Improved Quantum Multicollision-Finding Algorithm

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. The previous algorithm finds $l$-collisions by recursively calling the algorithm for finding $(l-1)$-collisions, and it achieves the query complexity of $O(N^{(3^{l-1}-1) / (2 \cdot 3^{l-1})})$. The new algorithm removes the redundancy of the previous recursive algorithm so that \tnote{different recursive calls} can share a part of computations. The new algorithm achieves the query complexity of $\tilde{O}(N^{(2^{l-1}-1) / (2^{l}-1)})$. Moreover, it finds multiclaws for random functions, which are harder to find than multicollisions.