Research And Implementation Of Elliptic Curve Cryptograghy

With the development and application of information technology, the problem of information security becomes more and more important. RS A cryptosystem, a public-key cryptosystem being used widely today, seems to have difficulty in meeting the users' need of higher security. So far, the Elliptic Curve Cryptosystem (ECC) provides the highest strength-per-bit of any cryptosystem known.In addition to its high security, ECC also has many other merits over other public-key cryptosystems such as less computation overheads,shorter key size, considerable bandwidth savings,and so on. All of these merits have made it the best public-key cryptosystem that is suitable for current use in future.This paper first analyses and summarizes the ststus quo and evolution trend of encryption, and introduces in detail the development of private-key cryptosystem and public-key cryptosystem, and provides the brief analysis of a few typical schemes. Second, the principle of ECC is discussed, including the math foundation of ECC, basic conception of elliptic curve, constructing idea of ECC, operation on the elliptic curve and so on. Meanwhile, the security and efficiency of ECC are analyzed and thus three conditions required by ECC are given. Third, an efficient algorithm to generate the secure elliptic curves which bases on the CM algorithm is presented. According to this algorithm, the rank of elliptic curve constructed is the product of two large prime numbers. And its correctness is proved in theory. Fourth, some key algorithms in ECC are implemented, including algorithm of generating the elliptic curve,algorithm of computing the kP of the ECC, algorithm of detecting prime number and lgorithm of operating between the big integers. Fifth, the variation of Elgamal signature scheme based on ECC is presented and its validity is proved.And the new scheme transplates the typical Elgamal signature scheme int ECC. At last, the evolution trend and reseach direction are discussed.