464 research outputs found
Cryptographic properties of Boolean functions defining elementary cellular automata
In this work, the algebraic properties of the local transition functions of elementary cellular automata (ECA) were analysed. Specifically, a classification of such cellular automata was done according to their algebraic degree, the balancedness, the resiliency, nonlinearity, the propagation criterion and the existence of non-zero linear structures. It is shown that there is not any ECA satisfying all properties at the same time
Heuristic search of (semi-)bent functions based on cellular automata
An interesting thread in the research of Boolean functions for cryptography and coding theory is the study of secondary constructions: given a known function with a good cryptographic profile, the aim is to extend it to a (usually larger) function possessing analogous properties. In this work, we continue the investigation of a secondary construction based on cellular automata (CA), focusing on the classes of bent and semi-bent functions. We prove that our construction preserves the algebraic degree of the local rule, and we narrow our attention to the subclass of quadratic functions, performing several experiments based on exhaustive combinatorial search and heuristic optimization through Evolutionary Strategies (ES). Finally, we classify the obtained results up to permutation equivalence, remarking that the number of equivalence classes that our CA-XOR construction can successfully extend grows very quickly with respect to the CA diameter
Pseudorandom sequence generation using binary cellular automata
Tezin basılısı İstanbul Şehir Üniversitesi Kütüphanesi'ndedir.Random numbers are an integral part of many applications from computer simulations,
gaming, security protocols to the practices of applied mathematics and physics. As
randomness plays more critical roles, cheap and fast generation methods are becoming a
point of interest for both scientific and technological use.
Cellular Automata (CA) is a class of functions which attracts attention mostly due to the
potential it holds in modeling complex phenomena in nature along with its discreteness
and simplicity. Several studies are available in the literature expressing its potentiality
for generating randomness and presenting its advantages over commonly used random
number generators.
Most of the researches in the CA field focus on one-dimensional 3-input CA rules. In
this study, we perform an exhaustive search over the set of 5-input CA to find out the
rules with high randomness quality. As the measure of quality, the outcomes of NIST
Statistical Test Suite are used.
Since the set of 5-input CA rules is very large (including more than 4.2 billions of rules),
they are eliminated by discarding poor-quality rules before testing.
In the literature, generally entropy is used as the elimination criterion, but we preferred
mutual information. The main motive behind that choice is to find out a metric for
elimination which is directly computed on the truth table of the CA rule instead of the
generated sequence. As the test results collected on 3- and 4-input CA indicate, all rules
with very good statistical performance have zero mutual information. By exploiting this
observation, we limit the set to be tested to the rules with zero mutual information. The
reasons and consequences of this choice are discussed.
In total, more than 248 millions of rules are tested. Among them, 120 rules show out-
standing performance with all attempted neighborhood schemes. Along with these tests,
one of them is subjected to a more detailed testing and test results are included.
