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In this paper we give techniques for recycling random bits both in the interactive and in the non-interactive model for perfect zero-knowledge proofs. Our first result is a non-interactive perfect zero-knowledge proof system for proving that at least one out of any given polynomial number of statements is true, in which the amount of public random bits used is the same as that for proving a single statement. Our second result is an interactive perfect zero-knowledge proof system for proving any given polynomial number of statements, in which the amount of private random bits used by the prover is, apart from a constant factor, the same as that for proving a single statement. In order to get a randomness-efficient proof system, we also reduce the random string of the verifier by using a multi-bit commitment scheme. The statements considered are of quadratic non residuosity modulo a Blum integer.

Recycling Random Bits in Composed Perfect Zero-Knowledge