Lecturer : Thanasis Bouganis
Office : CM 126A
Office Hours: Mondays 14:30 to 15:30 (or just write me an e-mail)
E-mail : athanasios.bouganis@durham.ac.uk
Lectures :Lecture Notes (It contains some more examples on elliptic curves over finite fields at the end.)
(This is just a summary of the lectures so far. It does not contain all the details done in the lectures.)Problem Sheets (Epiphany Term):
Problem Sheet 1 (due: Monday Jan 26 at 12:00 in CM116)Problem Sheet 2 (due: Monday Feb 9 at 12:00 in CM116)
Problem Sheet 3 (due: Monday Feb 23 at 12:00 in CM116)
Problem Sheet 4 (due: Monday Mar 9 at 12:00 in CM116)
Content of Lectures (Epiphany Term)
| Date | Content |
| Wednesday, 14 January 2015 (Week 11) | Revision of Quadratic Residues, Introduction to Diophantine Problems. |
| Friday, 16 January 2015 (Week 11) | Pythagorean triples, Characterization of Pythagorean triples. |
| Wednesday, 21 January 2015 (Week 12) | Characterization of Pythagorean triples(ctd), Fermat's Conjecture, and Fermat's Descent Method. |
| Friday, 23 January 2015 (Week 12) | Fermat's Descent Method, Fermat's Theorem in the case of n=4. |
| Wednesday, 28 January 2015 (Week 13) | Sum of Two Squares, Lemma of Thue, Theorem of Fermat. |
| Friday, 30 January 2015 (Week 13) | Uniqueness of Presentation as the Sum of Two Squares, Difference of Two Squares. |
| Wednesday, 4 February 2015 (Week 14) | Difference of Two Squares, Sum of three Squares, Sum of four Squares. |
| Friday, 6 February 2015 (Week 14) | Sum of three Squares, Sum of four Squares, Finite Continued Fractions (first definitions), . |
| Wednesday, 11 February 2015 (Week 15) | Properties of Finite Simple Continued Fractions. |
| Friday, 13 February 2015 (Week 15) | Infinite Continued Fractions, First Properties. |
| Wednesday, 18 February 2015 (Week 16) | Irrational Numbers represented by Infinite Continued Fractions, Approximation of Irrational Numbers. |
| Friday, 20 February 2015 (Week 16) | Approximation of Irrational Numbers, Pell's Equation. |
| Wednesday, 25 February 2015 (Week 17) | The Discrete Logarithm Problem (revisit), Background from Group Theory. |
| Friday, 27 February 2015 (Week 17) | The General Discrete Logarithm Problem, Introduction to Elliptic Curves. |
| Wednesday, 4 March 2015 (Week 18) | The Group Structure of Elliptic Curves. |
| Friday, 6 March 2015 (Week 18) | The Group of Rational Points, Elliptic Curves over Finite Fields |
| Wednesday, 11 March 2015 (Week 19) | Elliptic Curves over Finite Fields, Reduction modulo p. |
| Friday, 13 March 2015 (Week 19) | Elliptic Curve Cryptography. |
Problem Sheets (Michaelmas Term):
Problem Sheet 1 (due: Monday Oct 20 at 12:00 in CM116)Problem Sheet 2 (due: Monday Nov 3 at 12:00 in CM116)
Problem Sheet 3 (due: Monday Nov 17 at 12:00 in CM116)
Problem Sheet 4 (due: Monday Dec 1 at 12:00 in CM116)
Problem Sheet 5 (due: Friday Dec 12 at 12:00 in CM116)
Content of Lectures (Michaelmas Term)
| Date | Content |
| Tuesday, 7 October 2014 (Week 1) | Introduction to Elementary Number Theory and Cryptography, Basic properties of the integers and the rationals. |
| Wednesday, 8 October 2014 (Week 1) | The principle of Mathematical Induction, The notion of Divisibility, Division with Remainder. |
| Friday, 10 October 2014 (Week 1) | Examples using Mathematical Induction, The notion of Greatest Common Divisor (gcd) of two integers, Examples calculating the gcd. |
| Wednesday, 15 October 2014 (Week 2) | The Euclidean Algorithm, Definition of Prime and Composite Numbers, Basic Properties of Prime Numbers |
| Friday, 17 October 2014 (Week 2) | The Fundamental Theorem of Arithmetic, Euclid's Proof of the Infiniteness of Prime Numbers, Primes in Arithmetic Progression, Euler's Proof. |
| Wednesday, 22 October 2014 (Week 3) | Distribution of Primes, Congruences and Basic Properties, Residue Classes and Properties. |
| Friday, 24 October 2014 (Week 3) | Examples with Congruences, Linear Congruences, Euler's Phi function |
| Wednesday, 29 October 2014 (Week 4) | Euler's Theorem, Fermat's Little Theorem, Multiplicative Functions |
| Friday, 31 October 2014 (Week 4) | Chinese Remainder Theorem, Multiplicativity of Euler's Phi Function |
| Wednesday, 5 November 2014 (Week 5) | Primality Testing, |
| Friday, 7 November 2014 (Week 5) | Fast Exponentiation, Finding roots modulo n, Introduction to Primitive roots |
| Wednesday, 12 November 2014 (Week 6) | Questions on Primitive roots, The theorem of Lagrange on roots of polynomial modulo p, |
| Friday, 14 November 2014 (Week 6) | A theorem of Gauss on the Phi function, Existencwe of primitive roots modulo a prime p, |
| Wednesday, 19 November 2014 (Week 7) | The Theory of Indices, Main Properties and Examples, |
| Friday, 21 November 2014 (Week 7) | Applications to Cryptography, The Diffie-Hellman Key Exchange Protocol, |
| Wednesday, 26 November 2014 (Week 8) | Public Key Cryptography, RSA, |
| Friday, 28 November 2014 (Week 8) | Examples with RSA, Attacks on RSA (Fermat's Factorization Method), |
| Wednesday, 3 December 2014 (Week 9) | Factoring with high Probability, Introduction to Quadratic Residues |
| Friday, 5 December 2014 (Week 9) | The notions of Quadratic Residues (QR) and Quadratic Non-Residue (NR), Examples and basic Properties. |
| Wednesday, 10 December 2014 (Week 10) | The Legendre Symbol and its Properties, Euler's Criterion and Gauss' Lemma |
| Friday, 12 December 2014 (Week 10) | Proof of Gauss' Lemma and applications, The Quadratic Reciprocity Law. |