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We present zero-knowledge proofs and arguments for arithmetic circuits over finiteprime fields, namely given a circuit, show in zero-knowledge that inputs can beselected leading to a given output. For a field GF (q), where q is an n-bit prime, acircuit of size O(n), and error probability 2, our protocols require communication ofO(n) bits. This is the same worst-cast complexity as the trivial (non zero-knowledge)interactive proof where the prover just reveals the input values....

Zero-knowledge proofs for finite field arithmetic, or: Can zero-knowledge be for free?