X-Wing: general-purpose hybrid post-quantum KEM

Deirdre Connolly SandboxAQ Email: durumcrustulum@gmail.com · authenticated

def GenerateKeyPair():
  (pk_M, sk_M) = ML-KEM-768.KeyGen()
  sk_X = random(32)
  pk_X = X25519(sk_X, X25519_BASE)
  return concat(sk_M, sk_X, pk_X), concat(pk_M, pk_X)

def GenerateKeyPairDerand(seed):
  (pk_M, sk_M) = ML-KEM-768.KeyGenDerand(seed[0:64])
  sk_X = seed[64:96]
  pk_X = X25519(sk_X, X25519_BASE)
  return concat(sk_M, sk_X, pk_X), concat(pk_M, pk_X)

def Combiner(ss_M, ss_X, ct_X, pk_X):
  return SHA3-256(concat(
    XWingLabel,
    ss_M,
    ss_X,
    ct_X,
    pk_X
  ))

XWingLabel = concat(
    "\./",
    "/^\",
)

def Encapsulate(pk):
  pk_M = pk[0:1184]
  pk_X = pk[1184:1216]
  ek_X = random(32)
  ct_X = X25519(ek_X, X25519_BASE)
  ss_X = X25519(ek_X, pk_X)
  (ss_M, ct_M) = ML-KEM-768.Encaps(pk_M)
  ss = Combiner(ss_M, ss_X, ct_X, pk_X)
  ct = concat(ct_M, ct_X)
  return (ss, ct)

def EncapsulateDerand(pk, eseed):
  pk_M = pk[0:1184]
  pk_X = pk[1184:1216]
  ek_X = eseed[32:64]
  ct_X = X25519(ek_X, X25519_BASE)
  ss_X = X25519(ek_X, pk_X)
  (ss_M, ct_M) = ML-KEM-768.EncapsDerand(pk_M, eseed[0:32])
  ss = Combiner(ss_M, ss_X, ct_X, pk_X)
  ct = concat(ct_M, ct_X)
  return (ss, ct)

def Decapsulate(ct, sk):
  ct_M = ct[0:1088]
  ct_X = ct[1088:1120]
  sk_M = sk[0:2400]
  sk_X = sk[2400:2432]
  pk_X = sk[2432:2464]
  ss_M = ML-KEM-768.Decapsulate(ct_M, sk_M)
  ss_X = X25519(sk_X, ct_X)
  return Combiner(ss_M, ss_X, ct_X, pk_X)

def DeriveKeyPair(ikm):
  return GenerateKeyPairDerand(SHAKE128(ikm, 96))

# WARNING This is a specification of X-Wing; not a production-ready
# implementation. It is slow and does not run in constant time.

# Requires the CryptoDome for SHAKE, and pytest for testing. To install, run
#
#   pip install pycryptodome pytest

import binascii
import hashlib

import mlkem
import x25519

XWingLabel = br"""
                \./
                /^\
              """.replace(b'\n', b'').replace(b' ', b'')

assert len(XWingLabel) == 6
assert binascii.hexlify(XWingLabel) == b'5c2e2f2f5e5c'

def GenerateKeyPairDerand(seed):
    assert len(seed) == 96
    pkM, skM = mlkem.KeyGen(seed[0:64], mlkem.params768)
    skX = seed[64:96]
    pkX = x25519.X(skX, x25519.BASE)
    return skM + skX + pkX, pkM + pkX

def Combiner(ssM, ssX, ctX, pkX):
    return hashlib.sha3_256(
        XWingLabel +
        ssM +
        ssX +
        ctX +
        pkX
    ).digest()

def EncapsulateDerand(pk, eseed):
    assert len(eseed) == 64
    assert len(pk) == 1216
    pkM = pk[0:1184]
    pkX = pk[1184:1216]
    ekX = eseed[32:64]
    ctX = x25519.X(ekX, x25519.BASE)
    ssX = x25519.X(ekX, pkX)
    ctM, ssM = mlkem.Enc(pkM, eseed[0:32], mlkem.params768)
    ss = Combiner(ssM, ssX, ctX, pkX)
    return ss, ctM + ctX

def Decapsulate(ct, sk):
    assert len(ct) == 1120
    assert len(sk) == 2464
    ctM = ct[0:1088]
    ctX = ct[1088:1120]
    skM = sk[0:2400]
    skX = sk[2400:2432]
    pkX = sk[2432:2464]
    ssM = mlkem.Dec(skM, ctM, mlkem.params768)
    ssX = x25519.X(skX, ctX)
    return Combiner(ssM, ssX, ctX, pkX)

