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CSIDH on the Surface

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Post-Quantum Cryptography (PQCrypto 2020)

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

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Abstract

For primes \(p \equiv 3 \bmod 4\), we show that setting up CSIDH on the surface, i.e., using supersingular elliptic curves with endomorphism ring \(\mathbf {Z}[(1 + \sqrt{-p})/2]\), amounts to just a few sign switches in the underlying arithmetic. If \(p \equiv 7 \bmod 8\) then horizontal 2-isogenies can be used to help compute the class group action. The formulas we derive for these 2-isogenies are very efficient (they basically amount to a single exponentiation in \(\mathbf {F}_p\)) and allow for a noticeable speed-up, e.g., our resulting CSURF-512 protocol runs about \(5.68\%\) faster than CSIDH-512. This improvement is completely orthogonal to all previous speed-ups, constant-time measures and construction of cryptographic primitives that have appeared in the literature so far. At the same time, moving to the surface gets rid of the redundant factor \(\mathbf {Z}_3\) of the acting ideal-class group, which is present in the case of CSIDH and offers no extra security.

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Notes

  1. 1.

    Moreover, if \(p \equiv 3 \bmod 4\) then \(x^3 + Ax^2 - x\) is automatically square-free, allowing for a marginally simpler key validation. But this deserves a footnote, at most.

  2. 2.

    It has been pointed out, e.g. in [8, 17], that allowing for the action of \((4, \sqrt{-p} - 1)\) could lead to a minor improvement. See also Remark 2.

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Authors and Affiliations

  1. Cosic, Research Group at Imec and KU Leuven, Leuven, Belgium

    Wouter Castryck & Thomas Decru

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  1. Wouter Castryck
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  2. Thomas Decru
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Correspondence to Thomas Decru .

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Editors and Affiliations

  1. University of Cincinnati, Cincinnati, OH, USA

    Jintai Ding

  2. Inria, Paris, France

    Jean-Pierre Tillich

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Castryck, W., Decru, T. (2020). CSIDH on the Surface. 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_7

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  • DOI: https://doi.org/10.1007/978-3-030-44223-1_7

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