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mlkem.py
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mlkem.py
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# https://words.filippo.io/dispatches/kyber-math/
# https://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.203.ipd.pdf
from typing import List, Tuple
import hashlib
import os
from shakestream import ShakeStream
from functools import reduce
# ML-KEM-768 params:
N = 256
Q = 3329
K = 3
ETA1 = 2
ETA2 = 2
DU = 10
DV = 4
def bitrev7(n: int) -> int:
return int(f"{n:07b}"[::-1], 2) # gross but it works
# 17 is primitive 256th root of unity mod Q
ZETA = [pow(17, bitrev7(k), Q) for k in range(128)] # used in ntt and ntt_inv
GAMMA = [pow(17, 2*bitrev7(k)+1, Q) for k in range(128)] # used in ntt_mul
# can be reused for NTT representatives
def poly256_add(a: List[int], b: List[int]) -> List[int]:
return [(x + y) % Q for x, y in zip(a, b)]
def poly256_sub(a: List[int], b: List[int]) -> List[int]:
return [(x - y) % Q for x, y in zip(a, b)]
# naive O(n^2) multiplication algorithm for testing/comparison purposes.
# this is not used for the main impl.
def poly256_slow_mul(a: List[int], b: List[int]) -> List[int]:
c = [0] * 511
# textbook multiplication, without carry
for i in range(256):
for j in range(256):
c[i+j] = (c[i+j] + a[j] * b[i]) % Q
# now for reduction mod X^256 + 1
for i in range(255):
c[i] = (c[i] - c[i+256]) % Q
# we could explicitly zero c[i+256] here, but there's no need...
# ...because we're about to truncate c
return c[:256]
# by the way, this is O(n logn)
def ntt(f_in: List[int]) -> List[int]:
f_out = f_in.copy()
k = 1
for log2len in range(7, 0, -1):
length = 2**log2len
for start in range(0, 256, 2 * length):
zeta = ZETA[k]
k += 1
for j in range(start, start + length):
t = (zeta * f_out[j + length]) % Q
f_out[j + length] = (f_out[j] - t) % Q
f_out[j] = (f_out[j] + t) % Q
return f_out
# so is this
def ntt_inv(f_in: List[int]) -> List[int]:
f_out = f_in.copy()
k = 127
for log2len in range(1, 8):
length = 2**log2len
for start in range(0, 256, 2 * length):
zeta = ZETA[k]
k -= 1
for j in range(start, start + length):
t = f_out[j]
f_out[j] = (t + f_out[j + length]) % Q
f_out[j + length] = (zeta * (f_out[j + length] - t)) % Q
for i in range(256):
f_out[i] = (f_out[i] * 3303) % Q # 3303 == pow(128, -1, Q)
return f_out
ntt_add = poly256_add # it's just elementwise addition
# and this is just O(n)
def ntt_mul(a: List[int], b: List[int]) -> List[int]:
c = []
for i in range(128):
a0, a1 = a[2 * i: 2 * i + 2]
b0, b1 = b[2 * i: 2 * i + 2]
c.append((a0 * b0 + a1 * b1 * GAMMA[i]) % Q)
c.append((a0 * b1 + a1 * b0) % Q)
return c
# crypto functions
def mlkem_prf(eta: int, data: bytes, b: int) -> bytes:
return hashlib.shake_256(data + bytes([b])).digest(64 * eta)
def mlkem_xof(data: bytes, i: int, j: int) -> ShakeStream:
return ShakeStream(hashlib.shake_128(data + bytes([i, j])).digest)
def mlkem_hash_H(data: bytes) -> bytes:
return hashlib.sha3_256(data).digest()
def mlkem_hash_J(data: bytes) -> bytes:
return hashlib.shake_256(data).digest(32)
def mlkem_hash_G(data: bytes) -> bytes:
return hashlib.sha3_512(data).digest()
# encode/decode logic
def bits_to_bytes(bits: List[int]) -> bytes:
assert(len(bits) % 8 == 0)
return bytes(
sum(bits[i + j] << j for j in range(8))
for i in range(0, len(bits), 8)
)
def bytes_to_bits(data: bytes) -> List[int]:
bits = []
for word in data:
for i in range(8):
bits.append((word >> i) & 1)
return bits
def byte_encode(d: int, f: List[int]) -> bytes:
assert(len(f) == 256)
bits = []
for a in f:
for i in range(d):
bits.append((a >> i) & 1)
return bits_to_bytes(bits)
def byte_decode(d: int, data: bytes) -> List[int]:
bits = bytes_to_bits(data)
return [sum(bits[i * d + j] << j for j in range(d)) for i in range(256)]
def compress(d: int, x: List[int]) -> List[int]:
return [(((n * 2**d) + Q // 2 ) // Q) % (2**d) for n in x]
def decompress(d: int, x: List[int]) -> List[int]:
return [(((n * Q) + 2**(d-1) ) // 2**d) % Q for n in x]
# sampling
def sample_ntt(xof: ShakeStream):
res = []
while len(res) < 256:
a, b, c = xof.read(3)
d1 = ((b & 0xf) << 8) | a
d2 = c << 4 | b >> 4
if d1 < Q:
res.append(d1)
if d2 < Q and len(res) < 256:
res.append(d2)
return res
def sample_poly_cbd(eta: int, data: bytes) -> List[int]:
