PYI: Arithmetic and Geometric Methods in Computational Complexity

近年来，数论和代数几何思想的注入，使素性检验和整数分解的研究，乃至整个计算数论的研究，都取得了显著的进展。它计划继续探索算术和几何方法，作为研究计算问题的基本工具，特别是与数论有关的问题。目标是为它们开发有效的算法，研究它们之间的复杂性理论关系，探索它们在密码学中的应用，以及计算复杂性方面的其他问题。主要的焦点将是：(1)进一步研究素性检验和整数分解，以及计算数论中的其他基本问题。(2)开发高效的利用曲线和高维代数簇进行计算的算法，特别是当它们应用于上述计算问题时。(3)算术和几何技术在公钥密码系统安全性分析中的应用。基于几何思想的新密码系统的开发。

The injection of ideas from number theory and algebraic geometry has brought about remarkable progress in recent years in the study of primality testing and integer factoring, and in computational number theory in general. It is plan to continue the exploration of arithmetric and geometric methods as a fundamental tool in the study of computational problems, particularly those that are related to number theory. The goal is to develop efficient algorithms for them, to study complexity theoretic relation among them, and to explore application to cryptography, as well as other problems in computational complexity. The primary focus will be: (1) Further investigation into primality testing and integer factoring, and other fundamental problems in computational number theory. (2) Development of efficient algorithms for computing with curves and higher dimensional algebraic varieties, particularly as they apply to the computational problems mentioned above. (3) Application of arithmetic and geometric techniques in the analysis of security of public key cryptosystems. Development of new cryptosystems based on geometric ideas.