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

2011-06-15
Kirlar, Baris Bulent
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.