Four-dimensional GLV via the Weil restriction

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 double-struck Fp which have the property that the corresponding Jacobians are (2,2)-isogenous over an extension field to a product of elliptic curves defined over double-struck Fp2. 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 double-struck Fp2. © 2013 Springer-Verlag.