arXiv subjects

Daniele Taufer

Publications and source records attributed to Daniele Taufer.

14 recordsLinked to original sources

arXiv2026

The Hessian of elliptic curves as a Lattès map

We prove that the Hessian transformation of elliptic curves, both as an action on $j$-invariants and on the Hesse pencil, is a rigid Lattès map fitting into a reduced diagram, hence it lifts to a degree-$3$ endomorphism $ψ$ of a prescribed elliptic curve $E$. This result provides an effective tool to investigate the dynamics of the Hessian transformation, whose symmetries are inherited from those of $ψ$, which we characterize. In particular, over arbitrary fields of characteristic different from $2$ and $3$, the functional graphs of the Hessian and, more generally, of Lattès maps fitting into analogous reduced diagrams, are completely determined by the action of $ψ$ on the twists of $E$. When the underlying field is finite, we specialize these results to obtain a complete classification of Hessian functional graphs and derive an efficient method for computing iterated Hessians.

math.NT↗

arXiv2026

Determinantal computation of minimal local GADs

We study local generalized additive decompositions (GADs) of homogeneous polynomials and their associated points schemes through their local inverse systems. We verify that their construction and algebraic properties are independent of the chosen apolarity action. We propose a determinantal method for computing minimal local GADs by minimizing the rank of a symbolic inverse system. When the locus of minimal supports is finite, this procedure provides a practical tool to determine all minimal local decompositions without tensor extensions. We prove that this finiteness is guaranteed whenever the local GAD-rank of the form does not exceed its degree. We analyze both generic and special cases, provide computational evidence assessing the impact of different choices for minors in the determinantal algorithm, and compare our approach with existing algorithms for local apolar schemes.

math.AC↗

arXiv2025

Generalized Additive Decompositions of Symmetric Tensors

This article addresses the Generalized Additive Decomposition (GAD) of symmetric tensors, that is, degree-$d$ forms $f \in \mathcal{S}_d$. From a geometric perspective, a GAD corresponds to representing a point on a secant of osculating varieties to the Veronese variety, providing a compact and structured description of a tensor that captures its intrinsic algebraic properties. We provide a linear algebra method for measuring the GAD size and prove that the minimal achievable size, which we call the GAD-rank of the considered tensor, coincides with the rank of suitable Catalecticant matrices, under certain regularity assumptions. We provide a new explicit description of the apolar scheme associated with a GAD as the annihilator of a polynomial-exponential series. We show that if the Castelnuovo-Mumford regularity of this scheme is sufficiently small, then both the GAD and the associated apolar scheme are minimal and unique. Leveraging these results, we develop a numerical GAD algorithm for symmetric tensors that effectively exploits the underlying algebraic structure, extending existing algebraic approaches based on eigen computation to the treatment of multiple points. We illustrate the effectiveness and numerical stability of such an algorithm through several examples, including Waring and tangential decompositions.

math.AC↗

arXiv2024

Elliptic curves over Hasse pairs

We call a pair of distinct prime powers $(q_1,q_2) = (p_1^{a_1},p_2^{a_2})$ a Hasse pair if $|\sqrt{q_1}-\sqrt{q_2}| \leq 1$. For such pairs, we study the relation between the set $\mathcal{E}_1$ of isomorphism classes of elliptic curves defined over $\mathbb{F}_{q_1}$ with $q_2$ points, and the set $\mathcal{E}_2$ of isomorphism classes of elliptic curves over $\mathbb{F}_{q_2}$ with $q_1$ points. When both families $\mathcal{E}_i$ contain only ordinary elliptic curves, we prove that their isogeny graphs are isomorphic. When supersingular curves are involved, we describe which curves might belong to these sets. We also show that if both the $q_i$'s are odd and $\mathcal{E}_1 \cup \mathcal{E}_2 \neq \emptyset$, then $\mathcal{E}_1 \cup \mathcal{E}_2$ always contains an ordinary elliptic curve. Conversely, if $q_1$ is even, then $\mathcal{E}_1 \cup \mathcal{E}_2$ may contain only supersingular curves precisely when $q_2$ is a given power of a Fermat or a Mersenne prime. In the case of odd Hasse pairs, we could not rule out the possibility of an empty union $\mathcal{E}_1 \cup \mathcal{E}_2$, but we give necessary conditions for such a case to exist. In an appendix, Moree and Sofos consider how frequently Hasse pairs occur using analytic number theory, making a connection with Andrica's conjecture on the difference between consecutive primes.

