Abstract

Pseudo random number generators (PRNGs) are one of the most important components in security and cryptography applications. We propose an application of Hopfield Neural Networks (HNN) as pseudo random number generator. This research is done based on a unique property of HNN, i.e., its unpredictable behavior under certain conditions. Also, we propose an application of Fuzzy Hopfield Neural Networks (FHNN) as pseudo random number generator. We compare the main features of ideal random number generators with our proposed PRNGs. We use a battery of statistical tests developed by National Institute of Standards and Technology (NIST) to measure the performance of proposed HNN and FHNN. We also measure the performance of other standard PRNGs and compare the results with HNN and FHNN PRNG. We have shown that our proposed HNN and FHNN have good performance comparing to other PRNGs accordingly.

Highlights

  • 1.3.1 Pseudo (Deterministic) Random Number Generators A pseudo random number generator (PRNG) is a deterministic algorithm that generates a sequence of numbers

  • If we develop PRNG based on Neural Networks (NN) with good performance, it would be convertible to hardware format and it would have big usage in simulation projects

  • We have tried by combining fuzzy logic and Hopfield neural network, and developed a Fuzzy Hopfield Neural Networks (FHNN) PRNG that does not have the weakness of our Hopfield Neural Networks (HNN) PRNG

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Summary

History of Random Process

Literal meaning of Random in English language is disorder, unpredictable and without any purpose. The term randomness is used to emphasis on the well defined statistical properties, such as lack of bias or correlation. When a variable is said to be random, it means that the variable follows a given probability distribution. When a number is chosen arbitrarily from some specific distribution it can be called as a random number. Such numbers are almost expected to be independent with no correlations with successive numbers. Random numbers generated by computers are called pseudo random numbers [2] [3] [4] [5]. The history of random processes backs to ancient times. Most ancient cultures used various methods of divination to attempt to circumvent randomness and fate [5]

Usages of Random Numbers
Different Type of Random Number Generators (RNG)
Pseudo (Deterministic) Random Number Generators
True (Nondeterministic) Random Number Generators
Comparison of PRNGs to TRNGs
Importance of Pseudo Random Number Generators
Measuring the Quality of Pseudo Random Number Generators
Soft Computing Techniques for Developing PRNGs
Usage of Neural Networks in development of PRNGs
Usage of Hopfield Neural Networks in Development of PRNGs Hopfield Neural
Fuzzy Systems and Random Number Generator Lotfi
Motivation
Structure of the Thesis
Some well known PRNGs
Vulnerable correlation between successive and future outputs, that makes the output guessable
Linear Congruential
Quadratic Congruential 1, 2 Both Quadratic Congruential 1 and Quadratic
Modular Exponentiation
G using SHA-1 G function is a one way function that has been specified in
Blum Blum Shub Blum
Other random numbers There are other types of PRNGs such as Inversive
Randomness Tests
Some RNG testing issues There are two questions about RNG test suite
NIST RNG Tests Suite in a Glance
Deeper Look at NIST PRNG Test Suite
Conclusion
Frequency Test
Frequency Test within a block
Runs Test
Test for the Longest Run of Ones in a Block
Binary Matrix Rank Test
Discrete Fourier Transform (Spectral) Test
Non-overlapping Template Matching Test
Overlapping Template Matching Test
2.2.3.10 Linear Complexity Test The test’s focus is on the length of a Linear Feedback
2.2.3.11 Serial Test
2.2.3.14 Random Excursion Test Similar to the Cumulative
2.2.3.15 Random Excursion Variant Test
Modern PRNGs
Neural Networks for developing PRNGs
Neural Networks in Cryptosystems
Neural Network as PRNG
An Introduction to Neural Networks
An introduction to Hopfield Neural Network
Our Proposed Hopfield Neural Network as PRNG
Convergence Problem in Hopfield Neural Network
Our Bit Samplings Mechanism
An Introduction to Our Proposed Fuzzy Hopfield Neural Networks as PRNG
An Introduction to Fuzzy systems and Fuzzy Hopfiled Neural Networks
Our Proposed Fuzzy Hopfield Neural Network (FHNN) as PRNG
Fuzzy Logic System embedded each neurons of our FHNN
Reseeding mechanism for FHNN
Testing HNN PRNG by NIST Test Suite
Effect of Digit Sampling Mechanism on the HNN as PRNG
FHNN as PRNG
Comparing our HNN PRNG and FHNN PRNG with other PRNGs
Frequency and Block Frequency test result In Figure 14 the result for
Cumulative Sum-Forward/Reverse test result Figure 16 represents the results of
Run and Longest Run test result Figure represents the results of
Rank test result Rank Test result is shown in Figure 20
Universal test result Figure 23 depict Universal
4.4.10 Linear Complexity test result Linear
Findings

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