On the elliptic curves y2=x3−cy2=x3−c with embedding degree one

In this paper, we give a family of elliptic curves EE in the form y2=x3−cy2=x3−c over the prime field FpFp with embedding degree k=1k=1. This was carried out by computing the explicit formula of the number of points #E(Fp)#E(Fp) of the elliptic curve y2=x3−cy2=x3−c. Using this computation, we show that the elliptic curve y2=x3−1y2=x3−1 over FpFp for the primes pp of the form 27A2+127A2+1 has an embedding degree k=1k=1. Finally, we give examples of those primes pp for which the security level of the pairing-based cryptographic protocols on the curve y2=x3−1y2=x3−1 over FpFp is equivalent to 128-, 192-, or 256-bit AES keys.