path: root/aesgcmanalysis.py

import random, struct, hmac, itertools
from Crypto.Cipher import AES
import numpy as np

## Computation in GF(2^128)/(x^128 + x^7 + x^2 + x^1 + 1)
## Elements are represented as an integer n where (n & (1 << i)) is the coefficient for x^i.

class X:
    """A helper class for specifying elements of GF(2^n)."""
    def __pow__(self, y):
        return 1 << y
x = X()
gcm_modulus = x**128 + x**7 + x**2 + x**1 + 1
gcm_modulus_degree = 128

def gf128_deg(x):
    """Compute the degree of x.
    >>> assert gf128_deg(x**20 + x**5 + 1) == 20
    >>> assert gf128_deg(x**90) == 90
    >>> assert gf128_deg(x**1) == 1
    >>> assert gf128_deg(1) == 0
    >>> assert gf128_deg(0) == -1
    """
    return x.bit_length() - 1

def gf128_add(a, b):
    """Compute a + b.
    >>> assert gf128_add(0, 0) == 0
    >>> assert gf128_add(x**7 + x**2, 0) == x**7 + x**2
    >>> assert gf128_add(x**7 + x**2, x**2 + x**1) == x**7 + x**1
    >>> assert gf128_add(x**7 + x**2, x**7 + x**2) == 0
    """
    return a ^ b

def gf128_mul(a, b):
    """Compute ab.
    >>> assert gf128_mul(x**20 + x**5 + 1, x**13 + x**2) == x**33 + x**22 + x**18 + x**13 + x**7 + x**2
    >>> assert gf128_mul(x**80, x**90) == x**49 + x**44 + x**43 + x**42
    >>> assert gf128_mul(x**23 + x**5, 1) == x**23 + x**5
    """
    if a == 0 or b == 0:
        return 0
    if a > b:
        return gf128_mul(b, a)
    p = 0
    deg_b = gf128_deg(b)
    while a > 0:
        if a & 1:
            p ^= b
        a = a >> 1
        b = b << 1
        deg_b += 1
        if deg_b == gcm_modulus_degree:
            b = b ^ gcm_modulus
            deg_b = gf128_deg(b)
    return p

def gf128_divmod(a, b):
    """Compute q, r such that a = qb + r; gf128_deg(r) < gf128_deg(b).
    >>> assert gf128_divmod(x**20 + x**5 + 1, x**13 + x**2) == (x**7, x**9 + x**5 + 1)
    >>> assert gf128_divmod(x**90, x**80) == (x**10, 0)
    """
    q, r = 0, a
    while gf128_deg(r) >= gf128_deg(b):
        d = gf128_deg(r) - gf128_deg(b)
        q = q ^ (1 << d)
        r = r ^ (b << d)
    return q, r

def gf128_mod(a, m=gcm_modulus):
    """Compute b such that a = b; gf128_deg(b) < gf128_deg(m).
    >>> assert gf128_mod(x**220 + x**5 + 1, x**50 + x**4) == x**36 + x**5 + 1
    >>> assert gf128_mod(x**130 + x**64 + x**2) == x**64 + x**9 + x**4 + x**3
    """
    q, r = gf128_divmod(a, m)
    return r

