arXiv subjects

Ana Zumalacárregui

Publications and source records attributed to Ana Zumalacárregui.

8 recordsLinked to original sources

arXiv2017

Sidon sets and statistics of the ElGamal function

In the ElGamal signature and encryption schemes, an element $x$ of the underlying group $G = \mathbb{Z}_p^\times = \{1, \ldots, p-1 \}$ for a prime $p$ is also considered as an exponent, for example in $g^x$, where $g$ is a generator of G. This ElGamal map $x \mapsto g^x$ is poorly understood, and one may wonder whether it has some randomness properties. The underlying map from $G$ to $\mathbb{Z}_{p-1}$ with $x \mapsto x$ is trivial from a computer science point of view, but does not seem to have any mathematical structure. This work presents two pieces of evidence for randomness. Firstly, experiments with small primes suggest that the map behaves like a uniformly random permutation with respect to two properties that we consider. Secondly, the theory of Sidon sets shows that the graph of this map is equidistributed in a suitable sense. It remains an open question to prove more randomness properties, for example, that the ElGamal map is pseudorandom.

math.NT↗

arXiv2017

Threshold functions and Poisson convergence for systems of equations in random sets

We present a unified framework to study threshold functions for the existence of solutions to linear systems of equations in random sets which includes arithmetic progressions, sum-free sets, $B_{h}[g]$-sets and Hilbert cubes. In particular, we show that there exists a threshold function for the property "$\mathcal{A}$ contains a non-trivial solution of $M\cdot\textbf{x}=\textbf{0}$", where $\mathcal{A}$ is a random set and each of its elements is chosen independently with the same probability from the interval of integers $\{1,\dots,n\}$. Our study contains a formal definition of trivial solutions for any combinatorial structure, extending a previous definition by Ruzsa when dealing with a single equation. Furthermore, we study the behaviour of the distribution of the number of non-trivial solutions at the threshold scale. We show that it converges to a Poisson distribution whose parameter depends on the volumes of certain convex polytopes arising from the linear system under study as well as the symmetry inherent in the structures, which we formally define and characterize.

math.CO↗

arXiv2016

Sums of inverses in thin sets of finite fields

We obtain lower bounds for the cardinality of $k$-fold sum-sets of reciprocals of elements of suitable defined short intervals in high degree extensions of finite fields. Combining our results with bounds for multilinear character sums we obtain new results on incomplete multilinear Kloosterman sums in finite fields.

math.NT↗

arXiv2015

Salajan's conjecture on discriminating terms in an exponential sequence

Given a sequence of distinct positive integers $v_1,v_2,\ldots$ and any positive integer $n$, the discriminator $D_v(n)$ is defined as the smallest positive integer $m$ such $v_1,\ldots,v_n$ are pairwise incongruent modulo $m$. We consider the discriminator for the sequence $u_1,u_2,\ldots$, where $u_j$ equals the absolute value of $((-3)^j-5)/4$, that is $u_j=(3^j-5(-1)^j)/4$. We prove a 2012 conjecture of Sabin Salajan characterizing the discriminator of the sequence $u_1,u_2,\ldots$.

math.NT↗

arXiv2013

The least common multiple of random sets of positive integers

We study the typical behavior of the least common multiple of the elements of a random subset $A\subset \{1,\dots, n\}$. For example we prove that $\text{lcm}\{a:\ a\in A\}=2^{n(1+o(1))}$ for almost all subsets $A\subset\{1,\dots,n\}$.

math.NT↗

arXiv2012

Points on curves in small boxes en applications

We introduce several new methods to obtain upper bounds on the number of solutions of the congruences $f(x) \equiv y \pmod p$ and $f(x) \equiv y^2 \pmod p,$ with a prime $p$ and a polynomial $f$, where $(x,y)$ belongs to an arbitrary square with side length $M$. We use these results and methods to derive non-trivial upper bounds for the number of hyperelliptic curves $Y^2=X^{2g+1} + a_{2g-1}X^{2g-1} +...+ a_1X+a_0$ over the finite field $\F_p$ of $p$ elements, with coefficients in a $2g$-dimensional cube $ (a_0,..., a_{2g-1})\in [R_0+1,R_0+M]\times...\times [R_{2g-1}+1,R_{2g-1}+M]$ that are isomorphic to a given curve and give an almost sharp lower bound on the number of non-isomorphic hyperelliptic curves with coefficients in that cube. Furthermore, we study the size of the smallest box that contain a partial trajectory of a polynomial dynamical system over $\F_p$.

math.NT↗

arXiv2011

On the error term of the logarithm of the lcm of a quadratic sequence

We study the logarithm of the least common multiple of the sequence of integers given by $1^2+1, 2^2+1,..., n^2+1$. Using a result of Homma on the distribution of roots of quadratic polynomials modulo primes we calculate the error term for the asymptotics obtained by Cilleruelo.

math.NT↗

arXiv2011

Concentration of points on Modular Quadratic Forms

Let $Q(x,y)$ be a quadratic form with discriminant $D\neq 0$. We obtain non trivial upper bound estimates for the number of solutions of the congruence $Q(x,y)\equivλ\pmod{p}$, where $p$ is a prime and $x,y$ lie in certain intervals of length $M$, under the assumption that $Q(x,y)-λ$ is an absolutely irreducible polynomial modulo $p$. In particular we prove that the number of solutions to this congruence is $M^{o(1)}$ when $M\ll p^{1/4}$. These estimates generalize a previous result by Cilleruelo and Garaev on the particular congruence $xy\equiv λ\pmod{p}$.

math.NT↗