Recursion Polynomials of Unfolded Sequences

Watermarking digital media is one of the important chal- lenges 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 tech- nique, called frequency hopping. It requires ensembles of periodic binary sequences with low off-peak autocorrelation and cross-correlation. Un- fortunately, they are quite rare and difficult to find. The small Kasami, Kamaletdinov, and Extended Rational Cycle constructions are versatile, because they can also be converted into Costas-like arrays for frequency hopping. We study the implementation of such ensembles using linear feedback shift registers. This permits an efficient generation of sequences and arrays in real time in FPGAs. Such an implementation requires minimal memory usage and permits dynamic updating of sequences or arrays. The aim of our work was to broaden current knowledge of sets of se- quences with low correlation studying their implementation using linear feedback shift registers. A remarkable feature of these families is their similarities in terms of implementation and it may open new way to characterize sequences with low correlation, making it easier to gener- ate them. It also validates some conjectures made by Moreno and Tirkel about arrays constructed using the method of composition.Supported by Consejería de Universidades e Investigación, Medio Ambiente y Política Social, Gobierno de Cantabria (ref. VP34

Recovering zeros of polynomials modulo a prime

Let $p$ be a prime and $\mathbb{F}_p$ the finite field with $p$ elements. We show how, when given an irreducible bivariate polynomial $F \in \mathbb{F}_p[X,Y]$ 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 $X_1,\ldots ,X_m$ over the field $\mathbb{F}_p$. These results have been motivated by the predictability problem for nonlinear pseudorandom number generators and other potential applications to cryptography

Arithmetic Properties of Integers in Chains and Reflections of g-ary Expansions

During the preparation of this paper, the first author was partially supported by project MTM2014-55421-P from the Ministerio de Economia y Competitividad and the second author was partially supported by Australian Research Council Grant DP140100118

On the Expansion Complexity of Sequences over Finite Fields

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 socalled irreducible-expansion complexity which is more suitable for certain applications. We analyze both, the classical and the modified expansion complexity. Moreover, we also study the expansion complexity of the explicit inversive congruential generator.The research of the first author was supported by the Ministerio de Economia y Competitividad research project MTM2014-55421-P. The second was partially supported by the Austrian Science Fund FWF Project F5511-N26 which is part of the Special Research Program ”Quasi-Monte Carlo Methods: Theory and Application

On irreducible divisors of iterated polynomials

D. Gómez-Pérez, A. Ostafe, A.P. Nicolás and D. Sadornil have recently shown that for almost all polynomials f?Fq[X]f?Fq[X] over the finite field of qq elements, where qq is an odd prime power, their iterates eventually become reducible polynomials over FqFq. Here we combine their method with some new ideas to derive finer results about the arithmetic structure of iterates of ff. In particular, we prove that the nnth iterate of ff has a square-free divisor of degree of order at least n1+o(1)n1+o(1) as n?8n?8 (uniformly in qq)