Some techniques for faster scalar multiplication on GLS curves

• A more efficient way to implement quadratic extension field arithmetic is given for group arithmetic of GLS curves;
• An explicit and more efficient 4 GLV decomposition method is proposed for GLS curves with special CM;
• A fair comparison between the cost of 4 GLV based scalar multiplication on GLS curve with CM discriminant −8 and that on the Jacobian of its isogenous FKT genus 2 curve is given in our work.

Galbraith, Lin and Scott (EUROCRYPT 2009) [8] constructed a class of elliptic curves over Fp2Fp2 (a.k.a GLS curves) on which the Gallant–Lambert–Vanstone (GLV) method can be employed for fast scalar multiplication. In this work we give an alternative way to implement the quadratic extension field arithmetic for GLS curves, and exploit some explicit decomposition to support 4 dimensional GLV method on GLS curves with special complex multiplication (CM). Such techniques usually bring more computational benefits compared with previous methods. Specially, we give a fair comparison between the cost of 4 GLV based scalar multiplication on GLS curve with CM discriminant −8 and that on the Jacobian of its isogenous FKT genus 2 curve. Our implementations indicate that scalar multiplication on the Jacobian of hyperelliptic curve in Scholten model has competitive efficiency with that on its isogenous GLS curve in twisted Edwards model.