def gf128_egcd(a, b):
    """Compute g, x, y such that g | a, g | b, g = ax + by;
	for any h such that h | a and h | b, gf128_deg(g) >= gf128_deg(h).
    >>> assert gf128_egcd(x**2 + 1, x**1 + 1) == (x**1 + 1, 0, 1)
	>>> assert gf128_egcd(x**12 + x**4, x**5 + 1) == (x**1 + 1, x**3 + x**2 + 1, x**10 + x**9 + x**7 + x**5 + x**4 + x**1 + 1)
    >>> assert gf128_egcd(x**64 + x**2, x**37 + x**12 + 1) == (1, x**35 + x**34 + x**33 + x**31 + x**30 + x**28 + x**27 + x**25 + x**24 + x**21 + x**20 + x**18 + x**17 + x**15 + x**14 + x**12 + x**11 + x**9 + x**7 + x**6 + x**4 + x**3 + x**1 + 1, x**62 + x**61 + x**60 + x**58 + x**57 + x**55 + x**54 + x**52 + x**51 + x**48 + x**47 + x**45 + x**44 + x**42 + x**41 + x**39 + x**38 + x**37 + x**35 + x**34 + x**32 + x**31 + x**29 + x**28 + x**26 + x**25 + x**24 + x**22 + x**21 + x**19 + x**18 + x**16 + x**15 + x**13 + x**12 + x**11 + x**9 + x**8 + x**6 + x**5 + x**3 + x**2 + 1)
    """
    orig_a = a
    orig_b = b
    a_x = 1
    a_y = 0
    b_x = 0
    b_y = 1
    # Invariant = a_x*x + a_y*y = a, b_x*x + b_y*y = b, r_x*r + r_y*y = r
    r_x = 0
    r_y = 1
    if a == 0:
        return b, r_x, r_y
    while True:
        q, r = gf128_divmod(b, a)
        if r == 0:
            # FIX mul so mod isn't necessary; with zeros
            assert a == gf128_mod(gf128_add(gf128_mul(orig_a, r_x), gf128_mul(orig_b, r_y)))
            return a, r_x, r_y
        r_x = gf128_add(b_x, gf128_mul(q, a_x))
        r_y = gf128_add(b_y, gf128_mul(q, a_y))
        a, b = r, a
        b_x, b_y = a_x, a_y
        a_x, a_y = r_x, r_y

def gf128_inv(a):
    """Compute b such that ab = 1.
    >>> assert gf128_inv(x**1) == x**127 + x**6 + x**1 + 1
    >>> assert gf128_inv(x**4) == x**127 + x**125 + x**124 + x**6 + x**4 + x**3 + x**1 + 1
    """
    g, xx, yy = gf128_egcd(a, gcm_modulus)
    assert g == 1
    return xx

def gf128_exp(a, n):
    """Compute a^n.
    >>> assert gf128_exp(x**2 + 1, 0) == 1
    >>> assert gf128_exp(x**2 + 1, 1) == x**2 + 1
    >>> assert gf128_exp(x**2 + 1, 2) == x**4 + 1
    >>> assert gf128_exp(x**2 + 1, 10) == x**20 + x**16 + x**4 + 1
    """
    if n == 0:
        return 1
    rec = gf128_exp(a, n//2)
    rec = gf128_mul(rec, rec)
    if n % 2 == 1:
        rec = gf128_mul(rec, a)
    return rec

def gf128_sqrt(x):
    """Compute y such that y^2 = x: the inverse Frobenius automorphism.
    Reference: https://math.stackexchange.com/questions/943417/square-root-for-galois-fields-gf2m
    >>> assert gf128_sqrt(1) == 1
    >>> assert gf128_sqrt(x**2 + 1) == x**1 + 1
    >>> assert gf128_sqrt(x**3 + 1) == x**126 + x**123 + x**120 + x**117 + x**114 + x**111 + x**108 + x**105 + x**102 + x**99 + x**96 + x**93 + x**90 + x**87 + x**84 + x**81 + x**78 + x**75 + x**72 + x**69 + x**66 + x**63 + x**62 + x**60 + x**59 + x**57 + x**56 + x**54 + x**53 + x**51 + x**50 + x**48 + x**47 + x**45 + x**44 + x**42 + x**41 + x**39 + x**38 + x**36 + x**35 + x**33 + x**32 + x**30 + x**29 + x**27 + x**26 + x**24 + x**23 + x**21 + x**20 + x**18 + x**17 + x**15 + x**14 + x**12 + x**11 + x**9 + x**8 + x**6 + x**3 + 1
    """
    return gf128_exp(x, 2**127)

## Computation in GF(2^128)[X]/(x^128 + x^7 + x^2 + x^1 + 1)
## Elements are represented as arrays where the ith element is the coefficient for x^i.

