Course: Algorithmic Algebra

Final Exam Questions are up!

Registration Information.

Course Number and Index

Computer Science : 16:198:672:01 (index 63411)
Mathematics : 642:612:01 (index 72298)

Timing: Tuesdays and Thursdays, 6:40 PM to 8:00 PM.
Room Number: Hill 425.

Instructor

Neeraj Kayal.

Office Address:
Room 419
DIMACS Center
CoRE Building
Rutgers University
Piscataway, NJ 08854. USA.

Phone: +1-732-445-4577 (office),
+1-732-425-4533 (cell)
Email: kayaln at dimacs dot rutgers dot edu
Office hours: Fri 3pm-5pm

Lecture Notes

Lecture 1: Introduction to computational complexity, Integer Addition and Integer Multiplication. ( pdf ) ( jnt )
Lecture 8: Review of Linear Algebra. Scribe: Raga Velagapudi. ( pdf )

Homework Assignments

Assignment One. (pdf) Solution.
Assignment Two. (pdf) Solution.
Assignment Three. (pdf)
Assignment Four. (pdf)
Final Exam. (pdf) . (PS version is here ).

Overview

Computational problems come in two flavours - combinatorial or algebraic. Combinatorial problems lack structure and thus lead to intractability fairly quickly. Algebraic problems admit a lot of structure and often admit surprisingly efficient algorithms. In this course we will look at some basic concepts from algebra and the computational problems that arise naturally out of those concepts. We will see the algorithms used to solve these problems and the applications of these algorithms to some real-world problems. Indeed understanding and exploiting this structure is the key in many subdisciplines of computer science such as coding theory, cryptography and even subareas of combinatorics involving explicit constructions of combinatorially nice objects.

The course will consist of a series of topics; each topic will cover either one algebraic concept such as group, ring or field and a related algorithm or an algorithm and its application to a real world problem. Students are not expected to have any extensive background with either algebra or complexity theory. The initial lectures and assignments will allow the students to become familiar with the requisite algebra and notions from algorithmic complexity.

Goals

The course is designed with the following goals in mind:

Become familiar with some basic notions in algebra such as polynomials and finite fields.
Know the basic techniques used in devising efficient algorithms for problems such as finding the roots of a polynomial.
See some real-world applications of these algorithms.

Syllabus

Review of modular arithmetic, fields, rings and groups.
Motivating Application: Fast sequential and parallel multiplication.
Review of linear algebra: matrices and their eigenvalues.
Motivating Application: Fast parallel matrix inversion
Review of polynomials and their properties.
Motivating Application: Reed Solomon codes and Welch-Berlekamp decoding
Polynomial Identity Testing.
Motivating Application: Perfect matching in parallel.
Primality Testing and Integer Factorization.
Motivating Application: RSA cryptosystem
Factoring polynomials over finite fields.
Motivating Application: List decoding of Reed Solomon Codes
Hilbert's Nullstellensatz and Bezout's Theorem.
Motivating Application: Arithmetic circuit lower bounds
Decidability of the first order theory of reals.
Motivating Application: Decidability of theorems in euclidean geometry.

Schedule

References

Basic Algebra Text: I.N. Herstein's book Topics in Algebra.
Polynomial Factoring: Madhu Sudan's course notes on Algebra and Computation.
Victor Shoup's textbook: A computational introduction to number theory and algebra.
Primality and factoring: Manindra Agrawal's course notes on Computational Number Theory.
Algebraic Circuits: Avi Wigderson's lecture notes on arithmetic complexity.

Prerequisites

An undergraduate course on algebra or linear algebra.
An undergraduate course on algorithms.

Expected Workload

About 5 homework assignments. A take-home mid-term exam and an open notes end-sem exam of about three hours.

Grading

Homeworks (30%)
Mid-term (30%)
Final exam (40%)