Algorithmic Number Theory

642:587, Fall 2020

Course Info

Instructor: Swastik Kopparty (swastik.kopparty@gmail.com)
Class Time and Place: Tuesdays and Thursdays 1:40 am - 3:00 pm, on Zoom
Office Hours: TBD

IMPORTANT: If you are not registered for the course and want to attend, please email me for the zoom link!

Prerequisites: undergraduate level abstract algebra, number theory, algorithms, graduate level mathematical maturity
References: Lovasz (An Algorithmic Theory of Numbers, Graphs and Convexity), various recent papers available online

Syllabus

This course will be an introduction to basic algorithmic number theory (i.e., designing algorithms for number theoretic problems).

Algorithms for integer/polynomial arithmetic
Primality testing
Integer factorization, classical and quantum algorithms
Lattices
Diophantine approximation
Computing discrete logarithms
Polynomial factorization
Elliptic curve algorithms
Algorithmic problems important for cryptography
Undecidability of solving Diophantine equations

There will problem sets. Every one will have to do a project reading a recent paper and producing a writeup of some important theorem from that paper. A list of potential project topics will be put up here soon.

Lecture Schedule

September 1: course overview, GCDs
September 3: continued fractions, finding the best rational approximation with small denominator
September 10: linear algebra, solving systems of linear equations in integers
September 15: solving systems of linear equations in integers
September 17: solving univariate polynomial equations over the complex numbers
September 22: lattice generalities
September 24: lattice algorithms, 2 dimensional SVP
September 29: technical issues
October 1: LLL lattice reduction
October 6: LLL lattice reduction, factoring polynomials over rationals
October 8: factoring polynomials over the rationals
October 13: root finding over finite fields
October 15: irreducibility testing over finite fields, construction of finite fields
October 20: factoring polynomials over finite fields
October 22: factoring bivariate polynomials, power series solutions
October 27: square roots mod m
October 29: certifying primality, Agrawal-Biswas randomized primality test
November 3: Agrawal-Kayal-Saxena deterministic primality test
November 5: Agrawal-Kayal-Saxena deterministic primality test, discrete log
November 10: Adleman's algorithm for discrete log based on smooth numbers
November 12: Lehmann's deterministic algorithm for factoring integers
November 17: Dixon's algorithm for factoring integers
November 19: FFT, towards fast polynomial multiplication
November 24: fast polynomial multiplication, fast integer multiplication
December 1: Quantum algorithms, towards fast quantum factoring
December 3: fast quantum factoring
December 8: class presentations