Schnorr's proof of knowledge

Schnorr protocol to prove knowledge of 1 or more discrete logs in zero knowledge. Refer this for more details of Schnorr protocol.

Also implements the proof of knowledge of discrete log in pairing groups, i.e. given prover and verifier both know (A1, Y1), and prover additionally knows B1, prove that e(A1, B1) = Y1. Similarly, proving e(A2, B2) = Y2 when only prover knows A2 but both know (B2, Y2). See discrete_log_pairing

Also implements the proof of inequality of discrete log (a value committed in a Pedersen commitment), either with a public value or with another discrete log in Inequality. eg. Given a message m, its commitment C = g * m + h * r and a public value v, proving that m ≠ v. Or given 2 messages m1 and m2 and their commitments C1 = g * m1 + h * r1 and C2 = g * m2 + h * r2, proving m1 ≠ m2

Also implements the proof of inequality of discrete log when only one of the discrete log is known to the prover. i.e. given y = g * x and z = h * k, prover and verifier know g, h, y and z and prover additionally knows x but not k.

We outline the steps of Schnorr protocol. Prover wants to prove knowledge of x in y = g * x (y and g are public knowledge)
Step 1: Prover generates randomness r, and sends t = g * r to Verifier.
Step 2: Verifier generates random challenge c and send to Prover.
Step 3: Prover produces s = r + x*c, and sends s to Verifier.
Step 4: Verifier checks that g * s = (y * c) + t.

For proving knowledge of multiple messages like x_1 and x_2 in y = g_1*x_1 + g_2*x_2:
Step 1: Prover generates randomness r_1 and r_2, and sends t = g_1*r_1 + g_2*r_2 to Verifier
Step 2: Verifier generates random challenge c and send to Prover
Step 3: Prover produces s_1 = r_1 + x_1*c and s_2 = r_2 + x_2*c, and sends s_1 and s_2 to Verifier
Step 4: Verifier checks that g_1*s_1 + g_2*s_2 = y*c + t

Above can be generalized to more than 2 xs

There is another variant of Schnorr which gives shorter proof but is not implemented:

1. Prover creates r and then T = r * G.
2. Prover computes challenge as c = Hash(G||Y||T).
3. Prover creates response s = r + c*x and sends c and s to the Verifier as proof.
4. Verifier creates T' as T' = s * G - c * Y and computes c' as c' = Hash(G||Y||T')
5. Proof if valid if c == c'

The problem with this variant is that it leads to poorer failure reporting as in case of failure, it can't be pointed out which relation failed to verify. Eg. say there are 2 relations being proven which leads to 2 Ts T1 and T2 and 2 responses s1 and s2. If only the responses and challenge are sent then in case of failure, the verifier will only know that its computed challenge c' doesn't match prover's given challenge c but won't know which response s1 or s2 or both were incorrect. This is not the case with the implemented variant as verifier checks 2 equations s1 = r1 + x1*c and s2 = r2 + x2*c