We present a zero-knowledge proof system [19] for any NP language L, which<br />allows showing that x in L with error probability less than 2^−k using communication<br />corresponding to O(|x|^c) + k bit commitments, where c is a constant depending only<br />on L. The proof can be based on any bit commitment scheme with a particular set<br />of properties. We suggest an efficient implementation based on factoring.<br />We also present a 4-move perfect zero-knowledge interactive argument for any NP-language<br />L. On input x in L, the communication complexity is O(|x|^c) max(k; l)<br />bits, where l is the security parameter for the prover. Again, the protocol can be<br />based on any bit commitment scheme with a particular set of properties. We suggest<br />efficient implementations based on discrete logarithms or factoring.<br />We present an application of our techniques to multiparty computations, allowing<br />for example t committed oblivious transfers with error probability 2^−k to be done<br />simultaneously using O(t+k) commitments. Results for general computations follow<br />from this.<br />As a function of the security parameters, our protocols have the smallest known<br />asymptotic communication complexity among general proofs or arguments for NP.<br />Moreover, the constants involved are small enough for the protocols to be practical in<br />a realistic situation: both protocols are based on a Boolean formula Phi containing and-<br />, or- and not-operators which verifies an NP-witness of membership in L. Let n be<br />the number of times this formula reads an input variable. Then the communication<br />complexity of the protocols when using our concrete commitment schemes can be<br />more precisely stated as at most 4n + k + 1 commitments for the interactive proof<br />and at most 5nl +5l bits for the argument (assuming k <= l). Thus, if we use k = n,<br />the number of commitments required for the proof is linear in n.<br />Both protocols are also proofs of knowledge of an NP-witness of membership in<br />the language involved.
Linear Zero-Knowledgde. A Note on Efficient Zero-Knowledge Proofs and Arguments
