Improved Point 5P Formula for Twisted Edwards Curve in Projective Coordinate Over Prime Field

Abstract

Computer security is essential in today’s world of technological advancement. Providing high security for constrained devices with limited capacity, memory, and power consumption has become one of the prominent challenges in computer security. Elliptic curve cryptography (ECC) is a powerful cryptography approach that generates security keys using the mathematics of elliptic curve (EC) to achieve an equivalent level of security with smaller key sizes. The efficiency of ECC depends on an EC operation known as scalar multiplication (SM). SM in Affine coordinate that has an inversion can be accelerated by employing projective coordinates. At the point arithmetic level, this paper improved the precomputed point by proposing a new point 5P formula in projective coordinates using EN for the Twisted Edwards curve over prime field. By comparison to the Twisted Edwards curve with temporary variables method, the algorithm using the proposed 5P formula saved 3M with 16.3% cost reduction. The proposed method also saved 3.2M, 7.2M, and 4.2M with 17.2%, 31.9% and 21.4% cost reduction, when compared to 5P using the Weierstrass curve with point addition, Edwards curve with points addition and doubling, and Edwards curve with computable and point addition methods respectively. The proposed point 5P can be applied to improve the ECC of SM algorithm.