Paper 2021/676

Extending the GLS endomorphism to speed up GHS Weil descent using Magma

Jesús-Javier Chi-Domínguez, Francisco Rodríguez-Henríquez, and Benjamin Smith


Let \(q~=~2^n\), and let \(\mathcal{E} / \mathbb{F}_{q^{\ell}}\) be a generalized Galbraith--Lin--Scott (GLS) binary curve, with $\ell \ge 2$ and \((\ell, n) = 1\). We show that the GLS endomorphism on \(\mathcal{E} / \mathbb{F}_{q^{\ell}}\) induces an efficient endomorphism on the Jacobian \(\mathrm{Jac}_\mathcal{H}(\mathbb{F}_q)\) of the genus-\(g\) hyperelliptic curve \(\mathcal{H}\) corresponding to the image of the GHS Weil-descent attack applied to \(\mathcal{E} / \mathbb{F}_{q^\ell}\), and that this endomorphism yields a factor-$n$ speedup when using standard index-calculus procedures for solving the Discrete Logarithm Problem (DLP) on \(\mathrm{Jac}_\mathcal{H}(\mathbb{F}_q)\). Our analysis is backed up by the explicit computation of a discrete logarithm defined on a prime-order subgroup of a GLS elliptic curve over the field $\mathbb{F}_{2^{5\cdot 31}}$. A Magma implementation of our algorithm finds the aforementioned discrete logarithm in about $1,035$ CPU-days.

Note: Preprint accepted to journal Finite Field and their Applications. Acknowledgment extended

Available format(s)
Public-key cryptography
Publication info
Preprint. Minor revision.
GHS Weil descentextended GLS endomorphismindex-calculus algorithm
Contact author(s)
jesus dominguez @ tii ae
francisco @ cs cinvestav mx
smith @ lix polytechnique fr
2021-06-10: revised
2021-05-25: received
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Creative Commons Attribution


      author = {Jesús-Javier Chi-Domínguez and Francisco Rodríguez-Henríquez and Benjamin Smith},
      title = {Extending the GLS endomorphism to speed up GHS Weil descent using Magma},
      howpublished = {Cryptology ePrint Archive, Paper 2021/676},
      year = {2021},
      note = {\url{}},
      url = {}
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