Lecture 4: Understanding Lattices in Cryptography 🔐

Video lectures for Alfred Menezes's introductory course on the mathematics of lattice-based cryptography. Kyber (ML-KEM) and Dilithium (ML-DSA) are lattice-based cryptosystems that are resistant to attacks by quantum computers. Hardness of the Module-SIS and Module-LWE problems is the basis for the security of Kyber and Dilithium . These lectures describe the connection between lattices and the Module-SIS and Module-LWE problems, thus explaining why Kyber and Dilithium are considered to be lattice-based cryptosystems.

Topics covered: Definition of a lattice, fundamental parallelepiped, bases of a lattice, volume of a lattice, successive minima, LLL lattice basis reduction algorithm, shortest vector problem (SVP), shortest independent vectors problem (SIVP)

Lecture playlist:
https://www.youtube.com/playlist?list=PLA1qgQLL41STNFDvPJRqrHtuz0PIEJ4a8

Course web page: https://cryptography101.ca/lattice-based-cryptography/
The slides are available on the course web page.

Other cryptography courses: https://cryptography101.ca

Slides
00:00 Introduction
00:36 Slide 37: Lattice definition
01:56 Slide 38: Full-rank lattices
02:53 Slide 39: Example 1
03:40 Slide 40: Fundamental parallelepiped
04:31 Slide 41: Example 2
05:36 Slide 42: Example 3
06:26 Slide 43: One base is "nicer" than the other
06:53 Slide 44: Example 4
07:37 Slide 45: A lattice has infinitely many bases
08:55 Slide 46: Proof of the characterization of lattice bases
10:24 Slide 47: Volume of a lattice
11:02 Slide 48: Some bases are nicer than others
12:17 Slide 49: Successive minima
14:42 Slide 50: LLL lattice basis reduction algorithm
15:56 Slide 51: Cryptanalytic applications of LLL
17:16 Slide 52: SVP: a fundamental lattice problem
19:37 Slide 53: SIVP: another fundamental lattice problem