CPaceRistretto255" + + * G_Ristretto255.field_size_bytes = 32 + + * G_Ristretto255.group_size_bits = 252 + + * G_Ristretto255.group_order = 2^252 + + 27742317777372353535851937790883648493 + + CPace cipher suites using G_Ristretto255 MUST use a hash function + producing at least H.b_max_in_bytes >= 64 bytes of output. It is + RECOMMENDED to use G_Ristretto255 in combination with SHA-512. + + For decaf448 the following definitions apply: + + * G_Decaf448.DSI = b"CPaceDecaf448" + + * G_Decaf448.field_size_bytes = 56 + + * G_Decaf448.group_size_bits = 445 + + * G_Decaf448.group_order = l = 2^446 - + 1381806680989511535200738674851542 + 6880336692474882178609894547503885 + + CPace cipher suites using G_Decaf448 MUST use a hash function + producing at least H.b_max_in_bytes >= 112 bytes of output. It is + RECOMMENDED to use G_Decaf448 in combination with SHAKE-256. + + For both abstractions the following definitions apply: + + * It is RECOMMENDED to implement G.sample_scalar() as follows. + + - Set scalar = sample_random_bytes(G.group_size_bytes). + + - Then clear the most significant bits larger than + G.group_size_bits. + + - Interpret the result as the little-endian encoding of an + integer value and return the result. + + * Alternatively, if G.sample_scalar() is not implemented according + to the above recommendation, it SHALL be implemented using uniform + sampling between 1 and (G.group_order - 1). Note that the more + complex uniform sampling process can provide a larger side-channel + attack surface for embedded systems in hostile environments. + + * G.scalar_mult(y,_g) SHALL operate on a scalar y and a group + element _g in the internal representation of the group abstraction + environment. It returns the value Y = encode((_g) * y), i.e. it + returns a value using the public encoding. + + * G.I = is the public encoding representation of the identity + element. + + * G.scalar_mult_vfy(y,X) operates on a value using the public + encoding and a scalar and is implemented as follows. If the + decode(X) function fails, it returns G.I. Otherwise it returns + encode( decode(X) * y ). + + * The G.calculate_generator(H, PRS,sid,CI) function SHALL return a + decoded point and SHALL BE implemented as follows. + + - First gen_str = generator_string(G.DSI,PRS,CI,sid, + H.s_in_bytes) is calculated using the input block size of the + chosen hash function. + + - This string is then hashed to the required length gen_str_hash + = H.hash(gen_str, 2 * G.field_size_bytes). Note that this + implies that the permissible output length H.maxb_in_bytes MUST + BE larger or equal to twice the field size of the group G for + making a hash function suitable. + + - Finally the internal representation of the generator _g is + calculated as _g = element_derivation(gen_str_hash) using the + element derivation function from the abstraction. + + Note that with these definitions the scalar_mult function operates on + a decoded point _g and returns an encoded point, while the + scalar_mult_vfy(y,X) function operates on an encoded point X (and + also returns an encoded point). + +7.3.1. Verification tests + + For group abstractions verification tests according to Section 8 + SHALL check for proper handling of the abort conditions, when a party + is receiving encodings of the neutral element or receives an octet + string that does not decode to a valid group element. + +7.4. CPace group objects for curves in Short-Weierstrass representation + + The group environment objects G defined in this section for use with + Short-Weierstrass curves, are parametrized by the choice of an + elliptic curve and by choice of a suitable encode_to_curve function. + encode_to_curve must map an octet string to a point on the curve. + +7.4.1. Curves and associated functions + + Elliptic curves in Short-Weierstrass form are considered in + [IEEE1363]. [IEEE1363] allows for both, curves of prime and non- + prime order. However, for the procedures described in this section + any suitable group MUST BE of prime order. + + The specification for the group environment objects specified in this + section closely follow the ECKAS-DH1 method from [IEEE1363]. I.e. we + use the same methods and encodings and protocol substeps as employed + in the TLS [RFC5246] [RFC8446] protocol family. + + For CPace only the uncompressed full-coordinate encodings from [SEC1] + (x and y coordinate) SHOULD be used. Commonly used curve groups are + specified in [SEC2] and [RFC5639]. A typical representative of such + a Short-Weierstrass curve is NIST-P256. Point verification as used + in ECKAS-DH1 is described in Annex A.16.10. of [IEEE1363]. + + For deriving Diffie-Hellman shared secrets ECKAS-DH1 from [IEEE1363] + specifies the use of an ECSVDP-DH method. We use ECSVDP-DH in + combination with the identy map such that it either returns "error" + or the x-coordinate of the Diffie-Hellman result point as shared + secret in big endian format (fixed length output by FE2OSP without + truncating leading zeros). + +7.4.2. Suitable encode_to_curve methods + + All the encode_to_curve methods specified in [RFC9380] are suitable + for CPace. For Short-Weierstrass curves it is RECOMMENDED to use the + non-uniform variant of the SSWU mapping primitive from [RFC9380] if a + SSWU mapping is available for the chosen curve. (We recommend non- + uniform maps in order to give implementations the flexibility to opt + for x-coordinate-only scalar multiplication algorithms.) + +7.4.3. Definition of the group environment G for Short-Weierstrass + curves + + In this paragraph we use the following notation for defining the + group object G for a selected curve and encode_to_curve method: + + * With G.group_order we denote the order of the elliptic curve which + MUST BE a prime. + + * With is_valid(X) we denote a method which operates on an octet + stream according to [SEC1] of a point on the group and returns + true if the point is valid and returns false otherwise. This + is_valid(X) method SHALL be implemented according to Annex + A.16.10. of [IEEE1363]. I.e. it shall return false if X encodes + either the neutral element on the group or does not form a valid + encoding of a point on the group. + + * With encode_to_curve(str,DST) we denote a mapping function from + [RFC9380]. I.e. a function that maps octet string str to a point + on the group using the domain separation tag DST. [RFC9380] + considers both, uniform and non-uniform mappings based on several + different strategies. It is RECOMMENDED to use the nonuniform + variant of the SSWU mapping primitive within [RFC9380]. + + * G.DSI denotes a domain-separation identifier octet string. G.DSI + which SHALL BE obtained by the concatenation of b"CPace" and the + associated name of the cipher suite used for the encode_to_curve + function as specified in [RFC9380]. E.g. when using the map with + the name P384_XMD:SHA-384_SSWU_NU_ on curve NIST-P384 the + resulting value SHALL BE G.DSI = b"CPaceP384_XMD:SHA- + 384_SSWU_NU_". + + Using the above definitions, the CPace functions required for the + group object G are defined as follows. + + * G.DST denotes the domain-separation tag value to use in + conjunction with the encode_to_curve function from [RFC9380]. + G.DST shall be obtained by concatenating G.DSI and b"_DST". + + * G.sample_scalar() SHALL return a value between 1 and + (G.group_order - 1). The sampling SHALL BE indistinguishable from + uniform random selection between 1 and (G.group_order - 1). It is + RECOMMENDED to use a constant-time rejection sampling algorithm + for converting a uniform bitstring to a uniform value between 1 + and (G.group_order - 1). + + * G.calculate_generator(H, PRS,sid,CI) function SHALL be implemented + as follows. + + - First gen_str = generator_string(G.DSI,PRS,CI,sid, + H.s_in_bytes) is calculated. + + - Then the output of a call to encode_to_curve(gen_str, G.DST) is + returned, using the selected suite from [RFC9380]. + + * G.scalar_mult(s,X) is a function that operates on a scalar s and + an input point X. The input X shall use the same encoding as + produced by the G.calculate_generator method above. + G.scalar_mult(s,X) SHALL return an encoding of either the point + X*s or the point X*(-s) according to [SEC1]. Implementations + SHOULD use the full-coordinate format without compression, as + important protocols such as TLS 1.3 removed support for + compression. Implementations of scalar_mult(s,X) MAY output + either X*s or X*(-s) as both points X*s and X*(-s) have the same + x-coordinate and result in the same Diffie-Hellman shared secrets + K. (This allows implementations to opt for x-coordinate-only + scalar multiplication algorithms.) + + * G.scalar_mult_vfy(s,X) merges verification of point X according to + [IEEE1363] A.16.10. and the the ECSVDP-DH procedure from + [IEEE1363]. It SHALL BE implemented as follows: + + - If is_valid(X) = False then G.scalar_mult_vfy(s,X) SHALL return + "error" as specified in [IEEE1363] A.16.10 and 7.2.1. + + - Otherwise G.scalar_mult_vfy(s,X) SHALL return the result of the + ECSVDP-DH procedure from [IEEE1363] (section 7.2.1). I.e. it + shall either return "error" (in case that X*s is the neutral + element) or the secret shared value "z" (otherwise). "z" SHALL + be encoded by using the big-endian encoding of the x-coordinate + of the result point X*s according to [SEC1]. + + * We represent the neutral element G.I by using the representation + of the "error" result case from [IEEE1363] as used in the + G.scalar_mult_vfy method above. + +7.4.4. Verification tests + + For Short-Weierstrass curves verification tests according to + Section 8 SHALL check for proper handling of the abort conditions, + when a party is receiving an encoding of the point at infinity and an + encoding of a point not on the group. + +8. Implementation verification + + Any CPace implementation MUST be tested against invalid or weak point + attacks. Implementation MUST be verified to abort upon conditions + where G.scalar_mult_vfy functions outputs G.I. For testing an + implementation it is RECOMMENDED to include weak or invalid point + encodings within MSGa and MSGb and introduce this in a protocol run. + It SHALL be verified that the abort condition is properly handled. + + Moreover regarding the network format any implementation MUST be + tested with respect to invalid encodings of MSGa and MSGb. E.g. when + lv_cat is used as network format for encoding MSGa and MSGb, the sum + of the prepended lengths of the fields must be verified to match the + actual length of the message. Tests SHALL verify that a party aborts + in case that incorrectly encoded messages are received. + + Corresponding test vectors are given in the appendix for all + recommended cipher suites. + +9. Security Considerations + + A security proof of CPace is found in [AHH21]. This proof covers all + recommended cipher suites included in this document. In the + following sections we describe how to protect CPace against several + attack families, such as relay-, length extension- or side channel + attacks. We also describe aspects to consider when deviating from + recommended cipher suites. + +9.1. Party identifiers and relay attacks + + If unique strings identifying the protocol partners are included + either as part of the channel identifier CI, the session id sid or + the associated data fields ADa, ADb, the ISK will provide implicit + authentication also regarding the party identities. Incorporating + party identifier strings is important for fending off relay attacks. + Such attacks become relevant in a setting where several parties, say, + A, B and C, share the same password PRS. An adversary might relay + messages from a honest user A, who aims at interacting with user B, + to a party C instead. If no party identifier strings are used, and B + and C use the same PRS value, A might be establishing a common ISK + key with C while assuming to interact with party B. Including and + checking party identifiers can fend off such relay attacks. + +9.2. Network message encoding and hashing protocol transcripts + + It is RECOMMENDED to encode the (Ya,ADa) and (Yb,ADb) fields on the + network by using network_encode(Y,AD) = lv_cat(Y,AD). I.e. we + RECOMMEND to prepend an encoding of the length of the subfields. + Prepending the length of all variable-size input strings results in a + so-called prefix-free encoding of transcript strings, using + terminology introduced in [CDMP05]. This property allows for + disregarding length-extension imperfections that come with the + commonly used Merkle-Damgard hash function constructions such as + SHA256 and SHA512. + + Other alternative network encoding formats which prepend an encoding + of the length of variable-size data fields in the protocol messages + are equally suitable. This includes, e.g., the type-length-value + format specified in the DER encoding standard (X.690) or the protocol + message encoding