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

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

Evolving cellular automata to generate nonlinear sequences with desirable properties

This paper presents a new chromosomal representation and associated genetic operators for the evolution of highly nonlinear cellular automata that generate pseudorandom number sequences with desirable properties ensured. This chromosomal representation reduces the computational complexity of genetic operators to evolve valid solutions while facilitating fitness evaluation based on the DIEHARD statistical tests

Compressive Imaging Using RIP-Compliant CMOS Imager Architecture and Landweber Reconstruction

In this paper, we present a new image sensor architecture for fast and accurate compressive sensing (CS) of natural images. Measurement matrices usually employed in CS CMOS image sensors are recursive pseudo-random binary matrices. We have proved that the restricted isometry property of these matrices is limited by a low sparsity constant. The quality of these matrices is also affected by the non-idealities of pseudo-random number generators (PRNG). To overcome these limitations, we propose a hardware-friendly pseudo-random ternary measurement matrix generated on-chip by means of class III elementary cellular automata (ECA). These ECA present a chaotic behavior that emulates random CS measurement matrices better than other PRNG. We have combined this new architecture with a block-based CS smoothed-projected Landweber reconstruction algorithm. By means of single value decomposition, we have adapted this algorithm to perform fast and precise reconstruction while operating with binary and ternary matrices. Simulations are provided to qualify the approach.Ministerio de Economía y Competitividad TEC2015-66878-C3-1-RJunta de Andalucía TIC 2338-2013Office of Naval Research (USA) N000141410355European Union H2020 76586

A reversible system based on hybrid toggle radius-4 cellular automata and its application as a block cipher

The dynamical system described herein uses a hybrid cellular automata (CA) mechanism to attain reversibility, and this approach is adapted to create a novel block cipher algorithm called HCA. CA are widely used for modeling complex systems and employ an inherently parallel model. Therefore, applications derived from CA have a tendency to fit very well in the current computational paradigm where scalability and multi-threading potential are quite desirable characteristics. HCA model has recently received a patent by the Brazilian agency INPI. Several evaluations and analyses performed on the model are presented here, such as theoretical discussions related to its reversibility and an analysis based on graph theory, which reduces HCA security to the well-known Hamiltonian cycle problem that belongs to the NP-complete class. Finally, the cryptographic robustness of HCA is empirically evaluated through several tests, including avalanche property compliance and the NIST randomness suite.Comment: 34 pages, 12 figure