5 research outputs found
On Some Dynamical Systems in Finite Fields and Residue Rings
We use character sums to confirm several recent conjectures of V. I. Arnold
on the uniformity of distribution properties of a certain dynamical system in a
finite field. On the other hand, we show that some conjectures are wrong. We
also analyze several other conjectures of V. I. Arnold related to the orbit
length of similar dynamical systems in residue rings and outline possible ways
to prove them. We also show that some of them require further tuning
Full Orbit Sequences in Affine Spaces via Fractional Jumps and Pseudorandom Number Generation
Let be a positive integer. In this paper we provide a general theory to
produce full orbit sequences in the affine -dimensional space over a finite
field. For our construction covers the case of the Inversive Congruential
Generators (ICG). In addition, for we show that the sequences produced
using our construction are easier to compute than ICG sequences. Furthermore,
we prove that they have the same discrepancy bounds as the ones constructed
using the ICG.Comment: To appear in Mathematics of Computatio
On the Degree Growth in Some Polynomial Dynamical Systems and Nonlinear Pseudorandom Number Generators
In this paper we study a class of dynamical systems generated by iterations
of multivariate polynomials and estimate the degreegrowth of these iterations.
We use these estimates to bound exponential sums along the orbits of these
dynamical systems and show that they admit much stronger estimates than in the
general case and thus can be of use for pseudorandom number generation.Comment: Mathematics of Computation (to appear
Dynamical systems generated by rational functions
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)26436-1