On the elliptic curves y(2)=x(3)-c with embedding degree one

Kirlar B. B.

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, vol.235, no.16, pp.4724-4728, 2011 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 235 Issue: 16
  • Publication Date: 2011
  • Doi Number: 10.1016/j.cam.2010.08.020
  • Journal Name: JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.4724-4728
  • Keywords: Elliptic curves, Pairing-based cryptography, Embedding degree, Characters, Jacobi sums, DIFFIE-HELLMAN, CRYPTOSYSTEMS
  • Middle East Technical University Affiliated: Yes

Abstract

In this paper, we give a family of elliptic curves E in the form y(2) = x(3) - c over the prime field F-p with embedding degree k = 1. This was carried out by computing the explicit formula of the number of points #E(F-p) of the elliptic curve y(2) = x(3) - c. Using this computation, we show that the elliptic curve y(2) = x(3) - 1 over F-p for the primes p of the form 27A(2) + 1 has an embedding degree k = 1. Finally, we give examples of those primes p for which the security level of the pairing-based cryptographic protocols on the curve y(2) = x(3) - 1 over F-p is equivalent to 128-, 192-, or 256-bit AES keys. (C) 2010 Elsevier B.V. All rights reserved.