Efficient Arithmetic on Hessian Curves

Abstract

This paper considers a generalized form for Hessian curves. The family of generalized Hessian curves covers more isomorphism classes of elliptic curves. Over a finite field \(\mathbb{F}_q\) , it is shown to be equivalent to the family of elliptic curves with a torsion subgroup isomorphic to ℤ/3ℤ.

This paper provides efficient unified addition formulas for generalized Hessian curves. The formulas even feature completeness for suitably chosen parameters.

This paper also presents extremely fast addition formulas for generalized binary Hessian curves. The fastest projective addition formulas require 9M + 3S, where M is the cost of a field multiplication and S is the cost of a field squaring. Moreover, very fast differential addition and doubling formulas are provided that need only 5M + 4S when the curve is chosen with small curve parameters.