ANALYSIS OF ELEMENTARY CELLULAR AUTOMATA CHAOTIC RULES BEHAVIOR

International Journal of Security, Privacy and Trust Management ( IJSPTM) Vol 2, No 6, December 2013

ANALYSIS OF ELEMENTARY CELLULAR
AUTOMATA CHAOTIC RULES BEHAVIOR
K. Salman
Middle Tennessee State University, Murfreesboro, Tennessee, U.S.A.

ABSTRACT
We present detailed and in depth analysis of Elementary Cellular Automata (ECA) with periodic
cylindrical configuration. The focus is to determine whether Cellular Automata (CA) is suitable for the
generation of pseudo random number sequences (PRNs) of cryptographic strength. Additionally, we
identify the rules that are most suitable for such applications. It is found that only two sub-clusters of the
chaotic rule space are actually capable of producing viable PRNs. Furthermore, these two sub-clusters
consist of two majorly non-linear rules. Each sub-cluster of rules is derived from a cluster leader rule by
reflection or negation or the combined two transformations. It is shown that the members of each subcluster share the same dynamical behavior. Results of testing the ECA running under these rules for
comprehensively large number of lattice lengths using the Diehard Test suite have shown that apart from
some anomaly, the whole output sequence can be potentially utilized for cryptographic strength pseudo
random sequence generation with sufficiently large number of p-values pass rates.

KEYWORDS
Elementary Cellular Automata, Pseudo Random Number Sequences, Diehard Test suite, Chaotic Rules, pvalues.

1. INTRODUCTION
The subject of security of data from a transmitting agent to a receiving agent is well researched in
both academia and professional research entities and may fall under encryption/decryption
schemes. The basic idea is to find a means of delivering a message from one side to another side
using public transmission media such that the message when interrupted during the transmission
interval display zero information to the non-intended receiver. Obviously, the message must
undergo some transformation in order to hide the message in a noise like background. Of course
a persistent or interesting third party may resort to extremely powerful means such as super
computers and good algorithms, depending on the importance of the content of the message, to
try to recover the message from the distorting means used. Many excellent textbooks have been
written to explain the various techniques used to encrypt and decrypt the message, such as [1].
One common and established scheme used in the art is the synchronous stream cipher, the basic
structure of which is depicted in figure 1. The message to be transmitted by the sender is usually
referred to as Plaintext denoted by is mixed with the output
(Random Sequence) of the
random number generator (RNG) by the exclusive-or primitive represented by the operation
symbol  to yield the resultant which is the encrypted message and usually referred to as
ciphertext, thus: =  . The mixing operation, usually referred to as encryption, will make the
ciphertext look like a random sequence and therefore an unintended third party (the
Eavesdropper) should not be able to use it in order to extract the original message . However,
the intended Receiver can easily recover the original message by applying the same mixing
DOI : 10.5121/ijsptm.2013.2601

1

operation on the ciphertext using the exact replica of the random number generator , thus =
 . This operation is usually referred to as decryption. It should be noted that the mixing
operation, the exclusive-or, is a linear operation by means of which the recovery of the original
message is facilitated. In order for the two operations encryption and decryption to work properly
the random number generators at both sides should obviously be identical and use the exact same
seed. However, when the random number generator used is extracted from a natural noise source
then no matter what seed is used the two random sequences generated at both sides will not be the
same by virtue of the action of the random nature of the source. Therefore natural sources based
random number generators are precluded from utilization in stream ciphers. Mathematical based
algorithms can thus present themselves as a viable alternative. It should be easy to realize that any
algorithm based on mathematical means cannot be random, rather pseudo random and the
viability needs to be ascertained.
International Journal of Security, Privacy and Trust Management (IJSPTM) Vol 2, No 6, December 2013

Figure 1. Typical Synchronous Stream Cipher.

In order for the Pseudo Random Number Generator (PRNG) to be strong it must produce a
sequence that has at least the following salient attributes which are intrinsic in the natural RNGs:
•
•
•
•
•

Uniformity of the output digits
Long cycle length.
No repeated patterns can be detected in the sequence.
The sequence is easy to generate but hard to reproduce or best described by the so called
the next bit unpredictability, i.e. the next bit cannot be predicted or extrapolated even
with the a priori knowledge of the whole sequence.
The algorithm must be simple and both operationally and computationally efficient.

