Work

Finding a Basis of the Set of Isogenies between Two Ordinary Elliptic Curves Over a Finite Field

Year: 2024

Type: article

Abstract: Let E, E ′ be two isogenous ordinary elliptic curves defined over a finite field. The set Hom( E, E ′) of isogenies from E to E ′ is a free ℤ-module of rank two. In this paper, we give an algorithm for finding an explicit ℤ-basis of Hom( E, E ′). Firstly, we find a special pair of isogenies using the structure of ordinary isogeny graphs, called isogeny volcanoes. Secondly, we construct explicit generators of Hom( E, E ′) as a ℤ-module from the pair of isogenies. Finally, we remove the ℤ-linear dependency of the generators to obtain a ℤ-basis of Hom( E,E ′).

Source: ACM communications in computer algebra

DOI: 10.1145/3727336.3727337

Authors: Akira Katayama, Masaya Yasuda

Institution: Rikkyo University

Language: English

Topic: Cryptography and Residue Arithmetic

Subfield: Information Systems

Field: Computer Science

Domain: Physical Sciences

Open Access status: bronze

Funders: Japan Society for the Promotion of Science, Core Research for Evolutional Science and Technology

Awards: Investigation and exploration of mathematical solutions for isogeny problems with auxiliary information, G8972333614