Building Correlation Immune Functions from Sets of Mutually Orthogonal Cellular Automata

Correlation immune Boolean functions play an important role in the implementation of efficient masking countermeasures for side-channel attacks in cryptography. In this paper, we investigate a method to construct correlation immune functions through families of mutually orthogonal cellular automata (MOCA). First, we show that the orthogonal array (OA) associated to a family of MOCA can be expanded to a binary OA of strength at least 2. To prove this result, we exploit the characterization of MOCA in terms of orthogonal labelings on de Bruijn graphs. Then, we use the resulting binary OA to define the support of a second-order correlation immune function. Next, we perform some computational experiments to construct all such functions up to $n=12$ variables, and observe that their correlation immunity order is actually greater, always at least 3. We conclude by discussing how these results open up interesting perspectives for future research, with respect to the search of new correlation-immune functions and binary orthogonal arrays

Клеточно-автоматный алгоритм пермутации матриц

The article describes an algorithm to provide permutation of matrix elements through cyclic shifts of rows and columns and gives a formal description of a cellular automaton (CA) that implements this algorithm. For this, a square n × n lattice with closed boundaries and a neighbourhood of a von Neumann type cell are used.As a result of a computational experiment for the initial orders of the matrix n, it was found that after a sufficiently large number of steps, the algorithm transfers the matrix to the original one, i.e. has period N. For odd orders of the matrix, the growth of N as a function of n is faster than exponential.A movement of the individual elements of the matrix has been analysed to show that they move in a manner similar to billiard balls. The element moves to the matrix boundaries at an angle of 45 ° and changes direction when it reaches the bound. A defined explicit dependence of the element movement period on its initial position in the matrix allows us to prove that the global period N is equal to the least common multiple of all the odd numbers being less than 2n, i.e. N = LCM(3,5, ..., 2n-1).To analyse the permutation dynamics, two authors-introduced “metrics” that reflect a degree of randomizing were used. One of the metrics is introduced specifically for the matrix, the other for the linear array and depends on how the matrix is transformed into a one-dimensional array. By searching for permutations with extreme metric values, it was found that as a result of permutations, even-order matrices undergo block permutation. In the case of an odd order, the matrix undergoes a rotation of ± 90 ° and 180 ° (reflection relative to the centre). Moreover, the direction of rotation depends on the order n. For example, for n = 5, counter-clockwise rotation occurs, and for n = 7, it is clockwise.The algorithm can be used to generate pseudorandom numbers, and a value of its period indirectly argues for it. The period can be significantly increased through a slight complication of the algorithm, for example, by introducing different lengths for cycles of changing a direction of columns and rows. Stopping the algorithm at random time, one can consider the current permutation of the matrix as an array of random numbers or somehow transform it into one random number.В статье описывается алгоритм, осуществляющий перестановку элементов матрицы посредством циклических сдвигов строк и столбцов. Дано формальное описание клеточного автомата (КА), реализующего данный алгоритм. Для этого используется квадратная решётка размера n×n с замкнутыми границами и окрестность клетки типа фон Неймана.В результате вычислительного эксперимента для начальных порядков матрицы n установлено, что через достаточно большое число шагов алгоритм переводит матрицу в исходную, т.е. имеет период N. Для нечётных порядков матрицы рост N как функции n оказывается быстрее экспоненциального.Проведён анализ движения отдельных элементов матрицы. Показано, что они перемещаются по аналогии с бильярдными шарами. Элемент двигается под углом 45° к границам матрицы и меняет направление при достижении границы. Найдена явная зависимость периода движения элемента от его начального положения в матрице. На основе этой зависимости доказано, что глобальный период N равен наименьшему общему кратному всех нечетных чисел, меньших 2n, т.е. N=НОК(3,5,…,2n-1).Динамика пермутаций проанализирована с помощью введенных авторами двух «метрик», отражающих степень перемешанности. Одна из метрик вводится специально для матрицы, другая — для линейного массива и зависит от способа преобразования матрицы в одномерный массив. Поиском перестановок с экстремальными значениями метрик установлено, что в результате пермутаций матрицы чётных порядков подвергаются блочной перестановке. В случае же нечётного порядка матрица претерпевает поворот на ±90° и на 180° (отражение относительно центра). При этом направление вращение зависит от порядка n. Например, при n=5 вращение происходит против часовой стрелки, а при n=7 — по часовой стрелке.Алгоритм может быть использован для генерации псевдослучайных чисел, в пользу чего косвенно говорит величина его периода. Период может быть существенно увеличен небольшим усложнением алгоритма, например, введением различных длин для циклов смены направления столбцов и строк. Останавливая алгоритм в случайный момент времени, текущую перестановку матрицы можно рассматривать как массив случайных чисел или же преобразовать её каким-либо способом в одно случайное число

