arXiv subjects
Publications and source records attributed to Gessica Alecci.
We show that arithmetic definability of henselian valuations is preserved by the tilting correspondence. Moreover, we show that if a perfectoid valuation is arithmetically definable, then no parameters are needed. We also investigate whether these definitions can be chosen uniformly, and discuss the required quantifier complexity.
The assumed hardness of the Linear Code Equivalence problem (LCE) lies at the core of the security of the LESS signature scheme and other signature schemes with advanced functionalities. The LCE problem asks to determine whether two linear codes are equivalent. This equivalence is represented by a monomial matrix $ Q$, i.e. the product of a diagonal matrix $D$ and a permutation matrix $P$. The recovery of $Q=DP$ is known to be reduced to the recovery of the permutation matrix $ P$ alone. Exploiting this fact, we construct an algebraic model for LCE involving only the matrix $P$. To this end, we study the action of monomial matrices on linear codes using tools from algebraic geometry, including Plücker coordinates and fields of invariant rational functions. In particular, we analyse the action of diagonal matrices on linear codes, which can be interpreted as diagonal scaling of the coordinates of elements of the Grassmannian. We propose a method to determine algebraically independent generators of the field of rational functions invariant under this action, without relying on Reynolds operators or Gröbner basis computations. Furthermore, given two equivalent codes, we apply our results to explicitly construct, for each invariant function, a polynomial having $P$ as a root. However, the resulting polynomials are not of practical use: their degrees are high for cryptographically relevant parameters, and the number of monomials grows exponentially, making them infeasible to manipulate. Despite this limitation, our results are of theoretical interest, as they constitute the first application of these tools to the cryptanalysis of LCE and provide insight into how algebraic geometry and invariant theory can be employed in Cryptography.
The aim of these notes is to present an accessible overview of some topics in classical algebraic geometry which have applications to aspects of discrete integrable systems. Precisely, we focus on surface theory on the algebraic geometry side, which is applied to differential and discrete Painlevé equations on the integrable systems side. Along the way we also discuss the theory of resolution of indeterminacies, which is applied to the cohomological computation of algebraic entropy of birational transformations of projective spaces, which is closely related to the integrability of the discrete systems they define.
From around 2010 onward, Elsner et al.,developed and applied a method in which the algebraic independence of n quantities x_1,...,x_n over a field is transferred to further n quantities y_1,...,y_n by means of a system of polynomials in 2n variables X_1,...,X_n,Y_1,...,Y_n. In this paper, we systematically study and explain this criterion and its variants.
Lucas atoms are irreducible factors of Lucas polynomials and they were introduced in \cite{ST}. The main aim of the authors was to investigate, from an innovatory point of view, when some combinatorial rational functions are actually polynomials. In this paper, we see that the Lucas atoms can be introduced in a more natural and powerful way than the original definition, providing straightforward proofs for their main properties. Moreover, we fully characterize the $p$-adic valuations of Lucas atoms for any prime $p$, answering to a problem left open in \cite{ST}, where the authors treated only some specific cases for $p \in \{2, 3\}$. Finally, we prove that the sequence of Lucas atoms is not holonomic, contrarily to the Lucas sequence that is a linear recurrent sequence of order two.
Several operations can be defined on the set of all linear recurrent sequences, such as the binomial convolution (Hurwitz product) or the multinomial convolution (Newton product). Using elementary techniques, we prove that this set equipped with the termwise sum and the aforementioned products are R-algebras, given any commutative ring $R$ with identity. Moreover, we provide explicitly a characteristic polynomial of the Hurwitz product and Newton product of any two linear recurrent sequences. Finally, we also investigate whether these $R-$algebras are isomorphic, considering also the R-algebras obtained using the Hadamard product and the convolution product.
We present a study on the use of Pell hyperbolas in cryptosystems with security based on the discrete logarithm problem. Specifically, after introducing the group's structure over generalized Pell conics (and also giving the explicit isomorphisms with the classical Pell hyperbolas), we provide a parameterization with both an algebraic and a geometrical approach. The particular parameterization that we propose appears to be useful from a cryptographic point of view because the product that arises over the set of parameters is connected to the Rédei rational functions, which can be evaluated in a fast way. Thus, we exploit these constructions for defining three different public key cryptosystems based on the ElGamal scheme. We show that the use of our parameterization allows to obtain schemes more efficient than the classical ones based on finite fields.
Prempreesuk, Noppakaew, and Pongsriiam determined the Zeckendorf representation of the multiplicative inverse of $2$ modulo $F_n$, for every positive integer $n$ not divisible by $3$, where $F_n$ denotes the $n$th Fibonacci number. We determine the Zeckendorf representation of the multiplicative inverse of $a$ modulo $F_n$, for every fixed integer $a \geq 3$ and for all positive integers $n$ with $\gcd(a, F_n) = 1$. Our proof makes use of the so-called base-$φ$ expansion of real numbers.