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Iván Blanco-Chacón

Publications and source records attributed to Iván Blanco-Chacón.

17 recordsLinked to original sources
arXiv2026

On the computation of base-change lifts and lifts of Hida families

We derive an explicit formula for the Hecke eigenvalues of a Hilbert modular form which is a base-change lift of a classical newform to a totally real number field. We show that for a totally real Galois number field $F$ the $L$-function of a base-change lifted form can be factorized as a product of twists of the $L$-function of the underlying classical form over irreducible representations of $\mathrm{Gal}(F / \mathbb{Q})$. Moreover, we use the formula for the Hecke eigenvalues of a base-change lift to prove the existence of a base-change lift of a Hida family. In particular, we show that a Hida family of classical Hecke eigenforms can be lifted to a formal power series that specializes to the base-change lifts of the Hida family of classical cusp forms.

math.NT↗
arXiv2026

Extensions of root-based attacks against PLWE via isomorphisms of rings

In the present work we address some key questions regarding the generalization of root-based attacks presented in [2]. In particular, we analyze potential root-based attacks extensions via the construction of explicit isomorphisms from vulnerable instances, and provide a formal proof that this approach will not yield any new vulnerabilities. To do so, we first construct an explicit isomorphism between fully-split polynomial rings and polynomial rings where previous attacks apply and show that the application of such an isomorphism will always distort the samples in a way that the resulting samples cannot be used to distinguish. Then, we prove that any isomorphism between fully-split polynomial rings must be of the form of the constructed isomorphism. Lastly, we study the generalization beyond fully split settings and show that the same distortion happens, regardless of the degree of the field extension.

cs.CR↗
arXiv2025

Cryptanalysis of PLWE based on zero-trace quadratic roots

We extend two of the attacks on the PLWE problem presented in (Y. Elias, K. E. Lauter, E. Ozman, and K. E. Stange, Ring-LWE Cryptography for the Number Theorist, in Directions in Number Theory, E. E. Eischen, L. Long, R. Pries, and K. E. Stange, eds., vol. 3 of Association for Women in Mathematics Series, Cham, 2016, Springer International Publishing, pp. 271-290) to a ring $R_q=\mathbb{F}_q[x]/(f(x))$ where the irreducible monic polynomial $f(x)\in\mathbb{Z}[x]$ has an irreducible quadratic factor over $\mathbb{F}_q[x]$ of the form $x^2+ρ$ with $ρ$ of suitable multiplicative order in $\mathbb{F}_q$. Our attack exploits the fact that the trace of the root is zero, and has overwhelming success probability as a function of the number of samples taken as input. An implementation in Maple and some examples of our attack are also provided.

cs.CR↗
arXiv2025

Fast Multiplication and the PLWE-RLWE Equivalence for an Infinite Family of Maximal Real Subfields of Cyclotomic Fields

We prove the equivalence between the Ring Learning With Errors (RLWE) and the Polynomial Learning With Errors (PLWE) problems for the maximal totally real subfield of the $2^r 3^s$-th cyclotomic field for $r \geq 3$ and $s \geq 1$. Moreover, we describe a fast algorithm for computing the product of two elements in the ring of integers of these subfields. This multiplication algorithm has quasilinear complexity in the dimension of the field, as it makes use of the fast Discrete Cosine Transform (DCT). Our approach assumes that the two input polynomials are given in a basis of Chebyshev-like polynomials, in contrast to the customary power basis. To validate this assumption, we prove that the change of basis from the power basis to the Chebyshev-like basis can be computed with $\mathcal{O}(n \log n)$ arithmetic operations, where $n$ is the problem dimension. Finally, we provide a heuristic and theoretical comparison of the vulnerability to some attacks for the $p$-th cyclotomic field versus the maximal totally real subextension of the $4p$-th cyclotomic field for a reasonable set of parameters of cryptographic size.

cs.CR↗
arXiv2024

Modular supercuspidal lifts of weight $2$

Let $F/\mathbb{Q}$ be any totally real number field and $\frak{N}$ an ideal of its ring of integers of norm $N$ and define, for every even $n$, the $[F:\mathbb{Q}]$-dimensional multiweight $\textbf{n}=(n,...,n)$. We prove that for a non CM Hilbert cuspidal Hecke eigenform for $F$, say $f\in S_{\textbf{k}}(Γ_0(\frak{N}))$ with $k>2$ even, and a prime $p>\max\{k+1,6\}$ totally split in $F$ such that $p\nmid N$ and such that the residual mod $p$ representation $\overlineρ_f$ satisfies that $\mathrm{SL}_2(\mathbb{F}_p)\subseteq \mathrm{Im}(\overlineρ_f)$, there exists a lift $ρ_g$ associated to a Hilbert modular cuspform for $F$, say $g\in S_{\textbf{2}}(\frak{N}p^2,ε)$ for some Nebentypus character $ε$ which is supercuspidal at each prime of $F$ over $p$. We also observe that our techniques provide an alternative proof to the corresponding statement for classical Hecke cuspforms already proved by Khare \cite{khare} with classical techniques. Finally, we take the opportunity to include a corrigenda for \cite{dieulefait} which follows from our main result, which provides a congruence that puts the micro good dihedral prime in the level.

