# CSE206A: Lattices Algorithms and Applications (Fall 2021)

Schedule: Tue,Thu 11:00am-12:20am in CSE 4258
Instructor: Daniele Micciancio (CSE4214)

## Course description

Point lattices are powerful mathematical objects that can be 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, computational complexity, mathematical techniques, and applications to cryptography. Specific topics touched by the course include:

• Algorithms: approximation algorithms for the efficient solution of lattice approximation problems, exponential time algorithms for the exact solution of NP-hard lattice problems.
• Lattice based cryptanalysis: using lattice algorithms to break cryptographic functions.
• Computational Complexity: NP-hardness, reductions, connection between average-case and worst-case complexity
• Lattice Based Cryptography: the design of cryptographic functions that are as hard to break as solving hard lattice problems.
• Advanced mathematical techniques: Fourier/Harmonic analysis methods for the study of lattice problems. Lattice problems from Algebraic Number Theory and their application to the design of very efficient cryptographic functions.

## Prerequisites and Coursework

The main prerequisites for the course are general mathematical maturity, knowledge of basic mathematics (good linear algebra and probability theory, basic abstract algebra, and a little bit of calculus) and introductory level algorithms and complexity theory (mathematical models of computation, analysis of algorithms, polynomial time solvability, NP-hardness, etc.) Some prior knowledge of cryptography is useful, but not strictly required. No prior knowledge of advanced complexity theory, Fourier analysis, or algebraic number theory is assumed, but you should be prepared to learn a bit about all this through the course.

Coursework for students enrolled in the course will include a substantial amount of reading, 2 or 3 individual homework assignments, and a team project to be executed in groups of 2 or 3 students. Projects should be discussed with the instructor by the end of the 5th week of classes, and will be due during the 10th week. Each group should submit a written report, and possibly deliver a short presentation.

## Homework

Latex template for lecture notes.

The course will be based on a combination of lecture notes, research papers and other online material as posted below. The list will be updated as the course moves forward. You can find information about additional topics to be covered on the web pages of previous editions of this course (Fall 2019, Fall 2017, Winter 2016, Spring 2014, Winter 2012, Winter 2010, Spring 2007, Winter 2002)

A collection of Links to papers and other resources on lattice cryptography and algorithms (not very actively) maintained by Daniele Micciancio, mostly for his own personal use.

### Implementations and Libraries:

The two main libraries implementing the lattice reduction algorithms studied in this course are:

• fpLLL: state of the art impementation of lattice reduction and related algorithms using fast floating point arithmetics.
• Number Theory Library (NTL). A C++ library for lattice basis reduction, polynomial arithmetics and other algebraic problems.

There are many more libraries implementing cryptographic functions based on lattices, including:

• PALISADE. A general purpose C++ library for lattice cryptography.
• Lambda o lambda (Lol). A high level Haskell library for lattice based cryptography. See also paper.
• NFLlib. An NTT-based Fast Lattice Library for computations in power-of-two cyclotomic rings.
• HElib. A library for Fully Homomorphic Encryption.
• RingLWE C++ library: A C++implementation of the RingLWE Toolkit of Lyubashevsky, Peikert and Regev, as described in this MS Thesis.