Paper 2021/1578

On Quantum Query Complexities of Collision-Finding in Non-Uniform Random Functions

Abstract

Collision resistance and collision finding are now extensively exploited in Cryptography, especially in the case of quantum computing. For any function $f:[M]\to [N]$ with $f(x)$ uniformly distributed over $[N]$, Zhandry has shown that the number $\mathrm{\Theta }(N^{1/3})$ of queries is both necessary and sufficient for finding a collision in $f$ with constant probability. However, there is still a gap between the upper and the lower bounds of query complexity in general non-uniform distributions. In this paper, we investigate the quantum query complexity of collision-finding problem with respect to general non-uniform distributions. Inspired by previous work, we pose the concept of collision domain and a new parameter $$ that heavily depends on the underlying non-uniform distribution. We then present a quantum algorithm that uses $$ quantum queries to find a collision for any non-uniform random function. By making a transformation of a problem in non-uniform setting into a problem in uniform setting, we are also able to show that $$ quantum queries are necessary in collision-finding in any non-uniform random function. The upper bound and the lower bound in this work indicates that the proposed algorithm is nearly optimal with query complexity in general non-uniform case.