Zero-Knowledge Proofs for Finite Field Arithmetic or: Can Zero-Knowledge be for Free?

Ronald Cramer
Ivan B. Damgård

November 1997


We present zero-knowledge proofs and arguments for arithmetic circuits over finite prime fields, namely given a circuit, show in zero-knowledge that inputs can be selected leading to a given output. For a field GF(q), where q is an n-bit prime, a circuit of size O(n), and error probability tex2html_wrap_inline31, our protocols require communication of tex2html_wrap_inline33 bits. This is the same worst-cast complexity as the trivial (non zero-knowledge) interactive proof where the prover just reveals the input values. If the circuit involves n multiplications, the best previously known methods would in general require communication of tex2html_wrap_inline37 bits.

Variations of the technique behind these protocols lead to other interesting applications. We first look at the Boolean Circuit Satisfiability problem and give zero-knowledge proofs and arguments for a circuit of size n and error probability tex2html_wrap_inline31 in which there is an interactive preprocessing phase requiring communication of tex2html_wrap_inline33 bits. In this phase, the statement to be proved later need not be known. Later the prover can non-interactively prove any circuit he wants, i.e. by sending only one message, of size O(n) bits.

As a second application, we show that Shamirs (Shens) interactive proof system for the (IP-complete) QBF problem can be transformed to a zero-knowledge proof system with the same asymptotic communication complexity and number of rounds.

The security of our protocols can be based on any one-way group homomorphism with a particular set of properties. We give examples of special assumptions sufficient for this, including: the RSA assumption, hardness of discrete log in a prime order group, and polynomial security of Diffie-Hellman encryption.

We note that the constants involved in our asymptotic complexities are small enough for our protocols to be practical with realistic choices of parameters

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