def gf128poly_lead(x):
    """Compute the leading coefficient of x.
    >>> assert gf128poly_lead([x**3, x**1, 1]) == 1
    >>> assert gf128poly_lead([x**3, x**1, x**2, x**4]) == x**4
    >>> assert gf128poly_lead([1]) == 1
    """
    assert len(x) > 0
    return x[-1]

def collapse(x):
    if len(x) == 0:
        return []
    if gf128poly_lead(x) == 0:
        return collapse(x[:-1])
    return [gf128_mod(y, gcm_modulus) for y in x]

def gf128poly_deg(x):
    """Compute the degree of x.
    >>> assert gf128poly_deg([x**3, x**1, 1]) == 2
    >>> assert gf128poly_deg([x**2, x**3]) == 1
    >>> assert gf128poly_deg([x**15]) == 0
    >>> assert gf128poly_deg([1]) == 0
    >>> assert gf128poly_deg([]) == -1
    """
    return len(x) - 1

def gf128poly_add(a, b):
    """Compute a + b.
    >>> assert gf128poly_add([x**3, x**1, 1], [x**2 + 1, x**3]) == [x**3 + x**2 + 1, x**1 + x**3, 1]
    >>> assert gf128poly_add([x**2, x**3], [x**5, x**12, x**2, x**4]) == [x**5 + x**2, x**12 + x**3, x**2, x**4]
    >>> assert gf128poly_add([x**15], []) == [x**15]
    >>> assert gf128poly_add([x**15], [x**15]) == []
    """
    return collapse([gf128_add(x, y) for x, y in itertools.zip_longest(a, b, fillvalue=0)])

def gf128poly_mul(a, b):
    """Compute ab.
    >>> assert gf128poly_mul([x**3, x**1, 1], [x**2 + 1, x**3]) == [x**5 + x**3, x**6 + x**3 + x**1, x**4 + x**2 + 1, x**3]
    >>> assert gf128poly_mul([x**2, x**3], [x**5, x**12, x**2, x**4]) == [x**7, x**14 + x**8, x**15 + x**4, x**6 + x**5, x**7]
    >>> assert gf128poly_mul([x**15], []) == []
    >>> assert gf128poly_mul([x**15], [1]) == [x**15]
    >>> assert gf128poly_mul([0, x**15], [0, x**15]) == [0, 0, x**30]
    >>> assert gf128poly_mul([x**70], [1, x**70, 1]) == [x**70, x**19 + x**14 + x**13 + x**12, x**70]
    """
    p = []
    while gf128poly_deg(a) >= 0:
        p = gf128poly_add(p, [gf128_mul(x, a[0]) for x in b])
        a = a[1:]
        b = [0] + b
    return collapse(p)

def gf128poly_divmod(a, b):
    """Compute q, r such that a = qb + r; gf128poly_deg(r) < gf128poly_deg(b).
    >>> assert gf128poly_divmod([x**5 + x**3, x**6 + x**3 + x**1, x**4 + x**2 + 1, x**3], [x**2 + 1, x**3]) == ([x**3, x**1, 1], [])
    >>> assert gf128poly_divmod([x**7, x**14 + x**8, x**15 + x**4, x**6 + x**5, x**7], [x**2, x**3]) == ([x**5, x**12, x**2, x**4], [])
    >>> assert gf128poly_divmod([x**15], [1]) == ([x**15], [])
    >>> assert gf128poly_divmod([x**15], [x**15]) == ([1], [])
    >>> assert gf128poly_divmod([0, 0, x**30], [0, x**15]) == ([0, x**15], [])
    >>> assert gf128poly_divmod([x**70, x**19 + x**14 + x**13 + x**12, x**70], [1, x**70, 1]) == ([x**70], [])
    >>> assert gf128poly_divmod([x**18, x**18, x**3], [x**3, x**3]) == ([x**15 + 1, 1], [x**3])
    """
    assert b != [], 'divide by zero'
    q = []
    while gf128poly_deg(a) >= gf128poly_deg(b):
        highA = gf128poly_lead(a)
        highB = gf128poly_lead(b)
        highBinv = 1 if highB == 1 else gf128_inv(highB)
        qq = gf128_mul(highA, highBinv)
        de = gf128poly_deg(a) - gf128poly_deg(b)
        new_term = [0]*(de+1)
        new_term[de] = qq
        q = gf128poly_add(q, new_term)
        fact = gf128poly_mul(b, new_term)
        a = gf128poly_add(a, fact)
    return collapse(q), collapse(a)

