cmsc142 - yimg.com

The Shrinking Rule 30 Cellular
Automata Pseudorandom
Number Generator

University of the Philippines Cebu

Department of Computer Science

Cmsc142, Cmsc190, Cmsc199

Nico Martin A. Eñego

February 12, 2011

February 12, 2011

The Shrinking Rule 30 Cellular Automata
Pseudorandom Number Generator

2

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

February 12, 2011

The Shrinking Rule 30 Cellular Automata
Pseudorandom Number Generator

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Conceptual Framework

Random Numbers

PRNGs

Shrinking Rule 30 CA PRNG

Rule 30

CA PRNG

February 12, 2011

The Shrinking Rule 30 Cellular Automata
Pseudorandom Number Generator

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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

February 12, 2011

The Shrinking Rule 30 Cellular Automata
Pseudorandom Number Generator

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Randomness: What is it?

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

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

February 12, 2011

The Shrinking Rule 30 Cellular Automata
Pseudorandom Number Generator

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Randomness: What is it?


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?

February 12, 2011

The Shrinking Rule 30 Cellular Automata
Pseudorandom Number Generator

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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


February 12, 2011

The Shrinking Rule 30 Cellular Automata
Pseudorandom Number Generator

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int msa( int s, int d ){


square s; //s must be d
-
digit int


return 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

=
15
23990
25 and the next number is
23990


February 12, 2011

The Shrinking Rule 30 Cellular Automata
Pseudorandom Number Generator

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History of RNG

Lehmer’s Congruential Generator:

m = 31, a = 3, c = 0, x
0
= 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)

x
i

= (3
Xi
-
1
) mod 31

February 12, 2011

The Shrinking Rule 30 Cellular Automata
Pseudorandom Number Generator

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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



February 12, 2011

The Shrinking Rule 30 Cellular Automata
Pseudorandom Number Generator

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History of RNG


Linear Feedback Shift Registers:





Example:

Take p = 5; q = 2; r = 3 and b
1

= b
2

= b
3

= b
4

=
b
5

= 1. So, b
i

= b
i
-
5

XOR b
i
-
3

produces

b
6

= b
1

XOR b
3

= 1 XOR 1 = 0

b
7

= b
2

XOR b
4

= 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 b
1
…b
p

with the b
i

all being 0 or 1. Subsequent bit
values are produced via the recursion

b
i

= b
i
-
p

XOR b
i
-
p+q

February 12, 2011

The Shrinking Rule 30 Cellular Automata
Pseudorandom Number Generator

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History of RNG


Cellular Automata
Generators (1985):


Originates from simple
rules


Very large periods


Chaotic behavior


February 12, 2011

The Shrinking Rule 30 Cellular Automata
Pseudorandom Number Generator

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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)

February 12, 2011

The Shrinking Rule 30 Cellular Automata
Pseudorandom Number Generator

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Randomness: TRN vs PRN



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

February 12, 2011

The Shrinking Rule 30 Cellular Automata
Pseudorandom Number Generator

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Randomness: TRN vs PRN


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)

February 12, 2011

The Shrinking Rule 30 Cellular Automata
Pseudorandom Number Generator

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Conceptual Framework

Random Numbers

PRNGs

Shrinking Rule 30 CA PRNG

Rule 30

CA PRNG

February 12, 2011

The Shrinking Rule 30 Cellular Automata
Pseudorandom Number Generator

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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)].


February 12, 2011

The Shrinking Rule 30 Cellular Automata
Pseudorandom Number Generator

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February 12, 2011

The Shrinking Rule 30 Cellular Automata
Pseudorandom Number Generator

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February 12, 2011

The Shrinking Rule 30 Cellular Automata
Pseudorandom Number Generator

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Rule 30 CA PRNG


RNG used in Mathematica


2
n

repetition: insignificant according to
Andersson (2003)

function rule30CAPRNG(time seed, int n){


evolve seed n times;


take middle bits of each evolution;


}

February 12, 2011

The Shrinking Rule 30 Cellular Automata
Pseudorandom Number Generator

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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)

February 12, 2011

The Shrinking Rule 30 Cellular Automata
Pseudorandom Number Generator

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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


February 12, 2011

The Shrinking Rule 30 Cellular Automata
Pseudorandom Number Generator

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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.)

February 12, 2011

The Shrinking Rule 30 Cellular Automata
Pseudorandom Number Generator

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Conceptual Framework

Random Numbers

PRNGs

Shrinking Rule 30 CA PRNG

Rule 30

CA PRNG

February 12, 2011

The Shrinking Rule 30 Cellular Automata
Pseudorandom Number Generator

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Shrinking Rule 30 CA PRNG

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

February 12, 2011

The Shrinking Rule 30 Cellular Automata
Pseudorandom Number Generator

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Shrinking Rule 30 CA PRNG


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

February 12, 2011

The Shrinking Rule 30 Cellular Automata
Pseudorandom Number Generator

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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


February 12, 2011

The Shrinking Rule 30 Cellular Automata
Pseudorandom Number Generator

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Shrinking Rule 30 CA PRNG:
Methodology


Statistical

Test Suite

1.
Frequency or equidistribution test

2.
Serial test

3.
Gap test

4.
Poker test

5.
Coupon collector’s test

6.
Permutation test

7.
Runs up test

8.
Maximum
-
of
-
t test

9.
Avalanche effect

February 12, 2011

The Shrinking Rule 30 Cellular Automata
Pseudorandom Number Generator

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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()


February 12, 2011

The Shrinking Rule 30 Cellular Automata
Pseudorandom Number Generator

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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

February 12, 2011

The Shrinking Rule 30 Cellular Automata
Pseudorandom Number Generator

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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

February 12, 2011

The Shrinking Rule 30 Cellular Automata
Pseudorandom Number Generator

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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]

February 12, 2011

The Shrinking Rule 30 Cellular Automata
Pseudorandom Number Generator

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Q&A

February 12, 2011

The Shrinking Rule 30 Cellular Automata
Pseudorandom Number Generator

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Thank You!

February 12, 2011

The Shrinking Rule 30 Cellular Automata
Pseudorandom Number Generator

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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/in
dex.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.