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-l) -collisions, and it achieves the average quantum query complexity of (Formula presented), 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 (Formula presented), which improves the previous bound for all l ≥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 (Formula presented) for a random function (Formula presented) such that (Formula presented) for any (Formula presented). With the same query complexity, it also finds a multiclaw for random functions, which is harder to find than a multicollision.