Hard lattice problems
Lecturer: Damien Stehlé
In this course, we will overview the state of the art on the
computational hardness of problems on Euclidean lattices.
Euclidean lattices have implications in numerous areas in
computer science, communication theory and computational mathematics.
We will focus in particular on cryptographic applications of
the presumed computational intractability of lattice problems.
- 1- Euclidean lattices: definitions and computational aspects.
- Lattices and lattice invariants.
- The shortest vector problem (SVP).
- The LLL and BKZ algorithms.
- 2- Worst-case lattice problems.
- NP-hardness of the Closest Vector Problem.
- Approx-SVP, Unique-SVP and the Bounded Distance Decoding Problem.
- Limits to inapproximability.
- 3- Average-case lattice problems.
- Gaussian distributions over diverse supports.
- The SIS problem (a.k.a. average-case Approx-SVP).
- The LWE problem (a.k.a. average-case Bounded Distance Decoding).
- Cryptographic applications: encrypting and signing.
The student should be comfortable with algorithms, complexity reductions,
and probabilities. Having followed an introduction to cryptography is