Paper 2018/1122

Improved Quantum Multicollision-Finding Algorithm

Akinori Hosoyamada, Yu Sasaki, Seiichiro Tani, and Keita Xagawa


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 $|X| \geq l \cdot |Y| / c_N$ for any $1\le c_N \in o(N^{\frac{1}{2^l - 1}})$. 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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Publication info
Published elsewhere. PQCrypto 2019
post-quantum cryptographyquantum algorithmmulticlawmulticollision
Contact author(s)
hosoyamada akinori @ lab ntt co jp
2019-01-28: last of 6 revisions
2018-11-20: received
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      author = {Akinori Hosoyamada and Yu Sasaki and Seiichiro Tani and Keita Xagawa},
      title = {Improved Quantum Multicollision-Finding Algorithm},
      howpublished = {Cryptology ePrint Archive, Paper 2018/1122},
      year = {2018},
      note = {\url{}},
      url = {}
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