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 hashfunction based signature scheme HORS. The security of many hashbased signature schemes relies on this problem or a variant of this problem (e.g. HORS, SPHINCS, SPHINCS+, $\dots$). 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 $\Omega\left((k+1)^{\frac{2^{k}}{2^{k+1}1}}\cdot N^{\frac{2^{k}1}{2^{k+1}1}}\right)$ queries to the underlying hash functions with codomain size $N$ 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2022
 DOI:
 10.48550/arXiv.2210.15396
 arXiv:
 arXiv:2210.15396
 Bibcode:
 2022arXiv221015396B
 Keywords:

 Quantum Physics;
 Computer Science  Cryptography and Security