arXiv subjects

Vicenţiu Paşol

Publications and source records attributed to Vicenţiu Paşol.

11 recordsLinked to original sources

arXiv2026

Subgroup Packing for Batched PASTA Transciphering

With transciphering, a server converts symmetrically encrypted records into homomorphic ciphertexts without learning the records or the symmetric key. For the PASTA cipher, this conversion involves dense linear maps whose implementation depends on how record words are arranged in the ciphertext. We ask whether rearranging a fixed batch can reduce its conversion cost. Instead of storing each record's words in a contiguous block, our layout interleaves records so that cyclic word shifts preserve each record's positions, which form a coset of a cyclic subgroup. For direct evaluation as a sum of masked translations, we characterize the required displacements and relate their counts, 255 for the contiguous layout and 128 for the subgroup layout, to a prior transversal-difference invariant. We implement three equally batched schedules for complete PASTA-3 conversion and subsequent public subset-sum queries in HElib. Across twelve paired corpora under six homomorphic keys, all 24 direct and subgroup conversions and all 48 subsequent queries return the expected values and required zeroes. The median paired ratio of direct to subgroup server cost is 1.60, including fresh public generation, conversion and two queries. The reduction comes with less remaining noise capacity. These results establish a packing-dependent cost--noise tradeoff.

cs.CR↗

arXiv2026

An Eisenstein--Kronecker Modular Symbol and Beilinson--Kato Distributions

We construct an Eisenstein-Kronecker modular symbol with values in a completed bigraded quotient of modular forms by Eisenstein series. Its homogeneous finite-level specializations recover the Borisov-Gunnells symbols. The main theorem identifies its standard distribution, after the explicit transposition, orientation, finite symplectic Fourier, projection, and period normalizations, with Colmez's Eisenstein realization of the standard Busuioc-Park-Patashnick-Stevens Beilinson-Kato distribution. The comparison holds on the full BPPS test-function space, including the zero-torsion degenerations. The cyclic Manin relation is obtained from the Frobenius-Stickelberger identity and the theta heat equation; at level one its coefficient identities generate all linear relations among unordered products of two Eisenstein series whose weights are at least four. We also record compatible real-regulator and Poincare-bundle coefficient realizations with the same Fourier normalization.

math.NT↗

arXiv2026

On the second moment and non-vanishing of central values of Hecke $L$-functions of $r$-th order characters

In this paper, we establish asymptotic formulas for the first and second twisted moments of $r$-th order Hecke $L$-functions over global fields that contain the $2r$-th roots of unity, for $r\ge 3$. We focus primarily on algebraic number fields. As a consequence, we establish a positive proportion of non-vanishing central values for these $L$-functions, specifically for families of both square-free and $r$-th power-free ideals. Our approach is based on the machinery of multiple Dirichlet series.

math.NT↗

arXiv2026

Iterated extensions and the ramification dichotomy

Let $K/\mathbb Q_p$ be finite and let $f\in\mathcal O_K[X]$ be monic, of degree at least two, with $f'(X)\in\mathfrak m_K\mathcal O_K[X]$, equivalently $\bar f\in k[X^p]$. For a compatible inverse branch $f(t_{n+1})=t_n$ with $t_0\in\mathcal O_K$, put $K_n=K(t_n)$ and $K_\infty=\bigcup_nK_n$. We prove that $K_\infty/K$ is either unramified or deeply ramified. More precisely, once ramification appears, the ramification indices over the maximal unramified subfields tend to infinity and the finite-level differents are unbounded. In the Frobenius-type case $f(X)\equiv X^{p^a}\pmod{\mathfrak m_K}$ the unramified alternative is trivial, so $K_\infty=K$ or $K_\infty/K$ is deeply ramified. After completion, the non-unramified alternative gives perfectoid fields and examples show that APF property need not hold at the algebraic level.

math.NT↗

arXiv2026

Transversal Difference Numbers in Finite Abelian Quotients

Given $H\leq G$ finite abelian groups, a transversal $T\subseteq G$ for $G/H$ has fixed size $|G/H|$, but its ambient difference support $D(T)=T-T$ can vary with the embedding of $H$ in $G$. We call $ δ(G,H)=\min_T |D(T)| $ the transversal difference number of the pair $(G,H)$. This invariant is related to finite abelian factorisation, tiling complements, and small-sumset questions, and is motivated by recent work regarding ambient Galois labels in CRT transforms for cyclotomic-subfield homomorphic encryption. We prove various results regarding this invariant, including a general lower bound $δ(G,H)\geq 2|G/H|-m(G,H), $ where $m(G,H)$ is the largest order of a subgroup of $G$ disjoint from $H$. The bound is sharp for cyclic quotients, and Kneser's theorem gives a cross-transversal estimate leading to exact product families with one nonsplit cyclic coordinate and arbitrary split factors. These results isolate the first genuinely new residual obstruction, namely the same-prime square plane \[ G=(\mathbb Z/p^2\mathbb Z)^2,\qquad H=pG. \] For odd $p$, this case is the technical core of the paper. Here transversals are graphs of functions $\mathbb F_p^2\to \mathbb F_p^2$, and $D(T)$ decomposes into carry-corrected finite-field derivative images. We conjecture that \[ δ(G,H)=(2p-1)^2 \] for all odd primes $p$, prove the unconditional lower bound $3p^2-p-1$, and give small-prime, probabilistic, and fixed-polynomial evidence for the conjecture.

