Pseudorandom number generation with self programmable cellular automata

In this paper, we propose a new class of cellular automata – self programming cellular automata (SPCA) with specific application to pseudorandom number generation. By changing a cell's state transition rules in relation to factors such as its neighboring cell's states, behavioral complexity can be increased and utilized. Interplay between the state transition neighborhood and rule selection neighborhood leads to a new composite neighborhood and state transition rule that is the linear combination of two different mappings with different temporal dependencies. It is proved that when the transitional matrices for both the state transition and rule selection neighborhood are non-singular, SPCA will not exhibit non-group behavior. Good performance can be obtained using simple neighborhoods with certain CA length, transition rules etc. Certain configurations of SPCA pass all DIEHARD and ENT tests with an implementation cost lower than current reported work. Output sampling methods are also suggested to improve output efficiency by sampling the outputs of the new rule selection neighborhoods

Pseudorandom number generation based on controllable cellular automata

A novel Cellular Automata (CA) Controllable CA (CCA) is proposed in this paper. Further, CCA are applied in Pseudorandom Number Generation. Randomness test results on CCA Pseudorandom Number Generators (PRNGs) show that they are better than 1-d CA PRNGs and can be comparable to 2-d ones. But they do not lose the structure simplicity of 1-d CA. Further, we develop several different types of CCA PRNGs. Based on the comparison of the randomness of different CCA PRNGs, we find that their properties are decided by the actions of the controllable cells and their neighbors. These novel CCA may be applied in other applications where structure non-uniformity or asymmetry is desired

Incremental evolution of cellular automata for random number generation

Cellular automata (CA) have been used in pseudorandom number generation for over a decade. Recent studies show that controllable CA (CCA) can generate better random sequences than conventional one-dimensional (1-d) CA and compete with two-dimensional (2-d) CA. Yet the structural complexity of CCA is higher than that of 1-d PCA. It would be good if CCA can attain good randomness quality with the least structural complexity. In this paper, we evolve PCA/CCA to their lowest complexity level using genetic algorithms (GAs). Meanwhile, the randomness quality and output efficiency of PCA/CCA are also evolved. The evolution process involves two algorithms a multi-objective genetic algorithm (MOGA) and an algorithm for incremental evolution. A set of PCA/CCA are evolved and compared in randomness, complexity, and efficiency. The results show that without any spacing, CCA could generate good random number sequences that could pass DIEHARD. And, to obtain the same randomness quality, the structural complexity of CCA is not higher than that of 1-d CA. Furthermore, the methodology developed could be used to evolve other CA or serve as a yardstick to compare different types of CA

An Evolutionary Approach to the Design of Controllable Cellular Automata Structure for Random Number Generation

Cellular Automata (CA) has been used in pseudorandom number generation over a decade. Recent studies show that two-dimensional (2-d) CA Pseudorandom Number Generators (PRNGs) may generate better random sequences than conventional one-dimensional (1-d) CA PRNGs, but they are more complex to implement in hardware than 1-d CA PRNGs. In this paper, we propose a new class of 1-d CA Controllable Cellular Automata (CCA) without much deviation from the structure simplicity of conventional 1-d CA. We give a general definition of CCA first and then introduce two types of CCA – CCA0 and CCA2. Our initial study on them shows that these two CCA PRNGs have better randomness quality than conventional 1-d CA PRNGs but their randomness is affected by their structures. To find good CCA0/CCA2 structures for pseudorandom number generation, we evolve them using the Evolutionary Multi-Objective Optimization (EMOO) techniques. Three different algorithms are presented in this paper. One makes use of an aggregation function; the other two are based on the Vector Evaluated Genetic Algorithm (VEGA). Evolution results show that these three algorithms all perform well. Applying a set of randomness tests on the evolved CCA PRNGs, we demonstrate that their randomness is better than that of 1-d CA PRNGs and can be comparable to that of two-dimensional CA PRNGs

A Family of Controllable Cellular Automata for Pseudorandom Number Generation

In this paper, we present a family of novel Pseudorandom Number Generators (PRNGs) based on Controllable Cellular Automata (CCA) ─ CCA0, CCA1, CCA2 (NCA), CCA3 (BCA), CCA4 (asymmetric NCA), CCA5, CCA6 and CCA7 PRNGs. The ENT and DIEHARD test suites are used to evaluate the randomness of these CCA PRNGs. The results show that their randomness is better than that of conventional CA and PCA PRNGs while they do not lose the structure simplicity of 1-d CA. Moreover, their randomness can be comparable to that of 2-d CA PRNGs. Furthermore, we integrate six different types of CCA PRNGs to form CCA PRNG groups to see if the randomness quality of such groups could exceed that of any individual CCA PRNG. Genetic Algorithm (GA) is used to evolve the configuration of the CCA PRNG groups. Randomness test results on the evolved CCA PRNG groups show that the randomness of the evolved groups is further improved compared with any individual CCA PRNG

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

Investigations of cellular automata-based stream ciphers

In this thesis paper, we survey the literature arising from Stephan Wolfram\u27s original paper, “Cryptography with Cellular Automata” [WOL86] that first suggested stream ciphers could be constructed with cellular automata. All published research directly and indirectly quoting this paper are summarized up until the present. We also present a novel stream cipher design called Sum4 that is shown to have good randomness properties and resistance to approximation using linear finite shift registers. Sum4 is further studied to determine its effective strength with respect to key size given that an attack with a SAT solver is more efficient than a bruteforce attack. Lastly, we give ideas for further research into improving the Sum4 cipher

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

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum