Concurrent Zero-Knowledge with Timing, Revisited

Abstract

Following Dwork, Naor, and Sahai (30th STOC, 1998), we consider concurrent executions of protocols in a semi-synchronized network. Specifically, we assume that each party holds a local clock such that bounds on the relative rates of these clocks as well as on the message-delivery time are a-priori known, and consider protocols that employ time-driven operations (i.e., time-out in-coming messages and delay out-going messages).

We show that the constant-round zero-knowledge proof for ${\cal NP}$ of Goldreich and Kahan (Jour. of Crypto., 1996) preserves its security when polynomially-many independent copies are executed concurrently under the above timing model.

We stress that our main result refers to zero-knowledge of interactive proofs, whereas the results of Dwork et. al. are either for zero-knowledge arguments or for a weak notion of zero-knowledge (called epsilon-knowledge) proofs.

Our analysis identifies two extreme schedulings of concurrent executions under the above timing model: the first is the case of parallel execution of polynomially-many copies, and the second is of concurrent execution of polynomially-many copies such that only a small (i.e., constant) number of copies are simultaneously active at any time (i.e., bounded simultaneity). Dealing with each of these extreme cases is of independent interest, and the general result (regarding concurrent executions under the timing model) is obtained by combining the two treatments.

Preliminary version has appeared in the proceedings of the 34th ACM Symposium on the Theory of Computing, 2002. The current revision was prepared in memory of Shimon Even. I find it especially fitting that my wish to pay tribute to his memory has caused me to fulfill my duty (neglected for a couple of years) to produce a final version of the current work.