This course is taken by Part IA (50% Option) students.

Lecturer: Professor P. Robinson

No. of lectures: 12

This course is a prerequisite for all theory courses as well as Discrete Mathematics II, Security (Part IB and Part II), Artificial Intelligence (Part IB and Part II), Information Theory and Coding (Part II).

Aims

This course will develop the idea of formal proof by way of examples
involving simple objects such as integers and sets. The material
enables academic study of Computer Science and will be promoted with
examples from cryptography and the analysis of algorithms.

Factors.
Division: highest common factors and least common multiples.
Euclid's algorithm: solution in integers of ,
the complexity of Euclid's algorithm.
Euclid's proof of the infinity of primes.
Existence and uniqueness of prime factorisation.
Irrationality of .
[4 lectures]

Modular arithmetic.
Congruences.
Units modulo , Euler's totient function.
Chinese remainder theorem.
Wilson's theorem.
The Fermat-Euler theorems, testing for primes.
Public key cryptography, Diffie-Hellman and RSA.
[5 lectures]

Objectives

On completing the course, students should be able to

write a clear statement of a problem as a theorem in mathematical notation;
prove and disprove assertions using a variety of techniques

describe, analyse and use Euclid's algorithm;
explain and apply prime factorisation

perform calculations with modular arithmetic;
use number theory to explain public key cryptography

Recommended reading

* Humphreys, J.F. & Prest, M.Y. (1989). Numbers, groups and codes. Cambridge University Press.
Anderson, J.A. (2003). Discrete mathematics with combinatorics. Prentice Hall.
Conway, J.H. & Guy, R.K. (1996). The book of numbers. Springer-Verlag.
Davenport, H. (1992). The higher arithmetic (6th ed.). Cambridge University Press.
Giblin, P. (1993). Primes and programming. Cambridge University Press.
Pólya, G. (1980). How to solve it. Penguin.
Rosen, K.H. (1999). Discrete mathematics and its applications (4th ed.). McGraw-Hill.