Quantum Security of Subset Cover Problems

Authors Samuel Bouaziz-Ermann, Alex B. Grilo, Damien Vergnaud

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Samuel Bouaziz-Ermann
  • LIP6, Paris, France
  • Sorbonne Université, Paris, France
  • CNRS, Paris, France
Alex B. Grilo
  • LIP6, Paris, France
  • Sorbonne Université, Paris, France
  • CNRS, paris, France
Damien Vergnaud
  • LIP6, Paris, France
  • Sorbonne Université, Paris, France
  • CNRS, Paris, France


We thanks the anonymous reviewers for their valuable comments that helped improving the quality of this paper.

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Samuel Bouaziz-Ermann, Alex B. Grilo, and Damien Vergnaud. Quantum Security of Subset Cover Problems. In 4th Conference on Information-Theoretic Cryptography (ITC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 267, pp. 9:1-9:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


The subset cover problem for k ≥ 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 Ω((k+1)^{-(2^k)/(2^{k+1}-1})⋅ N^{(2^{k}-1})/(2^{k+1}-1)}) 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 ≥ 1). We prove that a generic quantum algorithm needs to make Ω(N^{k/5}) 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 (N^{k/(2+2r)}) queries to the k hash functions.

Subject Classification

ACM Subject Classification
  • Security and privacy → Cryptography
  • Cryptography
  • Random oracle model
  • Quantum information


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