The Group Structure of Bachet Elliptic Curves over Finite Fields F_{p}

TL;DR

This paper investigates the group structure of Bachet elliptic curves over finite fields, revealing that their groups are either cyclic of order p+1 or a product of two cyclic groups, depending on the prime's congruence class.

Contribution

It classifies the possible group structures of Bachet elliptic curves over finite fields based on the prime modulus, refining previous results with specific conditions.

Findings

Group E(F_p) is cyclic of order p+1 for p ≡ 5 (mod 6).

Group E(F_p) is a product of two cyclic groups for p ≡ 1 (mod 6).

If E(F_p) ≅ Z_n × Z_n, then p ≡ 7 (mod 12) and p = n^2 ± n + 1.

Abstract

Bachet elliptic curves are the curves y^2=x^3+a^3 and in this work the group structure E(F_{p}) of these curves over finite fields F_{p} is considered. It is shown that there are two possible structures E(F_{p}){\cong}C_{p+1} or E(F_{p}){\cong}C_{n}{\times}C_{nm}, for m,n{\in}{\mathbb{N}}, according to p{\equiv}5 (mod6) and p{\equiv}1 (mod6), respectively. A result of Washington is restated in a more specific way saying that if E(F_{p}){\cong}Z_{n}{\times}Z_{n}, then p{\equiv}7 (mod12) and p=n^2{\mp}n+1.