Introduction
Underneath many applications in real life like robotics, crytography, error correcting codes etc are certain basic computational tasks like gcd computation, factoring etc. This course is a study on such computational problems and the algorithms for them. We build on the necessary background from algebra and use it to give rigorous analysis of the performance of these algorithms.
Contents
The following topics will be covered in this course

Basic algorithms for arithmetic, addition, multiplication and modular arithmetic. (4 lectures)

Euclid's algorithm, Chinese Remainder theorem and applications to problems like determinant computation, Gaussian elimination. (6 lectures)

Fields, Finite fields, Field of rationals, Polynomial factorisation over finite fields (6 lectures)

Hensel lifting and application to bivariate factoring (4 lectures)

Shortest vector problem and LLL algorithm. Application to factorisation of polynomial over rationals. (5 lectures)

Applications of polynomial factorisation. (5 lectures)

Multivariate system of polynomial equation and Grobner basis (7 lectures)

Algorithms from number theory and cryptography (5 lectures).
Prerequisites
None. The course will start from the very basics.
Learning Outcomes

To state and analysis algorithms for tasks like arithmetic, factorisation.

To apply the algorithms learned to tasks in cryptography, number theory etc.
Textbooks
 Modern Computer Algebra. Joachim von zur Gathen and Jurgen Gerhard. Cambridge University press, 2013. ISBN: 9780521826464
References

Algorithmic Algebra. Bhubaneswar Mishra. Springer 1993, ISBN10: 0387940901, ISBN13: 9780387940908

Ideal, Varities and Algorithms. David A. Cox, John Little, Donal Oâ€™Shea. Springer 2008. ISBN: 9783319167213

An Introduction to Grobner Bases. Philippe Loustaunau, William W. Adams. American Mathematical Society, 1994 ISBN10: 0821838040, ISBN13: 9780821838044
Past Offerings
 Offered in JanMay, 2023 by Piyush P. Kurur
Course Metadata
Item  Details 

Course Title  Computational Algebra and Number Theory 
Course Code  CS5634 
Course Credits  3003 
Course Category  PME 
Proposing Faculty  Piyush P Kurur 
Approved on  Senate 22 of IIT Palakkad 
Course status  New 