Fast elliptic curve point multiplication based on window Non-Adjacent Form method

Abstract

This article presents window Non-Adjacent Form (wNAF) of scalar representation method for developing algorithms of computing scalar multiplication in groups of points on an elliptic curve over finite fields. A new efficient wNAF-based algorithm has been presented. The algorithm was developed based on simple and composite operations with a point and also based on affine and Jacobi coordinates systems taking into account the latest achievements in computing cost reduction. The theorem concerning its computational complexity is formulated and proved for this new algorithm. In the end of this article, an efficiency analysis of the proposed algorithm depending on various parameters is presented, namely, the characteristic of the field over which the curve is defined, the scalar length and the window width of the precomputations and the coordinates of the precomputed points. A comparative analysis of the new algorithm and previously known efficient algorithm is also presented.