used in the TLS protocol family for the TLS client- + hello or server-hello messages. + + In case that an application uses another form of network message + encoding which is not prefix-free, the guidance given in [CDMP05] + SHOULD BE considered (e.g. by replacing hash functions with the HMAC + constructions from[RFC2104]). + +9.3. Key derivation + + Although already K is a shared value, it MUST NOT itself be used as + an application key. Instead, ISK MUST BE used. Leakage of K to an + adversary can lead to offline dictionary attacks. + + As noted already in Section 6 it is RECOMMENDED to process ISK by use + of a suitable strong key derivation function KDF (such as defined in + [RFC5869]) first, before using the key in a higher-level protocol. + +9.4. Key confirmation + + In many applications it is advisable to add an explicit key + confirmation round after the CPace protocol flow. However, as some + applications might only require implicit authentication and as + explicit authentication messages are already a built-in feature in + many higher-level protocols (e.g. TLS 1.3) the CPace protocol + described here does not mandate use of a key confirmation on the + level of the CPace sub-protocol. + + Already without explicit key confirmation, CPace enjoys weak forward + security under the sCDH and sSDH assumptions [AHH21]. With added + explicit confirmation, CPace enjoys perfect forward security also + under the strong sCDH and sSDH assumptions [AHH21]. + + Note that in [ABKLX21] it was shown that an idealized variant of + CPace also enjoys perfect forward security without explicit key + confirmation. However this proof does not explicitly cover the + recommended cipher suites in this document and requires the stronger + assumption of an algebraic adversary model. For this reason, we + recommend adding explicit key confirmation if perfect forward + security is required. + + When implementing explicit key confirmation, it is recommended to use + an appropriate message-authentication code (MAC) such as HMAC + [RFC2104] or CMAC [RFC4493] using a key mac_key derived from ISK. + + One suitable option that works also in the parallel setting without + message ordering is to proceed as follows. + + * First calculate mac_key as mac_key = H.hash(b"CPaceMac" || ISK). + + * Then let each party send an authenticator tag Ta, Tb that is + calculated over the protocol message that it has sent previously. + I.e. let party A calculate its transmitted authentication code Ta + as Ta = MAC(mac_key, MSGa) and let party B calculate its + transmitted authentication code Tb as Tb = MAC(mac_key, MSGb). + + * Let the receiving party check the remote authentication tag for + the correct value and abort in case that it's incorrect. + +9.5. Sampling of scalars + + For curves over fields F_q where q is a prime close to a power of + two, we recommend sampling scalars as a uniform bit string of length + field_size_bits. We do so in order to reduce both, complexity of the + implementation and the attack surface with respect to side-channels + for embedded systems in hostile environments. The effect of non- + uniform sampling on security was demonstrated to be begnin in [AHH21] + for the case of Curve25519 and Curve448. This analysis however does + not transfer to most curves in Short-Weierstrass form. + + As a result, we recommend rejection sampling if G is as in + Section 7.4. Alternatively an algorithm designed allong the lines of + the hash_to_field() function from [RFC9380] can also be used. There + oversampling to an integer significantly larger than the curve order + is followed by a modular reduction to the group order. + +9.6. Preconditions for using the simplified CPace specification from + Section 7.2 + + The security of the algorithms used for the recommended cipher suites + for the Montgomery curves Curve25519 and Curve448 in Section 7.2 rely + on the following properties [AHH21]: + + * The curve has order (p * c) with p prime and c a small cofactor. + Also the curve's quadratic twist must be of order (p' * c') with + p' prime and c' a cofactor. + + * The cofactor c of the curve MUST BE EQUAL to or an integer + multiple of the cofactor c' of the curve's quadratic twist. Also, + importantly, the implementation of the scalar_mult and + scalar_mult_vfy functions must ensure that all scalars actually + used for the group operation are integer multiples of c (e.g. such + as asserted by the specification of the decodeScalar functions in + [RFC7748]). + + * Both field order q and group order p MUST BE close to a power of + two along the lines of [AHH21], Appendix E. Otherwise the + simplified scalar sampling specified in Section 7.2 needs to be + changed. + + * The representation of the neutral element G.I MUST BE the same for + both, the curve and its twist. + + * The implementation of G.scalar_mult_vfy(y,X) MUST map all c low- + order points on the curve and all c' low-order points on the twist + to G.I. + + Algorithms for curves other than the ones recommended here can be + based on the principles from Section 7.2 given that the above + properties hold. + +9.7. Nonce values + + Secret scalars ya and yb MUST NOT be reused. Values for sid SHOULD + NOT be reused since the composability guarantees established by the + simulation-based proof rely on the uniqueness of session ids [AHH21]. + + If CPace is used in a concurrent system, it is RECOMMENDED that a + unique sid is generated by the higher-level protocol and passed to + CPace. One suitable option is that sid is generated by concatenating + ephemeral random strings contributed by both parties. + +9.8. Side channel attacks + + All state-of-the art methods for realizing constant-time execution + SHOULD be used. Special care is RECOMMENDED specifically for + elliptic curves in Short-Weierstrass form as important standard + documents including [IEEE1363] describe curve operations with non- + constant-time algorithms. + + In case that side channel attacks are to be considered practical for + a given application, it is RECOMMENDED to pay special attention on + computing the secret generator G.calculate_generator(PRS,CI,sid). + The most critical substep to consider might be the processing of the + first block of the hash that includes the PRS string. The zero- + padding introduced when hashing the sensitive PRS string can be + expected to make the task for a side-channel attack somewhat more + complex. Still this feature alone is not sufficient for ruling out + power analysis attacks. + + Even though the calculate_generator operation might be considered to + form the primary target for side-channel attacks as information on + long-term secrets might be exposed, also the subsequent operations on + ephemeral values, such as scalar sampling and scalar multiplication + should be protected from side-channels. + +9.9. Quantum computers + + CPace is proven secure under the hardness of the strong computational + Simultaneous Diffie-Hellmann (sSDH) and strong computational Diffie- + Hellmann (sCDH) assumptions in the group G (as defined in [AHH21]). + These assumptions are not expected to hold any longer when large- + scale quantum computers (LSQC) are available. Still, even in case + that LSQC emerge, it is reasonable to assume that discrete-logarithm + computations will remain costly. CPace with ephemeral session id + values sid forces the adversary to solve one computational Diffie- + Hellman problem per password guess [ES21]. In this sense, using the + wording suggested by Steve Thomas on the CFRG mailing list, CPace is + "quantum-annoying". + +10. IANA Considerations + + No IANA action is required. + +11. Acknowledgements + + We would like to thank the participants on the CFRG list for comments + and advice. Any comment and advice is appreciated. + +12. References + +12.1. Normative References + + [I-D.draft-irtf-cfrg-ristretto255-decaf448] + de Valence, H., Grigg, J., Hamburg, M., Lovecruft, I., + Tankersley, G., and F. Valsorda, "The ristretto255 and + decaf448 Groups", Work in Progress, Internet-Draft, draft- + irtf-cfrg-ristretto255-decaf448-08, 5 September 2023, + . + + [I-D.irtf-cfrg-opaque] + Bourdrez, D., Krawczyk, H., Lewi, K., and C. A. Wood, "The + OPAQUE Augmented PAKE Protocol", Work in Progress, + Internet-Draft, draft-irtf-cfrg-opaque-14, 24 March 2024, + . + + [IEEE1363] "Standard Specifications for Public Key Cryptography, IEEE + 1363", 2000. + + [RFC2119] Bradner, S., "Key words for use in RFCs to Indicate + Requirement Levels", BCP 14, RFC 2119, + DOI 10.17487/RFC2119, March 1997, + . + + [RFC7748] Langley, A., Hamburg, M., and S. Turner, "Elliptic Curves + for Security", RFC 7748, DOI 10.17487/RFC7748, January + 2016, . + + [RFC8174] Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC + 2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174, + May 2017, . + + [SEC1] Standards for Efficient Cryptography Group (SECG), "SEC 1: + Elliptic Curve Cryptography", May 2009, + . + +12.2. Informative References + + [ABKLX21] Abdalla, M., Barbosa, M., Katz, J., Loss, J., and J. Xu, + "Algebraic Adversaries in the Universal Composability + Framework.", n.d., . + + [AHH21] Abdalla, M., Haase, B., and J. Hesse, "Security analysis + of CPace", n.d., . + + [CDMP05] Coron, J.-S., Dodis, Y., Malinaud, C., and P. Puniya, + "Merkle-Damgaard Revisited: How to Construct a Hash + Function", In Advances in Cryptology - CRYPTO 2005, + pages 430-448, DOI 10.1007/11535218_26, 2005, + . + + [ES21] Eaton, E. and D. Stebila, "The 'quantum annoying' property + of password-authenticated key exchange protocols.", n.d., + . + + [FIPS202] National Institute of Standards and Technology (NIST), + "SHA-3 Standard: Permutation-Based Hash and Extendable- + Output Functions", August 2015, + . + + [RFC2104] Krawczyk, H., Bellare, M., and R. Canetti, "HMAC: Keyed- + Hashing for Message Authentication", RFC 2104, + DOI 10.17487/RFC2104, February 1997, + . + + [RFC4493] Song, JH., Poovendran, R., Lee, J., and T. Iwata, "The + AES-CMAC Algorithm", RFC 4493, DOI 10.17487/RFC4493, June + 2006, . + + [RFC5246] Dierks, T. and E. Rescorla, "The Transport Layer Security + (TLS) Protocol Version 1.2", RFC 5246, + DOI 10.17487/RFC5246, August 2008, + . + + [RFC5639] Lochter, M. and J. Merkle, "Elliptic Curve Cryptography + (ECC) Brainpool Standard Curves and Curve Generation", + RFC 5639, DOI 10.17487/RFC5639, March 2010, + . + + [RFC5869] Krawczyk, H. and P. Eronen, "HMAC-based Extract-and-Expand + Key Derivation Function (HKDF)", RFC 5869, + DOI 10.17487/RFC5869, May 2010, + . + + [RFC6234] Eastlake 3rd, D. and T. Hansen, "US Secure Hash Algorithms + (SHA and SHA-based HMAC and HKDF)", RFC 6234, + DOI 10.17487/RFC6234, May 2011, + . + + [RFC8265] Saint-Andre, P. and A. Melnikov, "Preparation, + Enforcement, and Comparison of Internationalized Strings + Representing Usernames and Passwords", RFC 8265, + DOI 10.17487/RFC8265, October 2017, + . + + [RFC8446] Rescorla, E., "The Transport Layer Security (TLS) Protocol + Version 1.3", RFC 8446, DOI 10.17487/RFC8446, August 2018, + . + + [RFC9380] Faz-Hernandez, A., Scott, S., Sullivan, N., Wahby, R. S., + and C. A. Wood, "Hashing to Elliptic Curves", RFC 9380, + DOI 10.17487/RFC9380, August 2023, + . + + [SEC2] Standards for Efficient Cryptography Group (SECG), "SEC 2: + Recommended Elliptic Curve Domain Parameters", January + 2010, . + +Appendix A. CPace function definitions + +A.1. Definition and test vectors for string utility functions + +A.1.1. prepend_len function + + def prepend_len(data): + "prepend LEB128 encoding of length" + length = len(data) + length_encoded = b"" + while True: + if length < 128: + length_encoded += bytes([length]) + else: + length_encoded += bytes([(length & 0x7f) + 0x80]) + length = int(length >> 7) + if length == 0: + break; + return length_encoded + data + +A.1.2. prepend_len test vectors + + prepend_len(b""): (length: 1 bytes) + 00 + prepend_len(b"1234"): (length: 5 bytes) + 0431323334 + prepend_len(bytes(range(127))): (length: 128 bytes) + 7f000102030405060708090a0b0c0d0e0f101112131415161718191a1b + 1c1d1e1f202122232425262728292a2b2c2d2e2f303132333435363738 + 393a3b3c3d3e3f404142434445464748494a4b4c4d4e4f505152535455 + 565758595a5b5c5d5e5f606162636465666768696a6b6c6d6e6f707172 + 737475767778797a7b7c7d7e + prepend_len(bytes(range(128))): (length: 130 bytes) + 8001000102030405060708090a0b0c0d0e0f101112131415161718191a + 1b1c1d1e1f202122232425262728292a2b2c2d2e2f3031323334353637 + 38393a3b3c3d3e3f404142434445464748494a4b4c4d4e4f5051525354 + 55565758595a5b5c5d5e5f606162636465666768696a6b6c6d6e6f7071 + 72737475767778797a7b7c7d7e7f + +A.1.3. lv_cat function + + def lv_cat(*args): + result = b"" + for arg in args: + result += prepend_len(arg) + return result + +A.1.4. Testvector for lv_cat() + + lv_cat(b"1234",b"5",b"",b"6789"): (length: 13 bytes) + 04313233340135000436373839 + +A.1.5. Examples for messages not obtained from a lv_cat-based