math/ring-signatures.sage

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(object):
    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]