Improved Quantum Multicollision-Finding Algorithm

Abstract: The current paper improves the number of queries of the previous quantum multi-collision finding algorithms presented by Hosoyamada et al. at Asiacrypt 2017. Let an $l$-collision be a tuple of $l$ distinct inputs that result in the same output of a target function. In cryptology, it is important to study how many queries are required to find $l$-collisions for random functions of which domains are larger than ranges. The previous algorithm finds an $l$-collision for a random function by recursively calling the algorithm for finding $(l-1)$-collisions, and it achieves the average quantum query complexity of $$O(N^{{(3^{l-1}-1)} / (2 \cdot 3<sup>{l-1})})$$, where $N$ is the range size of target functions. The new algorithm removes the redundancy of the previous recursive algorithm so that different recursive calls can share a part of computations. The new algorithm finds an $l$-collision for random functions with the average quantum query complexity of $$O(N^{{(2^{l-1}-1)} / (2<sup>{l}-1)})$$, which improves the previous bound for all $l\ge 3$ (the new and previous algorithms achieve the optimal bound for $l=2$). More generally, the new algorithm achieves the average quantum query complexity of $$O\left(c^{3/2}_N N^{{\frac{2^{l-1}-1}{} 2<sup>{l}-1}}\right)$$ for a random function $f\colon X\to Y$ such that $|X| \geq l \cdot |Y| / c_N$ for any $$1\le c_N \in o(N^{{\frac{1}{2^l} - 1}})$$. With the same query complexity, it also finds a multiclaw for random functions, which is harder to find than a multicollision.