Families of sequences with good family complexity and cross-correlation measure

In this paper we study pseudorandomness of a family of sequences in terms of two measures, the family complexity ($f$-complexity) and the cross-correlation measure of order $\ell$. We consider sequences not only on binary alphabet but also on $k$-symbols ($k$-ary) alphabet. We first generalize some known methods on construction of the family of binary pseudorandom sequences. We prove a bound on the $f$-complexity of a large family of binary sequences of Legendre-symbols of certain irreducible polynomials. We show that this family as well as its dual family have both a large family complexity and a small cross-correlation measure up to a rather large order. Next, we present another family of binary sequences having high $f$-complexity and low cross-correlation measure. Then we extend the results to the family of sequences on $k$-symbols alphabet.Comment: 13 pages. Comments are welcome

Формувач послідовностей псевдовипадкових наборів заданої ваги

Кваліфікаційна робота включає пояснювальну записку (53 с., 30 рис. 17 табл.). Об’єкт розробки – створення комп’ютерної програми, відтворюючої роботу формувача псевдовипадкових двійкових послідовностей з заданою вагою на базі регістру зсуву з лінійним зворотним зв’язком. Комп’ютерна програма дозволяє: перевіряти ініціалізуючі параметри, задавати поліном та початновий стан для первинного генератора псевдовипадкових двійкових послідовностей, ініціалізувати генератор псевдовипадкових двійкових послідовностей з постійною вагою, конфігурувати обидва генератори, генерувати псевдовипадкові послідовності з постійною вагою та без, зберігати згенеровані послідовності, зберігати, аналізувати та виводити статистичні дані після генерації усіх можливих послідовностей. В ході розробки: • створено програму на базі алгоритму роботи принципової схеми генератора; • проведено аналіз методів побудови генераторів псевдовипадкових двійкових послідовностей; • проведено аналіз можливих модифікацій базового генератора для генерації послідовностей з постійною вагою; • розроблена принципова схема генератора псевдовипадкових послідовностей з постійною вагою; • створено алгоритм моделювання роботи принципової схеми; • розроблено застосунок для роботи з генератором;The qualifying work includes an explanatory note (53 p., 30 pic. 17 tables.). The object of development - the creation of a computer program reproducing the work of the scheme of a pseudorandom binary sequence generator with a given weight based on a shift register with linear feedback. The computer program allows you to: validate initializing parameters, initialize the primary pseudorandom binary generator, initialize the pseudorandom binary sequence generator with constant weight, configure both generators, generate pseudorandom sequences with constant weight and without, store generated sequences, output statistics after generation of all possible sequences. During development: • a custom application developed on the basis of the algorithm of the principal circuit for working with the generator; • An analysis of methods for constructing generators of pseudorandom binary sequences is carried out; • the analysis of the progress of the basic generator for fulfilling the requirements for the generation of sequences with constant weight; • the principal scheme of the generator of pseudorandom sequences of constant weight is developed; • the algorithm is developed on the basis of the principle scheme; • a custom application developed on the basis of the algorithm of the principal circuit for working with the generator

Yet Another Pseudorandom Number Generator

We propose a novel pseudorandom number generator based on R\"ossler attractor and bent Boolean function. We estimated the output bits properties by number of statistical tests. The results of the cryptanalysis show that the new pseudorandom number generation scheme provides a high level of data security.Comment: 5 pages, 7 figures; to be published in International Journal of Electronics and Telecommunications, vol.63, no.

On the Design of LIL Tests for (Pseudo) Random Generators and Some Experimental Results

NIST SP800-22 (2010) proposes the state of art testing suite for (pseudo) random generators to detect deviations of a binary sequence from randomness. On the one hand, as a counter example to NIST SP800-22 test suite, it is easy to construct functions that are considered as GOOD pseudorandom generators by NIST SP800-22 test suite though the output of these functions are easily distinguishable from the uniform distribution. Thus these functions are not pseudorandom generators by definition. On the other hand, NIST SP800-22 does not cover some of the important laws for randomness. Two fundamental limit theorems about random binary strings are the central limit theorem and the law of the iterated logarithm (LIL). Several frequency related tests in NIST SP800-22 cover the central limit theorem while no NIST SP800-22 test covers LIL. This paper proposes techniques to address the above challenges that NIST SP800-22 testing suite faces. Firstly, we propose statistical distance based testing techniques for (pseudo) random generators to reduce the above mentioned Type II errors in NIST SP800-22 test suite. Secondly, we propose LIL based statistical testing techniques, calculate the probabilities, and carry out experimental tests on widely used pseudorandom generators by generating around 30TB of pseudorandom sequences. The experimental results show that for a sample size of 1000 sequences (2TB), the statistical distance between the generated sequences and the uniform distribution is around 0.07 (with $0$ for statistically indistinguishable and $1$ for completely distinguishable) and the root-mean-square deviation is around 0.005

Non-binary m-sequences for more comfortable brain–computer interfaces based on c-VEPs

Producción CientíficaCode-modulated visual evoked potentials (c-VEPs) have marked a milestone in the scientific literature due to their ability to achieve reliable, high-speed brain–computer interfaces (BCIs) for communication and control. Generally, these expert systems rely on encoding each command with shifted versions of binary pseudorandom sequences, i.e., flashing black and white targets according to the shifted code. Despite the excellent results in terms of accuracy and selection time, these high-contrast stimuli cause eyestrain for some users. In this work, we propose the use of non-binary p-ary m-sequences, whose levels are encoded with different shades of gray, as a more pleasant alternative than traditional binary codes. The performance and visual fatigue of these p-ary m-sequences, as well as their ability to provide reliable c-VEP-based BCIs, are analyzed for the first time.Ministerio de Ciencia e Innovación/AEI- FEDER [TED2021-129915B-I00, RTC2019-007350-1 y PID2020-115468RB-I00