17 research outputs found
The Graph Structure of Chebyshev Polynomials over Finite Fields and Applications
We completely describe the functional graph associated to iterations of
Chebyshev polynomials over finite fields. Then, we use our structural results
to obtain estimates for the average rho length, average number of connected
components and the expected value for the period and preperiod of iterating
Chebyshev polynomials
Special Function Representation Of Dickson Polynomials
Generating functions and functional equations of Dickson polynomials of the
first and second kind are derived and continued analytically. These formulae
are expressed in terms of the incomplete gamma function over complex variables
of the parameters involved. Special cases are evaluated in terms of composite
incomplete gamma functions and mathematical constants
Reversed Dickson polynomials
We investigate fixed points and cycle types of permutation polynomials and
complete permutation polynomials arising from reversed Dickson polynomials of
the first kind and second kind over . We also study the
permutation behaviour of reversed Dickson polynomials of the first kind and
second kind over . Moreover, we prove two special cases of a
conjecture on the permutation behaviour of reversed Dickson polynomials over
.Comment: 40 page
Generalized Reciprocals, Factors of Dickson Polynomials and Generalized Cyclotomic Polynomials over Finite Fields
We give new descriptions of the factors of Dickson polynomials over finite fields in terms of cyclotomic factors. To do this generalized reciprocal polynomials are introduced and characterized. We also study the factorization of generalized cyclotomic polynomials and their relationship to the factorization of Dickson polynomials
Neural Networks as Pseudorandom Number Generators
This thesis brings two disparate fields of research together; the fields of artificial neural networks and pseudorandom number generation. In it, we answer variations on the following question: can recurrent neural networks generate pseudorandom numbers? In doing so, we provide a new construction of an -dimensional neural network that has period , for all . We also provide a technique for constructing neural networks based on the theory of shift register sequences. The randomness capabilities of these networks is then measured via the theoretical notion of computational indistinguishability and a battery of statistical tests. In particular, we show that neural networks cannot be pseudorandom number generators according to the theoretical definition of computational indistinguishability. We contrast this result by providing some neural networks that pass all of the tests in the SmallCrush battery of tests in the TestU01 testing suite
Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science (STACS'09)
The Symposium on Theoretical Aspects of Computer Science (STACS) is held alternately in France and in Germany. The conference of February 26-28, 2009, held in Freiburg, is the 26th in this series. Previous meetings took place in Paris (1984), Saarbr¨ucken (1985), Orsay (1986), Passau (1987), Bordeaux (1988), Paderborn (1989), Rouen (1990), Hamburg (1991), Cachan (1992), W¨urzburg (1993), Caen (1994), M¨unchen (1995), Grenoble (1996), L¨ubeck (1997), Paris (1998), Trier (1999), Lille (2000), Dresden (2001), Antibes (2002), Berlin (2003), Montpellier (2004), Stuttgart (2005), Marseille (2006), Aachen (2007), and Bordeaux (2008). ..