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Description
Computational Number Theory and Algebra (Prof. Nitin Saxena, IIT Kanpur). Instructor: Prof. Nitin Saxena, Department of Mathematics, IIT Kanpur.
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. (from nptel.ac.in)
Course Curriculum
- Lecture 01 – Introduction: Computation and Algebra Unlimited
- Lecture 02 – Basic Complexity Notation Unlimited
- Lecture 03 – GCD Algorithm and Chinese Remainder Theorem Unlimited
- Lecture 04 – Fast Polynomial Multiplication Unlimited
- Lecture 05 – Fast Polynomial Multiplication (cont.) Unlimited
- Lecture 06 – Fast Integer Multiplication and Division Unlimited
- Lecture 07 – Fast Integer Arithmetic and Matrix Multiplication Unlimited
- Lecture 08 – Matrix Multiplication Tensor Unlimited
- Lecture 09 – Polynomial Factoring over Finite Fields: Irreducibility Testing Unlimited
- Lecture 10 – Equi-degree Factorization and Idea of Berlekamp’s Algorithm Unlimited
- Lecture 11 – Berlekamp’s Algorithm as a Reduction Method Unlimited
- Lecture 12 – Factoring over Finite Fields: Cantor-Zassenhaus Algorithm Unlimited
- Lecture 13 – Reed Solomon Error Correcting Codes Unlimited
- Lecture 14 – List Decoding Unlimited
- Lecture 15 – Bivariate Factorization – Hensel Lifting Unlimited
- Lecture 16 – Bivariate Polynomial Factoring Unlimited
- Lecture 17 – Multivariate Polynomial Factorization Unlimited
- Lecture 18 – Multivariate Factoring – Hilbert’s Irreducibility Theorem Unlimited
- Lecture 19 – Multivariate Factoring (cont.) Unlimited
- Lecture 20 – Analysis of LLL Algorithm Unlimited
- Lecture 21 – Analysis of LLL Algorithm (cont.) Unlimited
- Lecture 22 – Analysis of LLL-reduced Basis Algorithm and Introduction to NTRU Cryptosystem Unlimited
- Lecture 23 – NTRU Cryptosystem (cont.) and Introduction to Primality Testing Unlimited
- Lecture 24 – Randomized Primality Testing: Solovay-Strassen and Miller-Rabin Tests Unlimited
- Lecture 25 – Deterministic Primality Test and RSA Cryptosystem Unlimited
- Lecture 26 – Integer Factoring: Smooth Numbers and Pollard’s RHO Method Unlimited
- Lecture 27 – Pollard’s P-1, Fermat, Morrison-Brillhart, Quadratic and Number Field Sieve Methods Unlimited
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