Domingo Gomez-Perez

University of Cantabria | UNICAN · Department of Mathematics, Statistics and Computing

Los simuladores son atractivos para la formación de los estudiantes, al mostrar un modelo de un sistema sin los riesgos que puede presentar el uso de plataformas de hardware real. Aunque existan soluciones comercia-les, no siempre cubren todos los sensores, actuadores y bibliotecas utilizados en clase. Por lo tanto, es el docente que tiene que adap...

Desde los inicios de la era de la información y el consiguiente impulso de las disciplinas STEM, la robótica es una de las áreas centrales en el currículo. Desafortunadamente, no existen herramientas docentes suficientemente maduras lo que hace que cada docente implemente su propia solución, con la consecuencia de que se cubre un número reducido de...

Costas arrays are fundamental to the operation of radar and sonar systems, yet it is still an open problem how to generate a Costas array for an arbitrary length. A way to improve computer search is to make use of a characterization of the structural properties in order to reduce the search space. In this work, we use techniques from uniform distri...

En un contexto de clase, el desarrollo de software sobre plataformas hardware, aunque es una herramienta pedagógica atractiva, presenta ciertos inconvenientes. En ocasiones el hardware puede ser defectuoso, estar mal configurado o ensamblado y/o realizarse un uso no esperado o correcto de éste. Otras veces, los problemas son externos, por ejemplo c...

The correlation measure of order k is an important measure of pseudorandomness for binary sequences. This measure tries to look for dependence between several shifted versions of a sequence. We study the relation between the correlation measure of order k and two other pseudorandom measures: the N th linear complexity and the N th maximum order com...

The star-discrepancy is a quantitative measure for the irregularity of distribution of a point set in the unit cube that is intimately linked to the integration error of quasi-Monte Carlo algorithms. These popular integration rules are nowadays also applied to very high-dimensional integration problems. Hence multi-dimensional point sets of reasona...

Correlation measure of order $k$ is an important measure of randomness in binary sequences. This measure tries to look for dependence between several shifted version of a sequence. We study the relation between the correlation measure of order $k$ and another two pseudorandom measures: the $N$th linear complexity and the $N$th maximum order complex...

Families of binary sequences with low correlation are required in applications such as wireless communications, ranging and time delay measurement and digital watermarking, among others. Many constructions have been proposed that employ m-sequences as basic building blocks, such as the Gordon-Mills-Welch (GMW) sequences. In this work we present a u...

Watermarking digital media is one of the important challenges for information hiding. Not only the watermark must be resistant to noise and against attempts of modification, legitimate users should not be aware that it is embedded in the media. One of the techniques for watermarking is using an special variant of spread-spectrum technique, called f...

We build on the work of Drakakis et al. (2011) on the maximal cross-correlation of the families of Welch and Golomb Costas permutations. In particular, we settle some of their conjectures. More precisely, we prove two results. First, for a prime p ≥ 5, the maximal cross-correlation of the family of the φ(p-1) different Welch Costas permutations of...

We build on the work of Drakakis et al. (2011) on the maximal cross-correlation of the families of Welch and Golomb Costas permutations. In particular, we settle some of their conjectures. More precisely, we prove two results. First, for a prime $p\ge 5$, the maximal cross-correlation of the family of the $\varphi(p-1)$ different Welch Costas permu...

This paper generalizes three constructions of families of sequences with bounded off peak correlation with application to Code Division Multiple Access (CDMA), frequency hopping, and Ultra Wide Band (UWB). These new families present flexible family sizes and sequence lengths, making them well suited to wireless communications and Multiple Input Mul...

The star-discrepancy is a quantitative measure for the irregularity of distribution of a point set in the unit cube that is intimately linked to the integration error of quasi-Monte Carlo algorithms. These popular integration rules are nowadays also applied to very high-dimensional integration problems. Hence multi-dimensional point sets of reasona...

