Beyond the Bootcamp:
CS 294-168 Lattices, Learning with Errors and Post-Quantum Cryptography
Course Description
The study of integer lattices, discrete additive subgroups of Rn, serves as a bridge between number theory and geometry and has for centuries received the attention of illustrious mathematicians including Lagrange, Gauss, Dirichlet, Hermite and Minkowski.The role of lattices in cryptography has been revolutionary and dramatic. In the 80s and the early 90s, lattices served as a destructive force, giving the cryptanalysts some of their most potent attack tools.
In the last two decades, the Learning with Errors (LWE) Problem, whose hardness is closely related to lattice problems, has revolutionized modern cryptography by giving us (a) a basis for "post-quantum" cryptography, (b) a dizzying variety of cryptographic primitives such as fully homomorphic encryption and signatures, attribute-based and functional encryption, a rich set of pseudorandom functions, various types of program obfuscation and much more; and finally, (c) a unique source of computational hardness with worst-case to average-case connections. This course explores the various facets of lattices, the LWE problem and their applications in cryptography.
The course is intended to complement the Spring'20 Simons Institute program on Lattices. In particular, to get more out of the course, the students are encouraged to attend the bootcamp and the workshops of the program.
Prerequisites: This is an advanced course, intended for beginning graduate students and upper level undergraduates in CS and Math. Mathematical maturity and comfort with linear algebraic notions is the most important pre-requisite, followed by courses in the theory of computation (CS 172 or equivalent) and cryptography (CS 171 or CS 276).
Course Information
INSTRUCTOR | Vinod Vaikuntanathan
E-mail: vinodv at berkeley dot edu |
LOCATION | Soda 320
|
TIME | M 5pm - 7:30pm (with a break in between),
Office Hours by appointment |
TEXTBOOK | There are no required textbooks. Instead, we will use lecture notes and papers from the references listed below, as well as instructor's notes. |
GRADING | Based on (a) class participation (come to class; attend a talk of your choice in each of the three workshops, and write a short email to the instructor with a review of the talk, including a summary, what you liked/disliked and any ideas you got from watching it) and (b) a final project (either come up with a new result -- talk to the instructor for project ideas -- or write a short survey on a topic that will be useful to future researchers). |
Announcements
- The first class is on Monday Jan 27, 2020.
- Starting March 15, we are moving the class online. Lectures will be on zoom.
- Lecture notes for the class are here (all in one file).
Schedule (tentative and subject to change)
Lecture | Topic | References |
Lecture 1 (Jan 27) | Introduction to SIS and LWE. Basic properties and cryptographic applications: public and private-key encryption and collision-resistant hashing. | Instructor notes |
Lecture 2 (Feb 3) | Algorithms for LWE. Introduction to Lattices. |
Instructor notes References |
Lecture 3 (Feb 10) | Smoothing Parameter and Worst-case to Average-case Reduction for SIS. |
Instructor notes References |
Feb 17 | President's Day (No Class) | Lecture 4 (Feb 24) | Reductions for LWE (Worst-to-average case, Search-to-decision) |
Instructor notes (updated 3/2) References |
Lecture 5 (Mar 2) | Pseudorandom Functions and Learning with Rounding. |
Instructor notes References |
Lecture 6 (Mar 9) |
Lattice Trapdoors and Discrete Gaussian Sampling. Digital Signatures. |
Instructor notes References |
Mar 16 | Class canceled | |
Mar 23 | Spring Break (No Class) | Lecture 7 (Mar 30) | Identity-based Encryption and Hierarchical IBE. |
Instructor notes References |
Lecture 8 (Apr 6) | Fully Homomorphic Encryption. The key equation and applications to attribute-based encryption, fully homomorphic signatures and constrained pseudorandom functions. |
Instructor notes References |
Lecture 9 (Apr 13) | The GGH15 Multilinear Map and Applications. |
Instructor notes References |
Lecture 10 (Apr 20) | Ideal Lattices and Ring Learning with Errors. |
Instructor notes References |
Lecture 11 (Apr 27) | AMA on lattices with Vinod. | Lecture 12 (May 4) | Lattices and Quantum Computing. |
Instructor notes References |
References
Courses Elsewhere:- A precursor of this course at MIT: Fall 2015 and Fall 2017 and Fall 2018.
- Oded Regev's course at Tel-Aviv University.
- Daniele Micciancio's course at UCSD.
- Cynthia Dwork's course at Stanford.
- Chris Peikert's course at Michigan/Georgia Tech.
Lecture 2
- The Arora-Ge paper: https://users.cs.duke.edu/~rongge/LPSN.pdf
- The original BKW paper (with an algorithm for LPN): https://arxiv.org/abs/cs/0010022
- Adaptation of BKW for LWE: https://eprint.iacr.org/2012/636.pdf
- Further improvements on BKW-LWE: https://arxiv.org/abs/1506.02717
- Lattice attacks against LWE (Lindner-Peikert): https://web.eecs.umich.edu/~cpeikert/pubs/lwe-analysis.pdf
Lecture 3
- Regev's lecture notes.
- Micciancio-Regev paper.
- Gentry-Peikert-Vaikuntanathan that improves the modulus q in the SIS reduction.
- Further improvements by Micciancio and Peikert.
- Forthcoming: Brakerski-Stephens-Davidowitz-Vaikuntanathan on exponential reductions from $\sqrt{n}$-approximate SIVP to $k$-SUM, a parameterized version of SIS.
Lecture 4
Lecture 5
- GGM (Goldreich-Goldwasser-Micali) paper.
- BPR12 (Banerjee-Peikert-Rosen) paper.
- BLMR (Boneh-Lewi-Montgomery-Raghunathan) paper.
- BP14 (Banerjee-Peikert) paper.
Lecture 6
- Ajtai99 paper on trapdoor sampling.
- Gentry-Peikert-Vaikuntanathan Gaussian sampling and signatures.
- Micciancio-Peikert trapdoor sampling paper.
Lecture 7
- Gentry-Peikert-Vaikuntanathan for IBE.
- Cash-Hofheinz-Kiltz-Peikert for standard model IBE.
- Agrawal-Boneh-Boyen for (a different) standard model IBE.
- Another ABB paper on hierarchical IBE.
Lecture 8
- Gentry-Sahai-Waters FHE Scheme.
- Boneh-Gorbunov-Gentry-Halevi-Nikolaenko-Segev-Vaikuntanathan-Vinayagamurthy (BGG+) ABE Scheme.
- Brakerski-Vaikuntanathan Constrained PRF.
- Gorbunov-Vaikuntanathan-Wichs Fully Homomorphic Signature.
- Hoeteck Wee's excellent tutorial on the key equation.
Lecture 9
- Gentry-Gorbunov-Halevi15 Multilinear maps.
- Canetti-Chen17.
- Chen-Vaikuntanathan-Wee18.
- Goyal-Koppula-Waters18 and Wichs-Zirdelis18.
Lecture 10
- NTRU paper.
- Micciancio cyclic lattices paper.
- Peikert-Rosen and Lyubashevsky-Micciancio on Ring-SIS.
- Lyubashevsky-Peikert-Regev 2010 Ring-LWE paper.
Lecture 12
- Kuperberg's Algorithm: Algorithm for dihedral HSP.
- Kuperberg's better algorithm
- Regev's "Quantum Computation and Lattice Problems" paper: Reduction from LWE to dihedral HSP.
- A modern version of Regev's reduction