Daniele Taufer

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Verified

Daniele verified their affiliation via an institutional email.

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• Ph. D.
• PostDoc Position at KU Leuven

Let {\mathbb{Q}}\left(\alpha ) and {\mathbb{Q}}\left(\beta ) be linearly disjoint number fields and let {\mathbb{Q}}\left(\theta ) be their compositum. We prove that the first-degree prime ideals (FDPIs) of {\mathbb{Z}}\left[\theta ] may almost always be constructed in terms of the FDPIs of {\mathbb{Z}}\left[\alpha ] and {\mathbb{Z}}\left[\beta ] ,...

We prove that the Hessian transformation of a plane projective cubic corresponds to a $3$-endomorphism of a model elliptic curve. By exploiting this result, we investigate a family of functional graphs -- called Hessian graphs -- defined by the Hessian transformation. We show that, over arbitrary fields of characteristics different from $2$ and $3$...

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 ellipti...

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 a...

We characterize the possible groups E ( Z ∕ N Z ) E\left({\mathbb{Z}}/N{\mathbb{Z}}) arising from elliptic curves over Z ∕ N Z {\mathbb{Z}}/N{\mathbb{Z}} in terms of the groups E ( F p ) E\left({{\mathbb{F}}}_{p}) , with p p varying among the prime divisors of N N . This classification is achieved by showing that the infinity part of any elliptic c...

For a given elliptic curve E over a finite local ring, we denote by E^∞ its subgroup at infinity. Every point P ∈ E^∞ can be described solely in terms of its x-coordinate Px, which can be therefore used to parameterize all its multiples nP. We refer to the coefficient of (Px)^i in the parameterization of (nP)x as the i-th multiplication polynomial....

Given a local ring (R,m) and an elliptic curve E(R/m), we define its elliptic loop as the points of P^2(R) projecting to E under the canonical modulo-m reduction, endowed with an operation that extends the curve’s addition. While its subset of points satisfying the curve’s Weierstrass equation is a group, this larger object is proved to be a power-...

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...

We survey the known group structures arising from elliptic curves defined by Weierstrass models over commutative rings with unity and satisfying a technical condition. For every considered base ring, the groups that may arise depending on the curve coefficients are recalled. When a complete classification is still out of reach, partial results abou...

Given a local ring $(R,\mathfrak{m})$ and an elliptic curve $E(R/\mathfrak{m})$, elliptic loops are defined 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. These objects are proved to be power associative abelian algebraic loops, w...

Let $\mathbb{Q}(\alpha)$ and $\mathbb{Q}(\beta)$ be linearly disjoint number fields and let $\mathbb{Q}(\theta)$ be their compositum. We prove that the first-degree prime ideals of $\mathbb{Z}[\theta]$ may almost always be constructed in terms of the first-degree prime ideals of $\mathbb{Z}[\alpha]$ and $\mathbb{Z}[\beta]$, and vice-versa. We also...

After a short revision 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 pseudo...

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...

We show how a small subgroup confinement-like attack may be mounted on the Bitcoin addresses generation protocol, by inspecting a special subgroup of the group associated to point multiplication. This approach does not undermine the system security but highlights the importance of using fair random sources during the private key selection.

We describe first-degree prime ideals of biquadratic extensions in terms of the first-degree prime ideals of two underlying quadratic fields. The identification of the prime divisors is given by numerical conditions involving their ideal norms. The correspondence between these ideals in the larger ring and those in the smaller ones extends to the d...

Blockchain technology has attracted a lot of research interest in the last few years. Originally, their consensus algorithm was Hashcash, which is an instance of the so-called Proof-of-Work. Nowadays, there are several competing consensus algorithms, not necessarily PoW. In this paper, we propose an alternative proof of work algorithm which is base...

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...

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 chang...

We provide a historical overview of proof-of-work techniques and the fields in which it plunges its roots. We are interested in PoW-techniques applied to blockchain technology and therefore we survey the state-of-the-art protocols employing these methods for consensus algorithms, emphasizing the differences between the efficient hashcash systems an...

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-t...

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 chang...

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 exten...

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-t...

(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 det...