The output of an association rule miner is often huge in practice. This is why several concise lossless representations have been proposed, such as the “essential” or “representative” rules. A previously known algorithm for mining representative rules relies on an incorrect mathematical claim, and can be seen to miss part of its intended output; in...

The linear complexity of a sequence is an important parameter for many applications, especially those related to information security, and hardware implementation. It is desirable to develop a corresponding measure and theory for multidimensional arrays that are consistent with those of sequences. In this paper we use Gröbner bases to develop a the...

Analyzing the security of cryptosystems under attacks based on the malicious modification of memory registers is a research topic of high importance. This type of attacks may affect the randomness of the secret parameters by forcing a limited number of bits to a certain value which can be unknown to the attacker. In this context, we revisit the att...

The nucleator is a method to estimate the volume of a particle, i.e. a compact subset of ℝ3, which is widely used in Stereology. It is based on geometric sampling and known to be unbiased. However, the prediction of the variance of this estimator is non-trivial and depends on the underlying sampling scheme. We propose well established tools from q...

In 2012, Diem introduced a new figure of merit for cryptographic sequences called expansion complexity. Recently, a series of paper has been published for analysis of expansion complexity and for testing sequences in terms of this new measure of randomness. In this paper, we continue this analysis. First we study the expansion complexity in terms o...

In this paper, we define the linear complexity for multidimensional sequences over finite fields, generalizing the one-dimensional case. We give some lower and upper bounds, valid with large probability, for the linear complexity and k-error linear complexity of multidimensional periodic sequences.

In this paper, we focus on linear complexity measures of multidimensional sequences over finite fields, generalizing the one-dimensional case and including that of multidimensional arrays (identified with multidimensional periodic sequence) as a particular instance. A cryptographically strong sequence or array should not only have a high linear com...

We study the problem of finding an optimum clustering, a problem known to be NP-hard. Existing literature contains algorithms running in time proportional to the number of points raised to a power that depends on the dimensionality and on the number of clusters. Published validations of some of these algorithms are unfortunately incomplete; besides...

Analyzing the security of cryptosystems under attacks based on the malicious modification of memory registers is a research topic of high importance. This type of attacks may affect the randomness of the secret parameters by forcing a limited number of bits to a certain value which can be unknown to the attacker. In this context, we revisit the att...

Studying randomness in different structures is important from the development of applications and theory. Dartyge, Mosaki and Sárközy (among others) have studied measures of randomness for families of subsets of integers. In this article, we improve results on the complexity of some families defined by polynomials, introducing new techniques from a...

We give an efficient algorithm to enumerate all sets of $r\ge 1$ quadratic polynomials over a finite field, which remain irreducible under iterations and compositions.

We give an efficient algorithm to enumerate all sets of $r\ge 1$ quadratic polynomials over a finite field, which remain irreducible under iterations and compositions.

We present a 3D array construction with application to video watermarking. This new construction uses two main ingredients: an extended rational cycle (ERC) as a shift sequence and a Legendre array as a base. This produces a family of 3D arrays with good auto and cross-correlation. We calculate exactly the values of the auto correlation and the cro...

In 2012, Diem introduced a new figure of merit for cryptographic sequences called expansion complexity. In this paper, we slightly modify this notion to obtain the so-called irreducible-expansion complexity which is more suitable for certain applications. We analyze both, the classical and the modified expansion complexity. Moreover, we also study...

In 2012, Diem introduced a new figure of merit for cryptographic sequences called expansion complexity. In this paper, we slightly modify this notion to obtain the so-called irreducible-expansion complexity which is more suitable for certain applications. We analyze both, the classical and the modified expansion complexity. Moreover, we also study...

In this paper, we introduce the linear complexity for multidimensional sequences over finite fields which generalizes the one dimensional case and includes that of multidimensional arrays (identified with multidimensional periodic sequence) as a special example. A cryptographically strong sequence or array should not only have a high linear complex...

Recently, there has been a sharp rise of interest in properties of digits primes. Here we study yet another question of this kind. Namely, we fix an integer base g ⩾ 2 and then for every infinite sequence of g-ary digits we consider the counting function of integers n ⩽ N for which ∑n − 1i = 0digⁱ is prime. We construct sequences for which grows f...

