the shrinking rule 30 cellular automata pseudorandom number generator

The Shrinking Rule 30 Cellular Automata Pseudorandom Number GeneratorUniversity of the Philippines Cebu

Department of Computer ScienceCmsc142, Cmsc190, Cmsc199Nico Martin A. EñegoFebruary 12, 2011

February 12, 2011 The Shrinking Rule 30 Cellular Automata Pseudorandom Number Generator

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Outline Randomness: What is it?

History True Randomness vs. Pseudo randomness

Rule 30 Cellular Automata RNG Problems with R30

The Problem and The Literature Trend Shrinking Rule 30 Cellular Automata RNG

Methodology Expected Results Recommendations

Q&A References


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Conceptual Framework

Random Numbers PRNGs

Shrinking Rule 30 CA PRNG

Rule 30 CA PRNG


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Randomness: What is it?

Random number generators (RNGs) have a myriad of real world applications games, experiments and statistics, gambling,

simulations, random search optimization etc. There is a need of a better random number

source for specific uses (more random, efficiency, size) Cryptology, security and

online gambling


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Some concepts: sporadic, irregular, nonuniform, a/periodic, Pattern?

How do we prove randomness when an exact universal definition is missing?


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What is more random, 9898 or 7878? Philosophical question: Physical phenomena (coin flipping, noise) are said to be random, but…

“God does NOT play dice with the universe.”-Albert Einstein

Is the universe deterministic?


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History of RNG

1941: ATT Machine generating random sequence

1946: Table of random numbers by Tippet and von Neumann’s Middle Square Approach

1951: Lehmer’s Congruential Generator


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int msa( int s, int d ){square s; //s must be d-digit intreturn middle d-digits;

}

History of RNG

Middle Square Approach by von Neumann:

Example: Suppose we want 5 digit numbers and start with 12345. Then, (12345)2 = 152399025 and the next number is 23990


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History of RNG

Lehmer’s Congruential Generator:

m = 31, a = 3, c = 0, x0 = 9.

Solution: 27; 19; 26; 16; 17; 20; 29; 25; 13; 8; 24; 10; 30; 28; 22; 4; 12; 5; 15; 14; 11; 2; 6; 18; 23; 7; 21; 1; 3; 9 (at which point series repeats)

xi = (3Xi-1) mod 31


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History of RNG

Lehmer’s congruential generator is also known as linear congruential generator Not so random!

Quadratic Congruential Generators Short periods occupies much space


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History of RNG

Linear Feedback Shift Registers:

Example:

Take p = 5; q = 2; r = 3 and b1 = b2 = b3 = b4 = b5 = 1. So, bi = bi-5 XOR bi-3 produces

b6 = b1 XOR b3 = 1 XOR 1 = 0

b7 = b2 XOR b4 = 1 XOR 1 = 0

Suppose that r-bit integers are to be generated. Then, for some integer p, start with a p-bit seed of the binary form b1…bp with the bi all being 0 or 1. Subsequent bit values are produced via the recursion

bi = bi-p XOR bi-p+q


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History of RNG

Cellular Automata Generators (1985):

Originates from simple rules Very large periods Chaotic behavior


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Randomness: TRN vs PRN

Coin Flipping: Truly random or difficult-to-describe system?

Simka et al. (2006): Randomness appears in the “instability” of the system.

Two types of random number generators Truly Random Number Generator (TRNG):

generates Truly random numbers (TRNs) Pseudo Random Number Generator (PRNG):

generates pseudo random numbers (PRNs)


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Truly random number (TRN): Cannot be subsequentially reliably reproduced

(nondeterministic) Unrepeatable even with same working conditions

(aperiodic) Needs external physical phenomena (inefficient)

Pseudo-random number (PRN) is a number that is generated by and algorithm or a pre-calculated table of values Deterministic, periodic, efficient


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TRN and PRN are both widely used today There are a lot of TRN sources (lava lamp) For some applications, PRNG are more

reasonable because of their properties A good PRNG usually needs a random

seed which would be good if it

comes from a TRNG

(Hybrid generator)


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Rule 30 CA PRNG


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Rule 30 CA PRNG

Introduced by S. Wolfram in 1983 & 1987 It is a class III rule: chaotic and aperiodic

x(n+1,i) = x(n,i-1) XOR

[x(n,i) OR x(n,i+1)].