def gf128poly_sqrt(p):
    """Compute q such that q^2 = p for p such that p' = 0.
    >>> assert gf128poly_sqrt([1, 0, 1, 0, 1, 0, 1]) == [1, 1, 1, 1]
    >>> assert gf128poly_sqrt([1, 0, 1, 0, 0, 0, 1]) == [1, 1, 0, 1]
    >>> assert gf128poly_sqrt([1, 0, x**4]) == [1, x**2]
    >>> assert gf128poly_sqrt([1, 0, x**3]) == [1, x**126 + x**123 + x**120 + x**117 + x**114 + x**111 + x**108 + x**105 + x**102 + x**99 + x**96 + x**93 + x**90 + x**87 + x**84 + x**81 + x**78 + x**75 + x**72 + x**69 + x**66 + x**63 + x**62 + x**60 + x**59 + x**57 + x**56 + x**54 + x**53 + x**51 + x**50 + x**48 + x**47 + x**45 + x**44 + x**42 + x**41 + x**39 + x**38 + x**36 + x**35 + x**33 + x**32 + x**30 + x**29 + x**27 + x**26 + x**24 + x**23 + x**21 + x**20 + x**18 + x**17 + x**15 + x**14 + x**12 + x**11 + x**9 + x**8 + x**6 + x**3]
    """
    assert gf128poly_formal_derivative(p) == []
    q = []
    for x in p:
        if x == 0 or x == 1:
            q.append(x)
        else:
            q.append(gf128_sqrt(x))
    for i in range(2, len(q), 2):
        q[i//2] = q[i]
        q[i] = 0
    return collapse(q)

def gf128poly_monic(p):
    """Compute p/p_lead for p such that gf128poly_deg(p) >= 0.
    >>> assert gf128poly_monic([x**54, x**2]) == [x**52, 1]
    >>> assert gf128poly_monic([x**2, x**3]) == [x**127 + x**6 + x**1 + 1, 1]
    """
    assert len(p) > 0
    lead = gf128poly_lead(p)
    if lead == 1:
        return p
    lead_inv = gf128_inv(lead)
    return [gf128_mul(x, lead_inv) for x in p]

def gf128poly_formal_derivative(p):
    """Compute p'.
    >>> assert gf128poly_formal_derivative([]) == []
    >>> assert gf128poly_formal_derivative([x**25]) == []
    >>> assert gf128poly_formal_derivative([x**25, x**3 + x**1, x**10, x**7]) == [x**3 + x**1, 0, x**7]
    """
    q = []
    for i, x in enumerate(p):
        if i == 0:
            continue
        if i % 2 == 0:
            e = 0
        else:
            e = x
        q.append(e)
    return collapse(q)

def gf128poly_modexp(a, n, m):
    """Compute a^n (mod m).
    >>> assert gf128poly_modexp([x**2, x**3, x**4], 0, [x**3, x**2, 1]) == [1]
    >>> assert gf128poly_modexp([x**2, x**3, x**4], 1, [x**3, x**2, 1, x**5]) == [x**2, x**3, x**4]
    >>> assert gf128poly_modexp([x**2, x**3, x**4], 5, [x**3, x**2, 1]) == [x**39 + x**38 + x**37 + x**36 + x**35 + x**33 + x**32 + x**30 + x**27 + x**26 + x**25 + x**24 + x**21 + x**20 + x**15 + x**10, x**38 + x**36 + x**35 + x**33 + x**32 + x**31 + x**26 + x**24 + x**23 + x**22 + x**21 + x**20 + x**14 + x**11]
    """
    def modmul(a, b, m):
        q, r = gf128poly_divmod(gf128poly_mul(a, b), m)
        return r

