Module Code - Title:

MA6011 - CRYPTOGRAPHIC MATHEMATICS

Year Last Offered:

2025/6

Hours Per Week:

Lecture

2

Lab

2

Tutorial

1

Other

0

Private

6

Credits

6

Grading Type:

N

Prerequisite Modules:

Rationale and Purpose of the Module:

To introduce the concepts of Number Theory that underpin cryptographic algorithm techniques and cryptanalysis and to develop skill in deductive reasoning. At the conclusion of the module a student should have the knowledge to handle the mathematics involved in public key cryptography and in the analysis of conventional key ciphers.

Syllabus:

Divisibility and Primes. Euclidean algorithm. Modular arithmetic: linear and polynomial congruences, Chinese remainder theorem. Euler phi function and Fermat's little theorem. Primality tests. Pseudoprimes, Carmichael numbers, strong pseudoprimes. Miller-Rabin test. Probabilistic primality testing. Primitive roots. Discrete logarithm. Quadratic reciprocity:Legendre symbol, Jacobi symbol. Square and cube roots mod p. Elliptic curves modulo p. Group law. Discrete logarithm revisited.

Learning Outcomes:

Cognitive (Knowledge, Understanding, Application, Analysis, Evaluation, Synthesis)

The student will be able to carry out simple calculations using modular arithmetic and solve equations in modular arithmetic. The student will be able to carry out simple primality tests and be aware of the problems that arise from pseudoprimes. The student will be able to use quadratic reciprocity to determine squares and non-squares. The student will understand the concept of order as related to the discrete logarithm problem. The student will be able to do calculations with elliptic curves mod p an determine the order of a point. The student will understand the discrete logarithm problem in the context of elliptic curves.

Affective (Attitudes and Values)

None

Psychomotor (Physical Skills)

None

How the Module will be Taught and what will be the Learning Experiences of the Students:

Lectures and tutorials

Research Findings Incorporated in to the Syllabus (If Relevant):

Prime Texts:

Silverman JH (2006) A Friendly introduction to Number Theory , Pearson

Other Relevant Texts:

Kumanduri, R and Romero,C (1997) Number Theory with Computer Applications , Prentice-Hall

Koblitz N (2001) A Course in Number Theory and Cryptography , Springer

Crandall, R and Pomerance, C (2001) Prime Numbers: A Computational Perspective , Springer

Programme(s) in which this Module is Offered:

Semester(s) Module is Offered:

Module Leader:

eberhard.mayerhofer@ul.ie