18 research outputs found
Multisequences with high joint nonlinear complexity
We introduce the new concept of joint nonlinear complexity for multisequences
over finite fields and we analyze the joint nonlinear complexity of two
families of explicit inversive multisequences. We also establish a
probabilistic result on the behavior of the joint nonlinear complexity of
random multisequences over a fixed finite field
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
Generalized explicit inversive generators of small p-weight degree
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
On lattice profile of the elliptic curve linear congruential generators
Lattice tests are quality measures for assessing the intrinsic structure of pseudorandom number generators. Recently a new lattice test has been introduced by Niederreiter and Winterhof. 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 elliptic curves (EC-LCG) under this new lattice test and prove that the EC-LCG passes it up to very high dimensions. We also use a result of Brandstätter and Winterhof on the linear complexity profile related to the correlation measure of order k to present lower bounds on the linear complexity profile of some binary sequences derived from the EC-LCG
Fractional jumps: complete characterisation and an explicit infinite family
In this paper we provide a complete characterisation of transitive fractional
jumps by showing that they can only arise from transitive projective
automorphisms. Furthermore, we prove that such construction is feasible for
arbitrarily large dimension by exhibiting an infinite class of projectively
primitive polynomials whose companion matrix can be used to define a full orbit
sequence over an affine space
Recent advances in the theory of nonlinear pseudorandom number generators
The classical linear congruential method for generating uniform pseudorandom numbers has some deficiencies that can render them useless for some simulation problems. This fact motivated the design and the analysis of nonlinear congruential methods for the generation of pseudorandom numbers. In this thesis, we aim to review the recent developments in the study of nonlinear congruential pseudorandom generators. Our exposition concentrates on inversive generators. We also describe the so-called power generator and the quadratic exponential generator which are particularly interesting for cryptographic applications. We give results on the period length and theoretical analysis of these generators. The emphasis is on the lattice structure, discrepancy and linear complexity of the generated sequences
Algebraic entropy, automorphisms and sparsity of algebraic dynamical systems and pseudorandom number generators
ABSTRACT: 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 pseu-dorandom number generators. We then give several concrete constructions that illustrate the applicability of these general results
Finite Fields: Theory and Applications
Finite fields are the focal point of many interesting geometric, algorithmic and combinatorial problems. The workshop was devoted to progress on these questions, with an eye also on the important applications of finite field techniques in cryptography, error correcting codes, and random number generation