    if n == 0:
        return [1]
    rec = gf128poly_modexp(a, n//2, m)
    rec = modmul(rec, rec, m)
    if n % 2 == 1:
        rec = modmul(rec, a, m)
    return rec

def gf128poly_monic_gcd(a, b):
    """Compute g such that g | a, g | b, gf128poly_lead(g) = 1;
	for any h such that h | a and h | b, gf128poly_deg(g) >= gf128poly_deg(h).
    >>> assert gf128poly_monic_gcd([x**3, x**2, x**4], [x**3, x**2, x**4]) == [x**127 + x**6 + x**1 + 1, x**127 + x**126 + x**6 + x**5 + x**1, 1]
    >>> assert gf128poly_monic_gcd([1], [1]) == [1]
    >>> assert gf128poly_monic_gcd([x**5 + x**2], []) == [x**5 + x**2]
    >>> assert gf128poly_monic_gcd([x**3, 0, 0, x**6], [x**3, x**8 + x**4, x**9 + x**7, x**8]) == [x**127 + x**6 + x**1 + 1, 1]
    """
    assert gf128poly_deg(a) >= 0 or gf128poly_deg(b) >= 0
    if gf128poly_deg(a) < 0:
        return b
    if gf128poly_deg(b) < 0:
        return a
    q, r = gf128poly_divmod(b, a)
    ret = gf128poly_monic_gcd(r, a)
    return gf128poly_monic(ret)

def gf128poly_square_free_factorization(f):
    """Compute the square-free factorization of a monic polynomial f.
    Reference: https://en.wikipedia.org/wiki/Factorization_of_polynomials_over_finite_fields#Square-free_factorization
    >>> a = [x**1, x**2, 1]
    >>> b = [x**2, x**3+1, 1]
    >>> c = [x**3, x**4, x**5, 1]
    >>> asq = gf128poly_mul(a, a)
    >>> acb = gf128poly_mul(asq, a)
    >>> bsq = gf128poly_mul(b, b)
    >>> acbbsq = gf128poly_mul(acb, bsq)
    >>> p = gf128poly_mul(acbbsq, c)
    >>> assert gf128poly_square_free_factorization([1]) == [([1], 1)]
    >>> assert gf128poly_square_free_factorization(p) == [([x**3, x**4, x**5, 1], 1), ([x**1, x**2, 1], 3), ([x**2, x**3 + 1, 1], 2)]
    """
    R = []
    assert f[-1] == 1, 'monic'
    if f == [1]:
        return [([1], 1)]
    fprime = gf128poly_formal_derivative(f)
    c = gf128poly_monic_gcd(f, fprime)
    w, r = gf128poly_divmod(f, c)
    assert r == []
    i = 1
    while w != [1]:
        y = gf128poly_monic_gcd(w, c)
        fac, r = gf128poly_divmod(w, y)
        assert r == []
        R.append((fac, i))
        w = y
        c, r = gf128poly_divmod(c, y)
        assert r == []
        i = i + 1
    if c != [1]:
        c = gf128poly_sqrt(c)
        c = gf128poly_monic(c)
        Rrec = gf128poly_square_free_factorization(c)
        R += [(fac, i*2) for (fac, i) in Rrec]
    return [(fac, mult) for fac, mult in R if fac != [1]]

def gf128poly_distinct_degree_factorization(f):
    """Compute the distinct-degree factorization of a square-free monic polynomial f.
    Reference: https://en.wikipedia.org/wiki/Factorization_of_polynomials_over_finite_fields#Distinct-degree_factorization
    >>> a = [x**1, 1]
    >>> b = [x**2, 1]
    >>> ab = gf128poly_mul(a, b)
    >>> assert gf128poly_distinct_degree_factorization(ab) == {((x**3, x**2 + x**1, 1), 1)}
    """
    i = 1
    S = set()
    f_ = f.copy()
    while gf128poly_deg(f_) >= 2*i:
        qq = gf128poly_modexp([0, 1], 2**(128*i), f_)
        qq = gf128poly_add(qq, [0, 1])
        g = gf128poly_monic_gcd(f_, qq)
        if g != [1]:
            S.add((tuple(g), i))
            q, r = gf128poly_divmod(f_, g)
            assert r == []
            f_ = q
        i += 1
    if f_ != [1]:
        S.add((tuple(f_), gf128poly_deg(f_)))
    if len(S) == 0:
        return {(tuple(f), 1)}
    else:
        return S

