Quantum algorithms for highly non-linear Boolean functions

TL;DR

This paper introduces quantum algorithms that efficiently solve hidden shift problems for highly non-linear Boolean functions called bent functions, demonstrating exponential separation from classical methods.

Contribution

It presents the first polynomial-time quantum algorithms for hidden shift problems of certain bent functions, highlighting their potential in cryptography.

Findings

Quantum algorithms solve hidden shift problems in polynomial time.

Classical query complexity for these problems is exponential.

Exponential separation between quantum and classical approaches is established.

Abstract

Attempts to separate the power of classical and quantum models of computation have a long history. The ultimate goal is to find exponential separations for computational problems. However, such separations do not come a dime a dozen: while there were some early successes in the form of hidden subgroup problems for abelian groups--which generalize Shor's factoring algorithm perhaps most faithfully--only for a handful of non-abelian groups efficient quantum algorithms were found. Recently, problems have gotten increased attention that seek to identify hidden sub-structures of other combinatorial and algebraic objects besides groups. In this paper we provide new examples for exponential separations by considering hidden shift problems that are defined for several classes of highly non-linear Boolean functions. These so-called bent functions arise in cryptography, where their property of having perfectly flat Fourier spectra on the Boolean hypercube gives them resilience against certain types of attack. We present new quantum algorithms that solve the hidden shift problems for several well-known classes of bent functions in polynomial time and with a constant number of queries, while the classical query complexity is shown to be exponential. Our approach uses a technique that exploits the duality between bent functions and their Fourier transforms.