Schnorr protocol to prove knowledge of 1 or more discrete logs in zero knowledge. Refer this for more details of Schnorr protocol.
We outline the steps here for your convenience, and to make this documentation more succinct.
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 yet:
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'
License: Apache-2.0