Implementation And Research On Secure Elliptic Curves

Elliptic curve cryptosystem is one of the most important cryptosystems, which is obtained by substituting finite group of elliptic curve on finite field for finite group of cryptosystem based on discrete logarithm problem. How to choose secure elliptic curves and fast implementation of Point multiplication have been always two direction of ECC's research. Fast generation of secure elliptic curves is precondition in the ECC's research and application. Only with the precondition that the curve we chose is secure, we can assure that the cryptosystem based on it may be secure, too.This paper first analyses and summarizes the ststus quo and evolution trend of encryption, some common used cryptograph are introduced, including the algorithms used in symmetric cryptosystem and asymmetirc cryptosystem. We describe the theory of each algorithms and compare the elliptic curve cryptosystem with the other two asymmetric cryptosystems to show the advantages of this algorithm. Second, the principle of ECC is discussed, including the math foundation of ECC, basic conception of elliptic curves, constructiong idea of ECC, operation on the elliptic curve and so on. Third, the current attacks of ECC were analyzed deeply, and an algorithm based on limited prime number field was constructed. We analyzed its realizability in theory, and implement it by using certain function of MIRACL software package. Latter half in this paper, the implementation model of a simple elliptic curve encryption system (SECES) which based on GF(p) has been introduced. The paper also put a deep analysis on the algorithm of point addition and point multiplication.