Several algorithms were used in the past to generate PRNs, notably Linear Feedback Shift
Registers (LFSRs). These registers proved to be simple and efficient and were used quite
extensively until an algorithm developed by [2], showed that the linear complexity of such
registers can be reduced drastically to just two span lengths of the register and rendered such
devices unsuitable for encryption/decryption of the synchronous stream ciphers. The breakdown
of these registers is in principle due to the linear primitive used exclusively in the feedback path.
Non-linear Feedback Shift Registers, on the other hand, could prove neither efficient nor viable
for such applications. Recently, however, cellular automata have been utilized to generate strong
PRNs, [3]. To the knowledge of the author, in spite of the wealth of available research on cellular
automata and the different configurations published, no thorough analysis of the strength of
2


periodic and uniform ECA for production of pseudo random number sequences seem to exist in
the open literature. This paper is intended to address this issue at the outset.
The focus of this paper is to test the quality of the randomness of the output of uniform
Elementary Cellular Automata to ascertain the strength of such novel means in utilization for
pseudo random sequence generation of cryptographic strength. Such strength is gauged in this
paper by the passing of the widely accredited state of the art Diehard Test Suite. The paper is
arranged as follows: section 2 will describe the theory of elementary cellular automata, followed
by the dynamics of ECA balanced chaotic rule equivalence in section 3, description of the
Diehard test suite use in testing the output data string in section 4 while section 5 gives an
account of the test data results with explanations, and finally the conclusions and suggestions for
further research.

2. THEORY OF ELEMENTARY CELLULAR AUTOMATA (ECA)
We
start
by
defining
our
ECA
structure 〈 , , , ( ), ,  , , , 〉, where:
•
•
•
•

•
•

by

a

nonuple

mathematical

∈  is the ECA span length and is made up of interconnected identical memory
cells
that are indexed spatially by ∈ and temporally by ∈  .
Σ is a finite set of states so that each memory cell
at location and time step
can occupy one element of the finite set.
∈  is the total time evolution of the ECA.
( ) ∈ is the neighbourhood of the memory cell where it expands equally and
spatially left and right at equal radius ∈  in which case it can be stated that
 = 2 + 1.
 is the global rule of the ECA which is defined by the mapping :  →  .
 ∈  is the local transition rule defined by the mapping  :  →  where ℜ is
the ECA rule space and ∈ 2 .
 is the global ECA state space defined by  =  .
={
} is the initial seed of the automaton at time step = 0 .
 is the output function that works on the output string of the data generated by the
ECA to form the resultant pseudo random string output used in the stream cipher.


•
•
•

The ECA under consideration is one dimensional (1-D) implying that it consists of a linear finite
lattice of interconnected and identical cells of length K , K ∈ N . Each cell denoted mathematically
by
is interconnected with one cell to the right hand side denoted by
and another cell to
the left hand side denoted by
. The alphabet is defined to be {0,1} over  , and the
neighbourhood id defined by  = 2 + 1 = 3 where = 1 that follows immediately from the
definition of binary elementary cellular automata. The interconnection is symmetrically valid for
1 ≤ ≤ − 2 leaving the two extreme end cells
and
to be made adjacently
interconnected as depicted in figure 2(a). The whole ECA is made effectively circular and the
3


shaded extreme end cells coloured blue and red are made adjacent as indicated by the thick black
demarcation line separating them in figure 2(b). The evolution of this circular (periodic)
automaton will eventually form cylindrical ECA automata.

c

t
0

c

t
K −1

c

t

c

K −2

t
k +1

c

t
k

c

t

c

k −1

t
1

c

t
0

c

t
K −1

(a)

(b)
The next state of a cell away from the two boundaries can be represented by
=
) under rule  for 1 ≤ ≤ − 2. It follows that the next state of the extreme
 (
, ,
end cells, the left hand cell and right hand cell or the boundary cells under the same rule are
) and
), respectively.
logically represented by
= ( ,
,
= ( , ,
The rule numbering used in this paper is implementation of the rule numbering scheme suggested
by Wolfram, [4], and is now widely used by those skilled in the art. The rule number is quoted in
the decimal number system and can be defined by = ∑
(2 ∗ ) , where
∈  denote
the minterms (as they are usually referred to in switching circuit theory) indexed by ∈ 2 or
∈ 2 which represents the row number in decimal of the truth table, as depicted in table 1. Since
the mapping is {0,1} → {0,1} it follows that each ECA rule will have a truth table the size of
which consists of 2 rows or 2 unique minters . It can be seen that the size of the ECA rule

space ℜ will therefore be 2 or 2 = 256 unique rules. They are  ,  , … ,  , … ,  , as
depicted in figure 3. Each rule is uniquely determined by these minterms { }, which can either
be a form of a rule table:
{ , , , , , , , }
Figure 2- Illustration of periodic (circular) ECA configuration.

Or a rule number
the above string:

=∑

(2 ∗

(

), where

∈  that gives the decimal representation of
)

4

x
t
ck 

1

t
ck

t
ck 

1

0

1

255

30

m0
m1
m2
m3
m4
m5

m6
m7

Figure 3- ECA rule space table.
Table 1- ECA Rule Minterms Table.

111 110 101 100 011 010 001 000

In order to realize the hardware of the ECA running under rule  the logical expression of this
rule must be derived in minimum form. One common approach is by means of Karnaugh
mapping as illustrated in figure 4 for the two rules  and  .

ckt −1

ckt +1

ckt −1

ckt +1

Figure 4- Karnaugh-Maps of the two chaotic rules ∅
30

45

and ∅ .

=
( +
) for rule ∅ and as
=
( +
) for rule ∅
example of the realization of this expression for rule ∅ is illustrated in figure 5.

