General Properties of Quantum Zero-Knowledge Proofs

TL;DR

This paper establishes fundamental equivalences and properties of various quantum zero-knowledge proof classes, showing they can be transformed into more efficient forms without relying on computational assumptions.

Contribution

It proves that several quantum zero-knowledge classes are equal and can be made to have public-coin, perfect completeness, and three-message protocols, unconditionally and without complete promise problems.

Findings

HVQZK = QZK

QZK has public-coin protocols

Existence of three-message protocols with small error

Abstract

This paper studies the complexity classes QZK and HVQZK of problems having a quantum computational zero-knowledge proof system and an honest-verifier quantum computational zero-knowledge proof system, respectively. The results proved in this paper include: (a) HVQZK = QZK, (b) any problem in QZK has a public-coin quantum computational zero-knowledge proof system, (c) any problem in QZK has a quantum computational zero-knowledge proof system of perfect completeness, and (d) any problem in QZK has a three-message public-coin quantum computational zero-knowledge proof system of perfect completeness with arbitrarily small constant error in soundness. All the results above are unconditional and do not rely any computational assumptions. For the classes QPZK, HVQPZK, and QSZK of problems having a quantum perfect zero-knowledge proof system, an honest-verifier quantum perfect zero-knowledge proof system, and a quantum statistical zero-knowledge proof system, respectively, the following new properties are proved: (e) HVQPZK = QPZK, (f) any problem in QPZK has a public-coin quantum perfect zero-knowledge proof system, (g) any problem in QSZK has a quantum statistical zero-knowledge proof system of perfect completeness, and (h) any problem in QSZK has a three-message public-coin quantum statistical zero-knowledge proof system of perfect completeness with arbitrarily small constant error in soundness. It is stressed that our proofs are direct and do not use complete promise problems or those equivalents. This gives a unified framework that works well for all of quantum perfect, statistical, and computational zero-knowledge proofs, and enables us to prove properties even on the computational and perfect zero-knowledge proofs for which no complete promise problems are known.