Recently, there has been a sharp rise of interest in properties of digits primes. Here we study yet another question of this kind. Namely, we fix an integer base $g \ge 2$ and then for every infinite sequence $${\mathcal D} = \{d_i\}_{i=0}^\infty \in \{0, \ldots, g-1\}^\infty $$ of $g$-ary digits we consider the counting function $\varpi_{{\mathcal...

We present a 3D array construction with application to video wa-termarking. This new construction uses two main ingredients: an extended rational cycle as a shift sequence and a Legendre array as a base. This produces a family of 3D arrays with good auto and cross-correlation. We calculate exactly the values of the auto correlation and the cross-co...

We study high-dimensional numerical integration in the worst-case setting. The subject of tractability is concerned with the dependence of the worst-case integration error on the dimension. Roughly speaking, an integration problem is tractable if the worst-case error does not explode exponentially with the dimension. Many classical problems are kno...

We study high-dimensional numerical integration in the worst-case setting. The subject of tractability is concerned with the dependence of the worst-case integration error on the dimension. Roughly speaking, an integration problem is tractable if the worst-case error does not explode exponentially with the dimension. Many classical problems are kno...

Population sizing from still aerial pictures is of wide applicability in ecological and social sciences. The problem is long standing because current automatic detection and counting algorithms are known to fail in most cases, and exhaustive manual counting is tedious, slow, difficult to verify and unfeasible for large populations. An alternative i...

Motivated by a question of van der Poorten about the existence of infinite chain of prime numbers (with respect to some base), in this paper we advance the study of sequences of consecutive polynomials whose coefficients are chosen consecutively from a sequence in a finite field of odd prime characteristic. We study the arithmetic of such sequences...

Multidimensional arrays have proven to be useful in watermarking, therefore interest in this subject has increased in the previous years along with the number of publications. For one dimensional arrays (sequences), linear complexity is regarded as standard measure of complexity. Although linear complexity of sequences has been widely studied, only...

An important number of academic tasks should be solved collaboratively by groups of learners. The Computer-Supported Collaborative Learning (CSCL) systems support this collaboration by means of shared workspaces and tools that enable communication and coordination between learners. Successful collaboration and interaction can depend on the criteria...

We study common composites of triangular polynomial and rational function systems with favorable effects under composition: polynomial degree growth. We construct classes of such systems that do not have common composites. This property makes them suitable for the construction of a recently proposed hash function. We give estimates for the number o...

D. Gómez-Perez, A. Ostafe, A.P. Nicol-Las and D. Sadornil have recently shown that for almost all polynomials f ε Fq[X] over the finite field of q elements, where q is an odd prime power, their iterates eventually become reducible polynomials over Fq. Here we combine their method with some new ideas to derive finer results about the arithmetic stru...

One of the main contributions which Harald Niederreiter made to mathematics is related to pseudorandom sequences theory. In this article, we improve on a bound on one of the pseudorandom number generators (PRNGs) proposed by Harald Niederreiter and Arne Winterhof and study its lattice structure. We obtain that this generator passes general lattice...

Let p be a prime and F-p the finite field with p elements. We show how, when given an irreducible bivariate polynomial F is an element of F-p[X, Y] and an approximation to a zero, one can recover the root efficiently, if the approximation is good enough. The strategy can be generalized to polynomials in the variables X-1,..., X-m over the field F-p...

Harald Niederreiter's pioneering research in the field of applied algebra and number theory has led to important and substantial breakthroughs in many areas. This collection of survey articles has been authored by close colleagues and leading experts to mark the occasion of his 70th birthday. The book provides a modern overview of different researc...

We present several general results that show how algebraic dynamical systems with a slow degree growth and also rational automorphisms can be used to construct stronger pseudorandom number generators. We then give several concrete constructions that illustrate the applicability of these general results.