The next state minimized logical expression can be derived from the K-Map shown in figure 4 as
. An

5


t
c k+1

Qk

Dk

t
ck

t
ck
45

t
c k +1

Figure 5- ∅

t
c k −1

logic circuit realization with memory cell at spatial index .

The evolution of the periodic ECA in a single time step is depicted in figure 6 while figure 7
illustrates the evolution of the periodic ECA running under ∅ with an arbitrary starting seed for
seven contiguous time steps. The random nature of the ECA output at the seventh time step is
apparent, which is not the case with all the ECA rule space.
t
cK 

1

t
cK 

2

t
ck 

1

t
ck

t
ck

1

t
c k 

1
1

c1t

t
c0

45

t
t
c K  11 c K  1 2

t
c k 

1
1

t
c k

1

c1t 

1

Figure 6- One time step evolution illustration using ∅
t

2

t

3

t

4

t

5

t

.

1

t

1

0

t

t
c 0

6

Figure 7- seven time-steps evolution of uniform periodic ECA with 8-cell span length using ∅
t

7

.
6


The ECA rule space has been conjectured by [4] to be classified into four different classes:
•
•
•
•

Class I:
Class II:
Class III
Class IV

Evolution leads to homogeneous fixed points.
Evolution leads to periodic configurations.
Evolution leads to chaotic, aperiodic patterns.
Evolution produces persistent, complex patterns of localized structures.

Although this classification was based primarily on the phenomenological nature of the
spatiotemporal patterns of the ECA dynamics, more in depth studies, such as that attributed to [5],
inferred almost the same classification. The only class that can be useful for possible strong PRN
generation is class III when the ECA is run uniformly under one local rule. However, in order for
the output binary sequence to be useful it has to have at least some common attributes as listed in
the previous section. The first attribute, Uniformity of the output digits, can only be satisfied
when the local rule used has balanced minters. This means that the number of asserted minterms
must equal to the number of unasserted minterms. It can be seen that the sub-space of the ECA
rule space that satisfy this condition reduces to



!∗







!

= 70 unique rules, to be denoted

by   . However, this sub-space, albeit satisfies the first randomness attribute as outlined
above does not contain purely chaotic rules of class III but rather other rules that belong to the
other three classes. It has been shown here that only some of these rules, exactly 16 rules and
denoted by  ,   , are actually chaotic, belong to class III and may prove to be useful for
PRNs generation. These rules are:

 , , , , , , ,

,

,

,

,

,

,

,

,

It can be seen that this sub-space consists of two groups, one group is dominated by non-linear
primitives in their logical expressions and will be denoted by 
  and consist of all
the constituent rules in the two cluster leaders  and  . The second half of the chaotic rules
sub-space relies only on linear primitives in their logical expressions, and is denoted
by 
:

 , ,

,

,

,

,

,

The following table 2 provides the Karnaugh Map based minimized logical expressions of the 16
rules based on the neighbourhood of the three cells,
, , and
. It can be seen that eight
rules are non-linear while the other eight rules are linear. The different colours associate the
clusters of the rules. The rules marked yellow belong to the cluster of  , those marked with
green belong to  , and these two clusters are the non-linear rules. The rules marked orange are
the cluster of rule  , while the blue colour show the two rules  and  . The last two linear
rules 
and  are coloured pink. It can be seen that the cluster rules of  and  consist
of just two rules.

7

Table 2-Chaotic Rules sub-space   Logic Expressions

Rule Cluster

Minterms































= {0,0,0,1,1,1,1,0}

Next State logical expression Rule Dynamics
=

= {0,1,0,1,0,1,1,0}

(

=

= {1,0,0,0,0,1,1,1}

+

(

=

(

= {1,0,0,1,0,1,0,1}

=

(

= {0,1,0,1,1,0,0,1}

=

+

(

= {0,0,1,0,1,1,0,1}

=

= {0,1,0,0,1,0,1,1}

(

=

= {0,1,1,0,0,1,0,1}

(

=

= {0,0,1,1,1,1,0,0}

= {0,1,1,0,0,1,1,0}
= {1,0,0,1,1,0,0,1}

=
=
=

= {1,1,0,0,0,0,1,1}

=

= {0,1,0,1,1,0,1,0}

=

= {0,0,0,1,1,1,1,0}

=

= {0,1,1,0,1,0,0,1}

=

= {1,0,0,1,0,1,1,0}

+

=

+

(



+

)
)
)

+

)

+

)

+

)
)

Non-Linear





)






Linear






 
 

Based on previous work, [6], the sixteen chaotic rules were shown to belong to clusters of rules
formed by the transformations, complementation, negation and reflection on any member rule of
the cluster as depicted in figure 8.
x

z
y

Figure 8- Basic transformations in Rule equivalence.
8


Suppose that the minterms of a generic rule  are:
{

,

,

,

,

,

,

Then by reflection transformation of the minterms, 
the minterms
{ ,
,
,
, ,
according to the scheme depicted in figure 9:
m7

m6

m5

m4

,

,

,

m2

m3

}



}

we obtain the rule 

m1

with

m0

Figure 9-ECA Rule Reflection Scheme

Note that rule reflection transformation undergoes the exchange or swap of the unsymmetrical
minterms, thus: (
→
) or (
→
) and (
→
) or (
→
) as
well as (
→
) or (
→
) and (
→
) or (
→
) for the
generic rule  . Similarly, rule negation transformation 
with the minterms:
{

