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

Lecture 06 - Fast Integer Multiplication and Division

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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