Arithmetic aspects of elliptic curve cryptography

基于椭圆曲线的加密方案自从20年前由Miller和Koblitz(独立地)发现以来，引起了研究界的极大兴趣。椭圆曲线系统在所有公钥系统中是独一无二的，因为在当今可用的所有此类系统中，它们成功地在最长时间内保持了已知的最高安全级别。我的研究旨在通过研究椭圆曲线算法的数学基础来提高我们对椭圆曲线密码体制的理解。我的方法将严谨的数学和密码学结合起来，以获得单独使用这两种方法都无法达到的结果。在较近的工作中，我首次证明了在假设广义黎曼假设的情况下，椭圆曲线密码体制的安全性在几乎所有情况下都纯粹取决于椭圆曲线的密钥大小。我打算在这项工作的基础上，将这些结果推广到超椭圆曲线密码系统，同时探索由我使用的技术产生的扩张图的新构造和公钥密码系统设计。

The study of encryption schemes based on elliptic curves has attracted considerable interestfrom the research community ever since their discovery 20 years ago by Miller and(independently) Koblitz. Elliptic curve systems are unique among all public key systems inthat they have successfully maintained the highest known security levels for the longestperiod of time among all such systems available today. My research aims to improve ourunderstanding of elliptic curve cryptosystems by investigating the mathematical foundationsof elliptic curve arithmetic.  My methods combine both rigorous mathematics and cryptologyin order to achieve results that are beyond the reach of either approach in isolation. Inearlier work, I show for the first time that the security of elliptic curve cryptosystemsdepends in almost all cases purely on the key size of the elliptic curve, assuming thegeneralized Riemann hypothesis. I intend to build upon this work and extend these results tohyperelliptic curve cryptosystems, while at the same time exploring new constructions ofexpander graphs and public key cryptosystem designs arising from the techniques that I used.