The role of help in Classical and Quantum Zero-Knowledge

TL;DR

This paper investigates the role of help in zero-knowledge protocols, establishing equivalences between help and interaction in both classical and quantum settings, and introducing new complete problems for related complexity classes.

Contribution

It proves help and interaction are equivalent in classical and quantum non-interactive zero-knowledge, and introduces new complete problems for these classes.

Findings

Help and interaction are equivalent in classical NIZK protocols.

Help and interaction are equivalent in quantum NIZK protocols.

New complete problems for classical and quantum zero-knowledge classes.

Abstract

We study the role of help in Non-Interactive Zero-Knowledge protocols and its relation to the standard interactive model. In the classical case, we show that help and interaction are equivalent, answering an open question of Ben-Or and Gutfreund. This implies a new complete problem for the class SZK, the Image Intersection Density. For this problem, we also prove a polarization lemma which is stronger than the previously known one. In the quantum setting, we define the notion of quantum help and show in a more direct way that help and interaction are again equivalent. Moreover, we define quantum Non-Interactive Zero-Knowledge with classical help and prove that it is equal to the class of languages that have classical honest-Verifier Zero Knowledge protocols secure against quantum Verifiers. Last, we provide new complete problems for all these quantum classes. Similar results were independently discovered by Dragos Florin Ciocan and Salil Vadhan.