A quantum lower bound for the collision problem

TL;DR

This paper improves the understanding of quantum lower bounds for the collision problem by extending Shi's original bound to cases with smaller range sizes, providing a more general proof.

Contribution

The authors modify Shi's proof to remove the restriction on the range size, establishing a more general quantum lower bound for the collision problem.

Findings

Extended Shi's quantum lower bound to smaller range sizes

Provided a more general proof technique for the collision problem

Confirmed the tightness of the bound in broader cases

Abstract

We extend Shi's 2002 quantum lower bound for collision in $r$-to-one functions with $n$ inputs. Shi's bound of $\Omega((n/r)^{1/3})$ is tight, but his proof applies only in the case where the range has size at least $3n/2$. We give a modified version of Shi's argument which removes this restriction.