A quantum Goldreich-Levin theorem with cryptographic applications

TL;DR

This paper extends the Goldreich-Levin Theorem into the quantum realm, demonstrating more efficient reductions between quantum one-way functions and hard-predicates, with applications to quantum cryptographic protocols.

Contribution

It introduces a quantum version of the Goldreich-Levin reduction with improved efficiency and applies it to construct quantum bit and qubit commitment schemes.

Findings

Quantum reduction is more efficient than classical by a factor of O(1/epsilon)

Classical reduction cannot have overhead less than O(n/epsilon^2)

Quantum bit commitment scheme can be derived from quantum one-way permutations

Abstract

We investigate the Goldreich-Levin Theorem in the context of quantum information. This result is a reduction from the computational problem of inverting a one-way function to the problem of predicting a particular bit associated with that function. We show that the quantum version of the reduction -- between quantum one-way functions and quantum hard-predicates -- is quantitatively more efficient than the known classical version. Roughly speaking, if the one-way function acts on n-bit strings then the overhead in the reduction is by a factor of O(n/epsilon^2) in the classical case but only by a factor of O(1/epsilon) in the quantum case, where 1/2 + epsilon is the probability of predicting the hard-predicate. Moreover, we prove via a lower bound that, in a black-box framework, the classical version of the reduction cannot have overhead less than order n/epsilon^2. We also show that, using this reduction, a quantum bit commitment scheme that is perfectly binding and computationally concealing can be obtained from any quantum one-way permutation. This complements a recent result by Dumais, Mayers and Salvail, where the bit commitment scheme is perfectly concealing and computationally binding. We also show how to perform qubit commitment by a similar approach.