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from hashlib import sha256
# Ring Signatures
# bLSAG: Back’s Linkable Spontaneous Anonymous Group signatures
# A Rust implementation of this scheme can be found at:
# https://github.com/arnaucube/ring-signatures-rs
def hashToPoint(a):
# TODO use a proper hash-to-point
h = sha256((str(a)).encode('utf-8'))
r = int(h.hexdigest(), 16) * g
return r
def hash(R, m, A, B, q):
h = sha256((
str(R) + str(m) + str(A) + str(B)
).encode('utf-8'))
r = int(h.hexdigest(), 16)
return int(mod(r, q))
def print_ring(a):
print("ring of c's:")
for i in range(len(a)):
print(i, a[i])
print("")
class Prover:
def __init__(self, F, g):
self.F = F # Z_p
self.g = g # elliptic curve generator
self.q = self.g.order() # order of group
def new_key(self):
self.w = int(self.F.random_element())
self.K = self.g * self.w
return self.K
def sign(self, m, R):
# determine pi (the position of signer's public key in R
pi = -1
for i in range(len(R)):
if self.K == R[i]:
pi = i
break
assert pi>=0
a = int(self.F.random_element())
r = [None] * len(R)
# for i \in {1, 2, ..., n} \ {i=pi}
for i in range(0, len(R)):
if i==pi:
continue
r[i] = int(mod(int(self.F.random_element()), self.q))
c = [None] * len(R)
# c_{pi+1}
pi1 = mod(pi + 1, len(R))
c[pi1] = hash(R, m, a * self.g, hashToPoint(R[pi]) * a, self.q)
key_image = self.w * hashToPoint(self.K)
# do c_{i+1} from i=pi+1 to pi-1:
for j in range(0, len(R)-1):
i = mod(pi1+j, len(R))
i1 = mod(pi1+j +1, len(R))
c[i1] = hash(R, m, r[i] * self.g + c[i] * R[i], r[i] * hashToPoint(R[i]) + c[i] * key_image, self.q)
# compute r_pi
r[pi] = int(mod(a - c[pi] * self.w, self.q))
print_ring(c)
return [c[0], r]
def verify(g, R, m, key_image, sig):
q = g.order()
c1 = sig[0]
r = sig[1]
assert len(R) == len(r)
# check that key_image \in G (EC), by l * key_image == 0
assert q * key_image == 0 # in sage 0 EC point is represented as (0:1:0)
# for i \in 1,2,...,n
c = [None] * len(R)
c[0] = c1
for j in range(0, len(R)):
i = mod(j, len(R))
i1 = mod(j+1, len(R))
c[i1] = hash(R, m, r[i] * g + c[i] * R[i], r[i] * hashToPoint(R[i]) + c[i] * key_image, q)
print_ring(c)
assert c1 == c[0]