**On Finding Quantum Multi-collisions**

*Qipeng Liu and Mark Zhandry*

**Abstract: **A $k$-collision for a compressing hash function $H$ is a set of $k$ distinct inputs that all map to the same output. In this work, we show that for any constant $k$, $\Theta\left(N^{\frac{1}{2}(1-\frac{1}{2^k-1})}\right)$ quantum queries are both necessary and sufficient to achieve a $k$-collision with constant probability. This improves on both the best prior upper bound (Hosoyamada et al., ASIACRYPT 2017) and provides the first non-trivial lower bound, completely resolving the problem.

**Category / Keywords: **foundations / quantum cryptography

**Original Publication**** (with minor differences): **IACR-EUROCRYPT-2019

**Date: **received 12 Nov 2018, last revised 26 Feb 2019

**Contact author: **qipengl at cs princeton edu

**Available format(s): **PDF | BibTeX Citation

**Version: **20190227:010300 (All versions of this report)

**Short URL: **ia.cr/2018/1096

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