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的方法，对于许多这些应用程序来说，最佳解决方案是使用“公钥密码学”。公钥密码系统通常涉及难以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质数乘积的问题有关。如果数字足够大，这个计算问题将需要大量的计算机资源来解决。要完全理解RSA密码系统，需要具备许多数学方面的知识。例如，有一些特殊的因子分解算法可以很好地处理某些类型的数字（例如，两个质数的乘积非常接近，或者可以被某种形式的质数整除的数字）。因此，为了确保系统的安全性，确定随机选择的公钥容易受到此类攻击的概率非常重要。幸运的是，RSA背后的许多基础数学理论（例如，素数定理）很久以前就已经由数论家开发出来了，所以我们对这些问题有很好的理解。本提案所涉及的研究是针对另一种类型的公钥加密，它基于一个称为“有限域上曲线的除数类群中的离散对数问题”的难数学问题。与RSA一样，要完全理解这些密码系统，需要了解许多数学问题。与RSA不同的是，这些问题中的许多在过去都没有被研究过。本提案的目的是对一些基础数学问题进行数学研究，这些问题对理解基于代数曲线的密码系统很重要。将研究的一组问题是上述RSA问题的类似物。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有多大就很重要了。这个问题还没有解决。研究计划包含解决该问题的方法的描述，该方法将由项目的首席研究员和博士后研究助理一起执行。进行的纯数学研究将导致算法设计和分析的改进。这些改进将对公钥加密的实际使用产生影响。该研究项目还将实现纯数学和实用密码学学科之间的知识转移。

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.