Lattice tests are quality measures for assessing the intrinsic structure of pseudorandom number generators. Recently a new lattice test has been introduced by Niederreiter andWinterhof. In this paper, we present a general inequality that is satisfied by any periodic sequence. Then, we analyze the behavior of the linear congruential generators on el...

In this note, we describe a large family of nonquadratic continued fractions in the field [TEX equation: \mathbb{F}_{3}((T^{-1}))] of power series over the finite field [TEX equation: \mathbb{F}_{3}] . These continued fractions are remarkable for two reasons: first, they satisfy an algebraic equation with coefficients in [TEX equation: \mathbb{F}_{...

For a large prime \(p\) , a rational function \(\psi \in {\mathbb F}_p(X)\) over the finite field \({\mathbb F}_p\) of \(p\) elements, and integers \(u\) and \(H\ge 1\) , we obtain a lower bound on the number consecutive values \(\psi (x)\) , \(x = u+1, \ldots , u+H\) that belong to a given multiplicative subgroup of \({\mathbb F}_p^*\) .

Using rational functions to generate pseudorandom number sequences is a popular research topic. In this paper, we study bounds on additive character sums of a new explicit generator based on rational functions with small p-weight degree. This extends the class of functions where a nontrivial character sum bound is known.

We consider a two polynomials analogue of the polynomial interpolation problem. Namely, we consider the Mixing Modular Operations (MMO) problem of recovering two polynomials $f\in \Z_p[x]$ and $g\in \Z_q[x]$ of known degree, where $p$ and $q$ are two (un)known positive integers, from the values of $f(t)\bmod p + g(t)\bmod q$ at polynomially many po...

One of the main contributions which Harald Niederreiter made to mathematics is related to the theory of pseudorandom sequences. In this paper we study several measures for asserting the quality of pseudorandom sequences, involving generalizations of linear complexity and lattice tests and relations between them.

Nowadays, most students take part in collaborative learning activities, which consist of carrying out academic tasks in groups. Computer-Supported Collaborative Learning (CSCL) systems offer tools to support these collective activities. The method used to form the learners groups can be a key element in achieving a successful collaboration. This pa...

For a large prime $p$, a polynomial $f\in\F_p[X]$ over a finite field $\F_p$ of $p$ elements, and integers $u$ and $H\ge 1$, we obtain a lower bound on the size of the multiplicative subgroup of $\F_p^*$ generated by the consecutive values $f(x)$, $x = u+1, \ldots, u+H$.

We study the security of the linear generator over a finite field. It is shown that the seed of a linear generator can be deduced from partial information of a short sequence of consecutive outputs of such generators.

L. Carlitz proved that any permutation polynomial ff over a finite field FqFq is a composition of linear polynomials and inversions. Accordingly, the minimum number of inversions needed to obtain ff is defined to be the Carlitz rank of ff by Aksoy et al. The relation of the Carlitz rank of ff to other invariants of the polynomial is of interest. He...

We study the problem of determining the probability that m vectors selected uniformly at random from the intersection of the full-rank lattice @L in R^n and the window [0,B)^n generate @L when B is chosen to be appropriately large. This problem plays ...

We describe a polynomial-time algorithm for deciding whether a given distance graph with a finite number of vertices is connected. This problem was conjectured to be NP-hard in Draque Penso et al. © 2012 Wiley Periodicals, Inc. NETWORKS, 2012

We use the theory of resultants of polynomials to study the stability of an arbitrary polynomial over a finite field, that is, the property of having all its iterates irreducible. This result partially generalises the quadratic polynomial case described by R. Jones and N. Boston. Moreover, for characteristic three, we show that certain polynomials...

We determine the linear complexity of p 2-periodic binary threshold sequences derived from polynomial quotient, which is defined by the function \((u^w-u^{wp})/p \pmod p\). When w = (p − 1)/2 and \(2^{p-1}\not\equiv 1 \pmod{p^2}\), we show that the linear complexity is equal to one of the following values \(\left\{p^2-1,\ p^2-p,\ (p^2+p)/2+1,\ (p^2...

