Computational Number Theory and Algebra

By Prof. Nitin Saxena | IIT Kanpur

Learners enrolled: 99

ABOUT THE COURSE :
Algebra plays an important role in both finding algorithms, and understanding the limitations of computation. This course will focus on some of the fundamental algebraic concepts that arise in computation, and the algebraic algorithms that have applications in real life. The course will cover the problems of fast integer (or polynomial) multiplication (or factoring), fast matrix multiplication, primality testing, computing discrete logarithm, error-correcting codes, lattice- based cryptography, etc. The course intends to introduce both basic concepts and practical applications.

INTENDED AUDIENCE : Computer Science & Engineering, Mathematics, Electronics, Physics, & similar disciplines.

PREREQUISITES : Preferable (but not necessary)-- Theory of Computation, Algorithms, Algebra

INDUSTRIES SUPPORT : Cryptography, Coding theory, Computer Algebra, Symbolic Computing Software, Cyber Security, Learning Software

Summary

Course Status :	Upcoming
Course Type :	Elective
Language for course content :	English
Duration :	12 weeks
Category :	Computer Science and Engineering Mathematics
Credit Points :	3
Level :	Undergraduate/Postgraduate
Start Date :	20 Jan 2025
End Date :	11 Apr 2025
Enrollment Ends :	27 Jan 2025
Exam Registration Ends :	14 Feb 2025
Exam Date :	27 Apr 2025 IST

Note: This exam date is subject to change based on seat availability. You can check final exam date on your hall ticket.

Page Visits

Course layout

Week 1:Outline. Notation. Background.

Week 2:GCD. Chinese remaindering. Fast polynomial multiplication.

Week 3: Fast integer multiplication. Fast integer division. Fast gcd.

Week 4:Fast matrix multiplication. Tensor rank.

Week 5:Factorization over finite fields.

Week 6:Berlekamp, Cantor-Zassenhaus factoring algorithms.

Week 7: Reed-Solomon code. List decoding. Bivariate polynomial factoring.

Week 8:Kaltofen's blackbox multivariate factoring.

Week 9:Integral polynomial factoring. LLL algo. Shortest vector in lattice.

Week 10:Lattice-based cryptography.

Week 11: Primality testing. RSA cryptosystem. Diffie-Hellman. Discrete Log.

Week 12:Integer factoring. Pollard, Fermat, Morrison-Brillhart, Quadratic sieve methods.

Books and references

https://www.cse.iitk.ac.in/users/nitin/courses/CS681-2019-20-II/index.html

Instructor bio

Prof. Nitin Saxena

IIT Kanpur

I completed my Bachelors in Computer Science from the Indian Institute of Technology, Kanpur in 2002 and completed my PhD under Manindra Agrawal in 2006. I am broadly interested in Computational Complexity Theory, Algebra, Geometry and Number Theory. I have been a visiting graduate student in Princeton University (2003-2004) and National University of Singapore (2004-2005); a postdoc at CWI, Amsterdam (2006-2008) and a Bonn Junior Fellow (W2 Professor) at Hausdorff Center for Mathematics, Bonn (2008-2013). Since April 2013, I have a faculty position in the department of CSE, IIT Kanpur.

Course certificate

The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.

The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).

Date and Time of Exams: April 27, 2025 Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.

Registration url: Announcements will be made when the registration form is open for registrations.

The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.

Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.

CRITERIA TO GET A CERTIFICATE

Average assignment score = 25% of average of best 8 assignments out of the total 12 assignments given in the course.
Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

Please note that assignments encompass all types (including quizzes, programming tasks, and essay submissions) available in the specific week.

YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF AVERAGE ASSIGNMENT SCORE >=10/25 AND EXAM SCORE >= 30/75. If one of the 2 criteria is not met, you will not get the certificate even if the Final score >= 40/100.

Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IIT Kanpur .It will be e-verifiable at nptel.ac.in/noc.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.

- NPTEL team

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