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

### Ronald Cramer and Ivan Damgård

We present a general method for constructing commitment schemes based on existence of $q$-one way group homomorphisms, in which elements in a finite prime field $GF(q)$ can be committed to. A receiver of commitments can non-interactively check whether committed values satisfy linear equations. Multiplicative relations can be verified interactively with exponentially small error, while communicating only a constant number of commitments. Particular assumptions sufficient for our commitment schemes include: the RSA assumption, hardness of discrete log in a prime order group, and polynomial security of Diffie-Hellman encryption. Based on these commitments, we give efficient zero-knowledge proofs and arguments for arithmetic circuits over finite prime fields, namely given such 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 $m$-bit prime, a circuit of size $O(m)$, and error probability $2^{-m}$, our protocols require communication of $O(m^2)$ bits. We then look at the Boolean Circuit Satisfiability problem and give non-interactive zero-knowledge proofs and arguments with preprocessing. In the proof stage, the prover can prove any circuit of size $n$ he wants by sending only one message of size $O(n)$ bits. As a final 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.

## BibTeX Citation

@inproceedings{CraDam98, author = {Ronald Cramer and Ivan Damgård}, title = {Zero-Knowledge for Finite Field Arithmetic or: Can Zero-Knowledge be for Free?}, editor = {Hugo Krawczyk}, booktitle = {Advances in Cryptology --- CRYPTO~'98}, pages = {424--441}, series = {Lecture Notes in Computer Science}, volume = {1462}, year = {1998}, month = {8}, publisher = {Springer-Verlag}, }