Lattice-based cryptography: the cryptosystems R/P-LWE, post quantum primitives and homomorphic encryption

Iván Blanco Chacón (Universidad de Alcalá y Aalto University)



30 oct 2023 - 11:11 CET

Profesores UCM responsables del curso: Ignacio Luengo y Mariemi Alonso
Duración: del 23/10/2023-20/11/2023, una sesión semanal de dos horas (total 10 horas).
Fechas: Lunes 23, 30 de octubre y 6, 13 y 20 de noviembre.
Horario: El 23/10 de 18h00 a 20h00. El resto de días de 17h00 a 19h00

Lugar: SEMINARIO 238

Summary:  Within post-quantum cryptography, the lattice-based approach supports three out of the four cryptosystems standardised by the NIST the past July 2022, not to speak that along the different rounds, more than a half of the proposals belonged to this category.

Despite the fact that no formal proof of NP-hardness is given to the mathematical problem which reduces to these cryptosystems, it enjoys a number of advantages, in particular, the reduced size of the keys to a given security threshold, the easiness to implement and deploy, as well as the fact that they support fully homomorphic encryption systems.

This course is an introduction (to a certain depth level though) to the lattice-based cryptography based on the problems Ring Learning With Errors (RLWE) and Polynomial Learning With Errors (PLWE). In particular, we will follow the articles [1], [2] and [3] at some point. One of the key issues of the course is to understand the reduction methods from supposedly hard problems like the Shortest Vector Problem with certain approximation factors.

The student is supposed to have followed a course on Galois Theory, and it is very advisable that the student has taken a course on Algebraic Number Theory, or has been exposed to this circle of ideas. We will give a brief summary of the required notions in any case.


[1] V. Lyubashevski, C. Peikert, O. Regev: On ideal lattices and Learning With Errors over Rings. Advances in Cryptology-EUROCRYPT 2010, 1-23 (2010)
[2] M. Rosca, D. Stehlé, A. Wallet: On the Ring-LWE and P-LWE problems. Advances in Cryptology-EUROCRYPT 2018, 146-173 (2010)
[3] I. Blanco-Chacón: Ring Learning With Errors: a crossroads between post-quantum cryptography, machine learning and number theory. Bulletin of the Irish mathematical Society, 86, 17-46 (2020)