math.NT↗

arXiv2024

On schemes evinced by generalized additive decompositions and their regularity

We define and explicitly construct schemes evinced by generalized additive decompositions (GADs) of a given $d$-homogeneous polynomial $F$. We employ GADs to investigate the regularity of $0$-dimensional schemes apolar to $F$, focusing on those satisfying some minimality conditions. We show that irredundant schemes to $F$ need not be $d$-regular, unless they are evinced by special GADs of $F$. Instead, we prove that tangential decompositions of minimal length are always $d$-regular, as well as irredundant apolar schemes of length at most $2d+1$.

math.AC↗

arXiv2024

First-degree prime ideals of composite extensions

Let $\mathbb{Q}(α)$ and $\mathbb{Q}(β)$ be linearly disjoint number fields and let $\mathbb{Q}(θ)$ be their compositum. We prove that the first-degree prime ideals of $\mathbb{Z}[θ]$ may almost always be constructed in terms of the first-degree prime ideals of $\mathbb{Z}[α]$ and $\mathbb{Z}[β]$, and vice-versa. We also classify the cases in which this correspondence does not hold, by providing explicit counterexamples. We show that for every pair of coprime integers $d,e \in \mathbb{Z}$, such a correspondence almost always respects the divisibility of principal ideals of the form $(e+dθ)\mathbb{Z}[θ]$, with a few exceptions that we characterize. Finally, we discuss the computational improvement of such an approach, and we verify the reduction in time needed for computing such primes for certain concrete cases.

math.NT↗

arXiv2023

The group structure of elliptic curves over Z/NZ

We characterize the possible groups $E(\mathbb{Z}/N\mathbb{Z})$ arising from elliptic curves over $\mathbb{Z}/N\mathbb{Z}$ in terms of the groups $E(\mathbb{F}_p)$, with $p$ varying among the prime divisors of $N$. This classification is achieved by showing that the infinity part of any elliptic curve over $\mathbb{Z}/p^e\mathbb{Z}$ is a $\mathbb{Z}/p^e\mathbb{Z}$-torsor, of which a generator is exhibited. As a first consequence, when $E(\mathbb{Z}/N\mathbb{Z})$ is a $p$-group, we provide an explicit and sharp bound on its rank. As a second consequence, when $N = p^e$ is a prime power and the projected curve $E(\mathbb{F}_p)$ has trace one, we provide an isomorphism attack to the ECDLP, which works only by means of finite rings arithmetic.

math.NT↗

arXiv2023

Multiplication polynomials for elliptic curves over finite local rings

For a given elliptic curve $E$ over a finite local ring, we denote by $E^{\infty}$ its subgroup at infinity. Every point $P \in E^{\infty}$ can be described solely in terms of its $x$-coordinate $P_x$, which can be therefore used to parameterize all its multiples $nP$. We refer to the coefficient of $(P_x)^i$ in the parameterization of $(nP)_x$ as the $i$-th multiplication polynomial. We show that this coefficient is a degree-$i$ rational polynomial without a constant term in $n$. We also prove that no primes greater than $i$ may appear in the denominators of its terms. As a consequence, for every finite field $\mathbb{F}_q$ and any $k\in\mathbb{N}^*$, we prescribe the group structure of a generic elliptic curve defined over $\mathbb{F}_q[X]/(X^k)$, and we show that their ECDLP on $E^{\infty}$ may be efficiently solved.

