SAC 2012: Selected Areas in Cryptography pp 135-148

# Efficient Arithmetic on Elliptic Curves over Fields of Characteristic Three

• Reza R. Farashahi
• Hongfeng Wu
• Chang-An Zhao
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7707)

## 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.

## Keywords

Elliptic curve Hessian curve scalar multiplication cryptography

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## Authors and Affiliations

• Reza R. Farashahi
• 1
• 2
• Hongfeng Wu
• 3
• Chang-An Zhao
• 4
1. 1.Dept. of ComputingMacquarie UniversitySydneyAustralia
2. 2.Dept. of Mathematical SciencesIsfahan University of TechnologyIsfahanIran
3. 3.College of SciencesNorth China University of TechnologyBeijingChina
4. 4.School of Computer Science and Educational SoftwareGuangzhou UniversityGuangzhouChina