A long view of curves in cryptography

电子通信(如互联网和移动电话)越来越多地被用于金融交易或发送敏感信息。因此，在这些情况下能够确保身份验证和机密性非常重要。‘密码学’的主题提供了保护通信安全的方法，对于这些应用中的许多，最好的解决方案是使用‘公钥密码学’。公钥密码系统通常与难以通过计算解决的数学问题有关。例如，RSA密码系统的安全性与将整数分解为素数的乘积的问题有关。如果数字足够大，这个计算问题将花费不可能的大量计算机资源来解决。完全理解RSA密码系统需要了解数学的许多部分。例如，最好的通用分解算法依赖于代数数论和代数几何等高级数学。幸运的是，RSA背后的许多基本数学理论在很久以前就已经由数学家们发展出来了，所以我们对这些问题有了很好的理解。在这个建议中涉及的研究是针对另一种类型的公钥密码学，该公钥密码学基于困难的数学问题，例如，有限域上的除数类曲线群中的离散对数问题或双线性Diffie-Hellman问题。与RSA一样，对这些密码系统的全面理解需要一些数学问题的知识。与RSA不同的是，这些问题中的许多过去都没有被研究过。这项建议的目的是对其中的一些问题进行数学研究。将研究的一组问题是如何在除数类曲线群上有效地计算被称为‘同源’的映射。这样的理论在密码学和计算数学中会有很多应用。另一组问题与最近的一个主题有关，该主题名为“基于配对的密码学”。这门学科是一个新学科，缺乏一个合适的框架来研究一些问题。该项目将加强基于配对的密码学的基础。所进行的纯数学研究将使人们对一些公钥密码体制背后的数学有更深的理解。这反过来将导致算法设计和分析的改进。这些改进将对公钥密码术的实际使用产生影响。

Electronic communications (such as the internet and mobile phones) are increasingly being used for financial transactions or for sending sensitive information. As a result, it is important to be able to ensure authentication and confidentiality in these situations. The subject of `cryptography' provides methods to secure communications, and for many of these applications the best solution is to use `public key cryptography'.Public key cryptosystems are usually related to mathematical problems which are difficult to solve computationally. For example, the security of the RSA cryptosystem is related to the problem of factorising an integer into a product of prime numbers. If the numbers are large enough this computational problem would take infeasibly large computer resources to solve.A full understanding of the RSA cryptosystem requires knowledge of many parts of mathematics. For example, the best general-purpose factoring algorithms rely on advanced mathematics such as algebraic number theory and algebraic geometry. Fortunately, a lot of the foundational mathematical theory behind RSA had been developed by mathematicians a long time ago, and so we have a good understanding of these issues.The research covered in this proposal is into a different type of public key cryptography, one which is based on hard mathematical problems such as the `discrete logarithm problem in divisor class groups of curves over finite fields' or the `bilinear Diffie-Hellman problem'. As with RSA, a full understanding of these cryptosystems requires knowledge about a number of mathematical questions. Unlike RSA, many of these questions have not been studied in the past. The aim of this proposal is to carry out mathematical research into some of these problems.One set of problems which will be studied is about how to efficiently compute with mappings called `isogenies' on divisor class groups of curves. There would be many applications of such a theory to cryptography and computational mathematics. Another set of problems relates to a very recent subject called `pairing based cryptography'. Being new, this subject lacks a suitable framework for studying some problems. The project will strengthen the foundations of pairing-based cryptography.The pure mathematical research performed will give a deeper understanding of the mathematics behind some public key cryptosystems. This will, in turn, lead to improvements in algorithm design and analysis. These improvements will have an impact on the practical use of public key cryptography.