On the elliptic curves y2=x3-c with embedding degree one

Authors:
Barış Bülent Kırlar
Affiliations:
Institute of Applied Mathematics, Middle East Technical University, 06531, Ankara, Turkey and Department of Mathematics, Süleyman Demirel University, 32260, Isparta, Turkey
Venue:
Journal of Computational and Applied Mathematics
Year:
2011

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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.