27 research outputs found
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 and a prime
such that , there exists an infinite family
such that for each , there is a
cusp form such that the Deligne
representation is a crystaline and potentially
diagonalizable lift of . When is -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 -representations in the
higher level case
Modular supercuspidal lifts of weight
We prove that for a non CM Hecke cuspform and a prime
and such that the residual -adic Deligne
representation is absolutely irreducible and
, there
exists a modular supercuspidal lift with for
some Nebentypus character . We apply this result to correct a mistake
in \cite{dieulefait}, where the micro good dihedral prime is introduced
to prove the automorphy of of level 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