ECC over the ring F3d[ε],ε4=0 by using two methods

Abstract

Let ${\mathbb{F}}_{3^d}$ is the finite field of order $3^d$ with $d$ be a positive integer, we consider $A_4:={\mathbb{F}}_{3^d}[\epsilon ]={\mathbb{F}}_{3^d}[X]/(X^4)$ is a finite quotient ring, where ${\epsilon }^4=0$. In this paper, we will show an example of encryption and decryption. Firstly, we will present the elliptic curve over this ring. In addition, we study the algorithmic properties by proposing effective implementations for representing the elements and the group law. Precisely, we give a numerical example of cryptography (encryption and decryption) by using two methods with a secret key.