In cryptography, the Feige–Fiat–Shamir identification scheme is a type of parallel zero-knowledge proof developed by Uriel Feige, Amos Fiat, and Adi Shamir in 1988. Like all zero-knowledge proofs, it allows one party, the Prover, to prove to another party, the Verifier, that they possess secret information without revealing to Verifier what that secret information is. The Feige–Fiat–Shamir identification scheme, however, uses modular arithmetic and a parallel verification process that limits the number of communications between Prover and Verifier.

Setup

Following a common convention, call the prover Peggy and the verifier Victor. Choose two large prime integers p and q and compute the product n = pq. Create secret numbers coprime to n. Compute. Peggy and Victor both receive n while p and q are kept secret. Peggy is then sent the numbers s_i. These are her secret login numbers. Victor is sent the numbers v_i by Peggy when she wishes to identify herself to Victor. Victor is unable to recover Peggy's s_i numbers from his v_i numbers due to the difficulty in determining a modular square root when the modulus' factorization is unknown.

Procedure

This procedure is repeated with different r and a_i values until Victor is satisfied that Peggy does indeed possess the modular square roots (s_i) of his v_i numbers.

Security

In the procedure, Peggy does not give any useful information to Victor. She merely proves to Victor that she has the secret numbers without revealing what those numbers are. Anyone who intercepts the communication between each Peggy and Victor would only learn the same information. The eavesdropper would not learn anything useful about Peggy's secret numbers. Suppose Eve has intercepted Victor's v_i numbers but does not know what Peggy's s_i numbers are. If Eve wants to try to convince Victor that she is Peggy, she would have to correctly guess what Victor's a_i numbers will be. She then picks a random y, calculates and sends x to Victor. When Victor sends a_i, Eve simply returns her y. Victor is satisfied and concludes that Eve has the secret numbers. However, the probability of Eve correctly guessing what Victor's a_i will be is 1 in 2^k. By repeating the procedure t times, the probability drops to 1 in 2^{k t}. For k = 5 and t = 4 the probability of successfully posing as Peggy is less than 1 in 1 million.