<abstract xmlns="http://www.w3.org/1999/xhtml">

Let 𝔽<italic>q</italic> be a finite field of <italic>q</italic> elements, where <italic>q</italic> is a power of an odd prime number. In this paper, we study the twisted Edwards curves denoted <italic>E</italic><italic>Ea,d</italic> over the local ring 𝔽<italic>q</italic>[<italic>e</italic>], where <italic>e</italic>2 = 0. In the first time, we study the arithmetic of the ring 𝔽<italic>q</italic>[<italic>e</italic>], <italic>e</italic>2 = 0. After that we define the twisted Edwards curves <italic>E</italic><italic>Ea,d</italic> over this ring and we give essential properties and we define the group <italic>E</italic><italic>Ea,d</italic> , these properties. Precisely, we give a bijection between the groups <italic>E</italic><italic>Ea,d</italic> and <italic>E</italic><italic>Ea,d</italic>0 × F<italic>q</italic>,where <italic>E</italic><italic>Ea,d</italic>0 is the twisted Edwards curves over the finite field 𝔽<italic>q</italic>.
</abstract>

Let 𝔽q be a finite field of q elements, where q is a power of an odd prime number. In this paper, we study the twisted Edwards curves denoted EEa,d over the local ring 𝔽q[e], where e2  = 0. In the first time, we study the arithmetic of the ring 𝔽q[e], e2  = 0. After that we define the twisted Edwards curves EEa,d over this ring and we give essential properties and we define the group EEa,d , these properties. Precisely, we give a bijection between the groups EEa,d and EEa,d0 × Fq,where EEa,d0 is the twisted Edwards curves over the finite field 𝔽q.

Twisted Edwards Curve Over the Ring

Tatra Mountains Mathematical Publications

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"Twisted Edwards Curve Over the Ring"}

Let 𝔽q be a finite field of q elements, where q is a power of an odd prime number. In this paper, we study the twisted Edwards curves denoted EEa,d over the...

Twisted Edwards Curve Over the Ring

Published Online: Mar 07, 2023

Page range: 43 - 50

Received: Nov 01, 2022

DOI: https://doi.org/10.2478/tmmp-2023-0004

Keywords
elliptic curves, twisted Edwards curves, cryptography

© 2023 Mathematical Institute, Slovak Academy of Sciences

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Twisted Edwards Curve Over the Ring

Published Online: Mar 07, 2023

Page range: 43 - 50

Received: Nov 01, 2022

DOI: https://doi.org/10.2478/tmmp-2023-0004

Keywordselliptic curves, twisted Edwards curves, cryptography

© 2023 Mathematical Institute, Slovak Academy of Sciences

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
elliptic curves, twisted Edwards curves, cryptography