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

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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).

Name	Last modified	Size

Parent Directory		-
codes/	23-Oct-2018 13:02	-
compiled_exercises.pdf	05-Feb-2019 14:26	45K
lecture_01.pdf	04-Feb-2019 11:27	53K
lecture_02.pdf	23-Oct-2018 12:38	45K
lecture_03.pdf	23-Oct-2018 12:37	44K
lecture_04.pdf	23-Oct-2018 14:29	51K
lecture_05.pdf	01-Nov-2018 17:41	55K
lecture_06.pdf	07-Dec-2018 21:12	55K
lecture_07.pdf	07-Jan-2019 08:28	55K
lecture_08.pdf	24-Nov-2018 18:16	56K
lecture_09.pdf	05-Feb-2019 15:09	67K
lecture_10.pdf	10-Dec-2018 15:09	44K
lecture_11.pdf	31-Dec-2018 15:18	50K
lecture_12.pdf	20-Jan-2019 11:12	52K
lecture_13.pdf	24-Jan-2019 13:09	52K
lecture_14.pdf	24-Jan-2019 13:19	47K
lecture_15.pdf	03-Feb-2019 13:48	64K