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DOI: 10.4230/LIPIcs.MFCS.2023.23

Cryptanalysis of a Generalized Subset-Sum Pseudorandom Generator

Authors: Charles Bouillaguet, Florette Martinez, and Damien Vergnaud

Published in: LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

Abstract

We present attacks on a generalized subset-sum pseudorandom generator, which was proposed by von zur Gathen and Shparlinski in 2004. Our attacks rely on a sub-quadratic algorithm for solving a vectorial variant of the 3SUM problem, which is of independent interest. The attacks presented have complexities well below the brute-force attack, making the generators vulnerable. We provide a thorough analysis of the attacks and their complexities and demonstrate their practicality through implementations and experiments.

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Charles Bouillaguet, Florette Martinez, and Damien Vergnaud. Cryptanalysis of a Generalized Subset-Sum Pseudorandom Generator. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 23:1-23:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bouillaguet_et_al:LIPIcs.MFCS.2023.23,
  author =	{Bouillaguet, Charles and Martinez, Florette and Vergnaud, Damien},
  title =	{{Cryptanalysis of a Generalized Subset-Sum Pseudorandom Generator}},
  booktitle =	{48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)},
  pages =	{23:1--23:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-292-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{272},
  editor =	{Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.23},
  URN =		{urn:nbn:de:0030-drops-185579},
  doi =		{10.4230/LIPIcs.MFCS.2023.23},
  annote =	{Keywords: Cryptography, pseudo-random generator, subset-sum problem, 3SUM problem, cryptanalysis}
}

Document

DOI: 10.4230/LIPIcs.ITC.2023.9

Quantum Security of Subset Cover Problems

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

Published in: LIPIcs, Volume 267, 4th Conference on Information-Theoretic Cryptography (ITC 2023)

Abstract

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.

Cite as

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)

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@InProceedings{bouazizermann_et_al:LIPIcs.ITC.2023.9,
  author =	{Bouaziz-Ermann, Samuel and Grilo, Alex B. and Vergnaud, Damien},
  title =	{{Quantum Security of Subset Cover Problems}},
  booktitle =	{4th Conference on Information-Theoretic Cryptography (ITC 2023)},
  pages =	{9:1--9:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-271-6},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{267},
  editor =	{Chung, Kai-Min},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITC.2023.9},
  URN =		{urn:nbn:de:0030-drops-183378},
  doi =		{10.4230/LIPIcs.ITC.2023.9},
  annote =	{Keywords: Cryptography, Random oracle model, Quantum information}
}

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Cryptanalysis of a Generalized Subset-Sum Pseudorandom Generator

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Quantum Security of Subset Cover Problems

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