,

,

,

,

,



,

The two transformations reflection followed by negation 
reverse order 

{



,

,

}

results in forming 
,

,

,

,

}

results in forming 



or the

with the minterms:

These transformations are illustrated in table 3 for the generic rule  generating rules  ,  ,
and 
noting that 
and 
are two representations of the same rule. Figure 10 shows
the formation of the rule cluster of generic rule  hypercube. Taking a specific example
is represented by the minterms (00011110)2 change under rule reflection
transformation to the following minterms (01010110)2 which represents 86 . Similarly, 






is represented by changing the minterms (00011110)2 under rule negation
transformation to the following minterms (10000111)2 and for 149 changing (00011110)2
into (10010101)2 under rule negation transformation followed by rule reflection transformation
or in reverse order, as illustrated in table 4.

9

Table 3- Generic Rule equivalence minterms modification

rules

minterms






x

x

x

N

x

R

xC

xC

xC

R N

N

xC

R

R N

Figure 10- Illustration of Rule equivalence Hypercube of the generic rule cluster 30
Table 4- 30 equivalence transformations minterms modification

rules

minterms






0
0
1
1

0
1
0
0

0
0
0
0

1
1
0
1

1
0
0
0

1
1
1
1

1
1
1
0

0
0
1
1

The cluster rules that are useful for PRN sequence generation are those of rules 30 , 45 , 60
and 90 as well as the two rules 105 and 150. However, the rules in the lower plane of the
hypercubes of rules 30 and 45 as illustrated in figure 11 that are formed by the complementation
transformation have been found to be inconsequential for the purposes of PRN sequence
generation except for the complementation transformation of rule 105 . Each of these two
hypercubes of rules 30 , and 45 consists of eight rules while the hypercube of 60 consists of
four rules and the last two rules 90 and 105 consists of just two rules. By virtue of the structure
of the minterms of rule 60 only two transformations and their succession can produce different
rules. The complementation transformation of rules 60 , 195 , 102 , 153 wrap around themselves.
The last two rules 90 and 105 produce different rules each with just one transformation. The
reduced number of rules in the rule cluster is referred to as collapsed clusters, [6].

10


Figure 11- Hypercube representation of balanced chaotic rule clusters.

3.

DYNAMICS OF
EQUIVALENCE

ECA

BALANCED

CHAOTIC

RULE

The dynamics of the 16 balanced chaotic rules identified in the previous section will be observed
by the phenomenological appearance of the spatio-temporal images of relatively short automaton
span length and limited time evolution. For example table 5 displays the spatio-temporal images
of the ECA running under the four chaotic rules of rule cluster  for a span length = 61 and
time-steps evolution = 200 for the same seed that consists of a single one on the extreme right
hand cell and all the other cells are set at zero. In these images a black cell represents the state of
the cell at “1” and a white cell when the cell state is “0”. The similarity of the dynamical behavior
is clear from inspection of the images. The mirror reflection transformation of  is clear by
inspecting the images of the two rules  and  or by inspecting the images of the two rules
 and  . Likewise the negation transformation of rule  is also apparent when inspecting
the image of the two rules  and 
or by inspecting the images of the two rules 
and  . The image of rule  is obtained by either reflection transformation followed by
negation transformation or the reverse order on the image of rule  . It can be seen that the
appearance of the self similar fractals are the same in the images of all the four rules. It can be
said that the four rules behave dynamically in the same manner starting from the same seed.
Although the seed consists of a single one on the extreme right hand side the evolution of less
than one hundred time steps yields a fairly complex image. It is also observable that the spreading
of the patterns takes place at maximum speed which is usually referred to as the speed of light
that corresponds to the ±45o diagonal spreading. Therefore starting from a single cell the whole
lattice span = 61 of the ECA fills up after /2 time-steps. These four rules exhibit good speed
of spreading amongst the 16 chaotic rules. Although the self similar or fractals of the cluster rule
 are different from the triangular like shapes of the self similar or fractals of cluster rule
 but the complexity of the dynamics are not less random. The three transformations on
 also behave similarly to the previous rule cluster  . The images shown in table 6 illustrate
the role of the transformations on  . The rule equivalence of rule cluster  for the same seed
and number of evolution time steps likewise demonstrate the similarity of the dynamical behavior
11


of the rules  ,  , 
and  . However, the dynamics are not as random as exhibited by
the rule equivalence of cluster rules  and  . Symmetrical behavior is quite prevalent in the
dynamics, a feature that is obviously undesirable for pseudo random sequence generation. The
same argument may apply to the rule equivalence of  and  . The symmetrical nature in the
dynamics of rule  and its reflection transformation 
highlight the possible linear
relationship in the mathematical expressions of the two rules. The same also applies to rule 
and its complement transformation equivalent rule  .
Table 5- Spatio-Temporal images of chaotic rule equivalence of 