# WARNING This is a specification of X25519; not a production-ready
# implementation. It is slow and does not run in constant time.

p = 2**255 - 19
a24 = 121665

BASE = b'\x09' + b'\x00'*31

def decode(bs):
    return sum(bs[i] << 8*i for i in range(32)) % p

def decodeScalar(k):
    bs = list(k)
    bs[0] &= 248
    bs[31] &= 127
    bs[31] |= 64
    return decode(bs)

# See rfc7748 §5.
def X(k, u):
    assert len(k) == 32
    assert len(u) == 32

    k = decodeScalar(k)
    u = decode(u)
    x1, x2, x3, z2, z3, swap = u, 1, u, 0, 1, 0

    for t in range(255, -1, -1):
        kt = (k >> t) & 1
        swap ^= kt
        if swap == 1:
            x3, x2 = x2, x3
            z3, z2 = z2, z3
        swap = kt

        A = x2 + z2
        AA = (A*A) % p
        B = x2 - z2
        BB = (B*B) % p
        E = AA - BB
        C = x3 + z3
        D = x3 - z3
        DA = (D*A) % p
        CB = (C*B) % p
        x3 = DA + CB
        x3 = (x3 * x3) % p
        z3 = DA - CB
        z3 = (x1 * z3 * z3) % p
        x2 = (AA * BB) % p
        z2 = (E * (AA + (a24 * E) % p)) % p

    if swap == 1:
        x3, x2 = x2, x3
        z2, z3 = z3, z2

    ret = (x2 * pow(z2, p-2, p)) % p
    return bytes((ret >> 8*i) & 255 for i in range(32))

# WARNING This is a specification of Kyber; not a production ready
# implementation. It is slow and does not run in constant time.

# Requires the CryptoDome for SHAKE. To install, run
#
#   pip install pycryptodome pytest
from Crypto.Hash import SHAKE128, SHAKE256

import io
import hashlib
import functools
import collections

from math import floor

q = 3329
nBits = 8
zeta = 17
eta2 = 2

n = 2**nBits
inv2 = (q+1)//2 # inverse of 2

params = collections.namedtuple('params', ('k', 'du', 'dv', 'eta1'))

params512  = params(k = 2, du = 10, dv = 4, eta1 = 3)
params768  = params(k = 3, du = 10, dv = 4, eta1 = 2)
params1024 = params(k = 4, du = 11, dv = 5, eta1 = 2)

def smod(x):
    r = x % q
    if r > (q-1)//2:
        r -= q
    return r

# Rounds to nearest integer with ties going up
def Round(x):
    return int(floor(x + 0.5))

def Compress(x, d):
    return Round((2**d / q) * x) % (2**d)

def Decompress(y, d):
    assert 0 <= y and y <= 2**d
    return Round((q / 2**d) * y)

def BitsToWords(bs, w):
    assert len(bs) % w == 0
    return [sum(bs[i+j] * 2**j for j in range(w))
            for i in range(0, len(bs), w)]

def WordsToBits(bs, w):
    return sum([[(b >> i) % 2 for i in range(w)] for b in bs], [])

def Encode(a, w):
    return bytes(BitsToWords(WordsToBits(a, w), 8))

def Decode(a, w):
    return BitsToWords(WordsToBits(a, 8), w)

def brv(x):
    """ Reverses a 7-bit number """
    return int(''.join(reversed(bin(x)[2:].zfill(nBits-1))), 2)

class Poly:
    def __init__(self, cs=None):
        self.cs = (0,)*n if cs is None else tuple(cs)
        assert len(self.cs) == n

    def __add__(self, other):
        return Poly((a+b) % q for a,b in zip(self.cs, other.cs))

    def __neg__(self):
        return Poly(q-a for a in self.cs)
    def __sub__(self, other):
        return self + -other

    def __str__(self):
        return f"Poly({self.cs}"

    def __eq__(self, other):
        return self.cs == other.cs

    def NTT(self):
        cs = list(self.cs)
        layer = n // 2
        zi = 0
        while layer >= 2:
            for offset in range(0, n-layer, 2*layer):
                zi += 1
                z = pow(zeta, brv(zi), q)

