module specification

This module, together with MA7010, Number Theory for Cryptography provides you with the essential grounding needed to complement the algorithms and techniques encountered in spring semester modules in cryptography and information security. It aims to demonstrate the theoretical underpinning that delivers security, particularly of asymmetric public key algorithms while also allowing you to appreciate the limitations and potential vulnerabilities of systems encountered elsewhere in their course. It is designed to complement, in particular, MA7011 -Applications in Cryptography and Cryptanalysis and can be taken either before or after this module depending on your start point.

You will explore techniques in discrete mathematics that have application in cryptography in a thorough way with emphasis on getting to a level where the mathematics of cryptography and the language used is accessible irrespective of your background and prior study. Beginning with a brief review of topics normally encountered in undergraduate mathematics and computer science courses, new topics will then be introduced from first principles with an emphasis on supported learning via problem classes, worked examples and formative assessment.

Key concepts in discrete mathematics: quotients, remainders and the greatest common divisor. Algorithms for finding the gcd equation. Operations on sets, closure, commutativity, associativity, inverses and identities. Polynomials and the gcd of polynomials, irreducible polynomials. The integers modulo p and methods for finding inverses modulo p for a natural number n. Solutions of systems of linear congruence equations. Euler totient function, primitive elements and the Euler Fermat Theorem. (LO2, LO3)

Connections between binary and hexadecimal numbers; simple calculations in hexadecimal arithmetic. (LO3)

Rings and Fields: axioms and key properties; examples of finite and infinite fields and the construction of GF(pn) as a quotient ring whose elements are polynomials. Minimum polynomials, Fermat’s Little Theorem. Homomorphisms and isomorphisms in rings and fields. (LO1, LO2)

Introduction to number fields and their construction; norms and irreducible elements. (LO3)

Elliptic curves: motivating definitions and geometric interpretations; elliptic curves over finite fields; tangents to elliptic curves and the addition and scalar multiplication of points on elliptic curves. The discrete logarithm problem for elliptic curves over finite fields. (LO1, LO4).

(Optional) Lattices: principles for construction of lattices over n-dimensional basis, examples, specification of a lattice using a finite set of basis elements. Description of the shortest vector problem and the closest vector problem for lattices. (LO4)

The module has 200 learning hours. 4 hours per week contact is divided approximately equally between lectures covering background, theory and examples and workshops which involve discussion of the ideas and an opportunity for students to practice the mathematical techniques. The remaining hours of private study will allow students to complete background reading, work on exercises and prepare for assessment.

This module aims to enable students to:

LO1: Describe, understand in detail and apply key concepts in discrete mathematics relevant to cryptography;

LO2: Implement algorithms and demonstrate an understanding of their complexity;

LO3: Perform hand calculations using numbers of appropriate size;

LO4: Understand and apply the properties of mathematical objects (e.g. elliptic curves, primitive roots) and how they aid the development of asymmetric cryptographic algorithms

https://londonmet.rl.talis.com/modules/ma7009.html