Number Theory and Cryptography

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 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, greatest common divisor, prime numbers, prime factorisation, primality testing, modular arithmetic, the Chinese remainder theorem, diophantine equations, sums of squares, Euler's function, Fermat's little theorem, primitive roots, quadratic residues, quadratic reciprocity, Pell's equation, continued fractions, Diophantine approximation. 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. The additional assessment will consist of assignments requiring deeper conceptual understanding and/or a project, and the final exam will contain alternative questions requiring deeper conceptual understanding.