                for j in range(offset, offset+layer):
                    t = (z * cs[j + layer]) % q
                    cs[j + layer] = (cs[j] - t) % q
                    cs[j] = (cs[j] + t) % q
            layer //= 2
        return Poly(cs)

    def RefNTT(self):
        # Slower, but simpler, version of the NTT.
        cs = [0]*n
        for i in range(0, n, 2):
            for j in range(n // 2):
                z = pow(zeta, (2*brv(i//2)+1)*j, q)
                cs[i] = (cs[i] + self.cs[2*j] * z) % q
                cs[i+1] = (cs[i+1] + self.cs[2*j+1] * z) % q
        return Poly(cs)

    def InvNTT(self):
        cs = list(self.cs)
        layer = 2
        zi = n//2
        while layer < n:
            for offset in range(0, n-layer, 2*layer):
                zi -= 1
                z = pow(zeta, brv(zi), q)

                for j in range(offset, offset+layer):
                    t = (cs[j+layer] - cs[j]) % q
                    cs[j] = (inv2*(cs[j] + cs[j+layer])) % q
                    cs[j+layer] = (inv2 * z * t) % q
            layer *= 2
        return Poly(cs)

    def MulNTT(self, other):
        """ Computes self o other, the multiplication of self and other
            in the NTT domain. """
        cs = [None]*n
        for i in range(0, n, 2):
            a1 = self.cs[i]
            a2 = self.cs[i+1]
            b1 = other.cs[i]
            b2 = other.cs[i+1]
            z = pow(zeta, 2*brv(i//2)+1, q)
            cs[i] = (a1 * b1 + z * a2 * b2) % q
            cs[i+1] = (a2 * b1 + a1 * b2) % q
        return Poly(cs)

    def Compress(self, d):
        return Poly(Compress(c, d) for c in self.cs)

    def Decompress(self, d):
        return Poly(Decompress(c, d) for c in self.cs)

    def Encode(self, d):
        return Encode(self.cs, d)

def sampleUniform(stream):
    cs = []
    while True:
        b = stream.read(3)
        d1 = b[0] + 256*(b[1] % 16)
        d2 = (b[1] >> 4) + 16*b[2]
        assert d1 + 2**12 * d2 == b[0] + 2**8 * b[1] + 2**16*b[2]
        for d in [d1, d2]:
            if d >= q:
                continue
            cs.append(d)
            if len(cs) == n:
                return Poly(cs)

def CBD(a, eta):
    assert len(a) == 64*eta
    b = WordsToBits(a, 8)
    cs = []
    for i in range(n):
        cs.append((sum(b[:eta]) - sum(b[eta:2*eta])) % q)
        b = b[2*eta:]
    return Poly(cs)

def XOF(seed, j, i):
    h = SHAKE128.new()
    h.update(seed + bytes([j, i]))
    return h

def PRF1(seed, nonce):
    assert len(seed) == 32
    h = SHAKE256.new()
    h.update(seed + bytes([nonce]))
    return h

def PRF2(seed, msg):
    assert len(seed) == 32
    h = SHAKE256.new()
    h.update(seed + msg)
    return h.read(32)

def G(seed):
    h = hashlib.sha3_512(seed).digest()
    return h[:32], h[32:]

def H(msg): return hashlib.sha3_256(msg).digest()

class Vec:
    def __init__(self, ps):
        self.ps = tuple(ps)

    def NTT(self):
        return Vec(p.NTT() for p in self.ps)

    def InvNTT(self):
        return Vec(p.InvNTT() for p in self.ps)

    def DotNTT(self, other):
        """ Computes the dot product <self, other> in NTT domain. """
        return sum((a.MulNTT(b) for a, b in zip(self.ps, other.ps)),
                   Poly())

    def __add__(self, other):
        return Vec(a+b for a,b in zip(self.ps, other.ps))

    def Compress(self, d):
        return Vec(p.Compress(d) for p in self.ps)

    def Decompress(self, d):
        return Vec(p.Decompress(d) for p in self.ps)

    def Encode(self, d):
        return Encode(sum((p.cs for p in self.ps), ()), d)

    def __eq__(self, other):
        return self.ps == other.ps

def EncodeVec(vec, w):
    return Encode(sum([p.cs for p in vec.ps], ()), w)
def DecodeVec(bs, k, w):
    cs = Decode(bs, w)
    return Vec(Poly(cs[n*i:n*(i+1)]) for i in range(k))
def DecodePoly(bs, w):
    return Poly(Decode(bs, w))