Keywords: Cellular Automata, Pseudorandom Number Generators, Randomness TestsContents
Declaration of Authorship ii
Abstract iii
Öz iv
Acknowledgments v
List of Figures ix
List of Tables x
1 Introduction 1
2 Random Number Sequences 4
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Theoretical Approaches to Randomness . . . . . . . . . . . . . . . . . . . 5
2.2.1 Information Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.2 Complexity Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.3 Computability Theory . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Random Number Generator Classification . . . . . . . . . . . . . . . . . . 7
2.3.1 Physical TRNGs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.2 Non-Physical TRNGs . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3.3 Pseudorandom Number Generators . . . . . . . . . . . . . . . . . . 10
2.3.3.1 Generic Design of Pseudorandom Number Generators . . 10
2.3.3.2 Cryptographically Secure Pseudorandom Number Gener- ators . . . . . . . . . . . . . .11
2.3.4 Hybrid Random Number Generators . . . . . . . . . . . . . . . . . 13
2.4 A Comparison between True and Pseudo RNGs . . . . . . . . . . . . . . . 14
2.5 General Requirements on Random Number Sequences . . . . . . . . . . . 14
2.6 Evaluation Criteria of PRNGs . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.7 Statistical Test Suites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.8 NIST Test Suite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.8.1 Hypothetical Testing . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.8.2 Tests in NIST Test Suite . . . . . . . . . . . . . . . . . . . . . . . . 20
2.8.2.1 Frequency Test . . . . . . . . . . . . . . . . . . . . . . . . 20
2.8.2.2 Block Frequency Test . . . . . . . . . . . . . . . . . . . . 20
2.8.2.3 Runs Test . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.8.2.4 Longest Run of Ones in a Block . . . . . . . . . . . . . . 21
2.8.2.5 Binary Matrix Rank Test . . . . . . . . . . . . . . . . . . 21
2.8.2.6 Spectral Test . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.8.2.7 Non-overlapping Template Matching Test . . . . . . . . . 22
2.8.2.8 Overlapping Template Matching Test . . . . . . . . . . . 22
2.8.2.9 Universal Statistical Test . . . . . . . . . . . . . . . . . . 23
2.8.2.10 Linear Complexity Test . . . . . . . . . . . . . . . . . . . 23
2.8.2.11 Serial Test . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.8.2.12 Approximate Entropy Test . . . . . . . . . . . . . . . . . 24
2.8.2.13 Cumulative Sums Test . . . . . . . . . . . . . . . . . . . . 24
2.8.2.14 Random Excursions Test . . . . . . . . . . . . . . . . . . 24
2.8.2.15 Random Excursions Variant Test . . . . . . . . . . . . . . 25
3 Cellular Automata 26 3.1 History of Cellular Automata . . . . . . . . . . . . . . . . . . . . . . . .26
3.1.1 von Neumann’s Work . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1.2 Conway’s Life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.1.3 Wolfram’s Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 Cellular Automata and the Definitive Parameters . . . . . . . . . . . . . . 31
3.2.1 Lattice Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.2 Cell Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.3 Guiding Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.4 Neighborhood Scheme . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3 A Formal Definition of Cellular Automata . . . . . . . . . . . . . . . . . . 37
3.4 Elementary Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.5 Rule Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.6 Producing Randomness via Cellular Automata . . . . . . . . . . . . . . . 42
3.6.1 CA-Based PRNGs . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.6.2 Balancedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.6.3 Mutual Information . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.6.4 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4 Test Results 47 4.1 Output of a Statistical Test . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2 Testing Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3 Interpretation of the Test Results . . . . . . . . . . . . . . . . . . . . . . . 49
4.3.1 Rate of success over all trials . . . . . . . . . . . . . . . . . . . . . 49
4.3.2 Distribution of P-values . . . . . . . . . . . . . . . . . . . . . . . . 50
4.4 Testing over a big space of functions . . . . . . . . . . . . . . . . . . . . . 50
4.5 Our Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.6 Results and Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.6.1 Change in State Width . . . . . . . . . . . . . . . . . . . . . . . . 53
4.6.2 Change in Neighborhood Scheme . . . . . . . . . . . . . . . . . . . 53
4.6.3 Entropy vs. Statistical Quality . . . . . . . . . . . . . . . . . . . . 58
4.6.4 Mutual Information vs. Statistical Quality . . . . . . . . . . . . . . 60
4.6.5 Entropy vs. Mutual Information . . . . . . . . . . . . . . . . . . . 62
4.6.6 Overall Test Results of 4- and 5-input CA . . . . . . . . . . . . . . 6
4.7 The simplest rule: 1435932310 . . . . . . . . . . . . . . . . . . . . . . . . . 68
5 Conclusion 74
A Test Results for Rule 30 and Rule 45 77
B 120 Rules with their Shortest Boolean Formulae 80
Bibliograph
SZYFRY BLOKOWE NA PODSTAWIE ODWRACALNYCH AUTOMATÓW KOMÓRKOWYCH
The given paper is devoted to the software development of block cipher based on reversible one-dimensional cellular automata and the study of its statistical properties. The software implementation of the proposed encryption algorithm is performed in C# programming language in Visual Studio 2017. The paper presents specially designed approach for key generation. To ensure desired cryptographic stability, the shared secret parameters can be adjusted to contain information needed for creating substitution tables, defining reversible rules, and hiding final data. For the first time, it is suggested to create substitution tables based on iterations of a cellular automaton that is initialized by the key data
Predicting Non-linear Cellular Automata Quickly by Decomposing Them into Linear Ones
We show that a wide variety of non-linear cellular automata (CAs) can be
decomposed into a quasidirect product of linear ones. These CAs can be
predicted by parallel circuits of depth O(log^2 t) using gates with binary
inputs, or O(log t) depth if ``sum mod p'' gates with an unbounded number of
inputs are allowed. Thus these CAs can be predicted by (idealized) parallel
computers much faster than by explicit simulation, even though they are
non-linear.
