The Research On Counting The Number Of Points On An Elliptic Curve

In 1985,Neal Koblitz and Victor Miller independently proposed the elliptic curve cryptosystems(ECC).after that, lots of paper about the secure and implementation of elliptic curve public key cryptography was published. Among the public key cryptosystems, Elliptic curve cryptography became the hotspot with its own advantage. Compare to the other public key cryptography, how to choose secure parameter of an Elliptic curve used for encrypt and decrypt is still a tough work. We need to know the points of a given elliptic curve which is important to judge whether the given curve is secure. So, in this paper I take Elliptic curve cryptography as my research object. It include to introduce related mathematic conception,encrypt and decrypt algorithms and the algorithm to counting the points of a given elliptic curve.The main research of this paper as follows:This paper is mainly discuss the Schoof algorithm and SEA algorithm, and take the Kangaroo algorithm and BSGS algorithm as the accelerate method to Schoof algorithm. The experiment shows that the acceletate method can short the running time of Schoof algorithm greatly. To SEA algorithm, I discussed the two implement method of it, when only use Elkies prime, the running time of SEA algorithm is shorter.