Chaotic Rule Equivalence of rule cluster 
= 61, = 100
= 0000000000000000000000000000000000000000000000000000000000001










= 61, = 100
= 0000000000000000000000000000000000000000000000000000000000001










12


= 61, = 100
= 0000000000000000000000000000000000000000000000000000000000001









and 

Chaotic Rule Equivalence of rule clusters  and 
= 61, = 100
= 0000000000000000000000000000000000000000000000000000000000001









4. DIEHARD TEST SUITE
As stated previously, the intended use of the ECA in this paper is for utilization as key generator
in a synchronous stream cipher. The key of the stream cipher is obtained from the seed originally
fed to the cellular automaton denoted by . The ECA then evolves with time and produce the
output stream that consists of the concatenation of { } conditioned by the output function  .
Let the output key generated by the ECA represented by  = ( { } ), then for this to be a
viable key for a cryptographic strength stream cipher we must be able to assess the strength of the
quality of its randomness. Fortunately, the Diehard test suite [?], has been available to use for
free and has been regarded as the most stringent test suite as of the writing of this paper. The
most recent version of this test suite consists of 22 individual tests as follows:
1.
2.
3.
4.
5.
6.
7.
8.
9.

The BIRTHDAY SPACINGS TEST (9+1)
The tough BIRTHDAY SPACINGS TEST (1)
The GCD TEST-(Big Crush)
The GORILLA TEST -(Big Crush)
The OVERLAPPING 5-PERMUTATION TEST-(Big Crush)
The BINARY RANK TEST for 31x31 matrices (1)
The BINARY RANK TEST for 32x32 matrices (1)
The BINARY RANK TEST for 6x8 matrices (25+1)
The BITSTREAM TEST (20)
13


10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.

The OPSO TEST means Overlapping-Pairs-Sparse-Occupancy (23)
The OQSO TEST means Overlapping-Quadruples-Sparse-Occupancy (28)
The DNA TEST (31)
The COUNT-THE-1's TEST on a stream of bytes (1)
The COUNT-THE-1's TEST for specific bytes (25)
The PARKING LOT TEST (10+1)
The MINIMUM DISTANCE TEST (10+1)
The 3-D SPHERES TEST (20+1)
The SQUEEZE TEST (1)
The OVERLAPPING SUMS TEST (10+1)
The UP-DOWN RUNS TEST (3)
The CRAPS TEST (2)
the CRAPS TEST 2 with different dice (2)

The tests vary in the number of p-values they yield, as described above. The Diehard test suite
requires that the output stream to be tested must be converted to a binary form that is suitable for
the test suite. It also requires that the size of the unconverted data must exceed 80 mega bits for
19 of the 22 tests, hence the binary file produced must be greater or equal to 10 Mbytes. The
other three tests, 3, 4, and 5 require much larger data size, namely exceeding 2.7 Giga bits. These
three tests are dubbed as Big Crush [22], a name borrowed to indicate the difficulty in passing the
test. The other 19 tests likewise can be dubbed as Small Crush. When Diehard runs the small
crush it produces 230 p-values. Some of the p-values are produced by the Kolmogorov-Smirnov
test, as indicated by a plus sign in the above list. There is also an overall Kolmogorov-Smirnov
test p-value which can be considered as an indication of Pass/Fail criteria, [7-9]. This paper
reports the findings of running the Small Crush test while the Big Crush test will be the subject
matter for future publication.

5. ECA RULE SPACE DYNAMICS ANALYSIS
Following the discussion of section II above, the number of rules that need to be tested is reduced
to the chaotic rules sub-space  ,  that consists of the 16 chaotic rules:

 , , , , , , ,

,

,

,

,

,

,

,

,

These rules belong to the chaotic rules of class III where the λ parameter =0.5, [5]. All the other
rules that belong to the other three classes stand no chance in the testing simply because either the
global dynamics collapse to a very small attraction cycle (named so since any transient of states
reaching this cycle will be trapped in this cycle for eternity), this case applies to the two classes I
and II, or the global dynamics are rich with repeated patterns, as is the case with class IV. This
paper is concerned with testing the whole data generated by the ECA running under one chaotic
rule in contrast to the approach adopted by [3], and explores the ability of the data collected to be
utilized as strong cryptographic data. Since the Diehard test suite for good reasons is the test suite
of choice, it is thus required to collect contiguous concatenated output sequences of size
exceeding 80 Mbits without modifications or decimation. The output function  is constructed to
output the data stream as illustrated in figure 12.

14


Figure 12 - Concatenation of the ECA output data stream by the output function  .

