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

Abstract

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 $\delta _s+\delta _z=1-\epsilon$, for any constant $\epsilon >0$, 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 $\mathsf {NP}$ 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 $\mathsf {NIZK}$ candidate can be turned into (roughly) $\delta ^n_s-$sound and $1-(1-\delta _{z})^n-$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 $\delta _s-$sound and $\delta _z-$zero-knowledge $\mathsf {NIZK}$ candidate can be turned into (roughly) $1-(1-\delta _s)^n-$sound and $2\cdot \delta ^n_{z}-$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 $\mathsf {NIZK}$ candidate satisfying $\delta _s,\delta _z<1/3 -1/\mathsf {poly}(\lambda )$ to a secure one.