def __init__(self, P, Q, s=1, strict=False):
'''
if strict == True:
use Carmichael Number
else:
use Euler Number
'''
N = P * Q
# Lam = lcm(P-1, Q-1)
G = field(N**s, "G") # n ** s == n if s = 1
# multiplicative group
MG = field(N**(s + 1), "N^{s+1}") # n ** (s +1 ) == n2 in pailer case
H = field(N, "H")
# https://crypto.stackexchange.com/questions/29591/lcm-versus-phi-in-rsa
if strict:
LG = field(rsa_lambda(P, Q), "PhiGroup")
else:
LG = field(rsa_phi(P, Q), "PhiGroup")
j = generate_prime(length(P))
assert gcd(j, N) == 1
x = randfield(H)
g = MG((MG(1 + N)**j) * x)
d = crt(a_list=[0, 1], n_list=[LG.P, G.P])
assert d % G.P == 1
assert d % LG.P == 0
self.s = s
self.privkey = d
self.N = N
self.G = g
self.pubkey = (N, g)
def setup(self, secret, k, n, poly_params = []):
if not poly_params:
poly_params = [randfield(self.F) for _ in range(k - 1)]
self.poly_proofs = [self.G ** p for p in [secret] + poly_params]
return super().setup(secret, k, n, poly_params)
def join(self, x=None):
assert x != 0
if not x:
x = randfield(self.F)
if not hasattr(self, 'f'):
raise Exception("Needs to encrypt first")
return (self.F(x), self.f(x))
def encrypt(cls, m, pub):
g, h = pub
y = randfield(field(g.N)).value
m_ = map_to_curve(m)
s = h ** y
c1 = g ** y
c2 = s * m_
return (c1, c2)
def commit(self, m, ak):
pk, vk, h = self.key
com = PedersonCommitment(H, G, pk, ak)
# (𝑐,𝑑)=𝐻𝑆𝑐𝑜𝑚𝑚𝑖𝑡𝑝𝑘(𝑎𝑘)
c, d = com.C, com.D
# α=ℎ(𝑐)
alpha = CF(to_sha256int(gen_address(c)))
# Now we simulate a proof of knowledge of a signature on 𝛼 with challenge 𝑚
S = PedersonCommitment(H, G, vk, alpha)
r1, r2 = randfield(vk.functor), randfield(vk.functor)
a = S.commit(r1, r2)
z = S.challenge(m)
self.S = S
# Finally, we compute 𝑚𝑎𝑐=𝑀𝐴𝐶𝑎𝑘(𝑎).
mac = hmac.new(str(ak.value).encode(), encode_pubkey(a).encode()).hexdigest()
self.c = (c, a, mac)
self.d = (m, d , z)
def encrypt(cls, m, pub, s=1, r=None):
N, G = pub
if hasattr(m, "value"):
m = m.value
if not r:
r = randfield(G.functor)
else:
r = G.functor(r)
return G**m * r**(N**s)
def test_ssss():
F = field(P)
s = SSSS(F)
k = random.randint(1, 100)
n = k * 3
secret = randfield(F)
s.setup(secret, k, n)
assert s.decrypt([s.join() for _ in range(k - 1)]) != secret
assert s.decrypt([s.join() for _ in range(k + 1)]) == secret
assert s.decrypt([s.join() for _ in range(k + 2)]) == secret
def setup(self, secret, k, n, poly_params=[]):
'''
k: threshold
'''
self.k = k
self.n = n
if not poly_params:
poly_params = [randfield(self.F) for _ in range(k - 1)]
self.f = lambda x: self.F(secret) + reduce(
add, [poly_params[i] * (x**(i + 1)) for i in range(k - 1)])
return self
def __init__(self, P, Q):
assert gcd(P * Q, (P - 1) * (Q - 1)) == 1
N = P * Q
Lam = lcm(P - 1, Q - 1)
F = field(N)
DF = field(N**2)
G = randfield(DF)
M = ~F(L(pow(G, Lam).value, N))
self.N = N
self.G = G
self.privkey = Lam
self.pubkey = (self.N, self.G)
def additive_share(secret, F, n):
shares = [randfield(F) for _ in range(n - 1)]
shares += [(F(secret) - reduce(add, shares))]
return shares
def keygen(F):
pk = randfield(F)
vk = F(to_sha256int(str(pk.value)))
h = gen_address
return (pk, vk, h)