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SQIsignHD: New Dimensions in Cryptography

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Advances in Cryptology – EUROCRYPT 2024 (EUROCRYPT 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14651))

Abstract

We introduce SQIsignHD, a new post-quantum digital signature scheme inspired by SQIsign. SQIsignHD exploits the recent algorithmic breakthrough underlying the attack on SIDH, which allows to efficiently represent isogenies of arbitrary degrees as components of a higher dimensional isogeny. SQIsignHD overcomes the main drawbacks of SQIsign. First, it scales well to high security levels, since the public parameters for SQIsignHD are easy to generate: the characteristic of the underlying field needs only be of the form \(2^{f}3^{f'}-1\). Second, the signing procedure is simpler and more efficient. Our signing procedure implemented in C runs in 28 ms, which is a significant improvement compared to SQISign. Third, the scheme is easier to analyse, allowing for a much more compelling security reduction. Finally, the signature sizes are even more compact than (the already record-breaking) SQIsign, with compressed signatures as small as 109 bytes for the post-quantum NIST-1 level of security. These advantages may come at the expense of the verification, which now requires the computation of an isogeny in dimension 4, a task whose optimised cost is still uncertain, as it has been the focus of very little attention. Our experimental sagemath implementation of the verification runs in around 600 ms, indicating the potential cryptographic interest of dimension 4 isogenies after optimisations and low level implementation.

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Notes

  1. 1.

    One could improve slightly the scheme by defining \(\ell ^e\)-good integers as integers q such that \(\ell ^e-q=sq'\), with s a smooth integer whose prime factors are all congruent to 1 modulo 4 and \(q'\) is a prime congruent to 1 modulo 4. Indeed, all we really need is that \(\ell ^e-q\) is easy to factor so Cornacchia’s algorithm can be applied efficiently. This alternate definition would improve a bit the search for \(\ell ^e\)-good integer, but we went for the simplest definition.

  2. 2.

    Actually, we will not have exactly \(D_\tau =D_\psi =\ell '^{2f'}\) but \(D_\tau \) and \(D_\psi \) will be divisors of \(\ell '^{2f'}\) close to \(\ell '^{2f'}\). It will be the same for \(D_{\psi '}\) (see Algorithm 1). We assume equality to simplify the exposition.

  3. 3.

    We can compute the norm of \(\sigma '\) on \(E'_2[m]\) (which is \(\deg (\sigma ') \mod m\)) for a bunch of small primes m and apply the Chinese remainder theroem.

  4. 4.

    Which essentially consists in computing a chain of 3-isogenies in dimension 1. The signature also needs to compute similar isogenies, and in this case the low level C implementation only takes 1–2 ms, which shows the potential of improvements of writing a low level implementation of the verification.

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Acknowledgements

We thank Luca De Feo for his advice all along this project and for suggesting the title of this paper. This project was supported by ANR grant CIAO (ANR-19-CE48-0008), PEPR PQ-TLS (the France 2030 program under grant agreement ANR-22-PETQ-0008 PQ-TLS) and the European Research Council under grant No. 101116169 (AGATHA CRYPTY).

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Authors and Affiliations

  1. Univ. Bordeaux, CNRS, INRIA, IMB, UMR 5251, 33400, Talence, France

    Pierrick Dartois & Damien Robert

  2. INRIA, IMB, UMR 5251, 33400, Talence, France

    Pierrick Dartois & Damien Robert

  3. DGA-MI, Bruz, France

    Antonin Leroux

  4. IRMAR - UMR 6625, Université de Rennes, Rennes, France

    Antonin Leroux

  5. ENS de Lyon, CNRS, UMPA, UMR 5669, Lyon, France

    Benjamin Wesolowski

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  1. Pierrick Dartois
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Correspondence to Pierrick Dartois .

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  1. Zama, Paris, France

    Marc Joye

  2. Ruhr University Bochum, Bochum, Germany

    Gregor Leander

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Dartois, P., Leroux, A., Robert, D., Wesolowski, B. (2024). SQIsignHD: New Dimensions in Cryptography. In: Joye, M., Leander, G. (eds) Advances in Cryptology – EUROCRYPT 2024. EUROCRYPT 2024. Lecture Notes in Computer Science, vol 14651. Springer, Cham. https://doi.org/10.1007/978-3-031-58716-0_1

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