Add the new version of dec

To replace the old dec when we are converting Presigning

PAPER.md

@@ -16,6 +16,8 @@ Replies from Nikos Makriyannis are marked as (NM)
 
 # Typos
 
+In `П_dec` (Fig. 28), Inputs, the condition `(1 + N)^z` should read `(1 + N)^y`. Also note that at the point where it is used, the first secret argument corresponds to `y`, and the second to `x`.
+
 The randomness derivation (`\mu` in Output, step 1, in Fig. 6) is overly complicated - `\mu = (C mod N)^(N^(-1) mod phi(N))` works just as well. Also, `(1 + N)^m mod N^2 == (1 + m * N) mod N^2`, which is much faster to compute, but I guess the exponential form is conventional.
 
 In `П^{fac}` (Fig. 28), step 2: `q` the curve order, not to be confused with `q` the RSA prime in the Inputs.

synedrion/src/cggmp21/sigma.rs

@@ -2,6 +2,7 @@
 
 mod aff_g;
 mod dec;
+mod dec_new;
 mod enc;
 mod fac;
 mod log_star;

synedrion/src/cggmp21/sigma/dec_new.rs

@@ -0,0 +1,294 @@
+//! Paillier Special Decryption in the Exponent ($\Pi^{dec}$, Section A.6, Fig. 28)
+
+#![allow(dead_code)]
+
+use alloc::{boxed::Box, vec::Vec};
+
+use rand_core::CryptoRngCore;
+use serde::{Deserialize, Serialize};
+
+use super::super::{
+    conversion::{scalar_from_signed, secret_scalar_from_signed},
+    SchemeParams,
+};
+use crate::{
+    curve::Point,
+    paillier::{Ciphertext, CiphertextWire, MaskedRandomizer, PaillierParams, PublicKeyPaillier, RPParams, Randomizer},
+    tools::{
+        bitvec::BitVec,
+        hashing::{Chain, Hashable, XofHasher},
+    },
+    uint::{PublicSigned, SecretSigned},
+};
+
+const HASH_TAG: &[u8] = b"P_dec";
+
+pub(crate) struct DecSecretInputs<'a, P: SchemeParams> {
+    /// $x \in \mathbb{I}$, that is $x \in \pm 2^\ell$ (see N.B. just before Section 4.1)
+    pub x: &'a SecretSigned<<P::Paillier as PaillierParams>::Uint>,
+    /// $y \in \mathbb{I}$, that is $x \in \pm 2^{\ell^\prime}$ (see N.B. just before Section 4.1)
+    pub y: &'a SecretSigned<<P::Paillier as PaillierParams>::Uint>,
+    /// $\rho$, a Paillier randomizer for the public key $N_0$.
+    pub rho: &'a Randomizer<P::Paillier>,
+}
+
+pub(crate) struct DecPublicInputs<'a, P: SchemeParams> {
+    /// Paillier public key $N_0$.
+    pub pk0: &'a PublicKeyPaillier<P::Paillier>,
+    /// Paillier ciphertext $K$ such that $enc_0(y, \rho) = K (*) x + D$.
+    // NOTE: paper says `enc_0(z, \rho)` which is a typo.
+    pub cap_k: &'a Ciphertext<P::Paillier>,
+    /// Point $X = g * x$, where $g$ is the curve generator.
+    pub cap_x: &'a Point,
+    /// Paillier ciphertext $D$, see the doc for `cap_k` above.
+    pub cap_d: &'a Ciphertext<P::Paillier>,
+    /// Point $S = g * y$, where $g$ is the curve generator.
+    pub cap_s: &'a Point,
+}
+
+/// ZK proof: Paillier decryption modulo $q$.
+#[derive(Debug, Clone, Serialize, Deserialize)]
+pub(crate) struct DecProof<P: SchemeParams> {
+    e: BitVec,
+    commitments: Box<[DecProofCommitment<P>]>,
+    elements: Box<[DecProofElement<P>]>,
+}
+
+struct DecProofEphemeral<P: SchemeParams> {
+    alpha: SecretSigned<<P::Paillier as PaillierParams>::Uint>,
+    beta: SecretSigned<<P::Paillier as PaillierParams>::Uint>,
+    r: Randomizer<P::Paillier>,
+}
+
+#[derive(Debug, Clone, Serialize, Deserialize)]
+pub(crate) struct DecProofCommitment<P: SchemeParams> {
+    cap_a: CiphertextWire<P::Paillier>,
+    cap_b: Point,
+    cap_c: Point,
+}
+
+#[derive(Debug, Clone, Serialize, Deserialize)]
+pub(crate) struct DecProofElement<P: SchemeParams> {
+    z: PublicSigned<<P::Paillier as PaillierParams>::Uint>,
+    omega: PublicSigned<<P::Paillier as PaillierParams>::Uint>,
+    nu: MaskedRandomizer<P::Paillier>,
+}
+
+impl<P: SchemeParams> DecProof<P> {
+    pub fn new(
+        rng: &mut impl CryptoRngCore,