This class includes any CA whose rule, when written as an algebra, is a
solvable group. We also show that CAs based on nilpotent groups can be
predicted in depth O(log t) or O(1) by circuits with binary or ``sum mod p''
gates respectively.
We use these techniques to give an efficient algorithm for a CA rule which,
like elementary CA rule 18, has diffusing defects that annihilate in pairs.
This can be used to predict the motion of defects in rule 18 in O(log^2 t)
parallel time
Cellular Automata in Cryptographic Random Generators
Cryptographic schemes using one-dimensional, three-neighbor cellular automata as a primitive have been put forth since at least 1985. Early results showed good statistical pseudorandomness, and the simplicity of their construction made them a natural candidate for use in cryptographic applications. Since those early days of cellular automata, research in the field of cryptography has developed a set of tools which allow designers to prove a particular scheme to be as hard as solving an instance of a well-studied problem, suggesting a level of security for the scheme. However, little or no literature is available on whether these cellular automata can be proved secure under even generous assumptions. In fact, much of the literature falls short of providing complete, testable schemes to allow such an analysis. In this thesis, we first examine the suitability of cellular automata as a primitive for building cryptographic primitives. In this report, we focus on pseudorandom bit generation and noninvertibility, the behavioral heart of cryptography. In particular, we focus on cyclic linear and non-linear automata in some of the common configurations to be found in the literature. We examine known attacks against these constructions and, in some cases, improve the results. Finding little evidence of provable security, we then examine whether the desirable properties of cellular automata (i.e. highly parallel, simple construction) can be maintained as the automata are enhanced to provide a foundation for such proofs. This investigation leads us to a new construction of a finite state cellular automaton (FSCA) which is NP-Hard to invert. Finally, we introduce the Chasm pseudorandom generator family built on this construction and provide some initial experimental results using the NIST test suite
Cellular Automata
Modelling and simulation are disciplines of major importance for science and engineering. There is no science without models, and simulation has nowadays become a very useful tool, sometimes unavoidable, for development of both science and engineering. The main attractive feature of cellular automata is that, in spite of their conceptual simplicity which allows an easiness of implementation for computer simulation, as a detailed and complete mathematical analysis in principle, they are able to exhibit a wide variety of amazingly complex behaviour. This feature of cellular automata has attracted the researchers' attention from a wide variety of divergent fields of the exact disciplines of science and engineering, but also of the social sciences, and sometimes beyond. The collective complex behaviour of numerous systems, which emerge from the interaction of a multitude of simple individuals, is being conveniently modelled and simulated with cellular automata for very different purposes. In this book, a number of innovative applications of cellular automata models in the fields of Quantum Computing, Materials Science, Cryptography and Coding, and Robotics and Image Processing are presented
Proceedings of AUTOMATA 2011 : 17th International Workshop on Cellular Automata and Discrete Complex Systems
International audienceThe proceedings contain full (reviewed) papers and short (non reviewed) papers that were presented at the workshop
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