Quantum Multicollision-Finding Algorithm
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 in a quantum setting. The tight bound of quantum query complexity for finding 2-collisions of random functions has been revealed to be \(\varTheta (N^{1/3})\), 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 l-collision of any function that has a domain size l times larger than the codomain size. A rigorous proof is given to guarantee that the expected number of quantum queries is \(O\left( N^{(3^{l-1}-1)/(2 \cdot 3^{l-1})} \right) \) for a small constant l, which matches the tight bound of \(\varTheta (N^{1/3})\) for \(l=2\) and improves the known bounds, say, the above simple bound of \(O(N^{1/2})\).