Cellular Automata Applications in Shortest Path Problem

Cellular Automata (CAs) are computational models that can capture the essential features of systems in which global behavior emerges from the collective effect of simple components, which interact locally. During the last decades, CAs have been extensively used for mimicking several natural processes and systems to find fine solutions in many complex hard to solve computer science and engineering problems. Among them, the shortest path problem is one of the most pronounced and highly studied problems that scientists have been trying to tackle by using a plethora of methodologies and even unconventional approaches. The proposed solutions are mainly justified by their ability to provide a correct solution in a better time complexity than the renowned Dijkstra's algorithm. Although there is a wide variety regarding the algorithmic complexity of the algorithms suggested, spanning from simplistic graph traversal algorithms to complex nature inspired and bio-mimicking algorithms, in this chapter we focus on the successful application of CAs to shortest path problem as found in various diverse disciplines like computer science, swarm robotics, computer networks, decision science and biomimicking of biological organisms' behaviour. In particular, an introduction on the first CA-based algorithm tackling the shortest path problem is provided in detail. After the short presentation of shortest path algorithms arriving from the relaxization of the CAs principles, the application of the CA-based shortest path definition on the coordinated motion of swarm robotics is also introduced. Moreover, the CA based application of shortest path finding in computer networks is presented in brief. Finally, a CA that models exactly the behavior of a biological organism, namely the Physarum's behavior, finding the minimum-length path between two points in a labyrinth is given.Comment: To appear in the book: Adamatzky, A (Ed.) Shortest path solvers. From software to wetware. Springer, 201

Четыре клеточно-автоматных алгоритма пермутаций матриц

Numerical calculation uses to describe the operation of matrix permutation algorithms based on cyclic shifts of rows and columns. This choice of discrete transformation algorithms justified by the convenience of the cellular automaton (CA) formulation, which is used. Obtained Empirical formulas for the permutation period and for the last algorithm, which period formula is recurrent. For a base scheme period has the asymptotics:  for a matrix  with pairwise different elements. Despite the complexity of the scheme, the other two modifications only give a polynomial growth of period, no higher than 3. Fourth scheme has a non-trivial period dependence, but no higher than the exponential. In some cases algorithms make special permutations: rotate, reflect, and rearrange blocks for the matrix . These formulas are closely related to individual cells paths. And paths connected with the influence of the boundaries that gives branching the matrix order by subtraction class modulo 3,4 or 12. Visualizations of these paths make in the extended CA-field. Two "mixing metrics" analyze as a parameter of CA dynamics on matrix permutations (compared to the initial). For all schemes and most branches, the behavior of these metrics shows in graphs and histograms (conditional density distribution) showing how often the permutation period occurs with the specified interval of metrics. The practical aim of this work is in the field of pseudorandom number generation and cryptography.С помощью численного расчета описывается работа алгоритмов пермутаций матриц, основанных на циклических сдвигах строк и столбцов. Такой выбор алгоритмов дискретного преобразования обоснован удобством клеточно-автоматных формулировок, которые и приводятся. Получены эмпирические формулы для периода пермутаций; для последнего алгоритма формула периода носит рекуррентный характер. Для базовой и наиболее простой схемы период N(n) имеет асимптотику exp(2n)/n для матрицы nxn с попарно различными элементами. Несмотря на усложнение схемы алгоритма, две другие модификации дают лишь полиномиальный рост степени не выше 3. Четвертая схема имеет нетривиальную зависимость периода, но не выше экспоненциальной. В ряде случаев алгоритмы порождают особые пермутации: поворот, отражение и перестановку блоков для матрицы 2kx2k. Эти формулы тесно связаны с индивидуальными траекториями элементов, а они – с влиянием границ, что дает ветвление порядка матрицы по классу вычета по модулю 3,4 или 12. Визуализации этих траекторий приводятся в расширенном поле КА. В качестве параметра динамики КА анализируются две «метрики перемешанности» на пермутациях матрицы (по сравнению с начальной). Для всех схем и большинства ветвей поведение этих метрик представлено на графиках и гистограммах (условно: плотности распределения), показывающих, как часто встречаются по периоду пермутации с заданным интервалом значений метрик. Практическое значение работы состоит в оценке применения КА в областях генерации псевдослучайных чисел и криптографии

Exploring Millions of 6-State FSSP Solutions: the Formal Notion of Local CA Simulation

