### Quantum security of subset cover problems

##### Abstract

The subset cover problem for $k \geq 1$ hash functions, which can be seen as an extension of the collision problem, was introduced in 2002 by Reyzin and Reyzin to analyse the security of their hash-function based signature scheme HORS. The security of many hash-based signature schemes relies on this problem or a variant of this problem (e.g. HORS, SPHINCS, SPHINCS+, ...). Recently, Yuan, Tibouchi and Abe (2022) introduced a variant to the subset cover problem, called restricted subset cover, and proposed a quantum algorithm for this problem. In this work, we prove that any quantum algorithm needs to make $\Omega\left(k^{-\frac{2^{k-1}}{2^k-1}}\cdot N^{\frac{2^{k-1}-1}{2^k-1}}\right)$ queries to the underlying hash functions to solve the restricted subset cover problem, which essentially matches the query complexity of the algorithm proposed by Yuan, Tibouchi and Abe. We also analyze the security of the general $(r,k)$-subset cover problem, which is the underlying problem that implies the unforgeability of HORS under a $r$-chosen message attack (for $r \geq 1$). We prove that a generic quantum algorithm needs to make $\Omega\left(N^{k/5}\right)$ queries to the underlying hash functions to find a $(1,k)$-subset cover. We also propose a quantum algorithm that finds a $(r,k)$-subset cover making $O\left(N^{k/(2+2r)}\right)$ queries to the $k$ hash functions.

Available format(s)
Category
Foundations
Publication info
Preprint.
Contact author(s)
samuel bouaziz @ ens-rennes fr
Alex Bredariol-Grilo @ lip6 fr
damien vergnaud @ lip6 fr
History
2022-10-28: approved
See all versions
Short URL
https://ia.cr/2022/1474

CC BY

BibTeX

@misc{cryptoeprint:2022/1474,
author = {Samuel Bouaziz--Ermann and Alex B. Grilo and Damien Vergnaud},
title = {Quantum security of subset cover problems},
howpublished = {Cryptology ePrint Archive, Paper 2022/1474},
year = {2022},
note = {\url{https://eprint.iacr.org/2022/1474}},
url = {https://eprint.iacr.org/2022/1474}
}

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