Complex Multiplication: Class invariants and cryptographic applications

随着计算机代数的出现和显式计算的需要，复数乘法理论已成为算法研究的对象，以期在算法代数数论和密码学中得到应用。其中特别有趣的应用包括主要证明、显式分类理论以及基于曲线和配对的密码学。算法CM理论的一个核心方面是通过分析方法描述离散对象的数值计算和符号计算之间的相互作用。本项目主要研究以下三个主题：二次和四次CM-域的类不变式，基于配对的密码学和算法数论方面。

With the emergence of computer algebra and the need for explicit calculations, the theory of complex multiplication has become subject to algorithmic investigations with a view towards applications in algorithmic algebraic number theory and cryptography. Among the applications are of particular interest primarily proving, explicit classified theory, and curve and pairing based cryptography. A central aspect of algorithmic CM theory is the interplay between numerical and symbolic computations describing discrete objects by analytic means. This project investigates the following three main topics: Class invariants for quadratic and quartic CM-fields, pairing based cryptography and aspects of algorithmic number theory.