Paper 2017/864

Quantum Multicollision-Finding Algorithm

Akinori Hosoyamada, Yu Sasaki, and Keita Xagawa

Abstract

The current paper presents a new quantum algorithm for finding multicollisions, often denoted by $l$-collisions, where an $l$-collision for a function is a set of $l$ distinct inputs having the same output value. Although it is fundamental in cryptography, the problem of finding multicollisions has not received much attention \emph{in a quantum setting}. The tight bound of quantum query complexity for finding $2$-collisions of random functions has been revealed to be $\mathrm{\Theta }(N^{1/3})$, where $N$ is the size of a codomain. However, neither the lower nor upper bound is known for $l$-collisions. The paper first integrates the results from existing research to derive several new observations, e.g.~$l$-collisions can be generated only with $O(N^{1/2})$ quantum queries for a small constant $l$. Then a new quantum algorithm is proposed, which finds an $$-collision of any function that has a domain size $$ times larger than the codomain size. A rigorous proof is given to guarantee that the expected number of quantum queries is $$ for a small constant $$, which matches the tight bound of $$ for $$ and improves the known bounds, say, the above simple bound of $$.