THE GROUP STRUCTURE OF BACHET ELLIPTIC CURVES OVER FINITE FIELDS F-p


Ikikardes N. Y. , DEMİRCİ M., Soydan G., CANGÜL İ. N.

MISKOLC MATHEMATICAL NOTES, vol.10, no.2, pp.129-136, 2009 (Peer-Reviewed Journal) identifier

  • Publication Type: Article / Article
  • Volume: 10 Issue: 2
  • Publication Date: 2009
  • Doi Number: 10.18514/mmn.2009.182
  • Journal Name: MISKOLC MATHEMATICAL NOTES
  • Journal Indexes: Science Citation Index Expanded, Scopus, Academic Search Premier, zbMATH
  • Page Numbers: pp.129-136
  • Keywords: elliptic curves over finite fields, rational points

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) congruent to Cp+1 or E(F-p) congruent to C-n x C-nm, for m, n is an element of N; according to p equivalent to 5 (mod 6) and p equivalent to 1 (mod 6), respectively. A result of Washington is restated in a more specific way saying that if E(F-p) congruent to Z(n) x Z(n) then p equivalent to 7 (mod 12) p = n(2) -/+ n + 1.