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Random Self-reducibility of Ideal-SVP via Arakelov Random Walks

  • Koen de BoerEmail author
  • Léo DucasEmail author
  • Alice Pellet-MaryEmail author
  • Benjamin WesolowskiEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12171)

Abstract

Fixing a number field, the space of all ideal lattices, up to isometry, is naturally an abelian group, called the Arakelov class group. This fact, well known to number theorists, has so far not been explicitly used in the literature on lattice-based cryptography. Remarkably, the Arakelov class group is a combination of two groups that have already led to significant cryptanalytic advances: the class group and the unit torus.

In the present article, we show that the Arakelov class group has more to offer. We start with the development of a new versatile tool: we prove that, subject to the Riemann Hypothesis for Hecke L-functions, certain random walks on the Arakelov class group have a rapid mixing property. We then exploit this result to relate the average-case and the worst-case of the Shortest Vector Problem in ideal lattices. Our reduction appears particularly sharp: for Hermite-SVP in ideal lattices of certain cyclotomic number fields, it loses no more than a \(\tilde{O}(\sqrt{n})\) factor on the Hermite approximation factor.

Furthermore, we suggest that this rapid-mixing theorem should find other applications in cryptography and in algorithmic number theory.

Notes

Acknowledgments

The authors are grateful to René Schoof for valuable feedback on a preliminary version of this work. Part of this work was done while the authors were visiting the Simons Institute for the Theory of Computing.

L.D. is supported by the European Union Horizon 2020 Research and Innovation Program Grant 780701 (PROMETHEUS), and by a Fellowship from the Simons Institute. K.d.B. was supported by the ERC Advanced Grant 740972 (ALGSTRONGCRYPTO) and by the European Union Horizon 2020 Research and Innovation Program Grant 780701 (PROMETHEUS). A.P. was supported in part by CyberSecurity Research Flanders with reference number VR20192203 and by the Research Council KU Leuven grant C14/18/067 on Cryptanalysis of post-quantum cryptography. Part of this work was done when A.P. was visiting CWI, under the CWI PhD internship program. Part of this work was done when B.W. was at the Cryptology Group, CWI, Amsterdam, The Netherlands, supported by the ERC Advanced Grant 740972 (ALGSTRONGCRYPTO).

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Copyright information

© International Association for Cryptologic Research 2020

Authors and Affiliations

  1. 1.Cryptology GroupCWIAmsterdamThe Netherlands
  2. 2.imec-COSIC, KU LeuvenLeuvenBelgium
  3. 3.Univ. Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251TalenceFrance
  4. 4.INRIA, IMB, UMR 5251TalenceFrance

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