A hybrid approach for elliptic scalar multiplication

Abstract

In this work, we proposed a new merged approach called the GLV-ISD scalar multiplication. This hybrid approach is based on the GLV (Gallant-Lambert-Vanstone) approach and the proposed integer sub-decomposition (ISD) approach to compute any multiple kP of a point P of order n lying on an elliptic curve E over prime field Fp. This approach consists of two stages to calculate kP, the GLV stage and secondly the ISD stage. The basic idea of the merged GLV-ISD method is to decompose the multiplier k ∈ [1, n] into the values k1 and k2, where k1, k2 < k. The application of GLV-ISD approach depends on the returned values k1 and k2 lying inside/outside the range ±n on the interval [1, n − 1]. This new insight, namely, GLV-ISD approach leads to improvement on scalar multiplication computation of elliptic curve cryptography through increase the percentage of successful computation of kP compared with the original GLV approach. Several theoretical aspects were proven and shown in this paper which bridge the gaps that we have encountered in the existing GLV method. The proposed ISD method is regarded as a complement of the existing GLV method and combining the GLV and the ISD method in one go will lead to an efficient implementation of scalar computation of kP. With the adaptation of ISD method in the existing system has brought in several theoretical extension which are discussed and proved in this paper.