We prove a bound on sums of products of multiplicative characters of shifted Fermat quotients modulo p. From this bound we derive results on the pseudorandomness of sequences of modular discrete logarithms of Fermat quotients modulo p: bounds on the well-distribution measure, the correlation measure of order ℓ, and the linear complexity.

We obtain a lower bound on the linear complexity of the Naor–Reingold sequence. This result solves an open problem proposed by Igor Shparlinski and improves known results in some cases.

We are concerned with power series in 1/T over a finite field of 3 elements $\F_3$. In a previous article, Alain Lasjaunias investigated the existence of particular power series of elements algebraic over $\F_3[T]$, having all partial quotients of degree 1 in their continued fraction expansion. Here, we generalize his result and we make a conjectur...

The output of an association rule miner is often huge in practice. This is why several concise lossless representations have been proposed, such as the "essential" or "representative" rules. We revisit the algorithm given by Kryszkiewicz (Int. Symp. Intelligent Data Analysis 2001, Springer-Verlag LNCS 2189, 350-359) for mining representative rules....

In this work we obtain a nontrivial estimate for the size of the set of triples (a,b,c)∈Fq∗×Fq×Fq which correspond to stable quadratic polynomials f(X)=aX2+bX+c over the finite field Fq with q odd. This estimate is an improvement of the bound O(q11/4) conjectured in a recent work of A. Ostafe and I. Shparlinski.

Nonlinear congruential pseudorandom number generators can have unexpectedly short periods. Shamir and Tsaban introduced the class of counter-dependent generators which admit much longer periods. In this paper we present a bound for multiplicative character sums for nonlinear sequences generated by counter-dependent generators.

The Naor–Reingold sequences with elliptic curves are used in cryptography due to their nice construction and good theoretical properties. Here we provide a new bound on the linear complexity of these sequences. Our result improves the previous one obtained by I.E. Shparlinski and J.H. Silverman and holds in more cases.

We study the distribution of s-dimensional points of digital explicit inversive pseudorandom numbers with arbitrary lags. We prove a discrepancy bound and derive results on the pseudorandomness of the binary threshold sequence derived from digital explicit inversive pseudorandom numbers in terms of bounds on the correlation measure of order k and t...

The nonlinear congruential method is an attractive alternative to the classical linear congruential method for pseudorandom number generation. In this paper we present new discrepancy bounds for sequences of s -tuples of successive nonlinear congruential pseudorandom numbers of higher orders modulo a composite integer M .

We show that the multiplicity of a prime p as a factor of the resultant of two polynomials with integer coefficients is at least the degree of their greatest common divisor modulo p. This answers an open question by Konyagin and Shparlinski.

We show how, when given an irreducible bivariate polynomial with coecients

Nonlinear congruential pseudorandom number generators can have unexpectedly short periods. Shamir and Tsaban introduced the class of counter-dependent generators which admit much longer periods. In this paper, using a technique developed by Niederreiter and Sh-parlinski, we present discrepancy bounds for sequences of s-tuples of successive pseudora...

We prove a new bound for multiplicative character sums of nonlinear recurring sequences with Rédei functions over a finite field of prime order. This result is motivated by earlier results on nonlinear recurring sequences and their application to the distribution of powers and primitive elements. The new bound is much stronger than the bound known...

Let p be a prime and a, c be integers such that a<>0 mod p. The quadratic generator is a sequence (u_n) of pseudorandom numbers defined by u_{n+1}=a*(u_n)^2+c mod p. In this article we probe that if we know sufficiently many of the most significant bits of two consecutive values u_n, u_{n+1}, then we can compute the seed u_0 except for a small numb...

We prove a new bound for multiplicative character sums with sequences of iterations of Dickson polynomials over a finite field of prime order. This result is motivated by earlier results on nonlinear recurring sequences and their application to the distribution of powers and primitive elements.

