Computational Complexity and its Relationship to Number Theory

研究的主要焦点是计算问题的数论方面。近年来，人们对这一领域的兴趣重新燃起。古老的问题已经解决了。素数测试)，新的分支学科已经产生（例如公钥密码学），深度数学工具已经出现（例如阿贝尔变种，类场论）。一个特别有趣的领域是新的“数字字段筛选”。由许多研究人员开发的这套算法集合，是过去四分之一世纪中整数分解最重要的进展。PI将继续研究该方法的改进和推广。数域筛法在离散对数问题中的应用是特别有趣的。第二个感兴趣的领域是将最近的结果应用于公钥加密。美国政府最近提出了一项国家“数字签名标准”（DSS）。提议的签名方案是有争议的，将研究几种方法的潜力，包括数字字段筛，用于破坏提议的系统或引入“活板门”。另一个研究领域涉及新的数学工具的发展。

The primary focus of the research is the number theoretic aspects of computational problems. In recent years there has been a resurgence of interest in this area. Ancient problems have been solved(e.g. primality testing), new subdisciplines have been spawned (e.g. public-key cryptography) and deep mathematical tools have emerged (e.g. Abelian varieties, class field theory). One area of particular interest is the new 'number field sieve'. This collection of algorithms, developed by many researchers, is the most significant advance in integer factoring in the last quarter century. The PI will continue to investigate improvements and generalizations of this method. The application of the number field sieve to the discrete logarithm problem is of particular interest. A second area of interest is the application of recent results to public-key cryptography. The United States Government has recently proposed a national 'Digital Signature Standard' (DSS). The proposed signature scheme is controversial and the potential of several methods will be investigated, including the number field sieve, for breaking the proposed system or for introducing 'trapdoors'. Another area of investigation involves the development of new mathematical tools.