Cryptography is an eminently applicable area of mathematics. The problems come in several different flavours and utilise distinct areas of mathematics, giving available projects in at least the following two areas.

a. Number-theoretic cryptography: Number-theoretic cryptosystems depend upon numerical base problems such as the discrete logarithm problem and integer factorisation problem. RSA is one example of such a cryptosystem. The aim of this project would be to conduct a survey of available number theoretic cryptographic schemes and the number theory behind them. You should then cover some attacks on one of the schemes. A discussion of sneaky attacks is expected, as well as a discussion of available mathematical techniques for breaking the applicable base problem. The student should aim to learn some in-depth number theory for this project;

b. Group-based cryptography: There are several available cryptosystems which rely upon group theoretic problems to protect the private key. Moreover, there are several distinct types of group over which the cryptosystems operate, and also some groups are known to have “difficult” base problems. The aim of this project is to briefly write about some distinct types of groups and associated proposed cryptographic schemes. Then a specific family of groups will be chosen and attacks upon the proposed schemes and base problems will be surveyed. A healthy command of algebra, and in particular, group theory, is assumed.

References:

- Koblitz, N., A Course in Number Theory and Cryptography, Graduate Texts in Mathematics 114, Springer (1994).
- Koblitz, N., Algebraic Aspects of Cryptography, Algorithms and Computation in Mathematics 3, Springer (1999).
- Menezes, A., Handbook of Applied Cryptography, downloadable from http://www.cacr.math.uwaterloo.ca/hac/ (1997).
- Schneier, B., Applied Cryptography, Wiley 1995.

Supervisor: Dr Matthew Craven