Applied Algebra and Cryptography

Overview

- (finite) fields and rings of polynomials over them.
- the division algorithm and splitting of polynomials.
- ideals and quotient rings, (principal) ideal domains, with examples from rings of polynomials.
- polynomials in several indeterminates, Hilbert’s basis theorem.
- applications of algebra to cryptography (such as affine Hill ciphers, RSA, lattice cryptography, Diophantine equations).
- optional topics may include Euclidean rings, unique factorisation domains, greatest common divisor domains.

Learning Objectives

It is intended that students shall, on successful completion of the module, be able to:
understand the concept of a ring of polynomials over a (finite field);
apply the factorisation algorithm;
understand ideals, quotient rings and the properties of quotient rings;
understand how algebra can be applied to cryptography and be able to encrypt messages using methods from the module.

Skills

Analytic argument skills, problem solving, analysis and construction of proofs.

Assessment

None