Claw Finding Algorithms Using Quantum Walk

TL;DR

This paper presents an optimal quantum walk algorithm for solving the claw finding problem, which is significant in cryptography, and extends to finding claws among multiple functions with different domain sizes.

Contribution

The paper introduces a novel optimal quantum walk algorithm for the claw finding problem and generalizes it to multiple functions with varying domain sizes.

Findings

Algorithm achieves optimal query complexity

Generalizes to k functions with different domain sizes

Advances quantum algorithms in cryptography

Abstract

The claw finding problem has been studied in terms of query complexity as one of the problems closely connected to cryptography. For given two functions, f and g, as an oracle which have domains of size N and M (N<=M), respectively, and the same range, the goal of the problem is to find x and y such that f(x)=g(y). This paper describes an optimal algorithm using quantum walk that solves this problem. Our algorithm can be generalized to find a claw of k functions for any constant integer k>1, where the domains of the functions may have different size.