A long view of curves in cryptography

电子通信（如互联网和移动的电话）正越来越多地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”这一学科提供了保障通信安全的方法，对于其中许多应用来说，最好的解决办法是使用“公钥密码学”，公钥密码系统通常涉及难以通过计算解决的数学问题。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，这个计算问题将需要不可行的大量计算机资源来解决。RSA密码系统的全面理解需要数学的许多部分的知识。例如，最好的通用因式分解算法依赖于代数数论和代数几何等高等数学。幸运的是，RSA背后的许多基础数学理论在很久以前就已经被数学家们开发出来了，所以我们对这些问题有了很好的理解。本提案中涵盖的研究是一种不同类型的公钥密码术，一个是基于硬数学问题，如“离散对数问题的除数类群的曲线在有限域”或“双线性Diffie-赫尔曼问题与RSA一样，要完全理解这些密码系统，需要了解一些数学问题。与RSA不同，这些问题中的许多在过去没有被研究过。本建议的目的是进行数学研究到这些问题中的一些。其中一组问题将被研究是关于如何有效地计算与映射称为'isogenies'的除数类组的曲线。这种理论在密码学和计算数学中有许多应用。另一组问题涉及一个非常新的主题，称为“基于配对的密码”。由于这门学科是一门新学科，对一些问题的研究还缺乏一个合适的框架。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更深入地理解某些公钥密码系统背后的数学。这将反过来导致算法设计和分析的改进。这些改进将对公钥加密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.