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

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.