Sequential iteration of interactive arguments and an efficient zero-knowledge argument for NP

Abstract

We study interactive arguments, in particular their error probability under sequential iteration. This problem is more complex than for interactive proofs, where the error trivially decreases exponentially in the number of iterations.

In particular, we study the typical efficient case where the iterated protocol is based on a single instance of a computational problem. This is not a special case of independent iterations of an entire protocol, and real exponential decrease of the error cannot be expected. Nevertheless, for practical applications, one needs concrete relations between the complexity and error probability of the underlying problem and the iterated protocol. We formalize and solve this problem using the theory of proofs of knowledge. We also seem to present the first definition of arguments in a fully uniform model of complexity.

We also prove that in non-uniform complexity, the error probability of independent iterations of an argument does decrease exponentially — to our knowledge this is the first result about a strictly exponentially small error probability in a computational cryptographic security property.

To illustrate our first result, we present a very efficient zero-knowledge argument for circuit satisfiability, and thus for any NP problem, based on any collision-intractable hash function.