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

A transformation sequencing approach to pseudorandom number generation

This paper presents a new approach to designing pseudorandom number generators based on cellular automata. Current cellular automata designs either focus on i) ensuring desirable sequence properties such as maximum length period, balanced distribution of bits and uniform distribution of n-bit tuples etc. or ii) ensuring the generated sequences pass stringent randomness tests. In this work, important design patterns are first identified from the latter approach and then incorporated into cellular automata such that the desirable sequence properties are preserved like in the former approach. Preliminary experiment results show that the new cellular automata designed have potential in passing all DIEHARD tests

Permutation and sampling with maximum length CA for pseudorandom number generation

In this paper, we study the effect of dynamic permutation and sampling on the randomness quality of sequences generated by cellular automata (CA). Dynamic permutation and sampling have not been explored in previous CA work and a suitable implementation is shown using a two CA model. Three different schemes that incorporate these two operations are suggested - Weighted Permutation Vector Sampling with Controlled Multiplexing, Weighted Permutation Vector Sampling with Irregular Decimation and Permutation Programmed CA Sampling. The experiment results show that the resulting sequences have varying degrees of improvement in DIEHARD results and linear complexity compared to the CA

Layered cellular automata for pseudorandom number generation

The proposed Layered Cellular Automata (L-LCA), which comprises of a main CA with L additional layers of memory registers, has simple local interconnections and high operating speed. The time-varying L-LCA transformation at each clock can be reduced to a single transformation in the set formed by the transformation matrix of a maximum length Cellular Automata (CA), and the entire transformation sequence for a single period can be obtained. The analysis for the period characteristics of state sequences is simplified by analyzing representative transformation sequences determined by the phase difference between the initial states for each layer. The L-LCA model can be extended by adding more layers of memory or through the use of a larger main CA based on widely available maximum length CA. Several L-LCA (L=1,2,3,4) with 10- to 48-bit main CA are subjected to the DIEHARD test suite and better results are obtained over other CA designs reported in the literature. The experiments are repeated using the well-known nonlinear functions and in place of the linear function used in the L-LCA. Linear complexity is significantly increased when or is used

A Search for Good Pseudo-random Number Generators : Survey and Empirical Studies

In today's world, several applications demand numbers which appear random but are generated by a background algorithm; that is, pseudo-random numbers. Since late $19^{th}$ century, researchers have been working on pseudo-random number generators (PRNGs). Several PRNGs continue to develop, each one demanding to be better than the previous ones. In this scenario, this paper targets to verify the claim of so-called good generators and rank the existing generators based on strong empirical tests in same platforms. To do this, the genre of PRNGs developed so far has been explored and classified into three groups -- linear congruential generator based, linear feedback shift register based and cellular automata based. From each group, well-known generators have been chosen for empirical testing. Two types of empirical testing has been done on each PRNG -- blind statistical tests with Diehard battery of tests, TestU01 library and NIST statistical test-suite and graphical tests (lattice test and space-time diagram test). Finally, the selected $29$ PRNGs are divided into $24$ groups and are ranked according to their overall performance in all empirical tests

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

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

Pseudorandom number generation with self-programmable cellular automata

10.1109/TCAD.2004.829808IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems2371095-1101ITCS