Research On Hyperelliptic Curve Cryptosystem Based On FPI

In 1989, Neal Koblitz proposed the theory of hyperelliptic curve cryptosystems (HECC) as a generalization of ECC (Elliptic Curve Cryptosystem). HECC is based on the discrete logarithm problem over hyper elliptic curve cryptosystem. Since HECC has many obvious advantages over the other cryptosystem, the research on the HECC has been drawing more and more researcher's attention. But, it is still in the stage of theoretical research, and many problems will be solved in the future.In this dissertation , the main works of the author is follows:(1)The method FPI (Fast Plaintext Imbedded to the divisor) is improved, which makes the probability of the plaintext not be imbedded into the divisor nearly to zero and its velocity is fast. This method can be applied to the ECC.(2)The author stresses investigation about the scalar multiplication of divisor, which is the core algorithm in the hyperelliptic curve cryptosystem. This chapter brings up a called SWNAF (Sliding Windows Non-Adjacent-Form) method of scalar multiplication whose velocity is fast. At the same time, the method how to calculate the optimum width of the window has been found;(3)A basic hyperelliptic curve cryptosystems integrated encryption system (for short HECIES) is implemented on the Java platform.(4)This paper expands a protocol of key exchange based on Diffie-Hellman,an ElGamal cryptosystems to hyperelliptic curve cryptosystem, and gives an expanding hyper elliptic curve cryptosystem based on DSA digital signature. Their security character bases on discrete logarithm problem over hyper elliptic curve cryptosystem.Finally, HECC's safe strength, complex degree, and realization are compared, HECC has smaller radix field at the same security, then, the direction of application of HECC is sketched, and this paper points out some important problems over HECC that will be resolved in the future.The meaning of this dissertation consists in which supplies reference value for theory research of the HECC, the realization of the HECC system and its standardization work.