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This paper presents new explicit formulas for the point doubling, tripling and addition for Hessian curves and their equivalent Weierstrass 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 formulas 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, the authors present several examples of ordinary elliptic curves in characteristic three for high security levels.
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