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International Conference on Post-Quantum Cryptography

PQCrypto 2020: Post-Quantum Cryptography pp 53–71Cite as

Isochronous Gaussian Sampling: From Inception to Implementation

With Applications to the Falcon Signature Scheme

Part of the Lecture Notes in Computer Science book series (LNSC,volume 12100)

Abstract

Gaussian sampling over the integers is a crucial tool in lattice-based cryptography, but has proven over the recent years to be surprisingly challenging to perform in a generic, efficient and provable secure manner. In this work, we present a modular framework for generating discrete Gaussians with arbitrary center and standard deviation. Our framework is extremely simple, and it is precisely this simplicity that allowed us to make it easy to implement, provably secure, portable, efficient, and provably resistant against timing attacks. Our sampler is a good candidate for any trapdoor sampling and it is actually the one that has been recently implemented in the Falcon signature scheme. Our second contribution aims at systematizing the detection of implementation errors in Gaussian samplers. We provide a statistical testing suite for discrete Gaussians called SAGA (Statistically Acceptable GAussian). In a nutshell, our two contributions take a step towards trustable and robust Gaussian sampling real-world implementations.

Keywords

  • Lattice based cryptography
  • Gaussian sampling
  • Isochrony
  • Statistical verification tools

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Fig. 1.

Notes

  1. 1.

    https://media.ccc.de/v/27c3-4087-en-console_hacking_2010.

  2. 2.

    We note that one could use [32] to speed up our base sampler; however this results in a huge code size (more than 50 kB). Since the running time of the base sampler was not a bottleneck for the usecase we considered, we instead relied on a straightforward, slightly less efficient CDT-based method.

  3. 3.

    We are thankful to Thomas Pornin for bringing up this fact.

  4. 4.

    Type I and type II errors are, respectively, rejection of a true null hypothesis and the non-rejection of a false null hypothesis.

  5. 5.

    Compilers may alter the design, thus one should always verify the design post-compilation.

  6. 6.

    The constant-time sampler in [59] may still reveal \(\sigma \).

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Acknowledgements

We thank Léo Ducas for helpful suggestions. We also thank Thomas Pornin and Mehdi Tibouchi for useful discussions. The first and second authors were supported by the project PQ Cybersecurity (Innovate UK research grant 104423). The third and fourth authors were supported by BPI-France in the context of the national project RISQ (P141580), and by the European Union PROMETHEUS project (Horizon 2020 Research and Innovation Program, grant 780701). The fourth author was also supported by ANRT under the program CIFRE N2016/1583.

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Authors and Affiliations

  1. PQShield, Oxford, UK

    James Howe & Thomas Prest

  2. Thales, Gennevilliers, France

    Thomas Ricosset & Mélissa Rossi

  3. ANSSI, Paris, France

    Mélissa Rossi

  4. École normale supérieure, CNRS, PSL University, Paris, France

    Mélissa Rossi

  5. Inria, Paris, France

    Mélissa Rossi

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  1. James Howe
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Correspondence to Mélissa Rossi .

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Editors and Affiliations

  1. University of Cincinnati, Cincinnati, OH, USA

    Dr. Jintai Ding

  2. Inria, Paris, France

    Jean-Pierre Tillich

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Howe, J., Prest, T., Ricosset, T., Rossi, M. (2020). Isochronous Gaussian Sampling: From Inception to Implementation. In: Ding, J., Tillich, JP. (eds) Post-Quantum Cryptography. PQCrypto 2020. Lecture Notes in Computer Science(), vol 12100. Springer, Cham. https://doi.org/10.1007/978-3-030-44223-1_4

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