In this paper, we study the problem of finding the shortest path in circulant graphs with an arbitrary number of jumps. We provide algorithms specifically tailored for weighted undirected and directed circulant graphs with two jumps which compute the shortest path. Our method only requires O(logN) arithmetic operations and the total bit complexity...

In this paper we present a new algorithm for finding small roots of multivariate polynomials over the integers based on lattice reduction techniques. Our simpler heuristic method is inspired in algorithms for predicting pseudorandom numbers, and it can be considered as another variant of Coppersmith's method for finding small solutions of integer b...

In the study of double-loop computer networks, the diagrams known as L-shapes arise as a graphical representation of an optimal routing for every graph’s node. The description of these diagrams provides an efficient method for computing the diameter and the average minimum distance of the corresponding graphs. We extend these diagrams to multiloop...

Let p be a prime and let c be an integer modulo p. The Pollard generator is a sequence (u_n) of pseudorandom numbers defined by the relation u_n+1equivu_n ²+c mod p. It is shown that if c and 9/14 of the most significant bits of two consecutive values u_n,u_n+1 of the Pollard generator ar...

Let q>1 be an integer and let a and b be elements of the residue ring Zq of integers modulo q. We show how, when given a polynomial f∈Zq[X] and approximations to v0,v1∈Zq such that v1≡f(v0)modq one can recover v0 and v1 efficiently. This result has direct applications to predicting the polynomial congruential generator: a sequence (vn) of pseudoran...

We give new bounds of exponential sums with sequences of iterations of Dickson polynomials over prime finite fields. This result is motivated by possible applications to polynomial generators of pseudorandom numbers.

Let p be a prime and let a and c be integers modulo p. The quadratic congruential generator (QCG) is a sequence (v n ) of pseudorandom numbers defined by the relation \(v_{n+1}\equiv av^{2}_{n}+c mod p\). We show that if sufficiently many of the most significant bits of several consecutive values v n of the QCG are given, one can recover in polynom...

It is known that there exists a Minimum Distance Diagram (MDD) for circulant digraphs of degree two (or double-loop computer networks) which is an L-shape. Its description provides the graph’s diameter and average distance on constant time. In this paper we clarify, justify and extend these diagrams to circulant digraphs of arbitrary degree by pres...

In this paper we present algorithms for finding a shortest path between two vertices of any weighted undirected and directed circulant graph with two jumps. Our shortest path algorithm only requires O(log N) arithmetic steps and the total bit complexity is O(log3 N), where N is the number of the graph’s vertices. This method has been derived from s...

Let p be a prime and let a and b be elements of the finite field Fp of p elements. The inversive congruential generator (ICG) is a sequence (un) of pseudorandom numbers defined by the relation un+1 ≡ au−1 n +b mod p .W e show that if sufficiently many of the most significant bits of several consecutive values un of the ICG are given, one can recove...

In this paper we present the first polynomial time deterministic algorithm to compute the shortest path between two vertices of a cir-culant graph of degree four. Our spectacular algorithm only requires O(log 3 N) bit operations, where N is the number of the vertices and it is based on shortest vector problems in a special class of lattices for L 1...

Let p be a prime and let a and b be integers modulo p. The inversive congruential generator (ICG) is a sequence (u n ) of pseudorandom numbers defined by the relation Un+1 º au-1n+b mod pU_{n+1}\equiv au{^{-1}_{n}}+b {\rm mod} p.We show that if b and sufficiently many of the most significant bits of three consecutive values u n of the ICG are giv...

In this paper we present an extension of a result in [2] about a discrepancy bound for sequences of s-tuples of successive nonlinear multiple recursive congruential pseudorandom numbers of higher orders. The key of this note is based on linear properties of the iterations of multivariate polynomials.

In this paper we present an heuristic algorithm and its im- plementation in C++ program for integer factoring with high-order bits known based on lattice reduction techniques. Our approach is inspired in algorithms for predicting pseudorandom numbers. Many very well known and important cryptographic protocols are based on the assumption that factor...

We give a brief survey of recent results about the discrepancy and the linear complexity profile of some pseudorandom sequences.