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**2017/864**

**Title:**

Quantum Multicollision-Finding Algorithm

**Authors:**
Akinori Hosoyamada, Yu Sasaki, Keita Xagawa

**Abstract:**

The current paper presents a new quantum algorithm for finding multicollisions, often denoted by l-collisions, where an l-collision for a function is a set of l distinct inputs having the same output value. Although it is fundamental in cryptography, the problem of finding multicollisions has not received much attention \emph{in a quantum setting}. The tight bound of quantum query complexity for finding 2-collisions of random functions has been revealed to be \Theta(N^{1/3}), where N is the size of a codomain. However, neither the lower nor upper bound is known for l-collisions. The paper first integrates the results from existing research to derive several new observations, e.g.~l-collisions can be generated only with O(N^{1/2}) quantum queries for a small constant l. Then a new quantum algorithm is proposed, which finds an l-collision of any function that has a domain size l times larger than the codomain size. A rigorous proof is given to guarantee that the expected number of quantum queries is O\left( N^{(3^{l-1}-1)/(2 \cdot 3^{l-1})} \right) for a small constant l, which matches the tight bound of \Theta(N^{1/3}) for l=2 and improves the known bounds, say, the above simple bound of O(N^{1/2}).

**ePrint:**
https://eprint.iacr.org/2017/864

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