Researches On Elliptic Curve Cryptosystems And The Related Algorithms

In this dissertation the elliptic curve cryptosystems and the related algorithms are investigated. With the emphasis of the results obtained by the author, the basic scheme of the elliptic curve cryptosystems theory is sketched. It is consists of six chapters. In chapter one, some introductive materials are presented, including the motivations and the developments of the elliptic curve cryptosystems, the inten- tions of the research work, the main contents of the paper and the list of the results obtained by the author. In chapter two, some necessary and basic materials of the elliptic curves theory are introduced. In chapter three, the elliptic curve discrete logarithm problems are discussed. In this chapter, with the approach of the prima- ry algorithms for the discrete logarithm problems, some general and special algo- rithms for the elliptic curve discrete logarithm problems are investigated. In chap- ter four, the fast implementation problems of the elliptic curve cryptosysterns are studied. In this chapter, the problems are firstly reduced to the computations of the scalar multiplication of the elliptic curves, by investigating the structures of all the convention elliptic cuves crvptosystems. Secondly, all the algorithms for the fast computations of the scalar multiplication of the elliptic curves, especially the method under the expansion of Frobenius, are studied intensively. Lastly, an im- provement of an elliptic curve cryptosysterns over the ring Zn is introduced. In chapter five, the SEA algorithm about the point counting problem of the elliptic curves over a finite field is thoroughly investigated~ including the mathematics background of the algorithm, the detailed procedures in implementing it and the possible improvement existed in it. For chapter six, it is relatively independent from the other chapters. In this chapter, a security analysis of a new public梜ey cryptosystems based on a third ?order linear feedback shift register sequences is brought out with the list of the weak private梜eys existed in the cryptosystems and the bit條evel security analysis of the cryptosystems under an attack proposed by the author.