ECC over the ring \(\mathbb{F}_{3^d}[\varepsilon]\), \(\varepsilon^4=0\) by using two methods (Q2143783)

The study of elliptic curves over finite rings plays an important role in cryptography. In this paper, authors present the elliptic curves over a local ring \(\mathbb{A}_4\) given by \(\mathbb{A}_4:=\mathbb{F}_{3^d}[\varepsilon]\) with its maximal ideal \(I_4=(\varepsilon)\), where \(\varepsilon^4=0\) and \(d\) is a positive integer. They consider the elliptic curve \[ E_{a,b}(\mathbb{A}_4)=\{[X,Y,Z]\in\mathbb{P}_2(\mathbb{A}_4):Y^2Z=X^3+aX^2Z+bZ^3\}, \] where \(a,b\in \mathbb{A}_4\) and the discriminant \(\Delta=-a^2b\) is invertible in the quotient ring \(\mathbb{A}_4\cong \mathbb{F}_{3^d}[X]/(X^4)\). Authors give a numerical example of cryptography (encryption-decryption) with a secret key based on Diffie-Hellman protocol over \(E_{a,b}(\mathbb{A}_4)\).