math.NT↗

arXiv2023

Quadratic Weyl group multiple Dirichlet series of Type $D_{\scriptscriptstyle 4}^{\scriptscriptstyle (1)}$

In this paper and its sequel \cite{DPP}, we investigate the precise relationship between the quadratic affine Weyl group multiple Dirichlet series in the sense of \cite{CG1, BD}, and those defined axiomatically by Whitehead \cite{White2} and \cite{White1}. In particular, we show that the axiomatic quadratic Weyl group multiple Dirichlet series of type $D_{\scriptscriptstyle 4}^{\scriptscriptstyle (1)}$ over rational function fields of odd characteristic admits meromorphic continuation to the interior of the corresponding complexified Tits cone. We shall also determine the polar divisor of this function, and compute the residue at each of its poles. As a consequence, we obtain an \emph{exact} formula for a weighted 4-th moment of quadratic Dirichlet $L$-functions over rational function fields; we shall also derive an asymptotic formula for this weighted moment that is expected to generalize to any global field.

math.NT↗

arXiv2023

Residues of quadratic Weyl group multiple Dirichlet series

We give explicit formulas for the residue of the Chinta-Gunnells average attached to a finite irreducible root system, at the polar divisor corresponding to a simple short root. The formula describes the residue in terms of the average attached to the root subsystem orthogonal to the relevant simple root. As a consequence, we obtain similar formulas for the residues of quadratic Weyl group multiple Dirichlet series over the rational function field and over the Gaussian field. The residue formula also allows us to obtain a new expression for the Chinta-Gunnells average of a finite irreducible root system, as an average over a maximal parabolic subgroup of a rational function that has an explicit description reflecting the combinatorics of the root system.

math.NT↗

arXiv2023

Composing Bridges

The present work builds on previous investigations of the authors (and their collaborators) regarding bridges, a certain type of morphisms between encryption schemes, making a step forward in developing a (category theory) language for studying relations between encryption schemes. Here we analyse the conditions under which bridges can be performed sequentially, formalizing the notion of composability. One of our results gives a sufficient condition for a pair of bridges to be composable. We illustrate that composing two bridges, each independently satisfying a previously established IND-CPA security definition, can actually lead to an insecure bridge. Our main result gives a sufficient condition that a pair of secure composable bridges should satisfy in order for their composition to be a secure bridge. We also introduce the concept of a complete bridge and show that it is connected to the notion of Fully composable Homomorphic Encryption (FcHE), recently considered by Micciancio. Moreover, we show that a result of Micciancio which gives a construction of FcHE schemes can be phrased in the language of complete bridges, where his insights can be formalised in a greater generality.

cs.CR↗

arXiv2022

Magnetic (quasi-)modular forms

A (folklore?) conjecture states that no holomorphic modular form $F(τ)=\sum_{n=1}^\infty a_nq^n\in q\mathbb Z[[q]]$ exists, where $q=e^{2πiτ}$, such that its anti-derivative $\sum_{n=1}^\infty a_nq^n/n$ has integral coefficients in the $q$-expansion. A recent observation of Broadhurst and Zudilin, rigorously accomplished by Li and Neururer, led to examples of meromorphic modular forms possessing the integrality property. In this note we investigate the arithmetic phenomenon from a systematic perspective and discuss related transcendental extensions of the differentially closed ring of quasi-modular forms.

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arXiv2018

Moduli of Hyperelliptic Curves and Multiple Dirichlet Series

In this paper we provide an explicit construction of a $distinctive$ multiple Dirichlet series associated to products of quadratic Dirichlet L-series, which we believe should be tightly connected to a generalized metaplectic Whittaker function on the double cover of a Kac-Moody group. To do so, we first impose a set of axioms, independent of any group of functional equations, which the aforementioned object should satisfy. As a consequence, we deduce that the coefficients of the $p$-parts of the multiple Dirichlet series satisfy certain recurrence relations. These relations lead to a family of identities, which turns out to be $encoded$ in the combinatorial structure of certain moduli spaces of admissible double covers. Finally, via this crucial connection, we apply Deligne's theory of weights to express inductively the coefficients of the $p$-parts in terms of the eigenvalues of Frobenius acting on the $\ell$-adic étale cohomology of local systems on the moduli $\mathscr{H}_{g}[2]$ of hyperelliptic curves of genus $g$ with level 2 structure.

math.NT↗

arXiv2018

A study of elliptic gamma function and allies

We study analytic and arithmetic properties of the elliptic gamma function $$ \prod_{m,n=0}^\infty\frac{1-x^{-1}q^{m+1}p^{n+1}}{1-xq^mp^n}, \qquad |q|,|p|<1, $$ in the regime $p=q$; in particular, its connection with the elliptic dilogarithm and a formula of S. Bloch. We further extend the results to more general products by linking them to non-holomorphic Eistenstein series and, via some formulae of D. Zagier, to elliptic polylogarithms.

math.NT↗