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Giacomo Micheli

Publications and source records attributed to Giacomo Micheli.

At least 37 records · Page 2Linked to original sources
arXiv2020

Local to global principle for expected values

This paper constructs a new local to global principle for expected values over free $\mathbb{Z}$-modules of finite rank. In our strategy we use the same philosophy as Ekedhal's Sieve for densities, later extended and improved by Poonen and Stoll in their local to global principle for densities. We show that under some additional hypothesis on the system of $p$-adic subsets used in the principle, one can use $p$-adic measures also when one has to compute expected values (and not only densities). Moreover, we show that our additional hypotheses are sharp, in the sense that explicit counterexamples exist when any of them is missing. In particular, a system of $p$-adic subsets that works in the Poonen and Stoll principle is not guaranteed to work when one is interested in expected values instead of densities. Finally, we provide both new applications of the method, and immediate proofs for known results.

math.NT↗
arXiv2020

Algebraic constructions of complete $m$-arcs

Let $m$ be a positive integer, $q$ be a prime power, and $\mathrm{PG}(2,q)$ be the projective plane over the finite field $\mathbb F_q$. Finding complete $m$-arcs in $\mathrm{PG}(2,q)$ of size less than $q$ is a classical problem in finite geometry. In this paper we give a complete answer to this problem when $q$ is relatively large compared with $m$, explicitly constructing the smallest $m$-arcs in the literature so far for any $m\geq 8$. For any fixed $m$, our arcs $\mathcal A_{q,m}$ satisfy $|\mathcal A_{q,m}|-q\rightarrow -\infty$ as $q$ grows. To produce such $m$-arcs, we develop a Galois theoretical machinery that allows the transfer of geometric information of points external to the arc, to arithmetic one, which in turn allows to prove the $m$-completeness of the arc.

math.CO↗
arXiv2020

Exceptional scatteredness in prime degree

Let $q$ be an odd prime power and $n$ be a positive integer. Let $\ell\in \mathbb F_{q^n}[x]$ be a $q$-linearised $t$-scattered polynomial of linearized degree $r$. Let $d=\max\{t,r\}$ be an odd prime number. In this paper we show that under these assumptions it follows that $\ell=x$. Our technique involves a Galois theoretical characterization of $t$-scattered polynomials combined with the classification of transitive subgroups of the general linear group over the finite field $\mathbb F_q$.

math.NT↗
arXiv2020

An equivariant isomorphism theorem for mod $\mathfrak p$ reductions of arboreal Galois representations

Let $ϕ$ be a quadratic, monic polynomial with coefficients in $\mathcal O_{F,D}[t]$, where $\mathcal O_{F,D}$ is a localization of a number ring $\mathcal O_F$. In this paper, we first prove that if $ϕ$ is non-square and non-isotrivial, then there exists an absolute, effective constant $N_ϕ$ with the following property: for all primes $\mathfrak p\subseteq\mathcal O_{F,D}$ such that the reduced polynomial $ϕ_\mathfrak p\in (\mathcal O_{F,D}/\mathfrak p)[t][x]$ is non-square and non-isotrivial, the squarefree Zsigmondy set of $ϕ_{\mathfrak p}$ is bounded by $N_ϕ$. Using this result, we prove that if $ϕ$ is non-isotrivial and geometrically stable then outside a finite, effective set of primes of $\mathcal O_{F,D}$ the geometric part of the arboreal representation of $ϕ_{\mathfrak p}$ is isomorphic to that of $ϕ$. As an application of our results we prove R. Jones' conjecture on the arboreal Galois representation attached to the polynomial $x^2+t$.

math.NT↗
arXiv2019

The Algebraic Theory of Fractional Jumps

In this paper we start by briefly surveying the theory of Fractional Jumps and transitive projective maps. Then, we give an efficient construction of a fractional jump of a projective map and we extend the compound generator construction for the Inversive Congruential Generator to Fractional jump sequences. In addition, we provide new results on the absolute jump index, on projectively primitive polynomials, and on the explicit description of fractional jump generators.

math.NT↗
arXiv2019

On the algebraic structure of $E_p^{(m)}$ and applications to cryptography

In this paper we show that the $\mathbb Z/p^{m}\mathbb Z$-module structure of the ring $E_p^{(m)}$ is isomorphic to a $\mathbb Z/p^{m}\mathbb Z$-submodule of the matrix ring over $\mathbb Z/p^{m}\mathbb Z$. Using this intrinsic structure of $E_p^{(m)}$, solving a linear system over $E_p^{(m)}$ becomes computationally equivalent to solving a linear system over $\mathbb Z/p^{m}\mathbb Z$. As an application we break the protocol based on the Diffie-Hellman Decomposition problem and ElGamal Decomposition problem over $E_p^{(m)}$. Our algorithm terminates in a provable running time of $O(m^{6})$ $\mathbb Z/p^{m}\mathbb Z$-operations.

cs.CR↗
arXiv2019

Constructions of Locally Recoverable Codes which are Optimal

Let $q$ be a prime power and $\mathbb F_q$ be the finite field of size $q$. In this paper we provide a Galois theoretical framework that allows to produce good polynomials for the Tamo and Barg construction of optimal locally recoverable codes (LRC). Using our approach we construct new good polynomials and then optimal LRCs with new parameters. The existing theory of good polynomials fits entirely in our new framework. The key advantage of our method is that we do not need to rely on arithmetic properties of the pair $(q,r)$, where $r$ is the locality of the code.

cs.IT↗
arXiv2019

On the selection of polynomials for the DLP quasi-polynomial time algorithm in small characteristic

In this paper we characterize the set of polynomials $f\in\mathbb F_q[X]$ satisfying the following property: there exists a positive integer $d$ such that for any positive integer $\ell$ less or equal than the degree of $f$, there exists $t_0$ in $\mathbb F_{q^d}$ such that the polynomial $f-t_0$ has an irreducible factor of degree $\ell$ over $\mathbb F_{q^d}[X]$. This result is then used to progress in the last step which is needed to remove the heuristic from one of the quasi-polynomial time algorithms for discrete logarithm problems (DLP) in small characteristic. Our characterization allows a construction of polynomials satisfying the wanted property. The method is general and can be used to tackle similar problems which involve factorization patterns of polynomials over finite fields.

math.NT↗
arXiv2019

Full Classification of permutation rational functions and complete rational functions of degree three over finite fields

Let $q$ be a prime power, $\mathbb F_q$ be the finite field of order $q$ and $\mathbb F_q(x)$ be the field of rational functions over $\mathbb F_q$. In this paper we classify all rational functions $φ\in \mathbb F_q(x)$ of degree 3 that induce a permutation of $\mathbb P^1(\mathbb F_q)$. Our methods are constructive and the classification is explicit: we provide equations for the coefficients of the rational functions using Galois theoretical methods and Chebotarev Density Theorem for global function fields. As a corollary, we obtain that a permutation rational function of degree 3 permutes $\mathbb F_q$ if and only if it permutes infinitely many of its extension fields. As another corollary, we derive the well-known classification of permutation polynomials of degree 3. As a consequence of our classification, we can also show that there is no complete permutation rational function of degree $3$ unless $3\mid q$ and $φ$ is a polynomial.

math.NT↗
arXiv2019

New Lower Bounds for Permutation Codes using Linear Block Codes

In this paper we prove new lower bounds for the maximal size of permutation codes by connecting the theory of permutation codes with the theory of linear block codes. More specifically, using the columns of a parity check matrix of an $[n,k,d]_q$ linear block code, we are able to prove the existence of a permutation code in the symmetric group of degree $n$, having minimum distance at least $d$ and large cardinality. With our technique, we obtain new lower bounds for permutation codes that enhance the ones in the literature and provide asymptotic improvements in certain regimes of length and distance of the permutation code.

cs.IT↗
arXiv2018

Full Orbit Sequences in Affine Spaces via Fractional Jumps and Pseudorandom Number Generation

Let $n$ be a positive integer. In this paper we provide a general theory to produce full orbit sequences in the affine $n$-dimensional space over a finite field. For $n=1$ our construction covers the case of the Inversive Congruential Generators (ICG). In addition, for $n>1$ we show that the sequences produced using our construction are easier to compute than ICG sequences. Furthermore, we prove that they have the same discrepancy bounds as the ones constructed using the ICG.

math.NT↗
arXiv2018

Fractional jumps: complete characterisation and an explicit infinite family

In this paper we provide a complete characterisation of transitive fractional jumps by showing that they can only arise from transitive projective automorphisms. Furthermore, we prove that such construction is feasible for arbitrarily large dimension by exhibiting an infinite class of projectively primitive polynomials whose companion matrix can be used to define a full orbit sequence over an affine space.

math.NT↗
arXiv2017

Irreducible polynomials over finite fields produced by composition of quadratics

For a set $S$ of quadratic polynomials over a finite field, let $C$ be the (infinite) set of arbitrary compositions of elements in $S$. In this paper we show that there are examples with arbitrarily large $S$ such that every polynomial in $C$ is irreducible. As a second result, we give an algorithm to determine whether all the elements in $C$ are irreducible, using only $O( \#S (\log q)^3 q^{1/2} )$ operations.

math.NT↗
arXiv2017

A local to global principle for densities over function fields

Let $d$ be a positive integer and $\mathbb H$ be an integrally closed subring of a global function field $F$. The purpose of this paper is to provide a general sieve method to compute densities of subsets of $\mathbb H^d$ defined by local conditions. The main advantage of the method relies on the fact that one can use results from measure theory to extract density results over $\mathbb H^{d}$. Using this method we are able to compute the density of the set of polynomials with coefficients in $\mathbb H$ which give rise to "good" totally ramified extensions of the global function field $F$. As another application, we give a closed expression for the density of rectangular unimodular matrices with coefficients in $\mathbb H$ in terms of the $L$-polynomial of the function field.

math.NT↗
arXiv2016

On sets of irreducible polynomials closed by composition

Let $\mathcal S$ be a set of monic degree $2$ polynomials over a finite field and let $C$ be the compositional semigroup generated by $\mathcal S$. In this paper we establish a necessary and sufficient condition for $C$ to be consisting entirely of irreducible polynomials. The condition we deduce depends on the finite data encoded in a certain graph uniquely determined by the generating set $\mathcal S$. Using this machinery we are able both to show examples of semigroups of irreducible polynomials generated by two degree $2$ polynomials and to give some non-existence results for some of these sets in infinitely many prime fields satisfying certain arithmetic conditions.

math.NT↗
arXiv2016

On the existence of infinite, non-trivial $F$-sets

In this paper we prove a conjecture of J. Andrade, S. J. Miller, K. Pratt and M. Trinh, showing the existence of a non trivial infinite $F$-set over $\mathbb F_q[x]$ for every fixed $q$. We also provide the proof of a refinement of the conjecture, involving the notion of width of an $F$-set, which is a natural number encoding the complexity of the set.

math.NT↗