M8190 Number Theoretic Algorithms

Faculty of Science
autumn 2021
Extent and Intensity
2/2/0. 6 credit(s). Type of Completion: zk (examination).
Teacher(s)
prof. RNDr. Radan Kučera, DSc. (lecturer)
Mgr. Pavel Francírek, Ph.D. (assistant)
Guaranteed by
prof. RNDr. Radan Kučera, DSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Thu 16:00–17:50 M6,01011
  • Timetable of Seminar Groups:
M8190/01: Fri 10:00–11:50 M6,01011, R. Kučera
Prerequisites
M3150 Algebra II or M6520 Elementary numbe theory
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
The aim of the lecture is to show that results of number theory can help to factorize a given large positive integer into the product of prime numbers. The importance of this task grows up because of applications in coding theory.
Learning outcomes
At the end of this course, students should be able to explain basic ideas of explained algorithms.
Syllabus
  • (1) Compositeness tests: Fermat test, Carmichael numbers, Rabin-Miller test.
  • (2) Primality tests: the Poclington-Lehmer n-1 test, the elliptic curve test.
  • (3) Agarwal-Kayal-Saxena test
  • (4) Factoring: Lehmann's method, Pollard's $\rho$ method, Pollard's p-1 method, the continued fraction method, the elliptic curve method, the quadratic sieve method, the number field sieve method.
Literature
  • COHEN, Henri. A Course in Computational Algebraic Number Theory. Springer-Verlag, 1993, 534 pp. Graduate Texts in Mathematics 138. ISBN 3-540-55640-0. info
Teaching methods
Lectures: theoretical explanation of necessary mathematical background, applications of the theory to construction of concrete algorithms. Exercises: solving problems with the aim to understand basic concepts and theorems, programming some algorithms.
Assessment methods
Examination consists of two parts: a written test and an oral examination. In the written part the students have to show that they are able to use the explained mathematical background (it is necessary to get at least 50% of points), in the oral part to prove the ability to explain the basic ideas of explained algorithms.
Language of instruction
Czech
Further comments (probably available only in Czech)
Study Materials
The course is taught once in two years.
Teacher's information
http://www.math.muni.cz/~kucera/texty/ATC10.pdf
The lessons are usually in Czech or in English as needed, and the relevant terminology is always given with English equivalents. The target skills of the study include the ability to use the English language passively and actively in their own expertise and also in potential areas of application of mathematics. Assessment in all cases may be in Czech and English, at the student's choice.
The course is also listed under the following terms Spring 2008 - for the purpose of the accreditation, Spring 2004, Spring 2006, Spring 2008, Spring 2010, Spring 2012, spring 2012 - acreditation, Spring 2014, Spring 2016, spring 2018, Spring 2020, Autumn 2023.
  • Enrolment Statistics (autumn 2021, recent)
  • Permalink: https://is.muni.cz/course/sci/autumn2021/M8190