Welcome to the resource topic for 2021/676
Title:
Extending the GLS endomorphism to speed up GHS Weil descent using Magma
Authors: Jesús-Javier Chi-Domínguez, Francisco Rodríguez-Henríquez, Benjamin Smith
Abstract: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.
ePrint: https://eprint.iacr.org/2021/676
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