# CSE206A: Lattices Algorithms and Applications (Winter 2010)

Istructor: Daniele Micciancio

Section ID: 672185
Schedule: TuTh 2:00pm-3:20pm in CENTER 201

## Course description

Point lattices are powerful mathematical objects that have been used to efficiently solve many important problems in computer science, most notably in the areas of cryptography and combinatorial optimization. This course gives a general introduction to the theory of point lattices, their algorithms, and a selection of applications of lattices to both cryptography and other areas of computer science, mathematics, and communication theory. Beside covering the best currently known algorithms to solve the most important lattice problems (both in their exact and approximate versions), we will touch several related areas:
• Combinatorial Optimization: The asymptotically fastest Integer Programming algorithm known to date is based on lattices.
• Cryptography: Lattices have been used to design a wide range of cryptographic primitives, including public key encryption, digital signatures, encryption resistant to key leakage attacks, identity based encryption, and fully homomorphic encryption.
• Complexity: Cryptographic applications of lattices have been in large part stimulated by a connection between the average-case and worst-case complexity of lattice problems that is very interesting on its own.
• Harmonic Analysis: n-dimensional Fourier analysis is one of the most powerful tools currently used in the study of lattices, both in algorithms, complexity and mathematics.
• Algebraic Number Theory: Lattices have been used to algorithmically solve many problems in algebraic number theory, and algebraic number theory is an imporant tool in the design of several lattices with special properties that turn out to be very useful in applications.

Prerequisites: The main prerequisites for the course are knowledge of basic math (e.g., linear algebra, finite fields, modular arithmetic, probability, some calculus, etc.) and introductory level algorithms and complexity theory (analysis of algorithms, polynomial time solvability, NP-hardness, etc.) In particular, no prior knowledge of cryptography, advanced complexity theory, Fourier analysis, or algebraic number theory is assumed. (Though in this course you will learn a little bit of all of this.)

## Lecture Notes

I am planning to cover a different selection of application from previous versions of this course. Revised lecture notes will be posted here as we cover the material in class. As a reference, see pointers to previous lecture notes and other courses in the externatl links section.

## Coursework

Coursework for students enrolled in the course will include 4 homework assignments, and an optional term project, and scribing one set of lecture notes if needed.

• Homework 2: due Feb 16 in class, hw2.pdf.
• Homework 1: due Jan 21 in class, hw1.pdf. If you need hints about the last problem, check out the first homework assignment from Spring 2007, where essentially the same problem is broken into parts giving out a solution guideline.

## External Links

### Schedule

 Date Class Topic Jan 5 Basic definitions, Gram-Schmidt orthogonalization, minimum distance Jan 7 Succesive minima, Minkowski's convex body theorem Jan 12 Basic Algorithms: Running time of Gram-Schmidt, Hermite Normal Form, Dual lattice Jan 14 Lattice approximation algorithms, LLL algorithm, nearest plane algorithm Jan 19 Cancelled Jan 21 Running time of LLL, Approximating CVP Jan 26 Integer Programming Jan 28 Cryptanalysis Feb 2 Approximating SVP within subexponential factors. Feb 4 Decoding algorithms for special lattices Feb 9 Exact algorithms for SVP Feb 11 Exact algorithms for CVP Feb 16 Feb 18 Feb 23 Feb 25 Mar 2 Mar 4 Mar 9 Mar 11 Daniele Micciancio
Last Modified: April 5th, 2007