WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 21, 2022
El Gamal Cryptosystem on a Montgomery
Curves Over Non Local Ring
Authors: , , ,
Abstract: Let $$\mathbb{F}_{q}$$ be the finite field of $$q$$ elements, where $$q$$ is a prime power. In this paper, we study the
Montgomery curves over the ring $$\frac{\mathbb{F}_{q} \left[ X\right]}{\left\langle X^{2}-X \right\rangle}$$, denoted by $$M_{A,B}\left( \frac{\mathbb{F}_{q} \left[ X\right]}{\left\langle X^{2}-X \right\rangle} \right); (A,B)\in \left( \frac{\mathbb{F}_{q} \left[ X\right]}{\left\langle X^{2}-X \right\rangle} \right)^{2}$$. Using the Montgomery equation, we define the Montgomery curves $$M_{A,B}\left( \frac{\mathbb{F}_{q} \left[ X\right]}{\left\langle X^{2}-X \right\rangle} \right)$$ and we give a bijection
between this curve and product of two Montgomery curves defined on $$\mathbb{F}_{q}$$. Furthermore, we study the
addition law of Montgomery curves over the ring $$\frac{\mathbb{F}_{q} \left[ X\right]}{\left\langle X^{2}-X \right\rangle}$$. We close this paper by introducing a public key
cryptosystem which is a variant of the ElGamal cryptosystem on a Montgomery curves over the same ring.
Search Articles
Pages: 85-89
DOI: 10.37394/23206.2022.21.13