encoding + + The following messages are examples which have invalid encoded length + fields. I.e. they are examples where parsing for the sum of the + length of subfields as expected for a message generated using + lv_cat(Y,AD) does not give the correct length of the message. + Parties MUST abort upon reception of such invalid messages as MSGa or + MSGb. + + Inv_MSG1 not encoded by lv_cat: (length: 3 bytes) + ffffff + Inv_MSG2 not encoded by lv_cat: (length: 3 bytes) + ffff03 + Inv_MSG3 not encoded by lv_cat: (length: 4 bytes) + 00ffff03 + Inv_MSG4 not encoded by lv_cat: (length: 4 bytes) + 00ffffff + +A.2. Definition of generator_string function. + + def generator_string(DSI,PRS,CI,sid,s_in_bytes): + # Concat all input fields with prepended length information. + # Add zero padding in the first hash block after DSI and PRS. + len_zpad = max(0,s_in_bytes - 1 - len(prepend_len(PRS)) + - len(prepend_len(DSI))) + return lv_cat(DSI, PRS, zero_bytes(len_zpad), + CI, sid) + +A.3. Definitions and test vector ordered concatenation + +A.3.1. Definitions for lexiographical ordering + + For ordered concatenation lexiographical ordering of byte sequences + is used: + + def lexiographically_larger(bytes1,bytes2): + "Returns True if bytes1 > bytes2 using lexiographical ordering." + min_len = min (len(bytes1), len(bytes2)) + for m in range(min_len): + if bytes1[m] > bytes2[m]: + return True; + elif bytes1[m] < bytes2[m]: + return False; + return len(bytes1) > len(bytes2) + +A.3.2. Definitions for ordered concatenation + + With the above definition of lexiographical ordering ordered + concatenation is specified as follows. + + def o_cat(bytes1,bytes2): + if lexiographically_larger(bytes1,bytes2): + return b"oc" + bytes1 + bytes2 + else: + return b"oc" + bytes2 + bytes1 + +A.3.3. Test vectors ordered concatenation + + string comparison for o_cat: + lexiographically_larger(b"\0", b"\0\0") == False + lexiographically_larger(b"\1", b"\0\0") == True + lexiographically_larger(b"\0\0", b"\0") == True + lexiographically_larger(b"\0\0", b"\1") == False + lexiographically_larger(b"\0\1", b"\1") == False + lexiographically_larger(b"ABCD", b"BCD") == False + + o_cat(b"ABCD",b"BCD"): (length: 9 bytes) + 6f6342434441424344 + o_cat(b"BCD",b"ABCDE"): (length: 10 bytes) + 6f634243444142434445 + +A.4. Decoding and Encoding functions according to RFC7748 + + def decodeLittleEndian(b, bits): + return sum([b[i] << 8*i for i in range((bits+7)/8)]) + + def decodeUCoordinate(u, bits): + u_list = [ord(b) for b in u] + # Ignore any unused bits. + if bits % 8: + u_list[-1] &= (1<<(bits%8))-1 + return decodeLittleEndian(u_list, bits) + + def encodeUCoordinate(u, bits): + return ''.join([chr((u >> 8*i) & 0xff) + for i in range((bits+7)/8)]) + +A.5. Elligator 2 reference implementation + + The Elligator 2 map requires a non-square field element Z which shall + be calculated as follows. + + def find_z_ell2(F): + # Find nonsquare for Elligator2 + # Argument: F, a field object, e.g., F = GF(2^255 - 19) + ctr = F.gen() + while True: + for Z_cand in (F(ctr), F(-ctr)): + # Z must be a non-square in F. + if is_square(Z_cand): + continue + return Z_cand + ctr += 1 + + The values of the non-square Z only depend on the curve. The + algorithm above results in a value of Z = 2 for Curve25519 and Z=-1 + for Ed448. + + The following code maps a field element r to an encoded field element + which is a valid u-coordinate of a Montgomery curve with curve + parameter A. + + def elligator2(r, q, A, field_size_bits): + # Inputs: field element r, field order q, + # curve parameter A and field size in bits + Fq = GF(q); A = Fq(A); B = Fq(1); + + # get non-square z as specified in the hash2curve draft. + z = Fq(find_z_ell2(Fq)) + powerForLegendreSymbol = floor((q-1)/2) + + v = - A / (1 + z * r^2) + epsilon = (v^3 + A * v^2 + B * v)^powerForLegendreSymbol + x = epsilon * v - (1 - epsilon) * A/2 + return encodeUCoordinate(Integer(x), field_size_bits) + +Appendix B. Test vectors + +B.1. Test vector for CPace using group X25519 and hash SHA-512 + +B.1.1. Test vectors for calculate_generator with group X25519 + + Inputs + H = SHA-512 with input block size 128 bytes. + PRS = b'Password' ; ZPAD length: 109 ; DSI = b'CPace255' + CI = b'\nAinitiator\nBresponder' + CI = 0a41696e69746961746f720a42726573706f6e646572 + sid = 7e4b4791d6a8ef019b936c79fb7f2c57 + Outputs + generator_string(G.DSI,PRS,CI,sid,H.s_in_bytes): + (length: 168 bytes) + 0843506163653235350850617373776f72646d000000000000000000 + 00000000000000000000000000000000000000000000000000000000 + 00000000000000000000000000000000000000000000000000000000 + 00000000000000000000000000000000000000000000000000000000 + 00000000000000000000000000000000160a41696e69746961746f72 + 0a42726573706f6e646572107e4b4791d6a8ef019b936c79fb7f2c57 + hash generator string: (length: 32 bytes) + 10047198e8c4cacf0ab8a6d0ac337b8ae497209d042f7f3a50945863 + 94e821fc + decoded field element of 255 bits: (length: 32 bytes) + 10047198e8c4cacf0ab8a6d0ac337b8ae497209d042f7f3a50945863 + 94e8217c + generator g: (length: 32 bytes) + 4e6098733061c0e8486611a904fe5edb049804d26130a44131a6229e + 55c5c321 + +B.1.2. Test vector for MSGa + + Inputs + ADa = b'ADa' + ya (little endian): (length: 32 bytes) + 21b4f4bd9e64ed355c3eb676a28ebedaf6d8f17bdc365995b3190971 + 53044080 + Outputs + Ya: (length: 32 bytes) + f970e36f37cfcd9a39e37dd2d1fbc9156d6d2f9ae422f4722cbd9d32 + e9b1e704 + MSGa = lv_cat(Ya,ADa): (length: 37 bytes) + 20f970e36f37cfcd9a39e37dd2d1fbc9156d6d2f9ae422f4722cbd9d + 32e9b1e70403414461 + +B.1.3. Test vector for MSGb + + Inputs + ADb = b'ADb' + yb (little endian): (length: 32 bytes) + 848b0779ff415f0af4ea14df9dd1d3c29ac41d836c7808896c4eba19 + c51ac40a + Outputs + Yb: (length: 32 bytes) + 0178bbbab0804a4455b8f02e5d6e7d80997c6470bfb3618d7e74c396 + 47af5a29 + MSGb = lv_cat(Yb,ADb): (length: 37 bytes) + 200178bbbab0804a4455b8f02e5d6e7d80997c6470bfb3618d7e74c3 + 9647af5a2903414462 + +B.1.4. Test vector for secret points K + + scalar_mult_vfy(ya,Yb): (length: 32 bytes) + 42ba4c6dc4c184a1cf405d4503f64bf7f015e2a0107450e38b9efff3 + bee52412 + scalar_mult_vfy(yb,Ya): (length: 32 bytes) + 42ba4c6dc4c184a1cf405d4503f64bf7f015e2a0107450e38b9efff3 + bee52412 + +B.1.5. Test vector for ISK calculation initiator/responder + + unordered cat of transcript : (length: 74 bytes) + 20f970e36f37cfcd9a39e37dd2d1fbc9156d6d2f9ae422f4722cbd9d + 32e9b1e70403414461200178bbbab0804a4455b8f02e5d6e7d80997c + 6470bfb3618d7e74c39647af5a2903414462 + DSI = G.DSI_ISK, b'CPace255_ISK': (length: 12 bytes) + 43506163653235355f49534b + lv_cat(DSI,sid,K)||MSGa||MSGb: (length: 137 bytes) + 0c43506163653235355f49534b107e4b4791d6a8ef019b936c79fb7f + 2c572042ba4c6dc4c184a1cf405d4503f64bf7f015e2a0107450e38b + 9efff3bee5241220f970e36f37cfcd9a39e37dd2d1fbc9156d6d2f9a + e422f4722cbd9d32e9b1e70403414461200178bbbab0804a4455b8f0 + 2e5d6e7d80997c6470bfb3618d7e74c39647af5a2903414462 + ISK result: (length: 64 bytes) + f5ef3c13fdb9dfe839bdbf8a9256e8cee7db8a8f1dfa74958a925450 + cf8089cd560d9a4e7956b7334b6f625c8559b75ea0764ac2be894b8f + 3d434b30e87797d5 + +B.1.6. Test vector for ISK calculation parallel execution + + ordered cat of transcript : (length: 76 bytes) + 6f6320f970e36f37cfcd9a39e37dd2d1fbc9156d6d2f9ae422f4722c + bd9d32e9b1e70403414461200178bbbab0804a4455b8f02e5d6e7d80 + 997c6470bfb3618d7e74c39647af5a2903414462 + DSI = G.DSI_ISK, b'CPace255_ISK': (length: 12 bytes) + 43506163653235355f49534b + lv_cat(DSI,sid,K)||o_cat(MSGa,MSGb): (length: 139 bytes) + 0c43506163653235355f49534b107e4b4791d6a8ef019b936c79fb7f + 2c572042ba4c6dc4c184a1cf405d4503f64bf7f015e2a0107450e38b + 9efff3bee524126f6320f970e36f37cfcd9a39e37dd2d1fbc9156d6d + 2f9ae422f4722cbd9d32e9b1e70403414461200178bbbab0804a4455 + b8f02e5d6e7d80997c6470bfb3618d7e74c39647af5a2903414462 + ISK result: (length: 64 bytes) + f4051edc63b2620e10d5ecf76d9f0c5ccd1447858a98d4bf847fafac + 737478c1350e14619bc0fcd4f028d10e4102dfca39f91fe9b829a503 + ab3e0549bd835edf + +B.1.7. Corresponding C programming language initializers + + const unsigned char tc_PRS[] = { + 0x50,0x61,0x73,0x73,0x77,0x6f,0x72,0x64, + }; + const unsigned char tc_CI[] = { + 0x0a,0x41,0x69,0x6e,0x69,0x74,0x69,0x61,0x74,0x6f,0x72,0x0a, + 0x42,0x72,0x65,0x73,0x70,0x6f,0x6e,0x64,0x65,0x72, + }; + const unsigned char tc_sid[] = { + 0x7e,0x4b,0x47,0x91,0xd6,0xa8,0xef,0x01,0x9b,0x93,0x6c,0x79, + 0xfb,0x7f,0x2c,0x57, + }; + const unsigned char tc_g[] = { + 0x4e,0x60,0x98,0x73,0x30,0x61,0xc0,0xe8,0x48,0x66,0x11,0xa9, + 0x04,0xfe,0x5e,0xdb,0x04,0x98,0x04,0xd2,0x61,0x30,0xa4,0x41, + 0x31,0xa6,0x22,0x9e,0x55,0xc5,0xc3,0x21, + }; + const unsigned char tc_ya[] = { + 0x21,0xb4,0xf4,0xbd,0x9e,0x64,0xed,0x35,0x5c,0x3e,0xb6,0x76, + 0xa2,0x8e,0xbe,0xda,0xf6,0xd8,0xf1,0x7b,0xdc,0x36,0x59,0x95, + 0xb3,0x19,0x09,0x71,0x53,0x04,0x40,0x80, + }; + const unsigned char tc_ADa[] = { + 0x41,0x44,0x61, + }; + const unsigned char tc_Ya[] = { + 0xf9,0x70,0xe3,0x6f,0x37,0xcf,0xcd,0x9a,0x39,0xe3,0x7d,0xd2, + 0xd1,0xfb,0xc9,0x15,0x6d,0x6d,0x2f,0x9a,0xe4,0x22,0xf4,0x72, + 0x2c,0xbd,0x9d,0x32,0xe9,0xb1,0xe7,0x04, + }; + const unsigned char tc_yb[] = { + 0x84,0x8b,0x07,0x79,0xff,0x41,0x5f,0x0a,0xf4,0xea,0x14,0xdf, + 0x9d,0xd1,0xd3,0xc2,0x9a,0xc4,0x1d,0x83,0x6c,0x78,0x08,0x89, + 0x6c,0x4e,0xba,0x19,0xc5,0x1a,0xc4,0x0a, + }; + const unsigned char tc_ADb[] = { + 0x41,0x44,0x62, + }; + const unsigned char tc_Yb[] = { + 0x01,0x78,0xbb,0xba,0xb0,0x80,0x4a,0x44,0x55,0xb8,0xf0,0x2e, + 0x5d,0x6e,0x7d,0x80,0x99,0x7c,0x64,0x70,0xbf,0xb3,0x61,0x8d, + 0x7e,0x74,0xc3,0x96,0x47,0xaf,0x5a,0x29, + }; + const unsigned char tc_K[] = { + 0x42,0xba,0x4c,0x6d,0xc4,0xc1,0x84,0xa1,0xcf,0x40,0x5d,0x45, + 0x03,0xf6,0x4b,0xf7,0xf0,0x15,0xe2,0xa0,0x10,0x74,0x50,0xe3, + 0x8b,0x9e,0xff,0xf3,0xbe,0xe5,0x24,0x12, + }; + const unsigned char tc_ISK_IR[] = { + 0xf5,0xef,0x3c,0x13,0xfd,0xb9,0xdf,0xe8,0x39,0xbd,0xbf,0x8a, + 0x92,0x56,0xe8,0xce,0xe7,0xdb,0x8a,0x8f,0x1d,0xfa,0x74,0x95, + 0x8a,0x92,0x54,0x50,0xcf,0x80,0x89,0xcd,0x56,0x0d,0x9a,0x4e, + 0x79,0x56,0xb7,0x33,0x4b,0x6f,0x62,0x5c,0x85,0x59,0xb7,0x5e, + 0xa0,0x76,0x4a,0xc2,0xbe,0x89,0x4b,0x8f,0x3d,0x43,0x4b,0x30, + 0xe8,0x77,0x97,0xd5, + }; + const unsigned char tc_ISK_SY[] = { + 0xf4,0x05,0x1e,0xdc,0x63,0xb2,0x62,0x0e,0x10,0xd5,0xec,0xf7, + 0x6d,0x9f,0x0c,0x5c,0xcd,0x14,0x47,0x85,0x8a,0x98,0xd4,0xbf, + 0x84,0x7f,0xaf,0xac,0x73,0x74,0x78,0xc1,0x35,0x0e,0x14,0x61, + 0x9b,0xc0,0xfc,0xd4,0xf0,0x28,0xd1,0x0e,0x41,0x02,0xdf,0xca, + 0x39,0xf9,0x1f,0xe9,0xb8,0x29,0xa5,0x03,0xab,0x3e,0x05,0x49, + 0xbd,0x83,0x5e,0xdf, + }; + +B.1.8. Test vectors for G_X25519.scalar_mult_vfy: low order points + + Test vectors for which G_X25519.scalar_mult_vfy(s_in,ux) must return + the neutral element or would return the neutral element if bit #255 + of field element representation was not correctly cleared. (The + decodeUCoordinate function from RFC7748 mandates clearing bit #255 + for field element representations for use in the X25519 function.). + + u0: 0000000000000000000000000000000000000000000000000000000000000000 + u1: 0100000000000000000000000000000000000000000000000000000000000000 + u2: ecffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff7f + u3: e0eb7a7c3b41b8ae1656e3faf19fc46ada098deb9c32b1fd866205165f49b800 + u4: 5f9c95bca3508c24b1d0b1559c83ef5b04445cc4581c8e86d8224eddd09f1157 + u5: edffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff7f + u6: daffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff + u7: eeffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff7f + u8: dbffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff + u9: d9ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff + ua: cdeb7a7c3b41b8ae1656e3faf19fc46ada098deb9c32b1fd866205165f49b880 + ub: 4c9c95bca3508c24b1d0b1559c83ef5b04445cc4581c8e86d8224eddd09f11d7 + + u0 ... ub MUST be verified to produce the correct results q0 ... qb: + + Additionally, u0,u1,u2,u3,u4,u5 and u7 MUST trigger the abort case + when included in MSGa or MSGb. + + s = af46e36bf0527c9d3b16154b82465edd62144c0ac1fc5a18506a2244ba449aff + qN = G_X25519.scalar_mult_vfy(s, uX) + q0: 0000000000000000000000000000000000000000000000000000000000000000 + q1: 0000000000000000000000000000000000000000000000000000000000000000 + q2: 0000000000000000000000000000000000000000000000000000000000000000 + q3: 0000000000000000000000000000000000000000000000000000000000000000 + q4: 0000000000000000000000000000000000000000000000000000000000000000 + q5: 0000000000000000000000000000000000000000000000000000000000000000 + q6: d8e2c776bbacd510d09fd9278b7edcd25fc5ae9adfba3b6e040e8d3b71b21806 + q7: 0000000000000000000000000000000000000000000000000000000000000000 + q8: c85c655ebe8be44ba9c0ffde69f2fe10194458d137f09bbff725ce58803cdb38 + q9: db64dafa9b8fdd136914e61461935fe92aa372cb056314e1231bc4ec12417456 + qa: e062dcd5376d58297be2618c7498f55baa07d7e03184e8aada20bca28888bf7a + qb: 993c6ad11c4c29da9a56f7691fd0ff8d732e49de6250b6c2e80003ff4629a175 + +B.2. Test vector for CPace using group X448 and hash