Algorithmic Problems in Number Theory

数域筛选是将整数分解成素数的最新算法。研究人员正在继续探索与数字域筛分相关的理论和实践问题。他还在继续研究一维交换代数中的基本计算工具，包括快速傅立叶变换和快速幂级数求幂。房利美副主席兼总统关键基础设施保护委员会咨询委员会联席主席杰米·S·戈雷里克(Jamie S.Gorelick)表示：“今天，一小群技术成熟的人可以扰乱我们的金融、电信、电力、供水系统和紧急服务系统所依赖的计算机和互联网连接。”“网络空间很可能是这个国家的下一个战场。”调查人员的工作与公钥密码学直接相关，包括公钥签名系统，公钥签名系统是保护互联网通信完整性的主要工具。调查人员的研究准确地显示了攻击者需要多长时间才能攻破最受欢迎的公钥系统；这一信息对于想要选择安全系统的用户来说至关重要。研究人员的研究还减少了使用这些系统所需的计算机时间。同样的研究在许多其他计算机问题上也很有用，包括医学图像处理。

9970409The number field sieve is the latest algorithm for factoring integers into primes. The investigator is continuing to explore theoretical and practical problems related to the number field sieve. He is also continuing his research into the fundamental computational tools in one-dimensional commutative algebra, including fast Fourier transforms and fast power series exponentiation."Today, a small group of technically sophisticated people could disrupt the computers and the Internet connections on which our finance, telecommunications, power, water systems, emergency service systems all depend," says Jamie S. Gorelick, Vice Chair of Fannie Mae and Co-Chair of the Advisory Committee of the President's Commission on Critical Infrastructure Protection. "Cyberspace is likely to be the next battlefield for this nation." The investigator's work is directly relevant to public-key cryptography, including public-key signature systems, the main tool for protecting the integrity of Internet communications. The investigator's research shows precisely how long it would take an attacker to break the most popular public-key systems; this information is crucial for users who want to choose safe systems. The investigator's research also reduces the amount of computer time necessary to use these systems. The same research is useful in many other computer problems, including medical image processing.