Algorithms in Number Theory

这是一个研究数论问题计算复杂性的更新项目。除了在复杂性理论中提供许多未解决的问题外，数论在随机数生成、密码学和计算机代数的系统设计中也很有用。调查分为三类：a)探索启发式假设的算法后果，如扩展的黎曼假设。B)分析和设计数论算法，特别注意素性、多项式因式分解、C)代数几何(特别是有限域上的变元理论)在构式中的应用技巧。

This is a renewal project to investigate the compuational complexity of number-theoretic problems. Besides providing a number of unsolved problems in complexity theory, number theory is useful in the design of systems for random number generation, cryptography, and computer algebra. The investigations fall into three categories: a) Exploration of the algorithmic consequences of heuristic assumptions such as the Extended Riemann Hypothesis. b) Analysis and design of number-theoretic algorithms, with special attention being paid to efficient randomized algorithms for primality, factorization of polynomials, etc. c) Applications of algebraic geometry (especially the theory of varieties over finite fields) to constructives techniques.