SHAKE-256 + +B.2.1. Test vectors for calculate_generator with group X448 + + Inputs + H = SHAKE-256 with input block size 136 bytes. + PRS = b'Password' ; ZPAD length: 117 ; DSI = b'CPace448' + CI = b'\nAinitiator\nBresponder' + CI = 0a41696e69746961746f720a42726573706f6e646572 + sid = 5223e0cdc45d6575668d64c552004124 + Outputs + generator_string(G.DSI,PRS,CI,sid,H.s_in_bytes): + (length: 176 bytes) + 0843506163653434380850617373776f726475000000000000000000 + 00000000000000000000000000000000000000000000000000000000 + 00000000000000000000000000000000000000000000000000000000 + 00000000000000000000000000000000000000000000000000000000 + 000000000000000000000000000000000000000000000000160a4169 + 6e69746961746f720a42726573706f6e646572105223e0cdc45d6575 + 668d64c552004124 + hash generator string: (length: 56 bytes) + 769e06d6c41c8cf1c87aa3df8e687167f6d0a2e41821e856276a0221 + d88272359d0b43204b546174c9179c83c107b707f296eafaa1c5a293 + decoded field element of 448 bits: (length: 56 bytes) + 769e06d6c41c8cf1c87aa3df8e687167f6d0a2e41821e856276a0221 + d88272359d0b43204b546174c9179c83c107b707f296eafaa1c5a293 + generator g: (length: 56 bytes) + 6fdae14718eb7506dd96e3f7797896efdb8db9ec0797485c9c48a192 + 2e44961da097f2908b084a5de33ab671630660d27d79ffd6ee8ec846 + +B.2.2. Test vector for MSGa + + Inputs + ADa = b'ADa' + ya (little endian): (length: 56 bytes) + 21b4f4bd9e64ed355c3eb676a28ebedaf6d8f17bdc365995b3190971 + 53044080516bd083bfcce66121a3072646994c8430cc382b8dc543e8 + Outputs + Ya: (length: 56 bytes) + 396bd11daf223711e575cac6021e3fa31558012048a1cec7876292b9 + 6c61eda353fe04f33028d2352779668a934084da776c1c51a58ce4b5 + MSGa = lv_cat(Ya,ADa): (length: 61 bytes) + 38396bd11daf223711e575cac6021e3fa31558012048a1cec7876292 + b96c61eda353fe04f33028d2352779668a934084da776c1c51a58ce4 + b503414461 + +B.2.3. Test vector for MSGb + + Inputs + ADb = b'ADb' + yb (little endian): (length: 56 bytes) + 848b0779ff415f0af4ea14df9dd1d3c29ac41d836c7808896c4eba19 + c51ac40a439caf5e61ec88c307c7d619195229412eaa73fb2a5ea20d + Outputs + Yb: (length: 56 bytes) + 53c519fb490fde5a04bda8c18b327d0fc1a9391d19e0ac00c59df9c6 + 0422284e593d6b092eac94f5aa644ed883f39bd4f04e4beb6af86d58 + MSGb = lv_cat(Yb,ADb): (length: 61 bytes) + 3853c519fb490fde5a04bda8c18b327d0fc1a9391d19e0ac00c59df9 + c60422284e593d6b092eac94f5aa644ed883f39bd4f04e4beb6af86d + 5803414462 + +B.2.4. Test vector for secret points K + + scalar_mult_vfy(ya,Yb): (length: 56 bytes) + e00af217556a40ccbc9822cc27a43542e45166a653aa4df746d5f8e1 + e8df483e9baff71c9eb03ee20a688ad4e4d359f70ac9ec3f6a659997 + scalar_mult_vfy(yb,Ya): (length: 56 bytes) + e00af217556a40ccbc9822cc27a43542e45166a653aa4df746d5f8e1 + e8df483e9baff71c9eb03ee20a688ad4e4d359f70ac9ec3f6a659997 + +B.2.5. Test vector for ISK calculation initiator/responder + + unordered cat of transcript : (length: 122 bytes) + 38396bd11daf223711e575cac6021e3fa31558012048a1cec7876292 + b96c61eda353fe04f33028d2352779668a934084da776c1c51a58ce4 + b5034144613853c519fb490fde5a04bda8c18b327d0fc1a9391d19e0 + ac00c59df9c60422284e593d6b092eac94f5aa644ed883f39bd4f04e + 4beb6af86d5803414462 + DSI = G.DSI_ISK, b'CPace448_ISK': (length: 12 bytes) + 43506163653434385f49534b + lv_cat(DSI,sid,K)||MSGa||MSGb: (length: 209 bytes) + 0c43506163653434385f49534b105223e0cdc45d6575668d64c55200 + 412438e00af217556a40ccbc9822cc27a43542e45166a653aa4df746 + d5f8e1e8df483e9baff71c9eb03ee20a688ad4e4d359f70ac9ec3f6a + 65999738396bd11daf223711e575cac6021e3fa31558012048a1cec7 + 876292b96c61eda353fe04f33028d2352779668a934084da776c1c51 + a58ce4b5034144613853c519fb490fde5a04bda8c18b327d0fc1a939 + 1d19e0ac00c59df9c60422284e593d6b092eac94f5aa644ed883f39b + d4f04e4beb6af86d5803414462 + ISK result: (length: 64 bytes) + 4030297722c1914711da6b2a224a44b53b30c05ab02c2a3d3ccc7272 + a3333ce3a4564c17031b634e89f65681f52d5c3d1df7baeb88523d2e + 481b3858aed86315 + +B.2.6. Test vector for ISK calculation parallel execution + + ordered cat of transcript : (length: 124 bytes) + 6f633853c519fb490fde5a04bda8c18b327d0fc1a9391d19e0ac00c5 + 9df9c60422284e593d6b092eac94f5aa644ed883f39bd4f04e4beb6a + f86d580341446238396bd11daf223711e575cac6021e3fa315580120 + 48a1cec7876292b96c61eda353fe04f33028d2352779668a934084da + 776c1c51a58ce4b503414461 + DSI = G.DSI_ISK, b'CPace448_ISK': (length: 12 bytes) + 43506163653434385f49534b + lv_cat(DSI,sid,K)||o_cat(MSGa,MSGb): (length: 211 bytes) + 0c43506163653434385f49534b105223e0cdc45d6575668d64c55200 + 412438e00af217556a40ccbc9822cc27a43542e45166a653aa4df746 + d5f8e1e8df483e9baff71c9eb03ee20a688ad4e4d359f70ac9ec3f6a + 6599976f633853c519fb490fde5a04bda8c18b327d0fc1a9391d19e0 + ac00c59df9c60422284e593d6b092eac94f5aa644ed883f39bd4f04e + 4beb6af86d580341446238396bd11daf223711e575cac6021e3fa315 + 58012048a1cec7876292b96c61eda353fe04f33028d2352779668a93 + 4084da776c1c51a58ce4b503414461 + ISK result: (length: 64 bytes) + 4cd30768e2f75f0583449614bce823b421c31163c5a3bde4eed1c664 + 284a32995ea3430b5c47fc7dd771b534ad38eaea5d8c8f97bd548966 + 7facfc044615075f + +B.2.7. Corresponding C programming language initializers + + const unsigned char tc_PRS[] = { + 0x50,0x61,0x73,0x73,0x77,0x6f,0x72,0x64, + }; + const unsigned char tc_CI[] = { + 0x0a,0x41,0x69,0x6e,0x69,0x74,0x69,0x61,0x74,0x6f,0x72,0x0a, + 0x42,0x72,0x65,0x73,0x70,0x6f,0x6e,0x64,0x65,0x72, + }; + const unsigned char tc_sid[] = { + 0x52,0x23,0xe0,0xcd,0xc4,0x5d,0x65,0x75,0x66,0x8d,0x64,0xc5, + 0x52,0x00,0x41,0x24, + }; + const unsigned char tc_g[] = { + 0x6f,0xda,0xe1,0x47,0x18,0xeb,0x75,0x06,0xdd,0x96,0xe3,0xf7, + 0x79,0x78,0x96,0xef,0xdb,0x8d,0xb9,0xec,0x07,0x97,0x48,0x5c, + 0x9c,0x48,0xa1,0x92,0x2e,0x44,0x96,0x1d,0xa0,0x97,0xf2,0x90, + 0x8b,0x08,0x4a,0x5d,0xe3,0x3a,0xb6,0x71,0x63,0x06,0x60,0xd2, + 0x7d,0x79,0xff,0xd6,0xee,0x8e,0xc8,0x46, + }; + const unsigned char tc_ya[] = { + 0x21,0xb4,0xf4,0xbd,0x9e,0x64,0xed,0x35,0x5c,0x3e,0xb6,0x76, + 0xa2,0x8e,0xbe,0xda,0xf6,0xd8,0xf1,0x7b,0xdc,0x36,0x59,0x95, + 0xb3,0x19,0x09,0x71,0x53,0x04,0x40,0x80,0x51,0x6b,0xd0,0x83, + 0xbf,0xcc,0xe6,0x61,0x21,0xa3,0x07,0x26,0x46,0x99,0x4c,0x84, + 0x30,0xcc,0x38,0x2b,0x8d,0xc5,0x43,0xe8, + }; + const unsigned char tc_ADa[] = { + 0x41,0x44,0x61, + }; + const unsigned char tc_Ya[] = { + 0x39,0x6b,0xd1,0x1d,0xaf,0x22,0x37,0x11,0xe5,0x75,0xca,0xc6, + 0x02,0x1e,0x3f,0xa3,0x15,0x58,0x01,0x20,0x48,0xa1,0xce,0xc7, + 0x87,0x62,0x92,0xb9,0x6c,0x61,0xed,0xa3,0x53,0xfe,0x04,0xf3, + 0x30,0x28,0xd2,0x35,0x27,0x79,0x66,0x8a,0x93,0x40,0x84,0xda, + 0x77,0x6c,0x1c,0x51,0xa5,0x8c,0xe4,0xb5, + }; + const unsigned char tc_yb[] = { + 0x84,0x8b,0x07,0x79,0xff,0x41,0x5f,0x0a,0xf4,0xea,0x14,0xdf, + 0x9d,0xd1,0xd3,0xc2,0x9a,0xc4,0x1d,0x83,0x6c,0x78,0x08,0x89, + 0x6c,0x4e,0xba,0x19,0xc5,0x1a,0xc4,0x0a,0x43,0x9c,0xaf,0x5e, + 0x61,0xec,0x88,0xc3,0x07,0xc7,0xd6,0x19,0x19,0x52,0x29,0x41, + 0x2e,0xaa,0x73,0xfb,0x2a,0x5e,0xa2,0x0d, + }; + const unsigned char tc_ADb[] = { + 0x41,0x44,0x62, + }; + const unsigned char tc_Yb[] = { + 0x53,0xc5,0x19,0xfb,0x49,0x0f,0xde,0x5a,0x04,0xbd,0xa8,0xc1, + 0x8b,0x32,0x7d,0x0f,0xc1,0xa9,0x39,0x1d,0x19,0xe0,0xac,0x00, + 0xc5,0x9d,0xf9,0xc6,0x04,0x22,0x28,0x4e,0x59,0x3d,0x6b,0x09, + 0x2e,0xac,0x94,0xf5,0xaa,0x64,0x4e,0xd8,0x83,0xf3,0x9b,0xd4, + 0xf0,0x4e,0x4b,0xeb,0x6a,0xf8,0x6d,0x58, + }; + const unsigned char tc_K[] = { + 0xe0,0x0a,0xf2,0x17,0x55,0x6a,0x40,0xcc,0xbc,0x98,0x22,0xcc, + 0x27,0xa4,0x35,0x42,0xe4,0x51,0x66,0xa6,0x53,0xaa,0x4d,0xf7, + 0x46,0xd5,0xf8,0xe1,0xe8,0xdf,0x48,0x3e,0x9b,0xaf,0xf7,0x1c, + 0x9e,0xb0,0x3e,0xe2,0x0a,0x68,0x8a,0xd4,0xe4,0xd3,0x59,0xf7, + 0x0a,0xc9,0xec,0x3f,0x6a,0x65,0x99,0x97, + }; + const unsigned char tc_ISK_IR[] = { + 0x40,0x30,0x29,0x77,0x22,0xc1,0x91,0x47,0x11,0xda,0x6b,0x2a, + 0x22,0x4a,0x44,0xb5,0x3b,0x30,0xc0,0x5a,0xb0,0x2c,0x2a,0x3d, + 0x3c,0xcc,0x72,0x72,0xa3,0x33,0x3c,0xe3,0xa4,0x56,0x4c,0x17, + 0x03,0x1b,0x63,0x4e,0x89,0xf6,0x56,0x81,0xf5,0x2d,0x5c,0x3d, + 0x1d,0xf7,0xba,0xeb,0x88,0x52,0x3d,0x2e,0x48,0x1b,0x38,0x58, + 0xae,0xd8,0x63,0x15, + }; + const unsigned char tc_ISK_SY[] = { + 0x4c,0xd3,0x07,0x68,0xe2,0xf7,0x5f,0x05,0x83,0x44,0x96,0x14, + 0xbc,0xe8,0x23,0xb4,0x21,0xc3,0x11,0x63,0xc5,0xa3,0xbd,0xe4, + 0xee,0xd1,0xc6,0x64,0x28,0x4a,0x32,0x99,0x5e,0xa3,0x43,0x0b, + 0x5c,0x47,0xfc,0x7d,0xd7,0x71,0xb5,0x34,0xad,0x38,0xea,0xea, + 0x5d,0x8c,0x8f,0x97,0xbd,0x54,0x89,0x66,0x7f,0xac,0xfc,0x04, + 0x46,0x15,0x07,0x5f, + }; + +B.2.8. Test vectors for G_X448.scalar_mult_vfy: low order points + + Test vectors for which G_X448.scalar_mult_vfy(s_in,ux) must return + the neutral element. This includes points that are non-canonicaly + encoded, i.e. have coordinate values larger than the field prime. + + Weak points for X448 smaller than the field prime (canonical) + + u0: (length: 56 bytes) + 0000000000000000000000000000000000000000000000000000000000 + 000000000000000000000000000000000000000000000000000000 + u1: (length: 56 bytes) + 0100000000000000000000000000000000000000000000000000000000 + 000000000000000000000000000000000000000000000000000000 + u2: (length: 56 bytes) + fefffffffffffffffffffffffffffffffffffffffffffffffffffffffe + ffffffffffffffffffffffffffffffffffffffffffffffffffffff + + Weak points for X448 larger or equal to the field prime (non- + canonical) + + u3: (length: 56 bytes) + fffffffffffffffffffffffffffffffffffffffffffffffffffffffffe + ffffffffffffffffffffffffffffffffffffffffffffffffffffff + u4: (length: 56 bytes) + 00000000000000000000000000000000000000000000000000000000ff + ffffffffffffffffffffffffffffffffffffffffffffffffffffff + + All of the above points u0 ... u4 MUST trigger the abort case + when included in the protocol messages MSGa or MSGb. + + Expected results for X448 resp. G_X448.scalar_mult_vfy + + scalar s: (length: 56 bytes) + af8a14218bf2a2062926d2ea9b8fe4e8b6817349b6ed2feb1e5d64d7a4 + 523f15fceec70fb111e870dc58d191e66a14d3e9d482d04432cadd + G_X448.scalar_mult_vfy(s,u0): (length: 56 bytes) + 0000000000000000000000000000000000000000000000000000000000 + 000000000000000000000000000000000000000000000000000000 + G_X448.scalar_mult_vfy(s,u1): (length: 56 bytes) + 0000000000000000000000000000000000000000000000000000000000 + 000000000000000000000000000000000000000000000000000000 + G_X448.scalar_mult_vfy(s,u2): (length: 56 bytes) + 0000000000000000000000000000000000000000000000000000000000 + 000000000000000000000000000000000000000000000000000000 + G_X448.scalar_mult_vfy(s,u3): (length: 56 bytes) + 0000000000000000000000000000000000000000000000000000000000 + 000000000000000000000000000000000000000000000000000000 + G_X448.scalar_mult_vfy(s,u4): (length: 56 bytes) + 0000000000000000000000000000000000000000000000000000000000 + 000000000000000000000000000000000000000000000000000000 + + Test vectors for scalar_mult with nonzero outputs + + scalar s: (length: 56 bytes) + af8a14218bf2a2062926d2ea9b8fe4e8b6817349b6ed2feb1e5d64d7a4 + 523f15fceec70fb111e870dc58d191e66a14d3e9d482d04432cadd + point coordinate u_curve on the curve: (length: 56 bytes) + ab0c68d772ec2eb9de25c49700e46d6325e66d6aa39d7b65eb84a68c55 + 69d47bd71b41f3e0d210f44e146dec8926b174acb3f940a0b82cab + G_X448.scalar_mult_vfy(s,u_curve): (length: 56 bytes) + 3b0fa9bc40a6fdc78c9e06ff7a54c143c5d52f365607053bf0656f5142 + 0496295f910a101b38edc1acd3bd240fd55dcb7a360553b8a7627e + + point coordinate u_twist on the twist: (length: 56 bytes) + c981cd1e1f72d9c35c7d7cf6be426757c0dc8206a2fcfa564a8e7618c0 + 3c0e61f9a2eb1c3e0dd97d6e9b1010f5edd03397a83f5a914cb3ff + G_X448.scalar_mult_vfy(s,u_twist): (length: 56 bytes) + d0a2bb7e9c5c2c627793d8342f23b759fe7d9e3320a85ca4fd61376331 + 50ffd9a9148a9b75c349fac43d64bec49a6e126cc92cbfbf353961 + +B.3. Test vector for CPace using group ristretto255 and hash SHA-512 + +B.3.1. Test vectors for calculate_generator with group ristretto255 + + Inputs + H = SHA-512 with input block size 128 bytes. + PRS = b'Password' ; ZPAD length: 100 ; + DSI = b'CPaceRistretto255' + CI = b'\nAinitiator\nBresponder' + CI = 0a41696e69746961746f720a42726573706f6e646572 + sid = 7e4b4791d6a8ef019b936c79fb7f2c57 + Outputs + generator_string(G.DSI,PRS,CI,sid,H.s_in_bytes): + (length: 168 bytes) + 11435061636552697374726574746f3235350850617373776f726464 + 00000000000000000000000000000000000000000000000000000000 + 00000000000000000000000000000000000000000000000000000000 + 00000000000000000000000000000000000000000000000000000000 + 00000000000000000000000000000000160a41696e69746961746f72 + 0a42726573706f6e646572107e4b4791d6a8ef019b936c79fb7f2c57 + hash result: (length: 64 bytes) + a5ce446f63a1ae6d1fee80fa67d0b4004a4b1283ec5549a462bf33a6 + c1ae06a0871f9bf48545f49b2a792eed255ac04f52758c9c60448306 + 810b44e986e3dcbb + encoded generator g: (length: 32 bytes) + 5e25411ca1ad7c9debfd0b33ad987a95cefef2d3f15dcc8bd26415a5 + dfe2e15a + +B.3.2. Test vector for MSGa + + Inputs + ADa = b'ADa' + ya (little endian): (length: 32 bytes) + da3d23700a9e5699258aef94dc060dfda5ebb61f02a5ea77fad53f4f + f0976d08 + Outputs + Ya: (length: 32 bytes) + 383a85dd236978f17f8c8545b50dabc52a39fcdab2cf8bc531ce040f + f77ca82d + MSGa = lv_cat(Ya,ADa): (length: 37 bytes) + 