Foundational problems in the arithmetic of curves and abelian varieties over finite fields

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，有一些特殊的因式分解算法，可以很好地处理某些类型的数(例如，两个素数的乘积非常接近，或者可以被某种形式的素数整除的数)。因此，为了确保系统的安全性，重要的是确定随机选择的公钥容易受到此类攻击的概率。幸运的是，RSA背后的许多基本数学理论(如素数定理)很久以前就由数学家发展起来了，所以我们对这些问题有很好的理解。该方案涉及的研究是一种不同类型的公钥密码学，该公钥密码学基于一个名为“有限域上除数类曲线群中的离散对数问题”的困难数学问题。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。这个建议的目的是对一些基本的数学问题进行数学研究，这些问题对于理解基于代数曲线的密码系统是重要的。将研究的一组问题是上面提到的RSA问题的类似。例如，如果在有限域上随机选择一条曲线，那么确定除数类组的大小可被大素数整除的可能性是很重要的。这个问题还没有解决。研究建议书描述了一种解决这个问题的方法，该方法将由该项目的首席研究员和一名博士后研究助理进行。所进行的纯数学研究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。该研究项目还将使纯数学和实用密码学学科之间的知识转移成为可能。

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 a knowledge of many parts of mathematics. For example, there are special factoring algorithms which work well on certain types of numbers (e.g., products of two primes which are very close together, or numbers divisible by primes of a certain form). Hence, to be sure of the security of a system it is important to determine the probability that a randomly chosen public key would be vulnerable to such attacks. Fortunately, a lot of the foundational mathematical theory behind RSA (e.g., the prime number theorem) had been developed by number theorists 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 a hard mathematical problem called the `discrete logarithm problem in divisor class groups of curves over finite fields'. 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 the foundational mathematical problems which are important for an understanding of cryptosystems based on algebraic curves.One set of problems which will be studied are the analogues of the problems mentioned above for RSA. For example, if a curve is chosen `randomly' over a finite field then it is important to determine how likely the divisor class group has size divisible by a large prime number. This problem has not yet been solved. The research proposal contains a description of an approach to solve this problem which will be carried out by the principal investigator of the project together with a postdoctoral research assistant.The pure mathematical research performed will lead to improvements in algorithm design and analysis. These improvements will have an impact on the practical use of public key cryptography. The research project will also enable a transfer of knowledge between the disciplines of pure mathematics and practical cryptography.