Implementation And Research On Choosing Secure Elliptic Curves

Elliptic Curve Cryptosystem (ECC) is the focus in cryptology, with the reasons of its short key and little resource requirement. It seemed that it's going to take the place of RSA in public-key cryptosystem. While in the implementation of ECC, the first thing we must do is to choose a secure curve. Only with the precondition that the curve we chose is secure, we can assure that the cryptosystem based on it may be secure, too.In this paper, we introduce Elliptic Curve Cryptosystem and some of its mathematic background first; and then some realizations of the encrypted scheme and signature scheme based on ECC as an example followed. Next, we discuss the algorithms of secure curve choosing in two classes named means based on finite fields and means based on p-adic circles. The algorithms author have implemented of the two class means is the Schoof's Algorithm, the SEA Algorithm, the Satoh's Algorithm and the AGM Algorithm; with each algorithm, we'll introduce its fundamental, software implement flow chart and the innovation points of the implementation separately. At last, we discuss some of the problems in implementation and operation and give the analyses and improvements, too.