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Computational Number Theory and Algebra

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)

Lecture 17 - Multivariate Polynomial Factorization

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Lecture 01 - Introduction: Computation and Algebra

Lecture 02 - Basic Complexity Notation

Lecture 03 - GCD Algorithm and Chinese Remainder Theorem

Lecture 04 - Fast Polynomial Multiplication

Lecture 05 - Fast Polynomial Multiplication (cont.)

Lecture 06 - Fast Integer Multiplication and Division

Lecture 07 - Fast Integer Arithmetic and Matrix Multiplication

Lecture 08 - Matrix Multiplication Tensor

Lecture 09 - Polynomial Factoring over Finite Fields: Irreducibility Testing

Lecture 10 - Equi-degree Factorization and Idea of Berlekamp's Algorithm

Lecture 11 - Berlekamp's Algorithm as a Reduction Method

Lecture 12 - Factoring over Finite Fields: Cantor-Zassenhaus Algorithm

Lecture 13 - Reed Solomon Error Correcting Codes

Lecture 14 - List Decoding

Lecture 15 - Bivariate Factorization - Hensel Lifting

Lecture 16 - Bivariate Polynomial Factoring

Lecture 17 - Multivariate Polynomial Factorization

Lecture 18 - Multivariate Factoring - Hilbert's Irreducibility Theorem

Lecture 19 - Multivariate Factoring (cont.)

Lecture 20 - Analysis of LLL Algorithm

Lecture 21 - Analysis of LLL Algorithm (cont.)

Lecture 22 - Analysis of LLL-reduced Basis Algorithm and Introduction to NTRU Cryptosystem

Lecture 23 - NTRU Cryptosystem (cont.) and Introduction to Primality Testing

Lecture 24 - Randomized Primality Testing: Solovay-Strassen and Miller-Rabin Tests

Lecture 25 - Deterministic Primality Test and RSA Cryptosystem

Lecture 26 - Integer Factoring: Smooth Numbers and Pollard's RHO Method

Lecture 27 - Pollard's P-1, Fermat, Morrison-Brillhart, Quadratic and Number Field Sieve Methods