20383a85dd236978f17f8c8545b50dabc52a39fcdab2cf8bc531ce04 + 0ff77ca82d03414461 + +B.3.3. Test vector for MSGb + + Inputs + ADb = b'ADb' + yb (little endian): (length: 32 bytes) + d2316b454718c35362d83d69df6320f38578ed5984651435e2949762 + d900b80d + Outputs + Yb: (length: 32 bytes) + a6206309c0e8e5f579295e35997ac4300ab3fecec3c17f7b604f3e69 + 8fa1383c + MSGb = lv_cat(Yb,ADb): (length: 37 bytes) + 20a6206309c0e8e5f579295e35997ac4300ab3fecec3c17f7b604f3e + 698fa1383c03414462 + +B.3.4. Test vector for secret points K + + scalar_mult_vfy(ya,Yb): (length: 32 bytes) + fa1d0318864e2cacb26875f1b791c9ae83204fe8359addb53e95a2e9 + 8893853f + scalar_mult_vfy(yb,Ya): (length: 32 bytes) + fa1d0318864e2cacb26875f1b791c9ae83204fe8359addb53e95a2e9 + 8893853f + +B.3.5. Test vector for ISK calculation initiator/responder + + unordered cat of transcript : (length: 74 bytes) + 20383a85dd236978f17f8c8545b50dabc52a39fcdab2cf8bc531ce04 + 0ff77ca82d0341446120a6206309c0e8e5f579295e35997ac4300ab3 + fecec3c17f7b604f3e698fa1383c03414462 + DSI = G.DSI_ISK, b'CPaceRistretto255_ISK': + (length: 21 bytes) + 435061636552697374726574746f3235355f49534b + lv_cat(DSI,sid,K)||MSGa||MSGb: (length: 146 bytes) + 15435061636552697374726574746f3235355f49534b107e4b4791d6 + a8ef019b936c79fb7f2c5720fa1d0318864e2cacb26875f1b791c9ae + 83204fe8359addb53e95a2e98893853f20383a85dd236978f17f8c85 + 45b50dabc52a39fcdab2cf8bc531ce040ff77ca82d0341446120a620 + 6309c0e8e5f579295e35997ac4300ab3fecec3c17f7b604f3e698fa1 + 383c03414462 + ISK result: (length: 64 bytes) + e91ccb2c0f5e0d0993a33956e3be59754f3f2b07db57631f5394452e + a2e7b4354674eb1f5686c078462bf83bec72e8743df440108e638f35 + 26d9b90e85be096f + +B.3.6. Test vector for ISK calculation parallel execution + + ordered cat of transcript : (length: 76 bytes) + 6f6320a6206309c0e8e5f579295e35997ac4300ab3fecec3c17f7b60 + 4f3e698fa1383c0341446220383a85dd236978f17f8c8545b50dabc5 + 2a39fcdab2cf8bc531ce040ff77ca82d03414461 + DSI = G.DSI_ISK, b'CPaceRistretto255_ISK': + (length: 21 bytes) + 435061636552697374726574746f3235355f49534b + lv_cat(DSI,sid,K)||o_cat(MSGa,MSGb): (length: 148 bytes) + 15435061636552697374726574746f3235355f49534b107e4b4791d6 + a8ef019b936c79fb7f2c5720fa1d0318864e2cacb26875f1b791c9ae + 83204fe8359addb53e95a2e98893853f6f6320a6206309c0e8e5f579 + 295e35997ac4300ab3fecec3c17f7b604f3e698fa1383c0341446220 + 383a85dd236978f17f8c8545b50dabc52a39fcdab2cf8bc531ce040f + f77ca82d03414461 + ISK result: (length: 64 bytes) + 1638fb6ff564a80a12af07c036870e10c4efb539fa847fdf3e9c4621 + 7bf52cd4df4ca0fe51146492a9ba6dd6a42ac402bc2d60adb4084c81 + 758d754d1d81482a + +B.3.7. Corresponding C programming language initializers + + const unsigned char tc_PRS[] = { + 0x50,0x61,0x73,0x73,0x77,0x6f,0x72,0x64, + }; + const unsigned char tc_CI[] = { + 0x0a,0x41,0x69,0x6e,0x69,0x74,0x69,0x61,0x74,0x6f,0x72,0x0a, + 0x42,0x72,0x65,0x73,0x70,0x6f,0x6e,0x64,0x65,0x72, + }; + const unsigned char tc_sid[] = { + 0x7e,0x4b,0x47,0x91,0xd6,0xa8,0xef,0x01,0x9b,0x93,0x6c,0x79, + 0xfb,0x7f,0x2c,0x57, + }; + const unsigned char tc_g[] = { + 0x5e,0x25,0x41,0x1c,0xa1,0xad,0x7c,0x9d,0xeb,0xfd,0x0b,0x33, + 0xad,0x98,0x7a,0x95,0xce,0xfe,0xf2,0xd3,0xf1,0x5d,0xcc,0x8b, + 0xd2,0x64,0x15,0xa5,0xdf,0xe2,0xe1,0x5a, + }; + const unsigned char tc_ya[] = { + 0xda,0x3d,0x23,0x70,0x0a,0x9e,0x56,0x99,0x25,0x8a,0xef,0x94, + 0xdc,0x06,0x0d,0xfd,0xa5,0xeb,0xb6,0x1f,0x02,0xa5,0xea,0x77, + 0xfa,0xd5,0x3f,0x4f,0xf0,0x97,0x6d,0x08, + }; + const unsigned char tc_ADa[] = { + 0x41,0x44,0x61, + }; + const unsigned char tc_Ya[] = { + 0x38,0x3a,0x85,0xdd,0x23,0x69,0x78,0xf1,0x7f,0x8c,0x85,0x45, + 0xb5,0x0d,0xab,0xc5,0x2a,0x39,0xfc,0xda,0xb2,0xcf,0x8b,0xc5, + 0x31,0xce,0x04,0x0f,0xf7,0x7c,0xa8,0x2d, + }; + const unsigned char tc_yb[] = { + 0xd2,0x31,0x6b,0x45,0x47,0x18,0xc3,0x53,0x62,0xd8,0x3d,0x69, + 0xdf,0x63,0x20,0xf3,0x85,0x78,0xed,0x59,0x84,0x65,0x14,0x35, + 0xe2,0x94,0x97,0x62,0xd9,0x00,0xb8,0x0d, + }; + const unsigned char tc_ADb[] = { + 0x41,0x44,0x62, + }; + const unsigned char tc_Yb[] = { + 0xa6,0x20,0x63,0x09,0xc0,0xe8,0xe5,0xf5,0x79,0x29,0x5e,0x35, + 0x99,0x7a,0xc4,0x30,0x0a,0xb3,0xfe,0xce,0xc3,0xc1,0x7f,0x7b, + 0x60,0x4f,0x3e,0x69,0x8f,0xa1,0x38,0x3c, + }; + const unsigned char tc_K[] = { + 0xfa,0x1d,0x03,0x18,0x86,0x4e,0x2c,0xac,0xb2,0x68,0x75,0xf1, + 0xb7,0x91,0xc9,0xae,0x83,0x20,0x4f,0xe8,0x35,0x9a,0xdd,0xb5, + 0x3e,0x95,0xa2,0xe9,0x88,0x93,0x85,0x3f, + }; + const unsigned char tc_ISK_IR[] = { + 0xe9,0x1c,0xcb,0x2c,0x0f,0x5e,0x0d,0x09,0x93,0xa3,0x39,0x56, + 0xe3,0xbe,0x59,0x75,0x4f,0x3f,0x2b,0x07,0xdb,0x57,0x63,0x1f, + 0x53,0x94,0x45,0x2e,0xa2,0xe7,0xb4,0x35,0x46,0x74,0xeb,0x1f, + 0x56,0x86,0xc0,0x78,0x46,0x2b,0xf8,0x3b,0xec,0x72,0xe8,0x74, + 0x3d,0xf4,0x40,0x10,0x8e,0x63,0x8f,0x35,0x26,0xd9,0xb9,0x0e, + 0x85,0xbe,0x09,0x6f, + }; + const unsigned char tc_ISK_SY[] = { + 0x16,0x38,0xfb,0x6f,0xf5,0x64,0xa8,0x0a,0x12,0xaf,0x07,0xc0, + 0x36,0x87,0x0e,0x10,0xc4,0xef,0xb5,0x39,0xfa,0x84,0x7f,0xdf, + 0x3e,0x9c,0x46,0x21,0x7b,0xf5,0x2c,0xd4,0xdf,0x4c,0xa0,0xfe, + 0x51,0x14,0x64,0x92,0xa9,0xba,0x6d,0xd6,0xa4,0x2a,0xc4,0x02, + 0xbc,0x2d,0x60,0xad,0xb4,0x08,0x4c,0x81,0x75,0x8d,0x75,0x4d, + 0x1d,0x81,0x48,0x2a, + }; + +B.3.8. Test case for scalar_mult with valid inputs + + s: (length: 32 bytes) + 7cd0e075fa7955ba52c02759a6c90dbbfc10e6d40aea8d283e407d88 + cf538a05 + X: (length: 32 bytes) + 2c3c6b8c4f3800e7aef6864025b4ed79bd599117e427c41bd47d93d6 + 54b4a51c + G.scalar_mult(s,decode(X)): (length: 32 bytes) + 7c13645fe790a468f62c39beb7388e541d8405d1ade69d1778c5fe3e + 7f6b600e + G.scalar_mult_vfy(s,X): (length: 32 bytes) + 7c13645fe790a468f62c39beb7388e541d8405d1ade69d1778c5fe3e + 7f6b600e + +B.3.9. Invalid inputs for scalar_mult_vfy + + For these test cases scalar_mult_vfy(y,.) MUST return the + representation of the neutral element G.I. When points Y_i1 or Y_i2 + are included in MSGa or MSGb the protocol MUST abort. + + s: (length: 32 bytes) + 7cd0e075fa7955ba52c02759a6c90dbbfc10e6d40aea8d283e407d88 + cf538a05 + Y_i1: (length: 32 bytes) + 2b3c6b8c4f3800e7aef6864025b4ed79bd599117e427c41bd47d93d6 + 54b4a51c + Y_i2 == G.I: (length: 32 bytes) + 00000000000000000000000000000000000000000000000000000000 + 00000000 + G.scalar_mult_vfy(s,Y_i1) = G.scalar_mult_vfy(s,Y_i2) = G.I + +B.4. Test vector for CPace using group decaf448 and hash SHAKE-256 + +B.4.1. Test vectors for calculate_generator with group decaf448 + + Inputs + H = SHAKE-256 with input block size 136 bytes. + PRS = b'Password' ; ZPAD length: 112 ; + DSI = b'CPaceDecaf448' + CI = b'\nAinitiator\nBresponder' + CI = 0a41696e69746961746f720a42726573706f6e646572 + sid = 5223e0cdc45d6575668d64c552004124 + Outputs + generator_string(G.DSI,PRS,CI,sid,H.s_in_bytes): + (length: 176 bytes) + 0d435061636544656361663434380850617373776f72647000000000 + 00000000000000000000000000000000000000000000000000000000 + 00000000000000000000000000000000000000000000000000000000 + 00000000000000000000000000000000000000000000000000000000 + 000000000000000000000000000000000000000000000000160a4169 + 6e69746961746f720a42726573706f6e646572105223e0cdc45d6575 + 668d64c552004124 + hash result: (length: 112 bytes) + 8955b426ff1d3a22032d21c013cf94134cee9a4235e93261a4911edb + f68f2945f0267c983954262c7f59badb9caf468ebe21b7e9885657af + b8f1a3b783c2047ba519e113ecf81b2b580dd481f499beabd401cc77 + 1d28915fb750011209040f5f03b2ceb5e5eb259c96b478382d5a5c57 + encoded generator g: (length: 56 bytes) + 682d1a4f49fc2a4834356ae4d7f58636bc9481521c845e66e6fb0b29 + 69341df45fbaeaea9e2221b3f5babc54c5f8ce456988ffc519defaeb + +B.4.2. Test vector for MSGa + + Inputs + ADa = b'ADa' + ya (little endian): (length: 56 bytes) + 33d561f13cfc0dca279c30e8cde895175dc25483892819eba132d58c + 13c0462a8eb0d73fda941950594bef5191d8394691f86edffcad6c1e + Outputs + Ya: (length: 56 bytes) + e233867540319ec86eaecc09a85dec233745db729f61c36bde14c034 + 200994fc4b6e8d263008c169585fd1d186d8ac560cb9f7ad0d166965 + MSGa = lv_cat(Ya,ADa): (length: 61 bytes) + 38e233867540319ec86eaecc09a85dec233745db729f61c36bde14c0 + 34200994fc4b6e8d263008c169585fd1d186d8ac560cb9f7ad0d1669 + 6503414461 + +B.4.3. Test vector for MSGb + + Inputs + ADb = b'ADb' + yb (little endian): (length: 56 bytes) + 2523c969f68fa2b2aea294c2539ef36eb1e0558abd14712a7828f16a + 85ed2c7e77e2bdd418994405fb1b57b6bbaadd66849892aac9d81402 + Outputs + Yb: (length: 56 bytes) + 5062a0f33478914bf162a80dad39b5b266c1dd02f408573b41827e38 + 599b682afbf7a0735adfd68c39ab4994fd1b034846270e38332b4da9 + MSGb = lv_cat(Yb,ADb): (length: 61 bytes) + 385062a0f33478914bf162a80dad39b5b266c1dd02f408573b41827e + 38599b682afbf7a0735adfd68c39ab4994fd1b034846270e38332b4d + a903414462 + +B.4.4. Test vector for secret points K + + scalar_mult_vfy(ya,Yb): (length: 56 bytes) + dc9edef7c127e79d32f2584f9fcd3269174fe32226c2082963879a6d + eafefb9c14efcee9fc1245917ad3658037d2d62aff2d3f76fa4fca99 + scalar_mult_vfy(yb,Ya): (length: 56 bytes) + dc9edef7c127e79d32f2584f9fcd3269174fe32226c2082963879a6d + eafefb9c14efcee9fc1245917ad3658037d2d62aff2d3f76fa4fca99 + +B.4.5. Test vector for ISK calculation initiator/responder + + unordered cat of transcript : (length: 122 bytes) + 38e233867540319ec86eaecc09a85dec233745db729f61c36bde14c0 + 34200994fc4b6e8d263008c169585fd1d186d8ac560cb9f7ad0d1669 + 6503414461385062a0f33478914bf162a80dad39b5b266c1dd02f408 + 573b41827e38599b682afbf7a0735adfd68c39ab4994fd1b03484627 + 0e38332b4da903414462 + DSI = G.DSI_ISK, b'CPaceDecaf448_ISK': (length: 17 bytes) + 435061636544656361663434385f49534b + lv_cat(DSI,sid,K)||MSGa||MSGb: (length: 214 bytes) + 11435061636544656361663434385f49534b105223e0cdc45d657566 + 8d64c55200412438dc9edef7c127e79d32f2584f9fcd3269174fe322 + 26c2082963879a6deafefb9c14efcee9fc1245917ad3658037d2d62a + ff2d3f76fa4fca9938e233867540319ec86eaecc09a85dec233745db + 729f61c36bde14c034200994fc4b6e8d263008c169585fd1d186d8ac + 560cb9f7ad0d16696503414461385062a0f33478914bf162a80dad39 + b5b266c1dd02f408573b41827e38599b682afbf7a0735adfd68c39ab + 4994fd1b034846270e38332b4da903414462 + ISK result: (length: 64 bytes) + a752612fe6dec542e96629a6eb68ecb9bfe2257224975e916035aee7 + 47c6aba32af2e6fe25eeb96261e6140100edcf95686e0aaa134026b4 + b5254fd271b7a4da + +B.4.6. Test vector for ISK calculation parallel execution + + ordered cat of transcript : (length: 124 bytes) + 6f6338e233867540319ec86eaecc09a85dec233745db729f61c36bde + 14c034200994fc4b6e8d263008c169585fd1d186d8ac560cb9f7ad0d + 16696503414461385062a0f33478914bf162a80dad39b5b266c1dd02 + f408573b41827e38599b682afbf7a0735adfd68c39ab4994fd1b0348 + 46270e38332b4da903414462 + DSI = G.DSI_ISK, b'CPaceDecaf448_ISK': (length: 17 bytes) + 435061636544656361663434385f49534b + lv_cat(DSI,sid,K)||o_cat(MSGa,MSGb): (length: 216 bytes) + 11435061636544656361663434385f49534b105223e0cdc45d657566 + 8d64c55200412438dc9edef7c127e79d32f2584f9fcd3269174fe322 + 26c2082963879a6deafefb9c14efcee9fc1245917ad3658037d2d62a + ff2d3f76fa4fca996f6338e233867540319ec86eaecc09a85dec2337 + 45db729f61c36bde14c034200994fc4b6e8d263008c169585fd1d186 + d8ac560cb9f7ad0d16696503414461385062a0f33478914bf162a80d + ad39b5b266c1dd02f408573b41827e38599b682afbf7a0735adfd68c + 39ab4994fd1b034846270e38332b4da903414462 + ISK result: (length: 64 bytes) + e6c79d30d4381a45bd47b14b769d41354211aff553ece937d4ac134f + 09844896c72a723b1f1b6da1ab281d759a15624d2bcd0e423b70b8b8 + 50a4d0ed126a3026 + +B.4.7. Corresponding C programming language initializers + + const unsigned char tc_PRS[] = { + 0x50,0x61,0x73,0x73,0x77,0x6f,0x72,0x64, + }; + const unsigned char tc_CI[] = { + 0x0a,0x41,0x69,0x6e,0x69,0x74,0x69,0x61,0x74,0x6f,0x72,0x0a, + 0x42,0x72,0x65,0x73,0x70,0x6f,0x6e,0x64,0x65,0x72, + }; + const unsigned char tc_sid[] = { + 0x52,0x23,0xe0,0xcd,0xc4,0x5d,0x65,0x75,0x66,0x8d,0x64,0xc5, + 0x52,0x00,0x41,0x24, + }; + const unsigned char tc_g[] = { + 0x68,0x2d,0x1a,0x4f,0x49,0xfc,0x2a,0x48,0x34,0x35,0x6a,0xe4, + 0xd7,0xf5,0x86,0x36,0xbc,0x94,0x81,0x52,0x1c,0x84,0x5e,0x66, + 0xe6,0xfb,0x0b,0x29,0x69,0x34,0x1d,0xf4,0x5f,0xba,0xea,0xea, + 0x9e,0x22,0x21,0xb3,0xf5,0xba,0xbc,0x54,0xc5,0xf8,0xce,0x45, + 0x69,0x88,0xff,0xc5,0x19,0xde,0xfa,0xeb, + }; + const unsigned char tc_ya[] = { + 0x33,0xd5,0x61,0xf1,0x3c,0xfc,0x0d,0xca,0x27,0x9c,0x30,0xe8, + 0xcd,0xe8,0x95,0x17,0x5d,0xc2,0x54,0x83,0x89,0x28,0x19,0xeb, + 0xa1,0x32,0xd5,0x8c,0x13,0xc0,0x46,0x2a,0x8e,0xb0,0xd7,0x3f, + 0xda,0x94,0x19,0x50,0x59,0x4b,0xef,0x51,0x91,0xd8,0x39,0x46, + 0x91,0xf8,0x6e,0xdf,0xfc,0xad,0x6c,0x1e, + }; + const unsigned char tc_ADa[] = { + 0x41,0x44,0x61, + }; + const unsigned char tc_Ya[] = { + 0xe2,0x33,0x86,0x75,0x40,0x31,0x9e,0xc8,0x6e,0xae,0xcc,0x09, + 0xa8,0x5d,0xec,0x23,0x37,0x45,0xdb,0x72,0x9f,0x61,0xc3,0x6b, + 0xde,0x14,0xc0,0x34,0x20,0x09,0x94,0xfc,0x4b,0x6e,0x8d,0x26, + 0x30,0x08,0xc1,0x69,0x58,0x5f,0xd1,0xd1,0x86,0xd8,0xac,0x56, + 0x0c,0xb9,0xf7,0xad,0x0d,0x16,0x69,0x65, + }; + const unsigned char tc_yb[] = { + 0x25,0x23,0xc9,0x69,0xf6,0x8f,0xa2,0xb2,0xae,0xa2,0x94,0xc2, + 0x53,0x9e,0xf3,0x6e,0xb1,0xe0,0x55,0x8a,0xbd,0x14,0x71,0x2a, + 0x78,0x28,0xf1,0x6a,0x85,0xed,0x2c,0x7e,0x77,0xe2,0xbd,0xd4, + 