Feige–Fiat–Shamir identification scheme

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, Peggy, to prove to another party, Victor, that she possesses secret information without revealing to Victor 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 Peggy and Victor.

Choose two large prime integers p and q and compute the product n = pq. Create secret numbers ${\displaystyle s_{1},\cdots ,s_{k}}$ coprime to n. Compute ${\displaystyle v_{i}\equiv s_{i}^{2}{\pmod {n}}}$. Peggy and Victor both receive ${\displaystyle n}$ while ${\displaystyle p}$ and ${\displaystyle q}$ are kept secret. Peggy is then sent the numbers ${\displaystyle s_{i}}$. These are her secret login numbers. Victor is sent the numbers ${\displaystyle v_{i}}$ by Peggy when she wishes to identify herself to Victor. Victor is unable to recover Peggy's ${\displaystyle s_{i}}$ numbers from his ${\displaystyle v_{i}}$ numbers due to the difficulty in determining a modular square root when the modulus' factorization is unknown.

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