It should be helpful if one could find an easy approach to qualify a sequence of pseudo random
numbers for testing prior to applying a suite of tests. It can be stated that one of the easiest tests is
the uniform distribution of 0’s and 1’s in the binary sequence under test which is attributed to
[10]. However, before attempting to find this distribution the length of the sequence or at least its
lower bound must be determined. Hence the data in the sequence must not repeat implying that
the period of the pseudo random sequence must be longer than the length of the sequence in use.
The LFSR can meet this requirement quite easily at the cost of low linear complexity. In
comparison the ECA can provide superior linear complexity but the length of the period is much
less than that provided by the LFSR. Hence the need to accommodate long periods is essential
when utilizing ECA in pseudo random number generation. The period
of the LFSR in its
maximum sequence mode is computed linearly and is represented by
= 2 -1, where ∈ 
is the span length of the LFSR. On the other hand the maximum period
attainable from ECA
is not an easy task to compute particularly when non-linear chaotic rules are used and does not
seem to follow any analytically discernible trend. One clear fact is that for the same span length
>
. Information about the periods of the ECA amongst a host of other features during
its time evolution are obtainable from study of the global dynamics of the ECA for arbitrary local
transition rule and span length. For example, figure 13 depicts the global dynamics and gives the
two state diagrams of the ECA running uniformly in a periodic boundary configurations under
 for
= 5. Two cycles of attraction can be identified the maximum is of size 5-states and
the minimum of size 1-state. The minimum attraction cycle is only reachable by the state 3110 =
111112, while the maximum attraction cycle is reachable by the five states 310, 610, 1210, 1710, and
2410. All these six states, coloured green, are usually referred to as Garden Of Eden (GOE) states
signifying that they are unreachable which means that they have no pre-images or predecessors.
Additionally, the minimum attraction cycle has a transient trajectory of just one state; it is the
GOD state 3110, which is typical with  , while the maximum attraction cycle can be reached by
any state in any one of the five transient branches. One can easily infer that the maximum number
of states to be traversed before the maximum attraction cycle begins to repeat is 5+5=10 states
while the minimum attraction cycle requires just one state before the ECA is locked in the
attraction cycle. Therefore the best scenario for this case of the ECA running uniformly under
rule  for
= 5 is to limit the seeds to a repertoire of five GOE states excluding the GOE
state 3110. In comparison, the LFSR of span length 5 in a maximum cycle feedback configuration
can be seeded with any state from the state space of 25-1, i.e. all the span length permutations
excluding the all 0’s state, and runs for 2 5-1 = 31states before repeating.
15


Figure 13- State Diagram of a periodic ECA running uniformly under rule 

for

= 5.

We can arguably state that we are interested in the major trajectory from a GOE state, let its
length be denoted by α, usually referred to as a transient, to the major attraction cycle, let the
length of this cycle be denoted by β, as well as the minimum trajectory from a GOE state to the
minimum attraction cycle. Obviously, the best scenario here is to form a seed repertoire that
consists of a collection of GOE states that has the major transient to the major attraction cycle.
Let the total number of states traversed before the major cycle repeats be denoted by  ,
giving  =  +  , then for viable cryptographic applications the output string of data generated
by the ECA, denoted by∆, must not exceed ζ, i.e.  ≥ . For large data requirements the ECA
must run with a sufficiently large span length , such that the above lower bounds can be
satisfied. Let represent the total number of the Garden of Eden states available for an arbitrary
chaotic rule  , then ⊊ represents the proper subset of the Garden of Eden states that lead to
the largest attraction cycle or cycles because it excludes those Garden of Eden states that transit to
shorter cycle or cycles including such states as all 0’s or all 1’s for both even and odd span
lengths and the two states 1010… or 0101… for the even span lengths when using  . This
subset should be the set available for use as seeds in any encryption/decryption setting that make
use of the whole ECA output to the stream cipher. This subset of states will be able to guarantee
the orbit of the longest transient evolution but at the cost of limited number of seeds, a case that
can be considered restrictive and inefficient for the utilization of cellular automata in
encryption/decryption applications.
The advantage of the LFSR cannot be overemphasized with the exception of computational
complexity where the ECA inherently gains the upper hand compared to the LFSR. All the
necessary information, such as the primitive polynomial and the feedback taps about the LFSR
can be derived from a collection of 2 ∗ contiguous bits from the output data stream, [**], the
ECA minimum complexity on the other hand asymptotically approaches ( ∗ 2 ∗ ), where 
represents the cardinal of the rule sub-space applicable to the periodic configuration. Several
authors have researched the global dynamics of the ECA rule space, [4,6,11], most prominently
was that due to [6] that culminated in the publication of the atlas of global dynamics of cellular
automat. The main goal of all these attempts is to give a detailed account of the state transition
graphs or diagrams or as it is referred to by [6], the field of basin of attraction of the ECA rule
16


