A Tight Quantum Algorithm for Multiple Collision Search
Résumé
Searching for collisions in random functions is a fundamental computational problem, with many applications in symmetric and asymmetric cryptanalysis. When one searches for a single collision, the known quantum algorithms match the query lower bound. This is not the case for the problem of finding multiple collisions, despite its regular appearance as a sub-component in sieving-type algorithms. At EUROCRYPT 2019, Liu and Zhandry gave a query lower bound Ω(2 m/3+2k/3) for finding 2^k collisions in a random function with m bit output. At EUROCRYPT 2023, Bonnetain et al. gave a quantum algorithm matching this bound for a large range of m and k, but not all admissible values. Like many previous collision-finding algorithms, theirs is based on the MNRS quantum walk framework, but it chains the walks by reusing the state after outputting a collision. In this paper, we give a new algorithm that tackles the remaining nonoptimal range, closing the problem. Our algorithm is tight (up to a polynomial factor) in queries, and also in time under a quantum RAM assumption. The idea is to extend the chained walk to a regime in which several collisions are returned at each step, and the "walks" themselves only perform a single diffusion layer.
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