# Efficient Arithmetic on Elliptic Curves over Fields of Characteristic Three

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Part of the Lecture Notes in Computer Science book series (LNSC,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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### Cite this paper

Farashahi, R.R., Wu, H., Zhao, CA. (2013). Efficient Arithmetic on Elliptic Curves over Fields of Characteristic Three. In: Knudsen, L.R., Wu, H. (eds) Selected Areas in Cryptography. SAC 2012. Lecture Notes in Computer Science, vol 7707. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35999-6_10