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Rule 30 CA PRNG

RNG used in Mathematica 2n repetition: insignificant according to

Andersson (2003)

function rule30CAPRNG(time seed, int n){

evolve seed n times;

take middle bits of each evolution;

}


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The Problem and The Literature Trend It is possible to crack Rule 30 CA Meier-Staffelbach (1998) Attack

Completion backwards Completion forwards Requires lots of resources (but possible)


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The Problem and The Literature Trend A Rule 30 CA based PRNG that can counter

the Meier-Staffelbach Attack PRNG that passes statistical test suite for

randomness PRNG that generates more randomness

compared to other PRNGs Considerable execution time


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The Problem and The Literature Trend

Wolfram’s rule 30 CA PRNG

Rule 30 CA linearity weaknesses

(Meier and Staffelbach)

Irregular Sampling(Clark and Essex)

Hybrid CA PRNG

Controllable CA PRNG

(Guan et al.)

Programmable CA PRNG

(Nandi et al.)


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Rule 30 CA PRNG


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The Shrinking Rule 30 CA suggested by Clark and Essex (2004)


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Important concepts of Clark and Essex model Storage requirement is a bit large Speed is relatively slower compared to other RNG Random and secure but not tested

The use of a non-CA controller

Non-CA RNG


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Shrinking Rule 30 CA PRNG: Methodology Aspects to test:

Intuitive description of execution time Statistical tests of randomness and the Avalanche

Effect Execution times and randomness of different

RNGs will be compared CPRNG vs. SR30CAPRNG WR30CAPRNG vs. SR30CAPRNG CESR30CAPRNG vs. SR30CAPRNG


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Shrinking Rule 30 CA PRNG: Methodology Statistical Test Suite

1. Frequency or equidistribution test2. Serial test3. Gap test4. Poker test5. Coupon collector’s test6. Permutation test7. Runs up test8. Maximum-of-t test9. Avalanche effect


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Shrinking Rule 30 CA PRNG: Methodology Intuitive execution time tests

Attach clock for every program Generate 1000 integers, 100 runs Average execution times of all 100 runs Compare significance of difference using statistics

All programs implemented in C CPRNG implemented using rand() All programs seeded with time()


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Shrinking Rule 30 CA PRNG: Expected Results In terms of intuitive execution time, the

researcher expects the following: CPRNG < WR30CAPRNG < SR30CAPRNG <

CESR30CAPRNG In terms of randomness, security and

avalanche statistics, the researcher expects the following: CPRNG < WR30CAPRNG < CESR30CAPRNG <

SR30CAPRNG


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Shrinking Rule 30 CA PRNG: Recommendations Fuse CCA and PCA concepts with shrinking

generator Use a more random generator (TRNG) for

the seed Devise a way to generate small integers Improve intuitive execution time tests for

programs to reflect optimal performance by using parallel programming (threading) and dedicated machines


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“We can only see a short distance ahead, but we can see plenty there

that needs to be done.”

Alan Turing, Father of Computer Science[p.460 of the Computing Machinery and Intelligence, 1950]


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Q&A


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Thank You!


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References Lawrence, A.P. (2003) Random Numbers. Available online:

http://aplawrence.com/Basics/randomnumbers.html/ [December 5, 2010] Park S. and Miller K. (1988) Random Number Generators: Good Ones Are

Hard to Find. Computing Practices. Communications of the ACM, vol. 31, p. 1192.

Bell, J. Fast Random Numbers. A Random Generator That is 10 Times Faster. Clinton South Carolina. Volume 8, Issue 3, Column Tag: Coding Efficiently. Available online: http://www.mactech.com/articles/mactech/Vol.08/08.03/RandomNumbers/index.html/ [December 5, 2010]

Haahr, M. Random.org. Introduction to Randomness and Random Numbers. Trinity College, School of Computer Science and Statistics, Trinity, Ireland. Available online: http://www.random.org/randomness/ [December 5, 2010]

Clark, J. and Essex, A. (2004) Real Time Encryption Using Cellular Automata. The University of Western Ontario, Department of Electrical and Computer Engineering. March 26, 2004.

Meier, W. and Staffelbach, O. (1998) Analysis of Pseudo Random Sequences by Cellular Automata. Springer Verlag. p.186-199

Andersson, K. (2003) Cellular Automata. Computer Science, Karlstad University.

the shrinking rule 30 cellular automata pseudorandom number generator

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