curve25519.py

"""A pure Python implementation of Curve25519

This module supports both a low-level interface through curve25519(base_point, secret)
and curve25519_base(secret) that take 32-byte blocks of data as inputs and a higher
level interface using the X25519PrivateKey and X25519PublicKey classes that are
compatible with the classes in cryptography.hazmat.primitives.asymmetric.x25519 with
the same names.
"""

# By Nicko van Someren, 2021. This code is released into the public domain.

#                                  #### WARNING ####

# Since this code makes use of Python's built-in large integer types, it is NOT EXPECTED
# to run in constant time. While some effort is made to minimise the time variations,
# the underlying math functions are likely to have running times that are highly
# value-dependent, leaving this code potentially vulnerable to timing attacks. If this
# code is to be used to provide cryptographic security in an environment where the start
# and end times of the execution can be guessed, inferred or measured then it is critical
# that steps are taken to hide the execution time, for instance by adding a delay so that
# encrypted packets are not sent until a fixed time after the _start_ of execution.


# Implements ladder multiplication as described in "Montgomery curves and the Montgomery
# ladder" by Daniel J. Bernstein and Tanja Lange. https://eprint.iacr.org/2017/293.pdf

# Curve25519 is a Montgomery curve defined by:
# y**2 = x**3 + A * x**2 + x  mod P
# where P = 2**255-19 and A = 486662

P = 2 ** 255 - 19
_A = 486662


def _point_add(point_n, point_m, point_diff):
    """Given the projection of two points and their difference, return their sum"""
    (xn, zn) = point_n
    (xm, zm) = point_m
    (x_diff, z_diff) = point_diff
    x = (z_diff << 2) * (xm * xn - zm * zn) ** 2
    z = (x_diff << 2) * (xm * zn - zm * xn) ** 2
    return x % P, z % P


def _point_double(point_n):
    """Double a point provided in projective coordinates"""
    (xn, zn) = point_n
    xn2 = xn ** 2
    zn2 = zn ** 2
    x = (xn2 - zn2) ** 2
    xzn = xn * zn
    z = 4 * xzn * (xn2 + _A * xzn + zn2)
    return x % P, z % P


def _const_time_swap(a, b, swap):
    """Swap two values in constant time"""
    index = int(swap) * 2
    temp = (a, b, b, a)
    return temp[index:index+2]


def _raw_curve25519(base, n):
    """Raise the point base to the power n"""
    zero = (1, 0)
    one = (base, 1)
    mP, m1P = zero, one

    for i in reversed(range(256)):
        bit = bool(n & (1 << i))
        mP, m1P = _const_time_swap(mP, m1P, bit)
        mP, m1P = _point_double(mP), _point_add(mP, m1P, one)
        mP, m1P = _const_time_swap(mP, m1P, bit)

    x, z = mP
    inv_z = pow(z, P - 2, P)
    return (x * inv_z) % P


def _unpack_number(s):
    """Unpack 32 bytes to a 256 bit value"""
    if len(s) != 32:
        raise ValueError('Curve25519 values must be 32 bytes')
    return int.from_bytes(s, "little")


def _pack_number(n):
    """Pack a value into 32 bytes"""
    return n.to_bytes(32, "little")


def _fix_secret(n):
    """Mask a value to be an acceptable exponent"""
    n &= ~7
    n &= ~(128 << 8 * 31)
    n |= 64 << 8 * 31
    return n


def curve25519(base_point_raw, secret_raw):
    """Raise the base point to a given power"""
    base_point = _unpack_number(base_point_raw)
    secret = _fix_secret(_unpack_number(secret_raw))
    return _pack_number(_raw_curve25519(base_point, secret))


def curve25519_base(secret_raw):
    """Raise the generator point to a given power"""
    secret = _fix_secret(_unpack_number(secret_raw))
    return _pack_number(_raw_curve25519(9, secret))


class X25519PublicKey:
    def __init__(self, x):
        self.x = x

    @classmethod
    def from_public_bytes(cls, data):
        return cls(_unpack_number(data))

    def public_bytes(self):
        return _pack_number(self.x)


class X25519PrivateKey:
    def __init__(self, a):
        self.a = a

    @classmethod
    def from_private_bytes(cls, data):
        return cls(_fix_secret(_unpack_number(data)))

    def private_bytes(self):
        return _pack_number(self.a)

    def public_key(self):
        return _pack_number(_raw_curve25519(9, self.a))

    def exchange(self, peer_public_key):
        if isinstance(peer_public_key, bytes):
            peer_public_key = X25519PublicKey.from_public_bytes(peer_public_key)
        return _pack_number(_raw_curve25519(peer_public_key.x, self.a))