Refinement of the four-dimensional GLV method on elliptic curves

Abstract

In this paper we refine the four-dimensional GLV method on elliptic curves presented by Longa and Sica (ASIACRYPT 2012). First we improve the twofold Cornacchia-type algorithm, and show that the improved algorithm possesses a better theoretic upper bound of decomposition coefficients. In particular, our proof is much simpler than Longa and Sica’s. We also apply the twofold Cornacchia-type algorithm to GLS curves over Fp4. Second in the case of curves with j-invariant 0, we compare this improved version with the almost optimal algorithm proposed by Hu, Longa and Xu in 2012 (Designs, Codes and Cryptography). Computational implementations show that they have almost the same performance, which provide further evidence that the improved version is a sufficiently good scalar decomposition approach.