Efficient arithmetic on elliptic curves over fields of characteristic three

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. © 2013 Springer-Verlag Berlin Heidelberg.