Ring Learning With Errors: A crossroads between postquantum cryptography, machine learning and number theory

The present survey reports on the state of the art of the different cryptographic functionalities built upon the ring learning with errors problem and its interplay with several classical problems in algebraic number theory. The survey is based to a certain extent on an invited course given by the author at the Basque Center for Applied Mathematics in September 2018.Comment: arXiv admin note: text overlap with arXiv:1508.01375 by other authors/ comment of the author: quotation has been added to Theorem 5.

Potentially diagonalizable modular lifts of large weight

We prove that for a Hecke cuspform $f\in S_k(\Gamma_0(N),\chi)$ and a prime $l>\max\{k,6\}$ such that $l\nmid N$, there exists an infinite family $\{k_r\}_{r\geq 1}\subseteq\mathbb{Z}$ such that for each $k_r$, there is a cusp form $f_{k_r}\in S_{k_r}(\Gamma_0(N),\chi)$ such that the Deligne representation $\rho_{f_{k_r,l}}$ is a crystaline and potentially diagonalizable lift of $\overline{\rho}_{f,l}$. When $f$ is $l$-ordinary, we base our proof on the theory of Hida families, while in the non-ordinary case, we adapt a local-to-global argument due to Khare and Wintenberger in the setting of their proof of Serre's modularity conjecture, together with a result on existence of lifts with prescribed local conditions over CM fields, a flatness result due to B\"ockle and a local dimension result by Kisin. We discuss the motivation and tentative future applications of our result in ongoing research on the automorphy of $\mathrm{GL}_{2n}$-representations in the higher level case

Modular supercuspidal lifts of weight 22

We prove that for a non CM Hecke cuspform $f\in S_k(\Gamma_0(N))$ and a prime $p>\max\{k+1,6\}$ and $p\nmid N$ such that the residual $p$-adic Deligne representation $\overline{\rho}_f$ is absolutely irreducible and $\mathrm{SL}_2(\mathbb{F}_p)\subseteq \mathrm{Im}(\overline{\rho}_f)$, there exists a modular supercuspidal lift $\rho_g$ with $g\in S_2(Np^2,\epsilon)$ for some Nebentypus character $\epsilon$. We apply this result to correct a mistake in \cite{dieulefait}, where the micro good dihedral prime $p=43$ is introduced to prove the automorphy of $\mathrm{Sym}^5$ of level $1$ modular forms. We also discuss how this result is versatile enough to prove other instances of Langlands functoriality like, for instance, in the safe chains introduced in \cite{luissara} and \cite{GL2GL2GL2}, where the automorphy of tensor products of certain modular or automorphic representations is established

Nonuniform Fuchsian codes for noisy channels

We develop a new transmission scheme for additive white Gaussian noisy (AWGN) channels based on Fuchsian groups from rational quaternion algebras. The structure of the proposed Fuchsian codes is nonlinear and nonuniform, hence conventional decoding methods based on linearity and symmetry do not apply. Previously, only brute force decoding methods with complexity that is linear in the code size exist for general nonuniform codes. However, the properly discontinuous character of the action of the Fuchsian groups on the complex upper half-plane translates into decoding complexity that is logarithmic in the code size via a recently introduced point reduction algorithm