Improved Quantum Multicollision-Finding Algorithm
(1811.08097)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 $O(N{(3{l-1}-1) / (2 \cdot 3{l-1})})$, 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 $O(N{(2{l-1}-1) / (2{l}-1)})$, 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 $O\left(c{3/2}_N N{\frac{2{l-1}-1}{ 2{l}-1}}\right)$ for a random function $f\colon X\to Y$ such that $|X| \geq l \cdot |Y| / cN$ for any $1\le cN \in o(N{\frac{1}{2l - 1}})$. With the same query complexity, it also finds a multiclaw for random functions, which is harder to find than a multicollision.
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