+        secret: DecSecretInputs<'_, P>,
+        public: DecPublicInputs<'_, P>,
+        setup: &RPParams<P::Paillier>,
+        aux: &impl Hashable,
+    ) -> Self {
+        secret.x.assert_exponent_range(P::L_BOUND);
+        secret.y.assert_exponent_range(P::LP_BOUND);
+        assert_eq!(public.cap_k.public_key(), public.pk0);
+        assert_eq!(public.cap_d.public_key(), public.pk0);
+
+        let (ephemerals, commitments): (Vec<_>, Vec<_>) = (0..P::SECURITY_PARAMETER)
+            .map(|_| {
+                let alpha = SecretSigned::random_in_exponent_range(rng, P::L_BOUND + P::EPS_BOUND);
+                let beta = SecretSigned::random_in_exponent_range(rng, P::LP_BOUND + P::EPS_BOUND);
+                let r = Randomizer::random(rng, public.pk0);
+
+                let cap_a =
+                    (public.cap_k * &-&alpha + Ciphertext::new_with_randomizer(&public.pk0, &beta, &r)).to_wire();
+                let cap_b = secret_scalar_from_signed::<P>(&beta).mul_by_generator();
+                let cap_c = secret_scalar_from_signed::<P>(&alpha).mul_by_generator();
+
+                let ephemeral = DecProofEphemeral::<P> { alpha, beta, r };
+                let commitment = DecProofCommitment { cap_a, cap_b, cap_c };
+
+                (ephemeral, commitment)
+            })
+            .unzip();
+
+        let mut reader = XofHasher::new_with_dst(HASH_TAG)
+            // commitments
+            .chain(&commitments)
+            // public parameters
+            .chain(public.pk0.as_wire())
+            .chain(&public.cap_k.to_wire())
+            .chain(&public.cap_x)
+            .chain(&public.cap_d.to_wire())
+            .chain(&public.cap_s)
+            .chain(&setup.to_wire())
+            .chain(aux)
+            .finalize_to_reader();
+
+        // Non-interactive challenge
+        let e = BitVec::from_xof_reader(&mut reader, P::SECURITY_PARAMETER);
+
+        let elements = ephemerals
+            .into_iter()
+            .zip(e.bits())
+            .map(|(ephemeral, e_bit)| {
+                let mut z = ephemeral.alpha;
+                if *e_bit {
+                    z = z + secret.x;
+                }
+
+                let mut omega = ephemeral.beta;
+                if *e_bit {
+                    omega = omega + secret.y;
+                }
+
+                let nu = if *e_bit {
+                    secret.rho.to_masked(&ephemeral.r, &PublicSigned::one())
+                } else {
+                    secret.rho.to_masked(&ephemeral.r, &PublicSigned::zero())
+                };
+
+                DecProofElement {
+                    z: z.to_public(),
+                    omega: omega.to_public(),
+                    nu,
+                }
+            })
+            .collect::<Vec<_>>();
+
+        Self {
+            e,
+            elements: elements.into(),
+            commitments: commitments.into(),
+        }
+    }
+
+    pub fn verify(&self, public: DecPublicInputs<'_, P>, setup: &RPParams<P::Paillier>, aux: &impl Hashable) -> bool {
+        let mut reader = XofHasher::new_with_dst(HASH_TAG)
+            // commitments
+            .chain(&self.commitments)
+            // public parameters
+            .chain(public.pk0.as_wire())
+            .chain(&public.cap_k.to_wire())
+            .chain(&public.cap_x)
+            .chain(&public.cap_d.to_wire())
+            .chain(&public.cap_s)
+            .chain(&setup.to_wire())
+            .chain(aux)
+            .finalize_to_reader();
+
+        // Non-interactive challenge
+        let e = BitVec::from_xof_reader(&mut reader, P::SECURITY_PARAMETER);
+
+        if e != self.e {
+            return false;
+        }
+
+        if e.bits().len() != self.commitments.len() || e.bits().len() != self.elements.len() {
