Rational approximations, multidimensional continued fractions and lattice reduction

We first survey the current state of the art concerning the dynamical properties of multidimensional continued fraction algorithms defined dynamically as piecewise fractional maps and compare them with algorithms based on lattice reduction. We discuss their convergence properties and the quality of the rational approximation, and stress the interest for these algorithms to be obtained by iterating dynamical systems. We then focus on an algorithm based on the classical Jacobi--Perron algorithm involving the nearest integer part. We describe its Markov properties and we suggest a possible procedure for proving the existence of a finite ergodic invariant measure absolutely continuous with respect to Lebesgue measure.Comment: 30 pages, 4 figure

Continued fraction for formal laurent series and the lattice structure of sequences

Besides equidistribution properties and statistical independence the lattice profile, a generalized version of Marsaglia's lattice test, provides another quality measure for pseudorandom sequences over a (finite) field. It turned out that the lattice profile is closely related with the linear complexity profile. In this article we give a survey of several features of the linear complexity profile and the lattice profile, and we utilize relationships to completely describe the lattice profile of a sequence over a finite field in terms of the continued fraction expansion of its generating function. Finally we describe and construct sequences with a certain lattice profile, and introduce a further complexity measure

Pseudorandomness and Dynamics of Fermat Quotients

We obtain some theoretic and experimental results concerning various properties (the number of fixed points, image distribution, cycle lengths) of the dynamical system naturally associated with Fermat quotients acting on the set $\{0, ..., p-1\}$. We also consider pseudorandom properties of Fermat quotients such as joint distribution and linear complexity

Linear Complexity Profiles: Hausdorff Dimensions for Almost Perfect Profiles and Measures for General Profiles

AbstractStream ciphers usually employ some sort of pseudorandomly generated bit strings to be added to the plaintext. The cryptographic properties of such a sequenceacan be stated in terms of the so-called linear complexity profile (l.c.p.),La(t),t∈ N. If the l.c.p. isLa(t) =t/2 +O(1), it is called (almost)perfect. This paper examines first those subsets Ad(q)of Fq∞where for fixedd∈ N the l.c.p. satisfies |2 ·La(t) −t| ≤dfor allt∈ N. It turns out that (after suitably mapping Ad(q)on [0, 1] ⊂ R) the Hausdorff dimension is1+logqϕd(q)2where ϕd(q)is the largest real root ofxd= (q− 1) · ∑i=0d−1xi. The second part deals with nondecreasing boundsd: N → N. Sinced(t) → ∞ ast→ ∞ always leads to a Hausdorff dimension 1, here we consider the measure of the set Ad(q)

Mathematics Yearbook 2021

The Deakin University Mathematics Yearbook publishes student reports and articles in all areas of mathematics with an aim of promoting interest and engagement in mathematics and celebrating student achievements. The 2021 edition includes 7 coursework articles, where students have extended upon submissions in their mathematics units, as well as 4 articles based on student research projects conducted throughout 2020 and 2021