On the elliptic curves [formula omitted] with embedding degree one

Abstract

In this paper, we give a family of elliptic curves E in the form y2=x3−c over the prime field Fp with embedding degree k=1. This was carried out by computing the explicit formula of the number of points #E(Fp) of the elliptic curve y2=x3−c. Using this computation, we show that the elliptic curve y2=x3−1 over Fp for the primes p of the form 27A2+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 y2=x3−1 over Fp is equivalent to 128-, 192-, or 256-bit AES keys.