On linear complexity of binary lattices, II

The Ramanujan Journal, 2013

The linear complexity is an important and frequently used measure of unpredictability and pseudorandomness of binary sequences. In this paper our goal is to extend this notion to two dimensions. We will define and study the linear complexity of binary lattices. The linear complexity of a truly random binary lattice will be estimated. Finally, we will analyze the connection between the linear complexity and the correlation measures, and we will utilize the inequalities obtained in this way for estimating the linear complexity of an important special binary lattice. Finally, we will study the connection between the linear complexity of binary lattices and of the associated binary sequences.

Publicationes Mathematicae Debrecen, 2011

In the last 15 years a new constructive theory of pseudorandomness of binary sequences has been developed. Later this theory was extended to n dimensions, i.e., to the study of pseudorandomness of binary lattices. In the applications it is not enough to consider single binary sequences, one also needs information on the structure of large families of binary sequences with strong pseudorandom properties. Thus the related notions of family complexity, collision and avalanche effect have been introduced. In this paper our goal is to extend these definitions to binary lattices, and we will present constructions of large families of binary lattices with strong pseudorandom properties such that these families also possess a nice structure.

Discrete Applied Mathematics, 2017

Pseudorandom binary sequences play a crucial role in cryptography. The classical approach to pseudorandomness of binary sequences is based on computational complexity. This approach has certain weak points thus in the last two decades years a new, more constructive and quantitative approach has been developed. Since multidimensional analogs of binary sequences (called binary lattices) also have important applications thus it is a natural idea to extend this new approach to the multidimensional case. This extension started with a paper published in 2006, and since that about 25 papers have been written on this subject. Here our goal is to present a survey of all these papers.

Acta Arithmetica, 2008

Acta Arithmetica, 2010

Finite fields and their applications, 2022

The correlation measure of order k is an important measure of pseudorandomness for binary sequences. This measure tries to look for dependence between several shifted versions of a sequence. We study the relation between the correlation measure of order k and two other pseudorandom measures: the N th linear complexity and the N th maximum order complexity. We simplify and improve several state-of-the-art lower bounds for these two measures using the Hamming bound as well as weaker bounds derived from it.

IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, 2023

In this survey we summarize properties of pseudorandomness and non-randomness of some number-theoretic sequences and present results on their behaviour under the following measures of pseudorandomness: balance, linear complexity, correlation measure of order , expansion complexity and 2-adic complexity. The number-theoretic sequences are the Legendre sequence and the two-prime generator, the Thue-Morse sequence and its sub-sequence along squares, and the prime omega sequences for integers and polynomials.

Lecture Notes in Computer Science, 2018

The cross-combined measure (which is a natural extension of cross-correlation measure) is introduced and important constructions of large families of binary lattices with optimal or nearly optimal cross-combined measures are presented. These results are also strongly related to the onedimensional case: An easy method is showed obtaining strong constructions of families of binary sequences with nearly optimal cross-correlation measures based on the previous constructions of families of lattices. The important feature of this result is that so far there exists only one type of constructions of very large families of binary sequences with small cross-correlation measure, and this only type of constructions was based on one-variable irreducible polynomials. Since it is very complicated to construct one-variable irreducible polynomials over Fp, it became necessary to show other types of constructions where the generation of sequences is much faster. Using binary lattices based on two-variable irreducible polynomials this problem can be avoided. (Since, contrary to one-variable polynomials, using Schöneman-Eisenstein criteria it is possible to generate two-variable irreducible polynomials over Fp fast.)

arXiv (Cornell University), 2021

Correlation measure of order k is an important measure of randomness in binary sequences. This measure tries to look for dependence between several shifted version of a sequence. We study the relation between the correlation measure of order k and another two pseudorandom measures: the N th linear complexity and the N th maximum order complexity. We simplify and improve several state-of-the-art lower bounds for these two measures using the Hamming bound as well as weaker bounds derived from it.

Applicable Algebra in Engineering, Communication and Computing, 2002

It is shown that a q-periodic sequence over the finite field F q passes an extended version of Marsaglia's lattice test for high dimensions if and only if its linear complexity is large. The consequences of this result for nonlinear and inversive pseudorandom number generators are worked out.