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

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

2011-06-29

arxiv.org/abs/1106.5851v1

Miskolc Mathematical Notes, Vol. 10 (2009), No. 2, 129-136

Mathematics

Number Theory

8 pages

Scientific paper

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.

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