On the elliptic curves with embedding degree one

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 27 A 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.