0x18,0x99,0x44,0x05,0xfb,0x1b,0x57,0xb6,0xbb,0xaa,0xdd,0x66, + 0x84,0x98,0x92,0xaa,0xc9,0xd8,0x14,0x02, + }; + const unsigned char tc_ADb[] = { + 0x41,0x44,0x62, + }; + const unsigned char tc_Yb[] = { + 0x50,0x62,0xa0,0xf3,0x34,0x78,0x91,0x4b,0xf1,0x62,0xa8,0x0d, + 0xad,0x39,0xb5,0xb2,0x66,0xc1,0xdd,0x02,0xf4,0x08,0x57,0x3b, + 0x41,0x82,0x7e,0x38,0x59,0x9b,0x68,0x2a,0xfb,0xf7,0xa0,0x73, + 0x5a,0xdf,0xd6,0x8c,0x39,0xab,0x49,0x94,0xfd,0x1b,0x03,0x48, + 0x46,0x27,0x0e,0x38,0x33,0x2b,0x4d,0xa9, + }; + const unsigned char tc_K[] = { + 0xdc,0x9e,0xde,0xf7,0xc1,0x27,0xe7,0x9d,0x32,0xf2,0x58,0x4f, + 0x9f,0xcd,0x32,0x69,0x17,0x4f,0xe3,0x22,0x26,0xc2,0x08,0x29, + 0x63,0x87,0x9a,0x6d,0xea,0xfe,0xfb,0x9c,0x14,0xef,0xce,0xe9, + 0xfc,0x12,0x45,0x91,0x7a,0xd3,0x65,0x80,0x37,0xd2,0xd6,0x2a, + 0xff,0x2d,0x3f,0x76,0xfa,0x4f,0xca,0x99, + }; + const unsigned char tc_ISK_IR[] = { + 0xa7,0x52,0x61,0x2f,0xe6,0xde,0xc5,0x42,0xe9,0x66,0x29,0xa6, + 0xeb,0x68,0xec,0xb9,0xbf,0xe2,0x25,0x72,0x24,0x97,0x5e,0x91, + 0x60,0x35,0xae,0xe7,0x47,0xc6,0xab,0xa3,0x2a,0xf2,0xe6,0xfe, + 0x25,0xee,0xb9,0x62,0x61,0xe6,0x14,0x01,0x00,0xed,0xcf,0x95, + 0x68,0x6e,0x0a,0xaa,0x13,0x40,0x26,0xb4,0xb5,0x25,0x4f,0xd2, + 0x71,0xb7,0xa4,0xda, + }; + const unsigned char tc_ISK_SY[] = { + 0xe6,0xc7,0x9d,0x30,0xd4,0x38,0x1a,0x45,0xbd,0x47,0xb1,0x4b, + 0x76,0x9d,0x41,0x35,0x42,0x11,0xaf,0xf5,0x53,0xec,0xe9,0x37, + 0xd4,0xac,0x13,0x4f,0x09,0x84,0x48,0x96,0xc7,0x2a,0x72,0x3b, + 0x1f,0x1b,0x6d,0xa1,0xab,0x28,0x1d,0x75,0x9a,0x15,0x62,0x4d, + 0x2b,0xcd,0x0e,0x42,0x3b,0x70,0xb8,0xb8,0x50,0xa4,0xd0,0xed, + 0x12,0x6a,0x30,0x26, + }; + +B.4.8. Test case for scalar_mult with valid inputs + + s: (length: 56 bytes) + dd1bc7015daabb7672129cc35a3ba815486b139deff9bdeca7a4fc61 + 34323d34658761e90ff079972a7ca8aa5606498f4f4f0ebc0933a819 + X: (length: 56 bytes) + 601431d5e51f43d422a92d3fb2373bde28217aab42524c341aa404ea + ba5aa5541f7042dbb3253ce4c90f772b038a413dcb3a0f6bf3ae9e21 + G.scalar_mult(s,decode(X)): (length: 56 bytes) + 388b35c60eb41b66085a2118316218681d78979d667702de105fdc1f + 21ffe884a577d795f45691781390a229a3bd7b527e831380f2f585a4 + G.scalar_mult_vfy(s,X): (length: 56 bytes) + 388b35c60eb41b66085a2118316218681d78979d667702de105fdc1f + 21ffe884a577d795f45691781390a229a3bd7b527e831380f2f585a4 + +B.4.9. Invalid inputs for scalar_mult_vfy + + For these test cases scalar_mult_vfy(y,.) MUST return the + representation of the neutral element G.I. When points Y_i1 or Y_i2 + are included in MSGa or MSGb the protocol MUST abort. + + s: (length: 56 bytes) + dd1bc7015daabb7672129cc35a3ba815486b139deff9bdeca7a4fc61 + 34323d34658761e90ff079972a7ca8aa5606498f4f4f0ebc0933a819 + Y_i1: (length: 56 bytes) + 5f1431d5e51f43d422a92d3fb2373bde28217aab42524c341aa404ea + ba5aa5541f7042dbb3253ce4c90f772b038a413dcb3a0f6bf3ae9e21 + Y_i2 == G.I: (length: 56 bytes) + 00000000000000000000000000000000000000000000000000000000 + 00000000000000000000000000000000000000000000000000000000 + G.scalar_mult_vfy(s,Y_i1) = G.scalar_mult_vfy(s,Y_i2) = G.I + +B.5. Test vector for CPace using group NIST P-256 and hash SHA-256 + +B.5.1. Test vectors for calculate_generator with group NIST P-256 + + Inputs + H = SHA-256 with input block size 64 bytes. + PRS = b'Password' ; ZPAD length: 23 ; + DSI = b'CPaceP256_XMD:SHA-256_SSWU_NU_' + DST = b'CPaceP256_XMD:SHA-256_SSWU_NU__DST' + CI = b'\nAinitiator\nBresponder' + CI = 0a41696e69746961746f720a42726573706f6e646572 + sid = 34b36454cab2e7842c389f7d88ecb7df + Outputs + generator_string(PRS,G.DSI,CI,sid,H.s_in_bytes): + (length: 104 bytes) + 1e4350616365503235365f584d443a5348412d3235365f535357555f + 4e555f0850617373776f726417000000000000000000000000000000 + 0000000000000000160a41696e69746961746f720a42726573706f6e + 6465721034b36454cab2e7842c389f7d88ecb7df + generator g: (length: 65 bytes) + 041b51433114e096c9d595f0955f5717a75169afb95557f4a6f51155 + 035dee19c76887bce5c7c054fa1fe48a4a62c7fb96dc75e34259d2f7 + 2b8d41f31b8e586bcd + +B.5.2. Test vector for MSGa + + Inputs + ADa = b'ADa' + ya (big endian): (length: 32 bytes) + 37574cfbf1b95ff6a8e2d7be462d4d01e6dde2618f34f4de9df869b2 + 4f532c5d + Outputs + Ya: (length: 65 bytes) + 04b75c1bcda84a0f324aabb7f25cf853ed7fb327c33f23db6aeb320d + 81df014649c2ac691925fce0eceac7dbc75eca25e6a1558066a610b4 + 021488279e3b989d52 + Alternative correct value for Ya: g*(-ya): + (length: 65 bytes) + 04b75c1bcda84a0f324aabb7f25cf853ed7fb327c33f23db6aeb320d + 81df0146493d5396e5da031f1415382438a135da195eaa7f9a59ef4b + fdeb77d861c46762ad + MSGa = lv_cat(Ya,ADa): (length: 70 bytes) + 4104b75c1bcda84a0f324aabb7f25cf853ed7fb327c33f23db6aeb32 + 0d81df014649c2ac691925fce0eceac7dbc75eca25e6a1558066a610 + b4021488279e3b989d5203414461 + +B.5.3. Test vector for MSGb + + Inputs + ADb = b'ADb' + yb (big endian): (length: 32 bytes) + e5672fc9eb4e721f41d80181ec4c9fd9886668acc48024d33c82bb10 + 2aecba52 + Outputs + Yb: (length: 65 bytes) + 04bb2783a57337e74671f76452876b27839c0ea9e044e3aadaad2e64 + 777ed27a90e80a99438e2f1c072462f2895c6dadf1b43867b92ffb65 + 562b78c793947dcada + Alternative correct value for Yb: g*(-yb): + (length: 65 bytes) + 04bb2783a57337e74671f76452876b27839c0ea9e044e3aadaad2e64 + 777ed27a9017f566bb71d0e3f9db9d0d76a392520e4bc79847d0049a + a9d487386c6b823525 + MSGb = lv_cat(Yb,ADb): (length: 70 bytes) + 4104bb2783a57337e74671f76452876b27839c0ea9e044e3aadaad2e + 64777ed27a90e80a99438e2f1c072462f2895c6dadf1b43867b92ffb + 65562b78c793947dcada03414462 + +B.5.4. Test vector for secret points K + + scalar_mult_vfy(ya,Yb): (length: 32 bytes) + 8fd12b283805750aeee6151bcd4211a6b71019e8fc416293ade24ed2 + bce12c39 + scalar_mult_vfy(yb,Ya): (length: 32 bytes) + 8fd12b283805750aeee6151bcd4211a6b71019e8fc416293ade24ed2 + bce12c39 + +B.5.5. Test vector for ISK calculation initiator/responder + + unordered cat of transcript : (length: 140 bytes) + 4104b75c1bcda84a0f324aabb7f25cf853ed7fb327c33f23db6aeb32 + 0d81df014649c2ac691925fce0eceac7dbc75eca25e6a1558066a610 + b4021488279e3b989d52034144614104bb2783a57337e74671f76452 + 876b27839c0ea9e044e3aadaad2e64777ed27a90e80a99438e2f1c07 + 2462f2895c6dadf1b43867b92ffb65562b78c793947dcada03414462 + DSI = G.DSI_ISK, b'CPaceP256_XMD:SHA-256_SSWU_NU__ISK': + (length: 34 bytes) + 4350616365503235365f584d443a5348412d3235365f535357555f4e + 555f5f49534b + lv_cat(DSI,sid,K)||MSGa||MSGb: (length: 225 bytes) + 224350616365503235365f584d443a5348412d3235365f535357555f + 4e555f5f49534b1034b36454cab2e7842c389f7d88ecb7df208fd12b + 283805750aeee6151bcd4211a6b71019e8fc416293ade24ed2bce12c + 394104b75c1bcda84a0f324aabb7f25cf853ed7fb327c33f23db6aeb + 320d81df014649c2ac691925fce0eceac7dbc75eca25e6a1558066a6 + 10b4021488279e3b989d52034144614104bb2783a57337e74671f764 + 52876b27839c0ea9e044e3aadaad2e64777ed27a90e80a99438e2f1c + 072462f2895c6dadf1b43867b92ffb65562b78c793947dcada034144 + 62 + ISK result: (length: 32 bytes) + 7ae1e916606e44652e3c0d7231198af6519226339c241e546afd0bbf + 48e1c96a + +B.5.6. Test vector for ISK calculation parallel execution + + ordered cat of transcript : (length: 142 bytes) + 6f634104bb2783a57337e74671f76452876b27839c0ea9e044e3aada + ad2e64777ed27a90e80a99438e2f1c072462f2895c6dadf1b43867b9 + 2ffb65562b78c793947dcada034144624104b75c1bcda84a0f324aab + b7f25cf853ed7fb327c33f23db6aeb320d81df014649c2ac691925fc + e0eceac7dbc75eca25e6a1558066a610b4021488279e3b989d520341 + 4461 + DSI = G.DSI_ISK, b'CPaceP256_XMD:SHA-256_SSWU_NU__ISK': + (length: 34 bytes) + 4350616365503235365f584d443a5348412d3235365f535357555f4e + 555f5f49534b + lv_cat(DSI,sid,K)||o_cat(MSGa,MSGb): (length: 227 bytes) + 224350616365503235365f584d443a5348412d3235365f535357555f + 4e555f5f49534b1034b36454cab2e7842c389f7d88ecb7df208fd12b + 283805750aeee6151bcd4211a6b71019e8fc416293ade24ed2bce12c + 396f634104bb2783a57337e74671f76452876b27839c0ea9e044e3aa + daad2e64777ed27a90e80a99438e2f1c072462f2895c6dadf1b43867 + b92ffb65562b78c793947dcada034144624104b75c1bcda84a0f324a + abb7f25cf853ed7fb327c33f23db6aeb320d81df014649c2ac691925 + fce0eceac7dbc75eca25e6a1558066a610b4021488279e3b989d5203 + 414461 + ISK result: (length: 32 bytes) + 5600a5c5bea5e92695dd68bd33d7f7b58326199c27c9b7326d76e4f9 + cb2fb276 + +B.5.7. Corresponding C programming language initializers + + const unsigned char tc_PRS[] = { + 0x50,0x61,0x73,0x73,0x77,0x6f,0x72,0x64, + }; + const unsigned char tc_CI[] = { + 0x0a,0x41,0x69,0x6e,0x69,0x74,0x69,0x61,0x74,0x6f,0x72,0x0a, + 0x42,0x72,0x65,0x73,0x70,0x6f,0x6e,0x64,0x65,0x72, + }; + const unsigned char tc_sid[] = { + 0x34,0xb3,0x64,0x54,0xca,0xb2,0xe7,0x84,0x2c,0x38,0x9f,0x7d, + 0x88,0xec,0xb7,0xdf, + }; + const unsigned char tc_g[] = { + 0x04,0x1b,0x51,0x43,0x31,0x14,0xe0,0x96,0xc9,0xd5,0x95,0xf0, + 0x95,0x5f,0x57,0x17,0xa7,0x51,0x69,0xaf,0xb9,0x55,0x57,0xf4, + 0xa6,0xf5,0x11,0x55,0x03,0x5d,0xee,0x19,0xc7,0x68,0x87,0xbc, + 0xe5,0xc7,0xc0,0x54,0xfa,0x1f,0xe4,0x8a,0x4a,0x62,0xc7,0xfb, + 0x96,0xdc,0x75,0xe3,0x42,0x59,0xd2,0xf7,0x2b,0x8d,0x41,0xf3, + 0x1b,0x8e,0x58,0x6b,0xcd, + }; + const unsigned char tc_ya[] = { + 0x37,0x57,0x4c,0xfb,0xf1,0xb9,0x5f,0xf6,0xa8,0xe2,0xd7,0xbe, + 0x46,0x2d,0x4d,0x01,0xe6,0xdd,0xe2,0x61,0x8f,0x34,0xf4,0xde, + 0x9d,0xf8,0x69,0xb2,0x4f,0x53,0x2c,0x5d, + }; + const unsigned char tc_ADa[] = { + 0x41,0x44,0x61, + }; + const unsigned char tc_Ya[] = { + 0x04,0xb7,0x5c,0x1b,0xcd,0xa8,0x4a,0x0f,0x32,0x4a,0xab,0xb7, + 0xf2,0x5c,0xf8,0x53,0xed,0x7f,0xb3,0x27,0xc3,0x3f,0x23,0xdb, + 0x6a,0xeb,0x32,0x0d,0x81,0xdf,0x01,0x46,0x49,0xc2,0xac,0x69, + 0x19,0x25,0xfc,0xe0,0xec,0xea,0xc7,0xdb,0xc7,0x5e,0xca,0x25, + 0xe6,0xa1,0x55,0x80,0x66,0xa6,0x10,0xb4,0x02,0x14,0x88,0x27, + 0x9e,0x3b,0x98,0x9d,0x52, + }; + const unsigned char tc_yb[] = { + 0xe5,0x67,0x2f,0xc9,0xeb,0x4e,0x72,0x1f,0x41,0xd8,0x01,0x81, + 0xec,0x4c,0x9f,0xd9,0x88,0x66,0x68,0xac,0xc4,0x80,0x24,0xd3, + 0x3c,0x82,0xbb,0x10,0x2a,0xec,0xba,0x52, + }; + const unsigned char tc_ADb[] = { + 0x41,0x44,0x62, + }; + const unsigned char tc_Yb[] = { + 0x04,0xbb,0x27,0x83,0xa5,0x73,0x37,0xe7,0x46,0x71,0xf7,0x64, + 0x52,0x87,0x6b,0x27,0x83,0x9c,0x0e,0xa9,0xe0,0x44,0xe3,0xaa, + 0xda,0xad,0x2e,0x64,0x77,0x7e,0xd2,0x7a,0x90,0xe8,0x0a,0x99, + 0x43,0x8e,0x2f,0x1c,0x07,0x24,0x62,0xf2,0x89,0x5c,0x6d,0xad, + 0xf1,0xb4,0x38,0x67,0xb9,0x2f,0xfb,0x65,0x56,0x2b,0x78,0xc7, + 0x93,0x94,0x7d,0xca,0xda, + }; + const unsigned char tc_K[] = { + 0x8f,0xd1,0x2b,0x28,0x38,0x05,0x75,0x0a,0xee,0xe6,0x15,0x1b, + 0xcd,0x42,0x11,0xa6,0xb7,0x10,0x19,0xe8,0xfc,0x41,0x62,0x93, + 0xad,0xe2,0x4e,0xd2,0xbc,0xe1,0x2c,0x39, + }; + const unsigned char tc_ISK_IR[] = { + 0x7a,0xe1,0xe9,0x16,0x60,0x6e,0x44,0x65,0x2e,0x3c,0x0d,0x72, + 0x31,0x19,0x8a,0xf6,0x51,0x92,0x26,0x33,0x9c,0x24,0x1e,0x54, + 0x6a,0xfd,0x0b,0xbf,0x48,0xe1,0xc9,0x6a, + }; + const unsigned char tc_ISK_SY[] = { + 0x56,0x00,0xa5,0xc5,0xbe,0xa5,0xe9,0x26,0x95,0xdd,0x68,0xbd, + 0x33,0xd7,0xf7,0xb5,0x83,0x26,0x19,0x9c,0x27,0xc9,0xb7,0x32, + 0x6d,0x76,0xe4,0xf9,0xcb,0x2f,0xb2,0x76, + }; + +B.5.8. Test case for scalar_mult_vfy with correct inputs + + s: (length: 32 bytes) + f012501c091ff9b99a123fffe571d8bc01e8077ee581362e1bd21399 + 0835643b + X: (length: 65 bytes) + 0424648eb986c2be0af636455cef0550671d6bcd8aa26e0d72ffa1b1 + fd12ba4e0f78da2b6d2184f31af39e566aef127014b6936c9a37346d + 10a4ab2514faef5831 + G.scalar_mult(s,X) (full coordinates): (length: 65 bytes) + 04f5a191f078c87c36633b78c701751159d56c59f3fe9105b5720673 + 470f303ab925b6a7fd1cdd8f649a21cf36b68d9e9c4a11919a951892 + 519786104b27033757 + G.scalar_mult_vfy(s,X) (only X-coordinate): + (length: 32 bytes) + f5a191f078c87c36633b78c701751159d56c59f3fe9105b572067347 + 0f303ab9 + +B.5.9. Invalid inputs for scalar_mult_vfy + + For these test cases scalar_mult_vfy(y,.) MUST return the + representation of the neutral element G.I. When including Y_i1 or + Y_i2 in MSGa or MSGb the protocol MUST abort. + + s: (length: 32 bytes) + f012501c091ff9b99a123fffe571d8bc01e8077ee581362e1bd21399 + 0835643b + Y_i1: (length: 65 bytes) + 0424648eb986c2be0af636455cef0550671d6bcd8aa26e0d72ffa1b1 + fd12ba4e0f78da2b6d2184f31af39e566aef127014b6936c9a37346d + 10a4ab2514faef5857 + Y_i2: (length: 1 bytes) + 00 + G.scalar_mult_vfy(s,Y_i1) = G.scalar_mult_vfy(s,Y_i2) = G.I + +B.6. Test vector for CPace using group NIST P-384 and hash SHA-384 + +B.6.1. Test vectors for calculate_generator with group NIST P-384 + + Inputs + H = SHA-384 with input block size 128 bytes. + PRS = b'Password' ; ZPAD length: 87 ; + DSI = b'CPaceP384_XMD:SHA-384_SSWU_NU_' + DST = b'CPaceP384_XMD:SHA-384_SSWU_NU__DST' + CI = b'\nAinitiator\nBresponder' + CI = 0a41696e69746961746f720a42726573706f6e646572 + sid = 5b3773aa90e8f23c61563a4b645b276c + Outputs + generator_string(PRS,G.DSI,CI,sid,H.s_in_bytes): + (length: 168 bytes) + 1e4350616365503338345f584d443a5348412d3338345f535357555f + 4e555f0850617373776f726457000000000000000000000000000000 + 00000000000000000000000000000000000000000000000000000000 + 00000000000000000000000000000000000000000000000000000000 + 00000000000000000000000000000000160a41696e69746961746f72 + 0a42726573706f6e646572105b3773aa90e8f23c61563a4b645b276c + generator g: (length: 97 bytes) + 04f35a925fe82e54350e80b084a8013b1960cb3f73c49b0c2ae9b523 + 997846ddd14c66f24f62223112cf35b866065f91ad86674cce2a2876 + 84904e49f01287b54666bb518df2ea53cec627fa6e1283f14c6ed4bc + d11b33fbb962da3e2e4ff1345c + +B.6.2. Test vector for MSGa + + Inputs + ADa = b'ADa' + ya (big endian): (length: 48 bytes) + ef433dd5ad142c860e7cb6400dd315d388d5ec5420c550e9d6f0907f + 375d988bc4d704837e43561c497e7dd93edcdb9d + Outputs + Ya: (length: 97 bytes) + 04fd864c1a81f0e657a8a3f8e4ebafa421da712b6fb98f0abfa139ff + 971718cab474fa74c6a44b80a46468699280dd5d271252f3b9c05acc + 93dbd8b939152987cd5a8d1fb7b70c45512c993ec5456cc10f1797c9 + 2fac2f1b7e363478a9ecd79e74 + Alternative correct value for Ya: g*(-ya): + (length: 97 bytes) + 04fd864c1a81f0e657a8a3f8e4ebafa421da712b6fb98f0abfa139ff + 971718cab474fa74c6a44b80a46468699280dd5d27edad0c463fa533 + 6c242746c6ead67832a572e04848f3baaed366c13aba933eefe86836 + cf53d0e481c9cb87571328618b + MSGa = lv_cat(Ya,ADa): (length: 102 bytes) + 6104fd864c1a81f0e657a8a3f8e4ebafa421da712b6fb98f0abfa139 + ff971718cab474fa74c6a44b80a46468699280dd5d271252f3b9c05a + cc93dbd8b939152987cd5a8d1fb7b70c45512c993ec5456cc10f1797 + c92fac2f1b7e363478a9ecd79e7403414461 + +B.6.3. Test vector for MSGb + + Inputs + ADb = b'ADb' + yb (big endian): (length: 48 bytes) + 50b0e36b95a2edfaa8342b843dddc90b175330f2399c1b36586dedda + 3c255975f30be6a750f9404fccc62a6323b5e471 + Outputs + Yb: (length: 97 bytes) + 04822b9874755c51adfdf624101eb4dc12a8ae433750be4fd6f4f7eb + f6954ddb57837752a4effa4a5b44627a64b62a2db9d3c9c031c4ad37 + dbe7bf180d6bcba54feb4e84eeb876ebfa64a85d4c5ac2063dc05ba7 + 26810824c41e1893faa9373a84 + Alternative correct value for Yb: g*(-yb): + (length: 97 bytes) + 04822b9874755c51adfdf624101eb4dc12a8ae433750be4fd6f4f7eb + f6954ddb57837752a4effa4a5b44627a64b62a2db92c363fce3b52c8 + 241840e7f294345ab014b17b11478914059b57a2b3a53df9c13fa458 + d87ef7db3be1e76c0656c8c57b + MSGb = lv_cat(Yb,ADb): (length: 102 bytes) + 6104822b9874755c51adfdf624101eb4dc12a8ae433750be4fd6f4f7 + ebf6954ddb57837752a4effa4a5b44627a64b62a2db9d3c9c031c4ad + 37dbe7bf180d6bcba54feb4e84eeb876ebfa64a85d4c5ac2063dc05b + a726810824c41e1893faa9373a8403414462 + +B.6.4. Test vector for secret points K + + scalar_mult_vfy(ya,Yb): (length: 48 bytes) + 374290a54e07015baad085b311b18fbae1a20652e137c7c4bd13d565 + 7d8b1ace028eb5acfba8c68d6211a79fff0965c9 + scalar_mult_vfy(yb,Ya): (length: 48 bytes) + 374290a54e07015baad085b311b18fbae1a20652e137c7c4bd13d565 + 7d8b1ace028eb5acfba8c68d6211a79fff0965c9 + +B.6.5. Test vector for ISK calculation initiator/responder + + unordered cat of transcript : (length: 204 bytes) + 6104fd864c1a81f0e657a8a3f8e4ebafa421da712b6fb98f0abfa139 + ff971718cab474fa74c6a44b80a46468699280dd5d271252f3b9c05a + cc93dbd8b939152987cd5a8d1fb7b70c45512c993ec5456cc10f1797 + c92fac2f1b7e363478a9ecd79e74034144616104822b9874755c51ad + fdf624101eb4dc12a8ae433750be4fd6f4f7ebf6954ddb57837752a4 + effa4a5b44627a64b62a2db9d3c9c031c4ad37dbe7bf180d6bcba54f + eb4e84eeb876ebfa64a85d4c5ac2063dc05ba726810824c41e1893fa + a9373a8403414462 + DSI = G.DSI_ISK, b'CPaceP384_XMD:SHA-384_SSWU_NU__ISK': + (length: 34 bytes) + 4350616365503338345f584d443a5348412d3338345f535357555f4e + 555f5f49534b + lv_cat(DSI,sid,K)||MSGa||MSGb: (length: 305 bytes) + 224350616365503338345f584d443a5348412d3338345f535357555f + 4e555f5f49534b105b3773aa90e8f23c61563a4b645b276c30374290 + a54e07015baad085b311b18fbae1a20652e137c7c4bd13d5657d8b1a + ce028eb5acfba8c68d6211a79fff0965c96104fd864c1a81f0e657a8 + a3f8e4ebafa421da712b6fb98f0abfa139ff971718cab474fa74c6a4 + 4b80a46468699280dd5d271252f3b9c05acc93dbd8b939152987cd5a + 8d1fb7b70c45512c993ec5456cc10f1797c92fac2f1b7e363478a9ec + d79e74034144616104822b9874755c51adfdf624101eb4dc12a8ae43 + 3750be4fd6f4f7ebf6954ddb57837752a4effa4a5b44627a64b62a2d + b9d3c9c031c4ad37dbe7bf180d6bcba54feb4e84eeb876ebfa64a85d + 4c5ac2063dc05ba726810824c41e1893faa9373a8403414462 + ISK result: (length: 48 bytes) + a62d337820ce9cc1195a1adfb3c1efc2d844c0d8c6bc44bd060fe3cd + d4ee8d2343aca0168c2b58478354a37d8d8856bd + +B.6.6. Test vector for ISK calculation parallel execution + + ordered cat of transcript : (length: 206 bytes) + 6f636104fd864c1a81f0e657a8a3f8e4ebafa421da712b6fb98f0abf + a139ff971718cab474fa74c6a44b80a46468699280dd5d271252f3b9 + c05acc93dbd8b939152987cd5a8d1fb7b70c45512c993ec5456cc10f + 1797c92fac2f1b7e363478a9ecd79e74034144616104822b9874755c + 51adfdf624101eb4dc12a8ae433750be4fd6f4f7ebf6954ddb578377 + 52a4effa4a5b44627a64b62a2db9d3c9c031c4ad37dbe7bf180d6bcb + a54feb4e84eeb876ebfa64a85d4c5ac2063dc05ba726810824c41e18 + 93faa9373a8403414462 + DSI = G.DSI_ISK, b'CPaceP384_XMD:SHA-384_SSWU_NU__ISK': + (length: 34 bytes) + 4350616365503338345f584d443a5348412d3338345f535357555f4e + 555f5f49534b + lv_cat(DSI,sid,K)||o_cat(MSGa,MSGb): (length: 307 bytes) + 224350616365503338345f584d443a5348412d3338345f535357555f + 4e555f5f49534b105b3773aa90e8f23c61563a4b645b276c30374290 + a54e07015baad085b311b18fbae1a20652e137c7c4bd13d5657d8b1a + ce028eb5acfba8c68d6211a79fff0965c96f636104fd864c1a81f0e6 + 57a8a3f8e4ebafa421da712b6fb98f0abfa139ff971718cab474fa74 + c6a44b80a46468699280dd5d271252f3b9c05acc93dbd8b939152987 + cd5a8d1fb7b70c45512c993ec5456cc10f1797c92fac2f1b7e363478 + a9ecd79e74034144616104822b9874755c51adfdf624101eb4dc12a8 + ae433750be4fd6f4f7ebf6954ddb57837752a4effa4a5b44627a64b6 + 2a2db9d3c9c031c4ad37dbe7bf180d6bcba54feb4e84eeb876ebfa64 + a85d4c5ac2063dc05ba726810824c41e1893faa9373a8403414462 + ISK result: (length: 48 bytes) + eebf988a62b5c854f0ba32822ab45d23329bd1c78c84a4a0e1b40704 + c99c0a6f6c01c29af5fc6943254b883ce8a65ea1 + +B.6.7. Corresponding C programming language initializers + + const unsigned char tc_PRS[] = { + 0x50,0x61,0x73,0x73,0x77,0x6f,0x72,0x64, + }; + const unsigned char tc_CI[] = { + 0x0a,0x41,0x69,0x6e,0x69,0x74,0x69,0x61,0x74,0x6f,0x72,0x0a, + 0x42,0x72,0x65,0x73,0x70,0x6f,0x6e,0x64,0x65,0x72, + }; + const unsigned char tc_sid[] = { + 0x5b,0x37,0x73,0xaa,0x90,0xe8,0xf2,0x3c,0x61,0x56,0x3a,0x4b, + 0x64,0x5b,0x27,0x6c, + }; + const unsigned char tc_g[] = { + 0x04,0xf3,0x5a,0x92,0x5f,0xe8,0x2e,0x54,0x35,0x0e,0x80,0xb0, + 0x84,0xa8,0x01,0x3b,0x19,0x60,0xcb,0x3f,0x73,0xc4,0x9b,0x0c, + 0x2a,0xe9,0xb5,0x23,0x99,0x78,0x46,0xdd,0xd1,0x4c,0x66,0xf2, + 0x4f,0x62,0x22,0x31,0x12,0xcf,0x35,0xb8,0x66,0x06,0x5f,0x91, + 0xad,0x86,0x67,0x4c,0xce,0x2a,0x28,0x76,0x84,0x90,0x4e,0x49, + 0xf0,0x12,0x87,0xb5,0x46,0x66,0xbb,0x51,0x8d,0xf2,0xea,0x53, + 0xce,0xc6,0x27,0xfa,0x6e,0x12,0x83,0xf1,0x4c,0x6e,0xd4,0xbc, + 0xd1,0x1b,0x33,0xfb,0xb9,0x62,0xda,0x3e,0x2e,0x4f,0xf1,0x34, + 0x5c, + }; + const unsigned char tc_ya[] = { + 0xef,0x43,0x3d,0xd5,0xad,0x14,0x2c,0x86,0x0e,0x7c,0xb6,0x40, + 0x0d,0xd3,0x15,0xd3,0x88,0xd5,0xec,0x54,0x20,0xc5,0x50,0xe9, + 0xd6,0xf0,0x90,0x7f,0x37,0x5d,0x98,0x8b,0xc4,0xd7,0x04,0x83, + 0x7e,0x43,0x56,0x1c,0x49,0x7e,0x7d,0xd9,0x3e,0xdc,0xdb,0x9d, + }; + const unsigned char tc_ADa[] = { + 0x41,0x44,0x61, + }; + const unsigned char tc_Ya[] = { + 0x04,0xfd,0x86,0x4c,0x1a,0x81,0xf0,0xe6,0x57,0xa8,0xa3,0xf8, + 0xe4,0xeb,0xaf,0xa4,0x21,0xda,0x71,0x2b,0x6f,0xb9,0x8f,0x0a, + 0xbf,0xa1,0x39,0xff,0x97,0x17,0x18,0xca,0xb4,0x74,0xfa,0x74, + 0xc6,0xa4,0x4b,0x80,0xa4,0x64,0x68,0x69,0x92,0x80,0xdd,0x5d, + 0x27,0x12,0x52,0xf3,0xb9,0xc0,0x5a,0xcc,0x93,0xdb,0xd8,0xb9, + 0x39,0x15,0x29,0x87,0xcd,0x5a,0x8d,0x1f,0xb7,0xb7,0x0c,0x45, + 0x51,0x2c,0x99,0x3e,0xc5,0x45,0x6c,0xc1,0x0f,0x17,0x97,0xc9, + 0x2f,0xac,0x2f,0x1b,0x7e,0x36,0x34,0x78,0xa9,0xec,0xd7,0x9e, + 0x74, + }; + const unsigned char tc_yb[] = { + 0x50,0xb0,0xe3,0x6b,0x95,0xa2,0xed,0xfa,0xa8,0x34,0x2b,0x84, + 0x3d,0xdd,0xc9,0x0b,0x17,0x53,0x30,0xf2,0x39,0x9c,0x1b,0x36, + 0x58,0x6d,0xed,0xda,0x3c,0x25,0x59,0x75,0xf3,0x0b,0xe6,0xa7, + 0x50,0xf9,0x40,0x4f,0xcc,0xc6,0x2a,0x63,0x23,0xb5,0xe4,0x71, + }; + const unsigned char tc_ADb[] = { + 0x41,0x44,0x62, + }; + const unsigned char tc_Yb[] = { + 0x04,0x82,0x2b,0x98,0x74,0x75,0x5c,0x51,0xad,0xfd,0xf6,0x24, + 0x10,0x1e,0xb4,0xdc,0x12,0xa8,0xae,0x43,0x37,0x50,0xbe,0x4f, + 0xd6,0xf4,0xf7,0xeb,0xf6,0x95,0x4d,0xdb,0x57,0x83,0x77,0x52, + 0xa4,0xef,0xfa,0x4a,0x5b,0x44,0x62,0x7a,0x64,0xb6,0x2a,0x2d, + 0xb9,0xd3,0xc9,0xc0,0x31,0xc4,0xad,0x37,0xdb,0xe7,0xbf,0x18, + 0x0d,0x6b,0xcb,0xa5,0x4f,0xeb,0x4e,0x84,0xee,0xb8,0x76,0xeb, + 0xfa,0x64,0xa8,0x5d,0x4c,0x5a,0xc2,0x06,0x3d,0xc0,0x5b,0xa7, + 0x26,0x81,0x08,0x24,0xc4,0x1e,0x18,0x93,0xfa,0xa9,0x37,0x3a, + 0x84, + }; + const unsigned char tc_K[] = { + 0x37,0x42,0x90,0xa5,0x4e,0x07,0x01,0x5b,0xaa,0xd0,0x85,0xb3, + 0x11,0xb1,0x8f,0xba,0xe1,0xa2,0x06,0x52,0xe1,0x37,0xc7,0xc4, + 0xbd,0x13,0xd5,0x65,0x7d,0x8b,0x1a,0xce,0x02,0x8e,0xb5,0xac, + 0xfb,0xa8,0xc6,0x8d,0x62,0x11,0xa7,0x9f,0xff,0x09,0x65,0xc9, + }; + const unsigned char tc_ISK_IR[] = { + 0xa6,0x2d,0x33,0x78,0x20,0xce,0x9c,0xc1,0x19,0x5a,0x1a,0xdf, + 0xb3,0xc1,0xef,0xc2,0xd8,0x44,0xc0,0xd8,0xc6,0xbc,0x44,0xbd, + 0x06,0x0f,0xe3,0xcd,0xd4,0xee,0x8d,0x23,0x43,0xac,0xa0,0x16, + 0x8c,0x2b,0x58,0x47,0x83,0x54,0xa3,0x7d,0x8d,0x88,0x56,0xbd, + }; + const unsigned char tc_ISK_SY[] = { + 0xee,0xbf,0x98,0x8a,0x62,0xb5,0xc8,0x54,0xf0,0xba,0x32,0x82, + 0x2a,0xb4,0x5d,0x23,0x32,0x9b,0xd1,0xc7,0x8c,0x84,0xa4,0xa0, + 0xe1,0xb4,0x07,0x04,0xc9,0x9c,0x0a,0x6f,0x6c,0x01,0xc2,0x9a, + 0xf5,0xfc,0x69,0x43,0x25,0x4b,0x88,0x3c,0xe8,0xa6,0x5e,0xa1, + }; + +B.6.8. Test case for scalar_mult_vfy with correct inputs + + s: (length: 48 bytes) + 6e8a99a5cdd408eae98e1b8aed286e7b12adbbdac7f2c628d9060ce9 + 2ae0d90bd57a564fd3500fbcce3425dc94ba0ade + X: (length: 97 bytes) + 045b4cd53c4506cc04ba4c44f2762d5d32c3e55df25b8baa5571b165 + 7ad9576efea8259f0684de065a470585b4be876748c7797054f3defe + f21b77f83d53bac57c89d52aa4d6dd5872bd281989b138359698009f + 8ac1f301538badcce9d9f4036e + G.scalar_mult(s,X) (full coordinates): (length: 97 bytes) + 0465c28db05fd9f9a93651c5cc31eae49c4e5246b46489b8f6105873 + 3173a033cda76c3e3ea5352b804e67fdbe2e334be8245dad5c8c993e + 63bacf0456478f29b71b6c859f13676f84ff150d2741f028f560584a + 0bdba19a63df62c08949c2fd6d + G.scalar_mult_vfy(s,X) (only X-coordinate): + (length: 48 bytes) + 65c28db05fd9f9a93651c5cc31eae49c4e5246b46489b8f610587331 + 73a033cda76c3e3ea5352b804e67fdbe2e334be8 + +B.6.9. Invalid inputs for scalar_mult_vfy + + For these test cases scalar_mult_vfy(y,.) MUST return the + representation of the neutral element G.I. When including Y_i1 or + Y_i2 in MSGa or MSGb the protocol MUST abort. + + s: (length: 48 bytes) + 6e8a99a5cdd408eae98e1b8aed286e7b12adbbdac7f2c628d9060ce9 + 2ae0d90bd57a564fd3500fbcce3425dc94ba0ade + Y_i1: (length: 97 bytes) + 045b4cd53c4506cc04ba4c44f2762d5d32c3e55df25b8baa5571b165 + 7ad9576efea8259f0684de065a470585b4be876748c7797054f3defe + f21b77f83d53bac57c89d52aa4d6dd5872bd281989b138359698009f + 8ac1f301538badcce9d9f40302 + Y_i2: (length: 1 bytes) + 00 + G.scalar_mult_vfy(s,Y_i1) = G.scalar_mult_vfy(s,Y_i2) = G.I + +B.7. Test vector for CPace using group NIST P-521 and hash SHA-512 + +B.7.1. Test vectors for calculate_generator with group NIST P-521 + + Inputs + H = SHA-512 with input block size 128 bytes. + PRS = b'Password' ; ZPAD length: 87 ; + DSI = b'CPaceP521_XMD:SHA-512_SSWU_NU_' + DST = b'CPaceP521_XMD:SHA-512_SSWU_NU__DST' + CI = b'\nAinitiator\nBresponder' + CI = 0a41696e69746961746f720a42726573706f6e646572 + sid = 7e4b4791d6a8ef019b936c79fb7f2c57 + Outputs + generator_string(PRS,G.DSI,CI,sid,H.s_in_bytes): + (length: 168 bytes) + 1e4350616365503532315f584d443a5348412d3531325f535357555f + 4e555f0850617373776f726457000000000000000000000000000000 + 00000000000000000000000000000000000000000000000000000000 + 00000000000000000000000000000000000000000000000000000000 + 00000000000000000000000000000000160a41696e69746961746f72 + 0a42726573706f6e646572107e4b4791d6a8ef019b936c79fb7f2c57 + generator g: (length: 133 bytes) + 0400dc927958f0b69ccad8fb67ef008905354b58c7c9c92ad50060a9 + e6afb10437d6ca8a26164e8573702b897275a25d05ed4407af2a3849 + 86dca7e243b92c5dd500d40057012121a9c8e34373fa619f918f7d47 + 9c23f85f0485379ef0f05284398de26653b49a155324c9d7b138be84 + d0b49bb58e232b7bf697798de6ee8afd6b92b6fa2f + +B.7.2. Test vector for MSGa + + Inputs + ADa = b'ADa' + ya (big endian): (length: 66 bytes) + 006367e9c2aeff9f1db19af600cca73343d47cbe446cebbd1ccd783f + 82755a872da86fd0707eb3767c6114f1803deb62d63bdd1e613f67e6 + 3e8c141ee5310e3ee819 + Outputs + Ya: (length: 133 bytes) + 04003701ec35caafa3dd416cad29ba1774551f9d2ed89f7e1065706d + ca230b86a11d02e4cee8b3fde64380d4a05983167d8a2414bc594ad5 + 286c068792ab7ca60ff6ea00919c41c00e789dabc2f42fd94178d7bf + d8fbe1aff1c1854b3dafb3a0ea13f5a5fc1703860f022bd271740469 + bb322b07c179c7c225499b31727c0ea3ee65578634 + Alternative correct value for Ya: g*(-ya): + (length: 133 bytes) + 04003701ec35caafa3dd416cad29ba1774551f9d2ed89f7e1065706d + ca230b86a11d02e4cee8b3fde64380d4a05983167d8a2414bc594ad5 + 286c068792ab7ca60ff6ea016e63be3ff18762543d0bd026be872840 + 27041e500e3e7ab4c2504c5f15ec0a5a03e8fc79f0fdd42d8e8bfb96 + 44cdd4f83e86383ddab664ce8d83f15c119aa879cb + MSGa = lv_cat(Ya,ADa): (length: 139 bytes) + 