Term	Trinity Term 2009  (8 (2 hour) lectures)

Overview

The goal is to introduce cryptography and the main mathematical tools used in cryptography. The main focus will be on public key cryptosystems. Each lecture will aim to illustrate a cryptographic concept and the related mathematics. Each week there will be a cryptographic challenge.

Synopsis

Summary of Lectures 

Lecture 1.

Terminology.
Classical cryptographic systems.
Examples.
Exercises + Crypto Challenge #1.

Lecture 2.

Elements of probability theory.
Information theory.
Entropy and mutual information.
Decision functions.
Cryptanalysis of classical cryptosystems.
Exercises + Crypto Challenge #2.

Lecture 3.

Two strong classical ciphers - a study.
Protocols.
Authentication - Privacy - Secrecy
The discrete log problem.
Index calculus.
Public Key Cryptography.
The key distribution and authentication problems.
Cryptographic uses of discrete logs.
Exercises + Crypto Challenge #3.

Lecture 4.

Computational complexity.
Turing machines.
P - NP and cryptographically hard problems.
Number lengths and run time estimates.
P-NP and NP completeness.
RP and BPP.
Arithmetic and the Fast Fourier Transform.
Exercises + Crypto Challenge #4.

Lecture 5.

Public Key Cryptosystems.
The key distribution problem revisitied.
The d'Urfe problem.
Trapdoor one way functions.
Finite fields : some theorems.
RSA.
Exercises + Crypto Challenge #5.

Lecture 6.

Elliptic curve cryptography.
Definition of Abelian varieties and Elliptic curves.
The group law.
The idea of Miller and Koblitz.
A review of finite fields and finite field arithmetic.
Elliptic curves over finite fields.
Exercises + Cryto Challenge #6.

Lecture 7.

The Elliptic curve discrete log problem.
Elliptic curve cryptography.
Examples.
Exercises + Crypto Challenge #7.

Lecture 8.

An introduction to hyperelliptic curve cryptography.
or
Quantum Cryptography
Exercises + Crypto Challenge #8.

Reading list

Complexity and cryptography : An introduction. John Talbot and Domininc Welsh. Cambridge University Press, 2006.

Elliptic Curves : Number theory and cryptography. Lawrence C. Washington. Chapman and Hall, 2003.

Taking our courses

This form is not to be used by students studying for a degree in the Department of Computer Science, or for Visiting Students who are registered for Computer Science courses

Other matriculated University of Oxford students who are interested in taking this, or other, courses in the Department of Computer Science, must complete this online form by 17.00 on Friday of 0th week of term in which the course is taught. Late requests, and requests sent by email, will not be considered. All requests must be approved by the relevant Computer Science departmental committee and can only be submitted using this form.