def gf128poly_equal_degree_factorization(f, d):
    """Compute the equal-degree factorization of a square-free monic polynomial f whose factors are all of degree d.
    Uses the Cantor-Zassenhaus algorithm.
    Reference: https://en.wikipedia.org/wiki/Factorization_of_polynomials_over_finite_fields#Equal-degree_factorization
    >>> a = [x**1, 1]
    >>> b = [x**2, 1]
    >>> ab = gf128poly_mul(a, b)
    >>> assert gf128poly_equal_degree_factorization(ab, 1) == {(x**2, 1), (x**1, 1)}
    """
    def gf128_rand():
        return random.randint(0, 2**128-1)

    def gf128poly_rand(degree):
        return [gf128_rand() for _ in range(degree+1)]

    exp = (2**128-1)//3
    n = gf128poly_deg(f)
    r = n//d
    S = {tuple(f)}
    while len(S) < r:
        h = gf128poly_rand(n - 1)
        g = gf128poly_monic_gcd(h, f)
        if g == [1]:
            g = gf128poly_add(gf128poly_modexp(h, exp, f), [1])
        for u in list(S):
            u = list(u)
            if gf128poly_deg(u) == d:
                continue
            gugcd = gf128poly_monic_gcd(g, u)
            if gugcd != [1] and gugcd != u:
                S.remove(tuple(u))
                S.add(tuple(gugcd))
                qq, rr = gf128poly_divmod(u, gugcd)
                assert rr == []
                S.add(tuple(qq))
    return S

def gf128poly_factorize(f):
    """Compute the factors of a polynomial f. Does not return multiplicity.
    """
    factors = set()
    f = gf128poly_monic(f)
    fs = gf128poly_square_free_factorization(f)
    for p, _ in fs:
        qs = gf128poly_distinct_degree_factorization(p)
        for q, d in qs:
            rs = gf128poly_equal_degree_factorization(list(q), d)
            factors |= rs
    return factors

## AES-GCM

def reverse_mask(b):
    # from https://stackoverflow.com/a/2602885
    b = (b & 0xF0) >> 4 | (b & 0x0F) << 4;
    b = (b & 0xCC) >> 2 | (b & 0x33) << 2;
    b = (b & 0xAA) >> 1 | (b & 0x55) << 1;
    return b;

def bytes_to_gf128(xs, st=None, en=None):
    if st is None and en is None:
        st = 0
        en = 16
    a, b = struct.unpack('<QQ', bytes(reverse_mask(xs[i]) for i in range(st, en)))
    return a + (b << 64)

def gf128_to_bytes(n):
    s = b''
    while n != 0:
        c = n & 0xff
        f = 0
        for i in range(8):
            if c & (1 << i):
                f += (1 << (7-i))
        s += int.to_bytes(f, 1, 'big')
        n >>= 8
    s += b'\x00' * (16 - len(s))
    assert len(s) == 16
    return s

def pad(xs):
    return xs + b'\x00' * ((16 - len(xs)) % 16)

def build_blocks(aad, c):
    padded_aad = pad(aad)
    padded_c = pad(c)
    length = struct.pack('>QQ', len(aad)*8, len(c)*8)
    return padded_aad + padded_c + length

def gmac(h, s, aad, c):
    blocks = build_blocks(aad, c)
    g = 0
    for i in range(len(blocks)//16):
        block = bytes_to_gf128(blocks, i*16, (i + 1)*16)
        g = gf128_add(g, block)
        g = gf128_mul(g, h)
    t = gf128_add(g, s)
    return gf128_to_bytes(t)

