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

Kirlar B. B.

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, cilt.235, sa.16, ss.4724-4728, 2011 (SCI-Expanded) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 235 Sayı: 16
  • Basım Tarihi: 2011
  • Doi Numarası: 10.1016/j.cam.2010.08.020
  • Dergi Adı: JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Sayfa Sayıları: ss.4724-4728
  • Anahtar Kelimeler: Elliptic curves, Pairing-based cryptography, Embedding degree, Characters, Jacobi sums, DIFFIE-HELLMAN, CRYPTOSYSTEMS
  • Orta Doğu Teknik Üniversitesi Adresli: Evet

Özet

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.