math.NT↗
arXiv2023

Trace-based cryptanalysis of cyclotomic $R_{q,0}\times R_q$-PLWE for the non-split case

We describe a decisional attack against a version of the PLWE problem in which the samples are taken from a certain proper subring of large dimension of the cyclotomic ring $\mathbb{F}_q[x]/(Φ_{p^k}(x))$ with $k>1$ in the case where $q\equiv 1\pmod{p}$ but $Φ_{p^k}(x)$ is not totally split over $\mathbb{F}_q$. Our attack uses the fact that the roots of $Φ_{p^k}(x)$ over suitable extensions of $\mathbb{F}_q$ have zero-trace and has overwhelming success probability as a function of the number of input samples. An implementation in Maple and some examples of our attack are also provided.

cs.CR↗
arXiv2023

Fast polynomial arithmetic in homomorphic encryption with cyclo-multiquadratic fields

We discuss the advantages and limitations of cyclotomic fields to have fast polynomial arithmetic within homomorphic encryption, and show how these limitations can be overcome by replacing cyclotomic fields by a family that we refer to as cyclo-multiquadratic. This family is of particular interest due to its arithmetic efficiency properties and to the fact that the Polynomial Learning with Errors (PLWE) and Ring Learning with Errors (RLWE) problems are equivalent for it. Likewise, we provide exact expressions for the condition number for any cyclotomic field, but under what we call the twisted power basis. As a tool for our result, we obtain refined polynomial upper bounds for the condition number of cyclotomic fields with up to 6 different primes dividing the conductor. From a more practical side, we also show that for this family, swapping between NTT and coefficient representations can be achieved at least twice faster than for the usual cyclotomic family.

cs.CR↗
arXiv2021

Potentially diagonalizable modular lifts of large weight

We prove that for a Hecke cuspform $f\in S_k(Γ_0(N),χ)$ 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 $f_{k_r}\in S_{k_r}(Γ_0(N),χ)$ such that the Deligne representation $ρ_{f_{k_r,l}}$ is a crystaline and potentially diagonalizable lift of $\overlineρ_{f,l}$. 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 Böckle 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.

math.NT↗
arXiv2018

Differential operators and hyperelliptic curves over finite fields

Boix, De Stefani and Vanzo have characterized ordinary/supersingular elliptic curves over $\mathbb{F}_p$ in terms of the level of the defining cubic homogenous polynomial. We extend their study to arbitrary genus, in particular we prove that every ordinary hyperelliptic curve $\mathcal{C}$ of genus $g\geq 2$ has level $2$. We provide a good number of examples and raise a conjecture.

math.NT↗
arXiv2014

Fuchsian codes with arbitrarily high code rate

Recently, so-called Fuchsian codes have been proposed in [I. Blanco-Chacón et al., "Nonuniform Fuchsian codes for noisy channels", J. of the Franklin Institute 2014] for communication over channels subject to additive white Gaussian noise (AWGN). The two main advantages of Fuchsian codes are their ability to compress information, i.e., high code rate, and their logarithmic decoding complexity. In this paper, we improve the first property further by constructing Fuchsian codes with arbitrarily high code rates while maintaining logarithmic decoding complexity. Namely, in the case of Fuchsian groups derived from quaternion algebras over totally real fields we obtain a code rate that is proportional to the degree of the base field. In particular, we consider arithmetic Fuchsian groups of signature (1;e) to construct explicit codes having code rate six, meaning that we can transmit six independent integers during one channel use.

cs.IT↗
arXiv2014

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.

cs.IT↗
arXiv2014

Fuchsian codes for AWGN channels

We develop a new transmission scheme for additive white Gaussian noisy (AWGN) single-input single-output (SISO) channels without fading based on arithmetic Fuchsian groups. The properly discontinuous character of the action of these groups on the upper half-plane translates into logarithmic decoding complexity.

cs.IT↗
arXiv2012

Cohomological interpretation of quadratic modular symbols

Bayer and Blanco-Chacón have recently defined quadratic modular symbols for the Shimura curves X(D,N) attached to Eichler orders of level N of an indefinite quaternion rational algebra of discriminant D. In this paper, we give a cohomological interpretation of these quadratic modular symbols. Explicit computations for the homology of some Shimura curves are also provided.

math.NT↗
arXiv2012

A higher order p-adic class number formula

We generalize a formula of Leopoldt which relates the p-adic regulator modulo p of a real abelian extension of Q with the value of the relative Dedekind zeta function at s=2-p. We use this generalization to give a statement on the non-vanishing modulo p of this relative zeta function at the point s=1 under a mild condition.

math.NT↗
arXiv2011

Quadratic modular symbols on Shimura curves

We introduce the concept of quadratic modular symbol and study how these symbols are related to quadratic p-adic L-functions. These objects were introduced in [3] in the case of modular curves. In this paper, we discuss a method to attach quadratic modular symbols and quadratic p-adic L-functions to more general Shimura curves.

math.NT↗