space. The information that can be extracted from such data is of particular pertinence when
trying to study the viability of cellular automata in generating cryptographic pseudo random
number sequences. The results published in the above referenced atlas, however, gave details of
the state transition diagrams for ≤ 15 possibly due to the prohibitive computational expense in
the attempt beyond = 15 because the ECA has to run for the entire state space 2 , but of
course larger sizes of can lead to better understanding of the behavior of the ECA and can also
justify the viability of the use of ECA in pseudo random number generation. Excerpts of the data
from the atlas of global dynamics [6] for  is tabulated and presented in the following table 9.
It can be seen that the periods of the maximum attraction cycles are inferior to the maximum
cycle lengths of the LFSR and do not follow a linear trend. In addition the LFSR can be seeded
with all states 2 except the all 0’s state while the ECA in the uniform periodic setting should be
seeded with those states that lead to the maximum attraction cycle in order to maximize the
period of the output sequence. Such conditions add severe restrictions to the collection of the
seeds available for the applications sought in this paper. For example the total number of states
available for seeding a uniform and periodic ECA running under  when = 11 is 1551
which is the addition of the maximum attraction cycle length and all the transients leading to the
cycle as compared to 2047 for the LFSR for the same span length. Therefore there are two main
factors that influence the required data size; the cellular automaton span length
∈  and the
number of steps of the evolution time ∈  . Let  denotes the output data size, then it is
required that the ECA under test should provide a data size of  ≥ ∗ . It can be seen that the
number of time steps follows the assignment of the span length . The ECA selected chaotic
rule space was subjected to run on cellular automaton span length range 27 ≤ ≤ 1503. The
minimum span length
= 27 was selected based on the possible cycle length of the ECA
matching the maximum cycle length of the Linear Feedback Shift Register, albeit the ECA
maximum cycle length cannot reach the maximum cycle length of the same span length of the
LFSR as will be explained shortly. For  = 80 Mbits the span length must satisfy ≥
/ 2. As explained previously, the ECA is a synchronous sequential circuit that evolves
according to the local transition rule  , ∈ 2 therefore for each  a group of state transition
diagrams must exist that covers all the states 2 for a span length ∈ . It can be seen that a
detailed compilation of such state diagrams for the range of used in this paper is presently
computationally too exhaustive.
The ECA was run for each span length in the range 27 ≤ ≤ 1503 for each of the 16 chaotic
rules mentioned above. Each output of size ≈ 80 M bits was tested individually by the Diehard
test suite to ensure the inclusion of all the 19 individual tests mentioned above. The results are
shown in the figures 14-19 and the following findings can be derived:
1. Cluster  seems to outperform the other 12 rules.
2. Cluster ∅ is the second best.
3. Cluster  , Rules 90 and 165 and Rules 105 and 150 seem to perform worst with low
passing rates compared to the two non-linear cluster rules 30 and 45.

The results of the four rules in cluster  seem to be almost identical and the same apply to the
other clusters therefore it should suffice to show the results of the leading rules in the clusters. In
depth study of the results of  shows that the range of results undergoes many regions. The
first region covers seed length range 27 ≤ ≤ 73 where the ECA failed to pass all the p-values.
However at
= 73 it passed 228 p-values and failed one p-value as well as the KS overall p17


value. In our criterion this is equivalent to failing the tests. In this way the first seed where the
ECA passed all the 230 p-values happened at = 109. If one considers the suitability of the
ECA for pseudo random generation based on passing all the p-values produced by the Diehard
test suite then the region just mentioned 27 ≤ ≤ 108 can be considered unsuitable for pseudo
random number generation when running uniformly under  in the periodic configuration. One
important observation is the frequent dip in the number of p-values passed at 32-bit intervals.
Starting from = 30 with 32 number of p-values passed with a neighbour span lengths passing
more p-values, a clear dip in the number of p-values passed takes place at = 62 where the pvalues passed by the neighbour span lengths are 123 and 92. It repeats after another 32-bit
increment, i.e. at = 94 and = 126 and so on. The number of p-values passed increase after
the dip at = 287with the number of p-values passed equal 180, the next dip takes place at
= 319 with 210 number of p-values passed while at 2-bit neighbouring seeds the ECA fails
around just 2 p-values. Similar patterns but more condensed also observable with rule cluster 45.
In this cluster the dip happens more frequently and at span length differences of 4, 8, 16 and
32bits. The increase of the rate of failures is quite apparent. The intervals of failures with the
other rules also take place at similar span length differences as with rule cluster 45. There is no
clear explanation for the reason why such failures happen at the differences in span lengths
described above. However, since the numbers are powers of two and that the Diehard test suite is
in fact based on testing binary numbers of 32-bit wide, it may have some sensitivity to the results
obtained. Further research is obviously warranted for this problem. By inspecting the results of
the rest of the chaotic rules it seems that none of the other rules behave better than  . From the
above results one can state that the data presented in table 9 may lead to the contention that
running the ECA with a random seed may probabilistically result in the failure of the output data
stream in passing the test suite. This should be clear from the results presented in this paper.
Table 9. State Diagrams Data for  for span length range 3 ≤ ≤ 14.
LFSR
(span length L)

Rule 30

ECA
span
length
K
3
4
5
6
7
8
9
10
11
12
13
14

#of attractors
producing
Less than
Maximum
maximum
period
period
1
3
1
1
1
3
1
8
4
1
1
2
4
2
2
1
8
4
4
1
17
1

