The need to protect information being transmitted electronically (such as the widespread use of electronic payment) has transformed the importance of cryptography. Most of the modern types of cryptosystems rely on (increasingly more sophisticated) number theory for their theoretical background. This course introduces elementary number theory, with an emphasis on those parts that have applications to cryptography, and shows how the theory can be applied to cryptography.

Number theory topics will be chosen from: the Euclidean algorithm, highest common factor, prime numbers, prime factorisation, primality testing, congruences, the Chinese remainder theorem, diophantine equations, sums of squares, Euler's function, Fermat's little theorem, quadratic residues, quadratic reciprocity, Pell's equation, continued fractions.

Cryptography topics will be chosen from: symmetric key cryptosystems, including classical examples and a brief discussion of modern systems such as DES and AES; public key systems such as RSA and discrete logarithm systems; cryptanalysis (code breaking) using some of the number theory developed.

Note: Graduate students attend joint classes with undergraduates but are assessed separately.

On satisfying the requirements of this course, students will have the knowledge and skills to:

1. Solve problems in elementary number theory
2. Apply elementary number theory to cryptography
3. Develop a deeper conceptual understanding of the theoretical basis of number theory and cryptography

Assessment will be based on:

• Three assignments (10%; LO 1, 2, 3)
• Final examination (70%; LO 1, 2, 3)