Cryptography and Group Theory

密码学和群论

• 批准号: 0625271
• 负责人: Gilbert Baumslag
• 金额: $ 0.6万
• 依托单位: CUNY City College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-10-01 至 2007-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0625271&HistoricalAwards=false
• 关键词: Cryptography; Group; Theory

The possibility that quantum computers might eventually make today's cryptographic protocols obsolete and thereby make hitherto private information public, has recently led to an ongoing and very active effort to develop cryptoysystems based on group theory. One such system, using the so--called braid group, was developed in part by Michael Anshel (a member of the City College of New York Computer Science Department and CAISS, the Center for Algorithms and Scientific Software, a research center at CCNY), in 1999, and can be viewed as a group--theoretic analogue of the famous key-exchange idea of Diffie--Hellman. Today, as already indicated, a great deal of research is being carried out in this general area of group theory and cryptography. The Graduate Center of the City University of New York hosts the famous New York Group Theory Seminar, where current research in group theory is discussed on a weekly basis. There is also now a seminar devoted to cryptography at the Graduate Center and the Stephens Institute in Hoboken has a team of researchers working in part on cryptography. So a conference at this time in this general area seems particularly appropriate and important.One tends to marvel at the intricate patterns that one sees in the tiled floors of St. Marks cathedral in Venice and the remarkable symmetry of everyday crystals. Such symmetry also arises in the study of elementary particles and also in knot theory, which surprisingly arises in physics. They also play a part in efforts to understand the intricacies of the three--dimensional world that we live in. There is an algebraic structure known as a group which captures the notion of symmetry in a highly sophisticated way, allowing for the classification of different types of crystals and provides a means for studying and describing the minute particles which make up the material of everyday life.These groups can encode in a remarkably simple way, a huge amount, indeed an infinite amount, of information. Such encodings are often hard to decode and so are extremely promising tools in obtaining secure new ways for hiding and transmitting information.Thus they offer novel means for safe electronic communication.With the birth of the internet and the explosion of commerce over the net, secure transmission of information is now a critical aspect of today's world. The proposed conference will be devoted to bringing some of the ideas involved to the wider mathematical community in the hope that it will lead to broader understanding and application of them.

量子计算机可能最终使今天的密码协议过时，从而使迄今为止的私人信息公开的可能性，最近导致了一个正在进行的和非常积极的努力，以开发基于群论的密码系统。一个这样的系统，使用所谓的辫子群，部分是由Michael Anshel（纽约城市学院计算机科学系和CAISS的成员，算法和科学软件中心，CCNY的一个研究中心）在1999年开发的，可以被视为Diffie-Hellman著名的密钥交换思想的群论模拟。今天，正如已经指出的那样，在群论和密码学的这个一般领域正在进行大量的研究。纽约城市大学的研究生中心主办了著名的纽约群论研讨会，每周讨论群论的最新研究。现在在研究生中心也有一个专门讨论密码学的研讨会，位于霍博肯的斯蒂芬斯研究所有一个研究小组，部分研究密码学。因此，在这个时候在这个地区举行会议似乎特别合适和重要。人们往往会惊叹于威尼斯圣马可大教堂瓷砖地板上的复杂图案和日常水晶的非凡对称性。这种对称性也出现在基本粒子的研究中，也出现在纽结理论中，这在物理学中令人惊讶地出现。它们也在努力理解我们生活的三维世界的复杂性方面发挥了作用。有一种被称为群的代数结构，它以一种高度复杂的方式捕捉对称性的概念，允许对不同类型的晶体进行分类，并为研究和描述构成日常生活材料的微小粒子提供了一种手段，这些群可以以一种非常简单的方式编码大量的信息，实际上是无限的信息。这种编码通常很难解码，因此在获得隐藏和传输信息的安全新方法方面是非常有前途的工具。因此，它们为安全的电子通信提供了新的手段。随着互联网的诞生和网上商业的爆炸式增长，信息的安全传输现在是当今世界的一个重要方面。拟议的会议将致力于把一些涉及到更广泛的数学界的想法，希望它将导致更广泛的理解和应用。