Simultaneous Amplification: The Case of Non-interactive Zero-Knowledge.

In this work, we explore the question of simultaneous privacy and soundness amplification for non-interactive zero-knowledge argument systems (NIZK). We show that any delta(s)-sound and delta(z)-zero-knowledge NIZK candidate satisfying delta(s) + delta(z) = 1 - C, for any constant C > 0, can be turned into a computationally sound and zero-knowledge candidate with the only extra assumption of a subexponentially secure public-key encryption. We develop novel techniques to leverage the use of leakage simulation lemma (Jetchev-Peitzrak TCC 2014) to argue amplification. A crucial component of our result is a new notion for secret sharing NP instances. We believe that this may be of independent interest. To achieve this result we analyze following two transformations: - Parallel Repetition: We show that using parallel repetition any delta(s)-sound and delta(z)-zero-knowledge NIZK candidate can be turned into (roughly) delta(n)(s)-soundand 1- (1- delta(z))(n)-zero-knowledge candidate. Here n is the repetition parameter. - MPC based Repetition: We propose a new transformation that amplifies zero-knowledge in the same way that parallel repetition amplifies soundness. We show that using this any ds-sound and delta(z)-zero-knowledge NIZK candidate can be turned into (roughly) 1(1 - delta(s))(n)-sound and 2 . delta(n)(z)-zero-knowledge candidate. Then we show that using these transformations in a zig-zag fashion we can obtain our result. Finally, we also present a simple transformation which directly turns any NIZK candidate satisfying delta(s), delta(z) < 1/3 - 1/poly(lambda) to a secure one.