Paper 2013/311

Four-dimensional GLV via the Weil restriction

Aurore Guillevic and Sorina Ionica

Abstract

The Gallant-Lambert-Vanstone (GLV) algorithm uses efficiently computable endomorphisms to accelerate the computation of scalar multiplication of points on an abelian variety. Freeman and Satoh proposed for cryptographic use two families of genus 2 curves defined over ${\text{F}}_p$ which have the property that the corresponding Jacobians are $(2,2)$-isogenous over an extension field to a product of elliptic curves defined over ${\text{F}}_{p^2}$. We exploit the relationship between the endomorphism rings of isogenous abelian varieties to exhibit efficiently computable endomorphisms on both the genus 2 Jacobian and the elliptic curve. This leads to a four dimensional GLV method on Freeman and Satoh's Jacobians and on two new families of elliptic curves defined over ${\text{F}}_{p^2}$.

Note: corrected some typos and added explanations about the endomorphism eigenvalues.