class Matrix:
    def __init__(self, cs):
        """ Samples the matrix uniformly from seed rho """
        self.cs = tuple(tuple(row) for row in cs)

    def MulNTT(self, vec):
        """ Computes matrix multiplication A*vec in the NTT domain. """
        return Vec(Vec(row).DotNTT(vec) for row in self.cs)

    def T(self):
        """ Returns transpose of matrix """
        k = len(self.cs)
        return Matrix((self.cs[j][i] for j in range(k))
                      for i in range(k))

def sampleMatrix(rho, k):
    return Matrix([[sampleUniform(XOF(rho, j, i))
            for j in range(k)] for i in range(k)])

def sampleNoise(sigma, eta, offset, k):
    return Vec(CBD(PRF1(sigma, i+offset).read(64*eta), eta)
               for i in range(k))

def constantTimeSelectOnEquality(a, b, ifEq, ifNeq):
    # WARNING! In production code this must be done in a
    # data-independent constant-time manner, which this implementation
    # is not. In fact, many more lines of code in this
    # file are not constant-time.
    return ifEq if a == b else ifNeq

def InnerKeyGen(seed, params):
    assert len(seed) == 32
    rho, sigma = G(seed)
    A = sampleMatrix(rho, params.k)
    s = sampleNoise(sigma, params.eta1, 0, params.k)
    e = sampleNoise(sigma, params.eta1, params.k, params.k)
    sHat = s.NTT()
    eHat = e.NTT()
    tHat = A.MulNTT(sHat) + eHat
    pk = EncodeVec(tHat, 12) + rho
    sk = EncodeVec(sHat, 12)
    return (pk, sk)

def InnerEnc(pk, msg, seed, params):
    assert len(msg) == 32
    tHat = DecodeVec(pk[:-32], params.k, 12)
    rho = pk[-32:]
    A = sampleMatrix(rho, params.k)
    r = sampleNoise(seed, params.eta1, 0, params.k)
    e1 = sampleNoise(seed, eta2, params.k, params.k)
    e2 = sampleNoise(seed, eta2, 2*params.k, 1).ps[0]
    rHat = r.NTT()
    u = A.T().MulNTT(rHat).InvNTT() + e1
    m = Poly(Decode(msg, 1)).Decompress(1)
    v = tHat.DotNTT(rHat).InvNTT() + e2 + m
    c1 = u.Compress(params.du).Encode(params.du)
    c2 = v.Compress(params.dv).Encode(params.dv)
    return c1 + c2

def InnerDec(sk, ct, params):
    split = params.du * params.k * n // 8
    c1, c2 = ct[:split], ct[split:]
    u = DecodeVec(c1, params.k, params.du).Decompress(params.du)
    v = DecodePoly(c2, params.dv).Decompress(params.dv)
    sHat = DecodeVec(sk, params.k, 12)
    return (v - sHat.DotNTT(u.NTT()).InvNTT()).Compress(1).Encode(1)

def KeyGen(seed, params):
    assert len(seed) == 64
    z = seed[32:]
    pk, sk2 = InnerKeyGen(seed[:32], params)
    h = H(pk)
    return (pk, sk2 + pk + h + z)

def Enc(pk, seed, params):
    assert len(seed) == 32

    K, r = G(seed + H(pk))
    ct = InnerEnc(pk, seed, r, params)
    return (ct, K)

def Dec(sk, ct, params):
    sk2 = sk[:12 * params.k * n//8]
    pk = sk[12 * params.k * n//8 : 24 * params.k * n//8 + 32]
    h = sk[24 * params.k * n//8 + 32 : 24 * params.k * n//8 + 64]
    z = sk[24 * params.k * n//8 + 64 : 24 * params.k * n//8 + 96]
    m2 = InnerDec(sk, ct, params)
    K2, r2 = G(m2 + h)
    ct2 = InnerEnc(pk, m2, r2, params)
    return constantTimeSelectOnEquality(
        ct2, ct,
        K2,                 # if ct == ct2
        PRF2(z, ct),        # if ct != ct2
    )

seed
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sk
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seed
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ct
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Internet-Draft	xwing	March 2024
Connolly, et al.	Expires 27 September 2024	[Page]