Faster Scalar Multiplication on Ordinary Weierstrass Elliptic Curves over Fields of Characteristic Three

This paper proposes new explicit formulae for the point doubling, tripling and addition on ordinary Weierstrass elliptic curves with a point of order 3 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. We also provide the unified and complete group laws. Finally, we present several examples of ordinary elliptic curves in characteristic three for high security levels.