math.NT↗

arXiv2023

Elliptic Loops

Given a local ring $(R,\mathfrak{m})$ and an elliptic curve $E(R/\mathfrak{m})$, we define elliptic loops as the points of $\mathbb{P}^2(R)$ projecting to $E$ under the canonical modulo-$\mathfrak{m}$ reduction, endowed with an operation that extends the curve's addition. While their subset of points satisfying the curve's Weierstrass equation is a group, these larger objects are proved to be power associative abelian algebraic loops, which are seldom completely associative. When an elliptic loop has no points of order $3$, its affine part is obtained as a stratification of a one-parameter family of elliptic curves defined over $R$, which we call layers. Stronger associativity properties are established when $\mathfrak{m}^e$ vanishes for small values of $e \in \mathbb{Z}$. When the underlying ring is $R = \mathbb{Z}/p^e\mathbb{Z}$, the infinity part of an elliptic loop is generated by two elements, the group structure of layers may be established and the points with the same projection and same order possess a geometric description.

math.AC↗

arXiv2022

Normal and pseudonormal numbers

After a short review of the historical milestones on normal numbers, we introduce the Borel numbers as the reals admitting a probability function on their different bases representations. In this setting, we provide two probabilistic characterizations of normality based on the stochastic independence of their digits. Finally, we define the pseudonormality condition, which is satisfied by normal numbers and may be evaluated in a finite number of steps.

math.NT↗

arXiv2021

First-Degree Prime Ideals of Biquadratic Fields dividing prescribed Principal Ideals

We describe first-degree prime ideals of biquadratic extensions in terms of first-degree prime ideals of two underlying quadratic fields. The identification of the prime divisors is given by numerical conditions involving their ideal norms. Interestingly, the correspondence between these ideals in the larger ring and those in the smaller ones extends to the divisibility of principal ideals in their respective rings, with some exceptions that we explicitly provide. Finally, we hint at possible applications of this correspondence.

math.NT↗

arXiv2020

Waring, tangential and cactus decompositions

(EN) We revise the famous algorithm for symmetric tensor decomposition due to Brachat, Comon, Mourrain and Tsidgaridas. Afterwards, we generalize it in order to detect possibly different decompositions involving points on the tangential variety of a Veronese variety. Finally, we produce an algorithm for cactus rank and decomposition, which also detects the support of the minimal apolar scheme and its length at each component. (FR) Nous revenons sur le fameux algorithme de Brachat, Comon, Mourrain et Tsidgaridas pour la dćomposition des tenseurs symétriques. Ensuite, nous le généralisons afin de détecter de possibles décompositions différentes impliquant des points sur la variété tangentielle d'une variété de Veronese. Enfin, nous proposons un algorithme pour le rang et la décomposition cactus, qui, lui aussi, détecte le support du schéma apolaire minimal ainsi que sa longueur sur chaque composante.

math.AC↗

arXiv2020

A new ECDLP-based PoW model

We lay the foundations for a blockchain scheme, whose consensus is reached via a proof of work algorithm based on the solution of consecutive discrete logarithm problems over the point group of elliptic curves. In the considered architecture, the curves are pseudorandomly determined by block creators, chosen to be cryptographically secure and changed every epoch. Given the current state of the chain and a prescribed set of transactions, the curve selection is fully rigid, therefore trust is needed neither in miners nor in the scheme proposers.

cs.CR↗

arXiv2019

A survey on efficient parallelization of blockchain-based smart contracts

The main problem faced by smart contract platforms is the amount of time and computational power required to reach consensus. In a classical blockchain model, each operation is in fact performed by each node, both to update the status and to validate the results of the calculations performed by others. In this short survey we sketch some state-of-the-art approaches to obtain an efficient and scalable computation of smart contracts. Particular emphasis is given to sharding, a promising method that allows parallelization and therefore a more efficient management of the computational resources of the network.

cs.CR↗