def ecb_encrypt(key, xs):
    c = AES.new(key, AES.MODE_ECB)
    return c.encrypt(xs)

def xor(a, b):
    return bytes([(x^y) for (x, y) in zip(a, b)])

def gctr(k, nonce, x):
    y = b''
    ctr = (int.from_bytes(nonce, 'big') << 32) + 2
    i = 0
    while len(y) < len(x):
        y += xor(x[i*16:(i + 1)*16], ecb_encrypt(k, ctr.to_bytes(16, 'big')))
        ctr += 1
        i += 1
    return y

def authentication_key(k):
    return bytes_to_gf128(ecb_encrypt(k, b'\x00'*16))

def blind(k, nonce):
    return bytes_to_gf128(ecb_encrypt(k, nonce + b'\x00\x00\x00\x01'))

def gcm_encrypt(k, nonce, aad, m, mac_bytes=16):
    c = gctr(k, nonce, m)
    mac = gmac(authentication_key(k), blind(k, nonce), aad, c)
    return c, mac[:mac_bytes]

def gcm_decrypt(k, nonce, aad, c, mac, mac_bytes=16):
    m = gctr(k, nonce, c)
    expected_mac = gmac(authentication_key(k), blind(k, nonce), aad, c)[:mac_bytes]
    assert hmac.compare_digest(mac, expected_mac)
    return m

## Forbidden Attack

def compute_forbidden_polynomial(aad1, aad2, c1, c2, mac1, mac2):
    bs1 = build_blocks(aad1, c1)
    bs2 = build_blocks(aad2, c2)
    if len(bs1) < len(bs2):
        bs1 = b'\x00'*(len(bs2)-len(bs1)) + bs1
    else:
        bs2 = b'\x00'*(len(bs1)-len(bs2)) + bs2
    assert len(bs1) == len(bs2)
    f = []
    N = len(bs1)//16
    for i in range(N):
        b1 = bytes_to_gf128(bs1[i*16:(i+1)*16])
        b2 = bytes_to_gf128(bs2[i*16:(i+1)*16])
        f.append(gf128_add(b1, b2))
    f.append(gf128_add(bytes_to_gf128(mac1), bytes_to_gf128(mac2)))
    return list(reversed(f))

def nonce_reuse_recover_secrets(nonce, aad1, aad2, c1, c2, mac1, mac2):
    f = compute_forbidden_polynomial(aad1, aad2, c1, c2, mac1, mac2)
    factors = gf128poly_factorize(f)
    secrets = []
    for factor in factors:
        if gf128poly_deg(factor) == 1:
            h = factor[0]
            zero_tag = gmac(h, 0, aad1, c1)
            s = gf128_add(bytes_to_gf128(mac1), bytes_to_gf128(zero_tag))
            secrets.append((h, s))
    return secrets

def demo():
    k = b"tlonorbistertius"
    nonce = b"jorgelborges"
    m1 = b"The universe (which others call the Library)"
    aad1 = b"The Anatomy of Melancholy"
    m2 = b"From any of the hexagons one can see, interminably"
    aad2 = b"Letizia Alvarez de Toledo"

    c1, mac1 = gcm_encrypt(k, nonce, aad1, m1)
    c2, mac2 = gcm_encrypt(k, nonce, aad2, m2)

    # Recover the authentication key and blind from public information
    possible_secrets = nonce_reuse_recover_secrets(nonce, aad1, aad2, c1, c2, mac1, mac2)

    # Forge the ciphertext
    m_forged = b"As was natural, this inordinate hope"
    assert len(m_forged) <= len(m1)
    c_forged = xor(c1, xor(m1, m_forged))
    aad_forged = b"You who read me, are You sure of understanding my language?"

    # Check possible candidates for authentication key
    succeeded = False
    for h, s in possible_secrets:
        mac_forged = gmac(h, s, aad_forged, c_forged)
        try:
            assert gcm_decrypt(k, nonce, aad_forged, c_forged, mac_forged) == m_forged
            succeeded = True
            print(c_forged.hex(), mac_forged.hex())
        except AssertionError:
            pass
    assert succeeded