+            return false;
+        }
+
+        for ((e_bit, commitment), element) in e
+            .bits()
+            .iter()
+            .cloned()
+            .zip(self.commitments.iter())
+            .zip(self.elements.iter())
+        {
+            // enc(\omega_j, \nu_j) (+) K (*) (-z_j) == A_j (+) D (*) e_j
+            let cap_a = commitment.cap_a.to_precomputed(public.pk0);
+            let lhs = Ciphertext::new_public_with_randomizer(public.pk0, &element.omega, &element.nu)
+                + public.cap_k * &-element.z;
+            let rhs = if e_bit { cap_a + public.cap_d } else { cap_a };
+            if lhs != rhs {
+                return false;
+            }
+
+            // g * z_j == C_j + X * e_j
+            let lhs = scalar_from_signed::<P>(&element.z).mul_by_generator();
+            let rhs = if e_bit {
+                commitment.cap_c + *public.cap_x
+            } else {
+                commitment.cap_c
+            };
+            if lhs != rhs {
+                return false;
+            }
+
+            // g * \omega_j == B_j + S * e_j
+            let lhs = scalar_from_signed::<P>(&element.omega).mul_by_generator();
+            let rhs = if e_bit {
+                commitment.cap_b + *public.cap_s
+            } else {
+                commitment.cap_b
+            };
+            if lhs != rhs {
+                return false;
+            }
+        }
+
+        true
+    }
+}
+
+#[cfg(test)]
+mod tests {
+    use rand_core::OsRng;
+
+    use super::{DecProof, DecPublicInputs, DecSecretInputs};
+    use crate::{
+        cggmp21::{conversion::secret_scalar_from_signed, SchemeParams, TestParams},
+        paillier::{Ciphertext, PaillierParams, RPParams, Randomizer, SecretKeyPaillierWire},
+        uint::SecretSigned,
+    };
+
+    #[test]
+    fn prove_and_verify() {
+        type Params = TestParams;
+        type Paillier = <Params as SchemeParams>::Paillier;
+
+        let sk = SecretKeyPaillierWire::<Paillier>::random(&mut OsRng).into_precomputed();
+        let pk = sk.public_key();
+
+        let setup = RPParams::random(&mut OsRng);
+
+        let aux: &[u8] = b"abcde";
+
+        let x = SecretSigned::random_in_exponent_range(&mut OsRng, Params::L_BOUND);
+        let y = SecretSigned::random_in_exponent_range(&mut OsRng, Params::LP_BOUND);
+        let rho = Randomizer::random(&mut OsRng, pk);
+
+        // We need $enc_0(y, \rho) = K (*) x + D$,
+        // so we can choose the plaintext of `K` at random, and derive the plaintext of `D`
+        // (not deriving `y` since we want it to be in a specific range).
+
+        let k = SecretSigned::random_in_exponent_range(&mut OsRng, Paillier::PRIME_BITS * 2 - 1);
+        let cap_k = Ciphertext::new(&mut OsRng, pk, &k);
+        let cap_d = Ciphertext::new_with_randomizer(pk, &y, &rho) + &cap_k * &-&x;
+
+        let cap_x = secret_scalar_from_signed::<Params>(&x).mul_by_generator();
+        let cap_s = secret_scalar_from_signed::<Params>(&y).mul_by_generator();
+
+        let proof = DecProof::<Params>::new(
+            &mut OsRng,
+            DecSecretInputs {
+                x: &x,
+                y: &y,
+                rho: &rho,
+            },
+            DecPublicInputs {
+                pk0: pk,
+                cap_k: &cap_k,
+                cap_x: &cap_x,
+                cap_d: &cap_d,
+                cap_s: &cap_s,
+            },
+            &setup,
+            &aux,
+        );
+        assert!(proof.verify(
+            DecPublicInputs {
+                pk0: pk,
+                cap_k: &cap_k,
+                cap_x: &cap_x,
+                cap_d: &cap_d,
+                cap_s: &cap_s,
+            },
+            &setup,
+            &aux
+        ));
+    }
+}