Arithmetic aspects of elliptic curve cryptography

自从米勒和Koblitz在20年前发现了基于椭圆曲线的加密方案以来，基于椭圆曲线的加密方案的研究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，因为它们在当今可用的所有此类系统中成功地保持了最高的已知安全级别。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系统的理解。 我的方法联合收割机结合了严格的数学和密码学，以实现任何一种方法都无法单独实现的结果。在更近的工作中，我第一次证明了椭圆曲线密码系统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，假设广义黎曼假设。我打算建立在这项工作的基础上，并将这些结果扩展到超椭圆曲线密码系统，同时探索新的构造ofexpander图和公钥密码系统的设计所产生的技术，我使用。

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.