The Research Of Elliptic Curve Cryptosystem Over Optimal Extension Fields

With the development and application of information technology, the information security becomes more and more important. RSA cryptosystem, a public-key cryptosystem being used widely today, seems to have difficulty in meeting the users' need of higher security. Elliptic curve cryptosystems are the basis for a relatively new class of public-key schemes. So far, the Elliptic Curve Cryptosystem (ECC) provides highest strength-per-bit of any cryptosystem known. Comparing with other public-key crypto schemes, the Elliptic Curve Cryptosystem (ECC) has many merits such as high security, shorter key size, less computation overheads, considerable bandwidth saving and so on. It can achieve high security with the least memory, the quickest speed and minimum resource consume, which causes international extensive attention. The research indicates that 160-bit key in ECC can get the same intensity of security as 1024-bit key in RSA or DSA but strong repellence on attack. All these make ECC be an ideal cryptosystem to the systems with limit resources (including processing power, time and storage space and so on).The elliptic curve encrypt scheme is an unsymmetry public key cryptosystem based on elliptic curve discrete logarithm. To ECC, all these including the t-ype of the finite fields, the algorithms for implementing the finite field arithmetic, the type of elliptic curve, a- lgorithms for implementing the elliptic curve group operation are necessary process and many of these selections have a major impact on overall performance. Different finite fields (including prime field, binary field and optimal extension field) have been analyzed and compared. The paper chooses OEF as the basis field of ECC.The paper first summarizes the public key cryptosystem and introduces in detail the status quo and evolution trend of elliptic curve cryptosystem. Second, the principle of ECC is discussed, including the math foundation of ECC, basic conception of elliptic curve, constructing idea of ECC. Meanwhile, the performance of ECC is analyzed and the applications of ECC in smart cards in which the resources are limit are introduced according to the merits of ECC .Third, the basic operation algorithms are introduced and a function library which can be used in any elliptic cryptosystem is constructed including the functions of implementing arithmetic operation in finite fields, the functions of implementing producing elliptic curve and embedding data into elliptic curve and the functions of implementing elliptic curve group operation. At last an improved scalar multiplication arithmetic is brought forward. Accordingly the efficiency of operation is improved. Meanwhile the evolution trend and research direction of elliptic curve cryptosystem are discussed.