assert(len(data) == 64 * eta)
bits = bytes_to_bits(data)
f = []
for i in range(256):
x = sum(bits[2*i*eta+j] for j in range(eta))
y = sum(bits[2*i*eta+eta+j] for j in range(eta))
f.append((x - y) % Q)
return f
# K-PKE
def kpke_keygen(seed: bytes=None) -> Tuple[bytes, bytes]:
d = os.urandom(32) if seed is None else seed
ghash = mlkem_hash_G(d)
rho, sigma = ghash[:32], ghash[32:]
ahat = []
for i in range(K):
row = []
for j in range(K):
row.append(sample_ntt(mlkem_xof(rho, i, j)))
ahat.append(row)
shat = [
ntt(sample_poly_cbd(ETA1, mlkem_prf(ETA1, sigma, i)))
for i in range(K)
]
ehat = [
ntt(sample_poly_cbd(ETA1, mlkem_prf(ETA1, sigma, i+K)))
for i in range(K)
]
that = [ # t = a * s + e
reduce(ntt_add, [
ntt_mul(ahat[j][i], shat[j])
for j in range(K)
] + [ehat[i]])
for i in range(K)
]
ek_pke = b"".join(byte_encode(12, s) for s in that) + rho
dk_pke = b"".join(byte_encode(12, s) for s in shat)
return ek_pke, dk_pke
def kpke_encrypt(ek_pke: bytes, m: bytes, r: bytes) -> bytes:
that = [byte_decode(12, ek_pke[i*128*K:(i+1)*128*K]) for i in range(K)]
rho = ek_pke[-32:]
# this is identical to as in kpke_keygen
ahat = []
for i in range(K):
row = []
for j in range(K):
row.append(sample_ntt(mlkem_xof(rho, i, j)))
ahat.append(row)
rhat = [
ntt(sample_poly_cbd(ETA1, mlkem_prf(ETA1, r, i)))
for i in range(K)
]
e1 = [
sample_poly_cbd(ETA2, mlkem_prf(ETA2, r, i+K))
for i in range(K)
]
e2 = sample_poly_cbd(ETA2, mlkem_prf(ETA2, r, 2*K))
u = [ # u = ntt-1(AT*r)+e1
poly256_add(ntt_inv(reduce(ntt_add, [
ntt_mul(ahat[i][j], rhat[j]) # note that i,j are reversed here
for j in range(K)
])), e1[i])
for i in range(K)
]
mu = decompress(1, byte_decode(1, m))
v = poly256_add(ntt_inv(reduce(ntt_add, [
ntt_mul(that[i], rhat[i])
for i in range(K)
])), poly256_add(e2, mu))
c1 = b"".join(byte_encode(DU, compress(DU, u[i])) for i in range(K))
c2 = byte_encode(DV, compress(DV, v))
return c1 + c2
def kpke_decrypt(dk_pke: bytes, c: bytes) -> bytes:
c1 = c[:32*DU*K]
c2 = c[32*DU*K:]
u = [
decompress(DU, byte_decode(DU, c1[i*32*DU:(i+1)*32*DU]))
for i in range(K)
]
v = decompress(DV, byte_decode(DV, c2))
shat = [byte_decode(12, dk_pke[i*384:(i+1)*384]) for i in range(K)]
# NOTE: the comment in FIPS203 seems wrong here?
# it says "NTT−1 and NTT invoked k times", but I think NTT−1 is only invoked once.
w = poly256_sub(v, ntt_inv(reduce(ntt_add, [
ntt_mul(shat[i], ntt(u[i]))
for i in range(K)
])))
m = byte_encode(1, compress(1, w))
return m
# KEM time
def mlkem_keygen(seed1=None, seed2=None):
z = os.urandom(32) if seed1 is None else seed1
ek_pke, dk_pke = kpke_keygen(seed2)
ek = ek_pke
dk = dk_pke + ek + mlkem_hash_H(ek) + z
return ek, dk
def mlkem_encaps(ek: bytes, seed=None) -> Tuple[bytes, bytes]:
# TODO !!!! input validation !!!!!!!
m = os.urandom(32) if seed is None else seed
ghash = mlkem_hash_G(m + mlkem_hash_H(ek))
k = ghash[:32]
r = ghash[32:]
c = kpke_encrypt(ek, m, r)
return k, c
def mlkem_decaps(c: bytes, dk: bytes) -> bytes:
# TODO !!!! input validation !!!!!!!
dk_pke = dk[:384*K]
ek_pke = dk[384*K : 768*K + 32]
h = dk[768*K + 32 : 768*K + 64]
z = dk[768*K + 64 : 768*K + 96]
mdash = kpke_decrypt(dk_pke, c)
ghash = mlkem_hash_G(mdash + h)
kdash = ghash[:32]
rdash = ghash[32:]
# NOTE: J() has unnecessary second argument in the spec???
kbar = mlkem_hash_J(z + c)
cdash = kpke_encrypt(ek_pke, mdash, rdash)
if cdash != c:
# I suppose this branch ought to be constant-time, but that's already out the window with this impl
#print("did not match") # XXX: what does implicit reject mean? I suppose it guarantees it fails in a not-attacker-controlled way?
return kbar
return kdash
if __name__ == "__main__":
a = list(range(256))
b = list(range(1024, 1024+256))
ntt_res = ntt_inv(ntt_add(ntt(a), ntt(b)))
poly_res = poly256_add(a, b)
assert(ntt_res == poly_res)
ntt_prod = ntt_inv(ntt_mul(ntt(a), ntt(b)))
poly_prod = poly256_slow_mul(a, b)
assert(ntt_prod == poly_prod)
ek_pke, dk_pke = kpke_keygen(b"SEED"*8)
msg = b"This is a demonstration message."
ct = kpke_encrypt(ek_pke, msg, b"RAND"*8)
pt = kpke_decrypt(dk_pke, ct)
print(pt)
assert(pt == msg)
ek, dk = mlkem_keygen()
k1, c = mlkem_encaps(ek)
print("encapsulated:", k1.hex())
k2 = mlkem_decaps(c, dk)
print("decapsulated:", k2.hex())
assert(k1 == k2)