Algorithmic Number Theory
# WS2018 Special Topic Lecture in : Algorithmic Number Theory

## Time and Place:

Every Mon. 16:15-18:00 (starts Oct. 1, 2018)

**Note:** We can adjust the regular schedule in the first meeting, depending on the needs of the registered students.

JKU S2046

View Larger Map

## Lecturer:

Jose Capco

## Content:

- Prime Proving Algorithms (including the deterministic AKS algorithm)
- Prime Decomposition Algorithms (including Lenstra's ECM)
- Basic Theories on Elliptic Curves over Rationals (including proof of Mordell Theorem)
- Elliptic Curve Point Counting Algorithms and Torsion Point Algorithms
- Theory of Heights
- Elliptic Curve Rank Conjectures (including Birch Swinnerton-Dyer Conjecture)

## Course Outline:

- On Rings and Groups
- Carmichael Numbers and Prime Recognition
- Wilson's Theorem and AKS Theorem
- AKS Theorem and AKS Algorithm
- Quadratic Extensions and Square Root mod p
- Sieving B-Smooth Numbers and Basic Quadratic Sieve
- Quadratic Sieves. Elliptic Curve Primer
- Algorithms on Elliptic Curve Group Structure
- Elliptic Curve Factorization Algorithm
- Valuation Theory
- Computing Heights
- Elliptic Curve Rank Conjectures and Elkies Curve
- Birch-Swinnerton-Dyer Conjecture

**References:**
*Algorithmische Zahlentheorie*, **O. Forster**, 2. Auflage, 2015 Springer
*Prime Numbers*, **R. Crandall, C. Pomerance**, Second Edition, 2005 Springer
*A Course in Computational Algebraic Number Theory*, **H. Cohen**, 1996 Springer

## Course Requirement:

Elementary Number Theory. Basic understanding of fields, rings and groups.

Open to all students with mathematical background (including computer science students).

**Note:** First part of the course will have more algorithms than the second part.
Below are the pdf of lecture (late uploads). Please take note of the time/date of the uploaded pdf. I could upload newer versions if I need to correct some mistake or add some additional text to the lectures (you might need to sometimes refresh (press F5) the browser to see the new files).