Algorithms and cryptography in global fields

全球领域的算法与密码学

• 批准号: 250246-2011
• 负责人: Scheidler, Renate
• 金额: $ 1.09万
• 依托单位: University of Calgary
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=570862
• 关键词: Algorithms; cryptography; global; fields

The last few decades have seen an unprecedented increase in the use of computing and communication technology. Coupled with this development is a crucial need for information protection: confidential data must be guarded against access or modification by unauthorized parties, users of online banking need to be authenticated to ensure that they are who they claim they are, and the origin of e-mails must be verifiable to ascertain if they are legitimate. Sadly, the public has become accustomed to news flashes about virus-infected computers, hacking, defaced web sites, cases of identity theft, and even potential threats of internet terrorism. Cryptography is an effective vehicle for preserving the confidentiality, integrity and authenticity of digital information. The security of many modern cryptosystems relies on certain mathematical problems that experts believe to be extremely difficult to solve. The idea underlying the design of such a system is that an adversary would have to solve an instance of one of these very hard problems in order to crack the scheme. It is therefore of great interest, both theoretical and practical, to thoroughly study these types of problems and devise new cryptosystems that make use of them. To this end, I propose to investigate a type of mathematical structure called a global field. Certain global fields provide an excellent framework for efficient and very secure cryptosystems; this includes the celebrated elliptic curve cryptosystems which have been deployed in a range of consumer electronics products such as BluRay technology and the BlackBerry. My research into fast arithmetic in global fields will make the corresponding cryptographic schemes more efficient. Moreover, my exploration into the difficulty of certain hard mathematical problems in global fields will lead to a better understanding of the security of such schemes. The outcomes of the proposed work will be a collection of well understood algorithms for number theoretic computations as well as a suite of secure and efficient cryptographic protocols that can be used to protect electronic communications.

过去几十年来，计算和通信的使用出现了前所未有的增长 技术。与这一发展相伴的是信息保护的关键需求：必须防止机密数据被未经授权的各方访问或修改，网上银行用户需要经过身份验证，以确保他们的真实身份，电子邮件的来源必须可验证，以确定它们是否合法。可悲的是，公众已经习惯了有关计算机感染病毒、黑客攻击、网站被破坏、身份盗窃案件，甚至网络恐怖主义潜在威胁的新闻报道。 密码学是保护数字信息的机密性、完整性和真实性的有效工具。许多现代密码系统的安全性依赖于专家认为极难解决的某些数学问题。这种系统设计的基本思想是，对手必须解决这些非常困难的问题之一的实例才能破解该计划。因此，深入研究这些类型的问题并设计利用它们的新密码系统，无论在理论上还是在实践上都非常有意义。为此，我建议研究一种称为全局域的数学结构。某些全球领域为高效且非常安全的密码系统提供了出色的框架；这包括著名的椭圆曲线密码系统，该系统已部署在蓝光技术和黑莓等一系列消费电子产品中。我对全球领域快速算术的研究将使相应的密码方案更加高效。此外，我对全球领域中某些数学难题的难度的探索将有助于更好地理解此类方案的安全性。拟议工作的成果将是一系列易于理解的数论计算算法以及一套可用于保护电子通信的安全高效的加密协议。