In this paper, we come back on the notion of local simulation allowing to transform a cellular automaton into a closely related one with different local encoding of information. This notion is used to explore solutions of the Firing Squad Synchronization Problem that are minimal both in time (2n -- 2 for n cells) and, up to current knowledge, also in states (6 states). While only one such solution was proposed by Mazoyer since 1987, 718 new solutions have been generated by Clergue, Verel and Formenti in 2018 with a cluster of machines. We show here that, starting from existing solutions, it is possible to generate millions of such solutions using local simulations using a single common personal computer

Performance of distributed multiscale simulations

Multiscale simulations model phenomena across natural scales using monolithic or component-based code, running on local or distributed resources. In this work, we investigate the performance of distributed multiscale computing of component-based models, guided by six multiscale applications with different characteristics and from several disciplines. Three modes of distributed multiscale computing are identified: supplementing local dependencies with large-scale resources, load distribution over multiple resources, and load balancing of small- and large-scale resources. We find that the first mode has the apparent benefit of increasing simulation speed, and the second mode can increase simulation speed if local resources are limited. Depending on resource reservation and model coupling topology, the third mode may result in a reduction of resource consumption

Artificial morphogenesis via 3D neural cellular automata

Morphogenesis is the biological process by which an organism grows and takes its form. It is responsible for the growth and shape of all living things and has historically been studied from the perspective of developmental biology. Accordingly, experiments have routinely been performed using real organisms and their cells. However, with continued advancements in computational capabilities, researchers have been able to simulate physical and biological systems with greater fidelity and accuracy. A recent example is the development of two-dimensional neural cellular automata, which have the ability to artificially model morphogenesis. Cellular automata, the antecedent to neural cellular automata, are a type of computational model uniquely suited to model physical systems due to their inherit discretization of space and time, locally rule-based dynamics, and embarrassingly parallel execution. Their ability to exhibit rich and sophisticated results from seemingly trivial implementations has lead to their continued use to study emergent properties found in nature. This thesis expands upon previous works by presenting a novel three-dimensional neural cellular automata which illustrates self-organizing, regenerative, and isotropic properties. Understanding the complexities of morphogenesis could provide beneficial insights for the development of regenerative medicines, self-organizing robots, and other bioengineering endeavors.Computer Scienc

Asynchronous cellular automata and applications - Special issue of Natural Computing

International audienceThis special issue contains four papers presented during the workshop "2nd International Workshop on Asynchronous Cellular Automata and Asynchronous Discrete Models" (ACA 2012), held at the 10th International Conference on Cellular Automata for Research and Industry (ACRI 2012) in Santorini Island (Greece) in the period September 24th-27th, 2012

Spatially explicit migration models of pike to support river management

De status van verschillende vissoorten in ons land, waaronder ook snoek (Esox lucius) voldoet niet aan de gestelde Europese vereisten. Behalve door een matige chemische waterkwaliteit komt dit voornamelijk door een ondermaatse habitatkwaliteit door habitatdegradatie, fragmentatie en obstructie. Rivierbeheerders plannen daarom maatregelen om het habitat te beschermen, te verbeteren of opnieuw toegankelijk te maken voor migrerende vissen. Habitatgeschiktheid- en soortverspreidingsmodellen kunnen helpen om het effect van deze maatregelen te voorspellen. Deze modellen zijn vaak niet in staat rekening te houden met factoren die gerelateerd zijn aan migratie en toegankelijkheid omdat ze niet ruimtelijk expliciet en dynamisch tegelijk zijn. In dit doctoraatsonderzoek evalueerden we de toepasbaarheid voor het simuleren van snoekmigratie van twee modelleertechnieken die wel geschikt lijken: Individueel Gebaseerde Modellen (IBMs) en Cellulaire Automaten (CAs). Daarnaast onderzochten we de migratiedynamiek, het habitatgebruik en de habitatpreferentie van volwassen snoeken ter ondersteuning van het rivierbeheer. Hiervoor werden veldgegevens verzameld van snoeken in de Ijzer (West-Vlaanderen) m.b.v. radiotelemetrie. De resultaten van dit onderzoek wijzen op een goede toepasbaarheid van IBMs en moeilijkheden bij het toepassen van de CAs voor de simulatie van snoekmigratie. De analyses van de veldgegevens tonen grote individuele verschillen in gedrag en onderlijnen het belang van habitatheterogeniteit en het toegankelijk maken van bestaande geschikte habitats voor volwassen snoeken. Dit onderzoek geeft meer inzicht in het ruimtelijk expliciet simuleren van snoekmigratie en levert kennis over de ecologie van snoek met directe suggesties voor rivierbeheerders