椭圆曲线离散对数研究

椭圆曲线密码（Elliptic Curve Cryptology,简称ECC）是密码学的一个重要研究方向。一类是基于有限域上标量乘的经典椭圆曲线密码，另一类是基于椭圆曲线上的配对的密码，是信息安全领域具有广泛应用价值的一种核心技术。而椭圆曲线离散对数求解困难性是椭圆曲线安全性的基础。本项目研究代数函数域的分解理论，kummer理论的算法实现，从而构造代数函数域上的指标演算法；研究椭圆曲面特值化理论，通过椭圆曲面把不同强度的椭圆曲线离散对数问题联系起来，从而由弱曲线的上的求解算法得到其他曲线上的离散对数解法；研究Semaev多项式的性质和计算方法，从而改进现有的椭圆曲线上指标演算法；研究Hilbert多项式的计算方法，从而实现对较大的复乘判别式情形下，各种性质的椭圆曲线的构造；把这些结果用于大规模的椭圆曲线离散对数问题的求解，挑战Certicom公司的公开问题。

Elliptic curve cryptology（ECC）is an important research area of cryptology. The traditional type of such cryptosystem is constructed by using the scalar multiplications of elliptic curve over finite fields, and pairing based cryptosystem is constructed by using bilinear pairing from an elliptic to a field. ECC are believed to be one of the core technology for their promising and widespread usage in information security area. However, an elliptic curve cryptosystem is secure only if when the discrete logarithm problem of the elliptic over finite fields is hard enouph. The main goal of this project will be analysis for ECC cryptosystem, or find high efficient algorithms for solving discrete logarithm problem for large scale elliptic curves, especially concerning for those open challenge problems proposed by the Certicom company. First, we devote to the factorization theory in algebraic function fields and the development for the arithmetic theory for Kummer theory, applying it to the index calculus on algebraic function fields. Another object is to make deep study in the specialization theorem about the elliptic surface, and find the relationship between different elliptic curves, that are different fibers on the surface, that makes it possible to solve DLP for one elliptic curve by the solutions of DLP for another weak elliptic curve. We will also study properties for Semaev's summation polynomials and develop their arithmetic theory，this is quite important for index calculus on elliptic curves. We are also interested to the improvement of calculations on Hilbert polynomial，that make it possible to construct elliptic curves by complex multiplications for little bit large discriminant, so that one can find elliptic curves with special properties more freely.