Finite Field F <sub> 2 </ Sub> On The Security Of Elliptic Curve Analysis And Its Implementation

In this dissertation the elliptic curve discrete logarithm problems and the secure elliptic curves for cryptography are investigated,including how to find and implement the secure elliptic curves for cryptography, and the related algorithms are presented. With the emphasis of the results obstained by the author, the crucial technological theory of the elliptic curve cryptosystem over the finite fields of characteristic two are discussed and studied in detail. It is consisted of five chapters, organized as follows. In chapter one, some introductive materials are presented, including the motivations and the developments of the elliptic curve cryptosystems, the intentions of the research work, the main contents of the paper and the results obstained by the author. In chapter two, the nessary and basic materials of the elliptic curve theory are introduced, including some hyperelliptic curve theory. In chapter three, the discrete logarithm problems of elliptic curves and hyperelliptic curves are discussed. In this chapter, with the approach of the primary algorithms for the two kind of the discrete logarithm problems, some general and special algorithms for the discrete logarithm problems are investigated. In particularly, the special algorithms GHS attack are studied detailedly. And the related conclusion obstained by the author are stated. In chapter four, the fast implementation problems of the point counting for elliptic curves over the finite fields are studied. Firstly, the l - adic point counting algorithms and the p - adic point counting algorithms are discussed and studied. Secondly, a new efficientive pointcounting for elliptic curves over the finite fields of characteristic two which based on the l-adic point counting algorithms are presented, by implementing the new algorithm, some results obstained by the algorithm are listed and are proven secure. At last, the AGM algorithm and the MSST algorithm are studied intensively. Their advantage and their disadvantage are discussed, their feasibilities are compared, the applications of the two algorithms are prosoed, and the researchful work of the author in the future are pointed out. In chapter five, the dissertation are end.