UWA Handbook 2026

Unit Overview

Description

Number theory lies at the heart of mathematics and underpins many of the cryptographic systems that secure modern digital communication. This unit introduces students to the fundamental structures and ideas of number theory, with a particular emphasis on their applications to cryptography. The course develops a rigorous understanding of integers, primes, and modular arithmetic, while demonstrating how deep theoretical results translate into practical cryptographic protocols.

Students will explore classical topics such as divisibility, prime numbers, modular arithmetic, and arithmetic functions, progressing to central results including Euler's Theorem, the Prime Number Theorem, and Dirichlet's Theorem on primes in arithmetic progressions. Alongside these theoretical foundations, the unit introduces modern cryptography, examining both classical ciphers and public-key cryptosystems. Core cryptographic ideas such as one-way functions, key exchange, and computational hardness assumptions are studied through concrete examples including RSA, Diffie-Hellman key exchange, and the Discrete Logarithm Problem, with algorithmic techniques such as Pollard's algorithm providing insight into practical attacks.

Students will appreciate both the elegance of number-theoretic theory and its central role in contemporary cryptography, gaining the mathematical tools needed to understand why modern cryptosystems work.

Key topics include: Prime numbers and factorisation; modular arithmetic and congruences; greatest common divisors and Bézout's Theorem; Euler's totient function and multiplicative functions; Möbius inversion; quadratic residues; perfect numbers and Mersenne primes; the Prime Number Theorem and prime gaps; Dirichlet's Theorem; classical cryptosystems; public-key cryptography; Diffie–Hellman key exchange; RSA cryptography; the Discrete Logarithm Problem; Pollard's algorithms; and an introduction to elliptic curve cryptography.

Credit

6 points

Offering

Availability	Location	Mode	First year of offer
Not available in 2026	UWA (Perth)	On-campus

Details for undergraduate courses

• Level 2 elective

Outcomes

Students are able to (1) define and apply fundamental concepts of number theory, including divisibility and factorisation, prime numbers, congruences, and modular arithmetic; (2) demonstrate an understanding of major theoretical results in number theory, such as Euler's Theorem, the Prime Number Theorem, and Dirichlet's Theorem, and explain their mathematical significance; (3) analyse and implement public-key cryptographic systems, including RSA and Diffie–Hellman, and explain the underlying number-theoretic principles that ensure their security; and (4) construct rigorous mathematical arguments and proofs involving modular arithmetic, primes, and arithmetic functions, and critically assess the correctness and security assumptions of cryptographic algorithms.

Assessment

Indicative assessments in this unit are as follows: (1) tests and (2) exam. Further information is available in the unit outline.

Student may be offered supplementary assessment in this unit if they meet the eligibility criteria.

Unit Coordinator(s)

Associate Professor John Bamberg

Unit rules

Prerequisites

MATH1721 Mathematics Foundations: Methods or equivalent
or Mathematics Methods ATAR or equivalent

Contact hours

lectures: 3 hours per week
practical classes: 1 hour per week