Efficient Arithmetic on Elliptic Curves over Fields of Characteristic Three

Abstract

Abstract. This paper presents new explicit formulae for the point doubling, tripling and addition for ordinary Weierstraß elliptic curves with a point of order 3 and their equivalent Hessian curves over finite fields of characteristic three. The cost of basic point operations is lower than that of all previously proposed ones. The new doubling, mixed addition and tripling formulae in projective coordinates require 3M + 2C, 8M + 1C + 1D and 4M + 4C + 1D respectively, where M, C and D is the cost of a field multiplication, a cubing and a multiplication by a constant. Finally, we present several examples of ordinary elliptic curves in characteristic three for high security levels.

Point Multiplication using Integer Sub-Decomposition for Elliptic Curve Cryptography

“…Since then, the elliptic curve cryptosystem has been invested in the field of public key cryptography because of its low bandwidth and small space storage requirements [3]- [6]. In the main operation of public key schemes, scalar multiplication can be accomplished using elliptic curves [7].…”

Isogenies on twisted Hessian curves

“…Twisted Hessian curves were used in [10] to provide a complete unified addition formula and improve efficiency for point doubling and tripling over fields of arbitrary characteristic. Other works that optimized arithmetic on (twisted) Hessian curves include [25][26][27]. Definition 1.…”

New Parallel Approaches for Scalar Multiplication in Elliptic Curve over Fields of Small Characteristic

“…• The scaled projective coordinate system (X, Y, T ) was proposed in [4]. The affine coordinates (x, y) of a point P can be derived from its scaled projective coordinates (X, Y, T ) as x = X/bT and y = Y /bT .…”

“…The complexities of the curve operations in each coordinate system are given in Table 2. Farashahi et al showed in [4] that the most advantageous coordinate system is the scaled projective system for curves admitting a point of order three. Smart and Westwood proved in [15] that these curves are isomorphic to curves given by a Weierstrass equation of the form…”

“…We now replace y P and x P by their respective expression in terms of x i and y i : 6 and 4 which appear on each side of the equation cancel. We then move the term 3 to the left side of the equation and this gives:…”