850104003701ec35caafa3dd416cad29ba1774551f9d2ed89f7e1065 + 706dca230b86a11d02e4cee8b3fde64380d4a05983167d8a2414bc59 + 4ad5286c068792ab7ca60ff6ea00919c41c00e789dabc2f42fd94178 + d7bfd8fbe1aff1c1854b3dafb3a0ea13f5a5fc1703860f022bd27174 + 0469bb322b07c179c7c225499b31727c0ea3ee6557863403414461 + +B.7.3. Test vector for MSGb + + Inputs + ADb = b'ADb' + yb (big endian): (length: 66 bytes) + 009227bf8dc741dacc9422f8bf3c0e96fce9587bc562eaafe0dc5f6f + 82f28594e4a6f98553560c62b75fa4abb198cecbbb86ebd41b0ea025 + 4cde78ac68d39a240ae7 + Outputs + Yb: (length: 133 bytes) + 0400f5cb68bf0117bd1a65412a2bc800af92013f9969cf546e1ea6d3 + bcf08643fdc482130aec1eecc33a2b5f33600be51295047fa3399fa2 + 82cc1a78de91f3a4e30b5d01a085b453f22bf3dc947386b042e5fc4e + c691fee47fe3c3ec6408c22a17c26bc0ab73940910614d6fcee32daf + bfd2d340d6e382d71b1fc763d7cec502fbcbcf93b4 + Alternative correct value for Yb: g*(-yb): + (length: 133 bytes) + 0400f5cb68bf0117bd1a65412a2bc800af92013f9969cf546e1ea6d3 + bcf08643fdc482130aec1eecc33a2b5f33600be51295047fa3399fa2 + 82cc1a78de91f3a4e30b5d005f7a4bac0dd40c236b8c794fbd1a03b1 + 396e011b801c3c139bf73dd5e83d943f548c6bf6ef9eb290311cd250 + 402d2cbf291c7d28e4e0389c28313afd0434306c4b + MSGb = lv_cat(Yb,ADb): (length: 139 bytes) + 85010400f5cb68bf0117bd1a65412a2bc800af92013f9969cf546e1e + a6d3bcf08643fdc482130aec1eecc33a2b5f33600be51295047fa339 + 9fa282cc1a78de91f3a4e30b5d01a085b453f22bf3dc947386b042e5 + fc4ec691fee47fe3c3ec6408c22a17c26bc0ab73940910614d6fcee3 + 2dafbfd2d340d6e382d71b1fc763d7cec502fbcbcf93b403414462 + +B.7.4. Test vector for secret points K + + scalar_mult_vfy(ya,Yb): (length: 66 bytes) + 00503e75e38e012a6dc6f3561980e4cf540dbcff3de3a4a6f09d79c3 + 2cc45764d3a6605eb45df1dc63fb7937b7879f2820da1b3266b69fa0 + 99bf8720dd8f6a07e8ed + scalar_mult_vfy(yb,Ya): (length: 66 bytes) + 00503e75e38e012a6dc6f3561980e4cf540dbcff3de3a4a6f09d79c3 + 2cc45764d3a6605eb45df1dc63fb7937b7879f2820da1b3266b69fa0 + 99bf8720dd8f6a07e8ed + +B.7.5. Test vector for ISK calculation initiator/responder + + unordered cat of transcript : (length: 278 bytes) + 850104003701ec35caafa3dd416cad29ba1774551f9d2ed89f7e1065 + 706dca230b86a11d02e4cee8b3fde64380d4a05983167d8a2414bc59 + 4ad5286c068792ab7ca60ff6ea00919c41c00e789dabc2f42fd94178 + d7bfd8fbe1aff1c1854b3dafb3a0ea13f5a5fc1703860f022bd27174 + 0469bb322b07c179c7c225499b31727c0ea3ee655786340341446185 + 010400f5cb68bf0117bd1a65412a2bc800af92013f9969cf546e1ea6 + d3bcf08643fdc482130aec1eecc33a2b5f33600be51295047fa3399f + a282cc1a78de91f3a4e30b5d01a085b453f22bf3dc947386b042e5fc + 4ec691fee47fe3c3ec6408c22a17c26bc0ab73940910614d6fcee32d + afbfd2d340d6e382d71b1fc763d7cec502fbcbcf93b403414462 + DSI = G.DSI_ISK, b'CPaceP521_XMD:SHA-512_SSWU_NU__ISK': + (length: 34 bytes) + 4350616365503532315f584d443a5348412d3531325f535357555f4e + 555f5f49534b + lv_cat(DSI,sid,K)||MSGa||MSGb: (length: 397 bytes) + 224350616365503532315f584d443a5348412d3531325f535357555f + 4e555f5f49534b107e4b4791d6a8ef019b936c79fb7f2c574200503e + 75e38e012a6dc6f3561980e4cf540dbcff3de3a4a6f09d79c32cc457 + 64d3a6605eb45df1dc63fb7937b7879f2820da1b3266b69fa099bf87 + 20dd8f6a07e8ed850104003701ec35caafa3dd416cad29ba1774551f + 9d2ed89f7e1065706dca230b86a11d02e4cee8b3fde64380d4a05983 + 167d8a2414bc594ad5286c068792ab7ca60ff6ea00919c41c00e789d + abc2f42fd94178d7bfd8fbe1aff1c1854b3dafb3a0ea13f5a5fc1703 + 860f022bd271740469bb322b07c179c7c225499b31727c0ea3ee6557 + 86340341446185010400f5cb68bf0117bd1a65412a2bc800af92013f + 9969cf546e1ea6d3bcf08643fdc482130aec1eecc33a2b5f33600be5 + 1295047fa3399fa282cc1a78de91f3a4e30b5d01a085b453f22bf3dc + 947386b042e5fc4ec691fee47fe3c3ec6408c22a17c26bc0ab739409 + 10614d6fcee32dafbfd2d340d6e382d71b1fc763d7cec502fbcbcf93 + b403414462 + ISK result: (length: 64 bytes) + ed208a15af3ef8a67a5cac4acb360d03154570e3b1b1c54867f53a72 + 53cb919d13aa47efc647375be2250cb39ad965afa4ddfcb6be47d586 + d28c7eef6d654525 + +B.7.6. Test vector for ISK calculation parallel execution + + ordered cat of transcript : (length: 280 bytes) + 6f6385010400f5cb68bf0117bd1a65412a2bc800af92013f9969cf54 + 6e1ea6d3bcf08643fdc482130aec1eecc33a2b5f33600be51295047f + a3399fa282cc1a78de91f3a4e30b5d01a085b453f22bf3dc947386b0 + 42e5fc4ec691fee47fe3c3ec6408c22a17c26bc0ab73940910614d6f + cee32dafbfd2d340d6e382d71b1fc763d7cec502fbcbcf93b4034144 + 62850104003701ec35caafa3dd416cad29ba1774551f9d2ed89f7e10 + 65706dca230b86a11d02e4cee8b3fde64380d4a05983167d8a2414bc + 594ad5286c068792ab7ca60ff6ea00919c41c00e789dabc2f42fd941 + 78d7bfd8fbe1aff1c1854b3dafb3a0ea13f5a5fc1703860f022bd271 + 740469bb322b07c179c7c225499b31727c0ea3ee6557863403414461 + DSI = G.DSI_ISK, b'CPaceP521_XMD:SHA-512_SSWU_NU__ISK': + (length: 34 bytes) + 4350616365503532315f584d443a5348412d3531325f535357555f4e + 555f5f49534b + lv_cat(DSI,sid,K)||o_cat(MSGa,MSGb): (length: 399 bytes) + 224350616365503532315f584d443a5348412d3531325f535357555f + 4e555f5f49534b107e4b4791d6a8ef019b936c79fb7f2c574200503e + 75e38e012a6dc6f3561980e4cf540dbcff3de3a4a6f09d79c32cc457 + 64d3a6605eb45df1dc63fb7937b7879f2820da1b3266b69fa099bf87 + 20dd8f6a07e8ed6f6385010400f5cb68bf0117bd1a65412a2bc800af + 92013f9969cf546e1ea6d3bcf08643fdc482130aec1eecc33a2b5f33 + 600be51295047fa3399fa282cc1a78de91f3a4e30b5d01a085b453f2 + 2bf3dc947386b042e5fc4ec691fee47fe3c3ec6408c22a17c26bc0ab + 73940910614d6fcee32dafbfd2d340d6e382d71b1fc763d7cec502fb + cbcf93b403414462850104003701ec35caafa3dd416cad29ba177455 + 1f9d2ed89f7e1065706dca230b86a11d02e4cee8b3fde64380d4a059 + 83167d8a2414bc594ad5286c068792ab7ca60ff6ea00919c41c00e78 + 9dabc2f42fd94178d7bfd8fbe1aff1c1854b3dafb3a0ea13f5a5fc17 + 03860f022bd271740469bb322b07c179c7c225499b31727c0ea3ee65 + 57863403414461 + ISK result: (length: 64 bytes) + e7b10b6da531d9a8fd47fdd08441e8bb803d16c59a93e366d5cd9a10 + 277bbc543d943182889154704d80f2b0756ed62da87e0eb4e6d07920 + 480100d5e800ca85 + +B.7.7. Corresponding C programming language initializers + + const unsigned char tc_PRS[] = { + 0x50,0x61,0x73,0x73,0x77,0x6f,0x72,0x64, + }; + const unsigned char tc_CI[] = { + 0x0a,0x41,0x69,0x6e,0x69,0x74,0x69,0x61,0x74,0x6f,0x72,0x0a, + 0x42,0x72,0x65,0x73,0x70,0x6f,0x6e,0x64,0x65,0x72, + }; + const unsigned char tc_sid[] = { + 0x7e,0x4b,0x47,0x91,0xd6,0xa8,0xef,0x01,0x9b,0x93,0x6c,0x79, + 0xfb,0x7f,0x2c,0x57, + }; + const unsigned char tc_g[] = { + 0x04,0x00,0xdc,0x92,0x79,0x58,0xf0,0xb6,0x9c,0xca,0xd8,0xfb, + 0x67,0xef,0x00,0x89,0x05,0x35,0x4b,0x58,0xc7,0xc9,0xc9,0x2a, + 0xd5,0x00,0x60,0xa9,0xe6,0xaf,0xb1,0x04,0x37,0xd6,0xca,0x8a, + 0x26,0x16,0x4e,0x85,0x73,0x70,0x2b,0x89,0x72,0x75,0xa2,0x5d, + 0x05,0xed,0x44,0x07,0xaf,0x2a,0x38,0x49,0x86,0xdc,0xa7,0xe2, + 0x43,0xb9,0x2c,0x5d,0xd5,0x00,0xd4,0x00,0x57,0x01,0x21,0x21, + 0xa9,0xc8,0xe3,0x43,0x73,0xfa,0x61,0x9f,0x91,0x8f,0x7d,0x47, + 0x9c,0x23,0xf8,0x5f,0x04,0x85,0x37,0x9e,0xf0,0xf0,0x52,0x84, + 0x39,0x8d,0xe2,0x66,0x53,0xb4,0x9a,0x15,0x53,0x24,0xc9,0xd7, + 0xb1,0x38,0xbe,0x84,0xd0,0xb4,0x9b,0xb5,0x8e,0x23,0x2b,0x7b, + 0xf6,0x97,0x79,0x8d,0xe6,0xee,0x8a,0xfd,0x6b,0x92,0xb6,0xfa, + 0x2f, + }; + const unsigned char tc_ya[] = { + 0x00,0x63,0x67,0xe9,0xc2,0xae,0xff,0x9f,0x1d,0xb1,0x9a,0xf6, + 0x00,0xcc,0xa7,0x33,0x43,0xd4,0x7c,0xbe,0x44,0x6c,0xeb,0xbd, + 0x1c,0xcd,0x78,0x3f,0x82,0x75,0x5a,0x87,0x2d,0xa8,0x6f,0xd0, + 0x70,0x7e,0xb3,0x76,0x7c,0x61,0x14,0xf1,0x80,0x3d,0xeb,0x62, + 0xd6,0x3b,0xdd,0x1e,0x61,0x3f,0x67,0xe6,0x3e,0x8c,0x14,0x1e, + 0xe5,0x31,0x0e,0x3e,0xe8,0x19, + }; + const unsigned char tc_ADa[] = { + 0x41,0x44,0x61, + }; + const unsigned char tc_Ya[] = { + 0x04,0x00,0x37,0x01,0xec,0x35,0xca,0xaf,0xa3,0xdd,0x41,0x6c, + 0xad,0x29,0xba,0x17,0x74,0x55,0x1f,0x9d,0x2e,0xd8,0x9f,0x7e, + 0x10,0x65,0x70,0x6d,0xca,0x23,0x0b,0x86,0xa1,0x1d,0x02,0xe4, + 0xce,0xe8,0xb3,0xfd,0xe6,0x43,0x80,0xd4,0xa0,0x59,0x83,0x16, + 0x7d,0x8a,0x24,0x14,0xbc,0x59,0x4a,0xd5,0x28,0x6c,0x06,0x87, + 0x92,0xab,0x7c,0xa6,0x0f,0xf6,0xea,0x00,0x91,0x9c,0x41,0xc0, + 0x0e,0x78,0x9d,0xab,0xc2,0xf4,0x2f,0xd9,0x41,0x78,0xd7,0xbf, + 0xd8,0xfb,0xe1,0xaf,0xf1,0xc1,0x85,0x4b,0x3d,0xaf,0xb3,0xa0, + 0xea,0x13,0xf5,0xa5,0xfc,0x17,0x03,0x86,0x0f,0x02,0x2b,0xd2, + 0x71,0x74,0x04,0x69,0xbb,0x32,0x2b,0x07,0xc1,0x79,0xc7,0xc2, + 0x25,0x49,0x9b,0x31,0x72,0x7c,0x0e,0xa3,0xee,0x65,0x57,0x86, + 0x34, + }; + const unsigned char tc_yb[] = { + 0x00,0x92,0x27,0xbf,0x8d,0xc7,0x41,0xda,0xcc,0x94,0x22,0xf8, + 0xbf,0x3c,0x0e,0x96,0xfc,0xe9,0x58,0x7b,0xc5,0x62,0xea,0xaf, + 0xe0,0xdc,0x5f,0x6f,0x82,0xf2,0x85,0x94,0xe4,0xa6,0xf9,0x85, + 0x53,0x56,0x0c,0x62,0xb7,0x5f,0xa4,0xab,0xb1,0x98,0xce,0xcb, + 0xbb,0x86,0xeb,0xd4,0x1b,0x0e,0xa0,0x25,0x4c,0xde,0x78,0xac, + 0x68,0xd3,0x9a,0x24,0x0a,0xe7, + }; + const unsigned char tc_ADb[] = { + 0x41,0x44,0x62, + }; + const unsigned char tc_Yb[] = { + 0x04,0x00,0xf5,0xcb,0x68,0xbf,0x01,0x17,0xbd,0x1a,0x65,0x41, + 0x2a,0x2b,0xc8,0x00,0xaf,0x92,0x01,0x3f,0x99,0x69,0xcf,0x54, + 0x6e,0x1e,0xa6,0xd3,0xbc,0xf0,0x86,0x43,0xfd,0xc4,0x82,0x13, + 0x0a,0xec,0x1e,0xec,0xc3,0x3a,0x2b,0x5f,0x33,0x60,0x0b,0xe5, + 0x12,0x95,0x04,0x7f,0xa3,0x39,0x9f,0xa2,0x82,0xcc,0x1a,0x78, + 0xde,0x91,0xf3,0xa4,0xe3,0x0b,0x5d,0x01,0xa0,0x85,0xb4,0x53, + 0xf2,0x2b,0xf3,0xdc,0x94,0x73,0x86,0xb0,0x42,0xe5,0xfc,0x4e, + 0xc6,0x91,0xfe,0xe4,0x7f,0xe3,0xc3,0xec,0x64,0x08,0xc2,0x2a, + 0x17,0xc2,0x6b,0xc0,0xab,0x73,0x94,0x09,0x10,0x61,0x4d,0x6f, + 0xce,0xe3,0x2d,0xaf,0xbf,0xd2,0xd3,0x40,0xd6,0xe3,0x82,0xd7, + 0x1b,0x1f,0xc7,0x63,0xd7,0xce,0xc5,0x02,0xfb,0xcb,0xcf,0x93, + 0xb4, + }; + const unsigned char tc_K[] = { + 0x00,0x50,0x3e,0x75,0xe3,0x8e,0x01,0x2a,0x6d,0xc6,0xf3,0x56, + 0x19,0x80,0xe4,0xcf,0x54,0x0d,0xbc,0xff,0x3d,0xe3,0xa4,0xa6, + 0xf0,0x9d,0x79,0xc3,0x2c,0xc4,0x57,0x64,0xd3,0xa6,0x60,0x5e, + 0xb4,0x5d,0xf1,0xdc,0x63,0xfb,0x79,0x37,0xb7,0x87,0x9f,0x28, + 0x20,0xda,0x1b,0x32,0x66,0xb6,0x9f,0xa0,0x99,0xbf,0x87,0x20, + 0xdd,0x8f,0x6a,0x07,0xe8,0xed, + }; + const unsigned char tc_ISK_IR[] = { + 0xed,0x20,0x8a,0x15,0xaf,0x3e,0xf8,0xa6,0x7a,0x5c,0xac,0x4a, + 0xcb,0x36,0x0d,0x03,0x15,0x45,0x70,0xe3,0xb1,0xb1,0xc5,0x48, + 0x67,0xf5,0x3a,0x72,0x53,0xcb,0x91,0x9d,0x13,0xaa,0x47,0xef, + 0xc6,0x47,0x37,0x5b,0xe2,0x25,0x0c,0xb3,0x9a,0xd9,0x65,0xaf, + 0xa4,0xdd,0xfc,0xb6,0xbe,0x47,0xd5,0x86,0xd2,0x8c,0x7e,0xef, + 0x6d,0x65,0x45,0x25, + }; + const unsigned char tc_ISK_SY[] = { + 0xe7,0xb1,0x0b,0x6d,0xa5,0x31,0xd9,0xa8,0xfd,0x47,0xfd,0xd0, + 0x84,0x41,0xe8,0xbb,0x80,0x3d,0x16,0xc5,0x9a,0x93,0xe3,0x66, + 0xd5,0xcd,0x9a,0x10,0x27,0x7b,0xbc,0x54,0x3d,0x94,0x31,0x82, + 0x88,0x91,0x54,0x70,0x4d,0x80,0xf2,0xb0,0x75,0x6e,0xd6,0x2d, + 0xa8,0x7e,0x0e,0xb4,0xe6,0xd0,0x79,0x20,0x48,0x01,0x00,0xd5, + 0xe8,0x00,0xca,0x85, + }; + +B.7.8. Test case for scalar_mult_vfy with correct inputs + + s: (length: 66 bytes) + 0182dd7925f1753419e4bf83429763acd37d64000cd5a175edf53a15 + 87dd986bc95acc1506991702b6ba1a9ee2458fee8efc00198cf0088c + 480965ef65ff2048b856 + X: (length: 133 bytes) + 0400dc5078b24c4af1620cc10fbecc6cd8cf1cab0b011efb73c782f2 + 26dc21c7ca7eb406be74a69ecba5b4a87c07cfc6e687b4beca9a6eda + c95940a3b4120573b26a80005e697833b0ba285fce7b3f1f25243008 + 860b8f1de710a0dcc05b0d20341efe90eb2bcca26797c2d85ae6ca74 + c00696cb1b13e40bda15b27964d7670576647bfab9 + G.scalar_mult(s,X) (full coordinates): (length: 133 bytes) + 040122f88ce73ec5aa2d1c8c5d04148760c3d97ba87daa10d8cb8bb7 + c73cf6e951fc922721bf1437995cfb13e132a78beb86389e60d3517c + df6d99a8a2d6db19ef27bd0055af9e8ddcf337ce0a7c22a9c8099bc4 + a44faeded1eb72effd26e4f322217b67d60b944b267b3df5046078fd + 577f1785728f49b241fd5e8c83223a994a2d219281 + G.scalar_mult_vfy(s,X) (only X-coordinate): + (length: 66 bytes) + 0122f88ce73ec5aa2d1c8c5d04148760c3d97ba87daa10d8cb8bb7c7 + 3cf6e951fc922721bf1437995cfb13e132a78beb86389e60d3517cdf + 6d99a8a2d6db19ef27bd + +B.7.9. Invalid inputs for scalar_mult_vfy + + For these test cases scalar_mult_vfy(y,.) MUST return the + representation of the neutral element G.I. When including Y_i1 or + Y_i2 in MSGa or MSGb the protocol MUST abort. + + s: (length: 66 bytes) + 0182dd7925f1753419e4bf83429763acd37d64000cd5a175edf53a15 + 87dd986bc95acc1506991702b6ba1a9ee2458fee8efc00198cf0088c + 480965ef65ff2048b856 + Y_i1: (length: 133 bytes) + 0400dc5078b24c4af1620cc10fbecc6cd8cf1cab0b011efb73c782f2 + 26dc21c7ca7eb406be74a69ecba5b4a87c07cfc6e687b4beca9a6eda + c95940a3b4120573b26a80005e697833b0ba285fce7b3f1f25243008 + 860b8f1de710a0dcc05b0d20341efe90eb2bcca26797c2d85ae6ca74 + c00696cb1b13e40bda15b27964d7670576647bfaf9 + Y_i2: (length: 1 bytes) + 00 + G.scalar_mult_vfy(s,Y_i1) = G.scalar_mult_vfy(s,Y_i2) = G.I + +Authors' Addresses + + Michel Abdalla + DFINITY - Zurich + Email: michel.abdalla@gmail.com + + + Bjoern Haase + Endress + Hauser Liquid Analysis - Gerlingen + Email: bjoern.m.haase@web.de + + + Julia Hesse + IBM Research Europe - Zurich + Email: JHS@zurich.ibm.com diff --git a/draft-irtf-cfrg-cpace-11/index.html b/draft-irtf-cfrg-cpace-11/index.html new file mode 100644 index 0000000..b49a3ed --- /dev/null +++ b/draft-irtf-cfrg-cpace-11/index.html @@ -0,0 +1,45 @@ + + + + cfrg/draft-irtf-cfrg-cpace draft-irtf-cfrg-cpace-11 preview + + + + +

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