Maximum
Attractor
period

Total # of states in
the maximum
attractor state
diagrams

Total # of
GOE
states

8
5
63
40
171
15
154
102
832
1428

12
30
77
224
414
420
1551
975
2600
13818

3
5
6
12
10
33
57
56
136
91
964
1478

Maximum cycle
length
2 −1
7
15
31
63
127
255
511
1023
2047
4095
8191
16383

18


#p-values passed

Rule 30
250
200
150
100
50
0
0

500

1000

1500

Seed Length K
Figure 14 – Diehard Results of ECA running uniformly under 

2000

for span length range 27 ≤

≤ 1503

#p-values passed

Rule 45
250
200
150
100
50
0
0

500

1000

1500

2000

Seed Length K


≤ 1503

Rule 60

#p-values passed

250
200
150
100
50
0
0

200

400

600

800

1000

1200

1400

1600

Seed Length K


≤ 1503

19


#p-values passed

Rule 90
250
200
150
100
50
0
0

200

400

600

800

1000

1200

1400

1600

Seed Length K


≤ 1503

Rule 105
250

#p-values passed

200
150
100
50
0
0

200

400

600

800

1000

1200

1400

1600

Seed Length K


≤ 1503

#p-values passed

Rule 150
250
200
150
100
50
0
0

200

400

600

800

1000

1200

1400

1600

Seed Length K


≤ 1503

20


6. CONCLUSIONS
It has been shown that if passing all the p-values generated by the Diehard test suite is considered
a good criterion to qualify a sequence of pseudo random numbers as being cryptographically
strong then only a sub-space   of the ECA rule space ℜ can be considered useful for this
application. The ECA under test is designed to run uniformly on one local transition rule in a
periodic boundary configuration. The ECA was run for a wide range of span lengths 27 ≤ ≤
1503. A random seed was used for each span length and the same seed was used by all the 16
rules in the sub-space  . In the case that the ECA fails any of the 230 p-values by the Diehard
test suite, 9 more random seeds are generated and the ECA is re-run on all these seeds to create a
new average of the p-values passed and then recorded. The results evidently show that cluster
 outperforms all the other rules followed by cluster ∅ . Although the linear rules sub-space

  can pass all the Diehard 230 p-values, it is clear that in addition to the fact that the
passing rate is visibly lower than the results obtained from the rules of the non-linear sub-space

  the dependencies of these rules on the linear primitives operations makes them
inherently amenable to cracking and render them weak in cryptographic applications. The results
also show that all the rules exhibit some periodic patterns of seed length differences of 32, 16, 8
and 4, or in terms of powers of two. {2} . Since the common denominator is a power of two up to
power of 5, it has little to do with the span lengths since the failures take place at odd numbers of
the span length as well as even numbers in addition to prime numbers. One of the possible
causes of this trend may be attributed to the tests in the Diehard test suite itself. The Diehard
deals with numbers of 32-bit wide and therefore some sensitivity to this number may propagate to
the results and possible inclusion of divisible numbers as well. In conclusion it can be deduced
for recommendation that when the ECA is intended for use to generate pseudo random numbers
of high or cryptographic quality based on the criterion stated above, namely that the whole data
output is used as key number generator for the synchronous stream and that the output must pass
all the 230 p-values generated by the Diehard test suite, then the best choice would be cluster rule
30, i.e. any one of the four rules  ,  ,  ,  , running with a repertoire of seeds collected
as discussed in the body of this paper. Any seed used outside this carefully compiled repertoire
would leave the output data to the probabilistic failure. A recommendation for future work would
be to construct an efficient algorithm that allows the computation in polynomial time of the state
diagrams of the useful rules for the range of span lengths  . The results will facilitate the
compilation of a reliable repertoire of a set of seeds to be used for the applications in question.

REFERENCES
[1]

[2]
[3]
[4]
[5]
[6]

Menezes, J. Alfred, van Oorschot, C. Paul, & Vanstone A. Scott (1996), “Handbook of Applied
Cryptography”, Editors: Kenneth H. Rosen, CRC Press, ISBN: 0-8493-8523-7, Fifth Printing Edition,
816 pages.
Berlekamp, E.R. (1968), "Algebraic Coding Theory", McGraw-Hill, New York, chapter 7.
Wolfram, Stephen (1986), "Random Sequence Generation by Cellular Automata", Advances In
Applied Mathematics 7, pp 123-169.
Wolfram, S. (2002), “A New Kind of Science”, Champaign, IL: Wolfram Media, ISBN: 1579550088.
Langton, Chris G. (1990), “Computation at the edge of chaos: Phase transitions and emergent
computation”, Physica D, 42:12-37.
Wuensche Andrew, and Mike Lesser (1992), “The Global Dynamics of Cellular Automata”,
Reference Volume I, Addison Wesley Publishing Company, ISBN: 0-201-55740-1.
21

[7]

K. Salman (2013), “Elementary Cellular Automata (ECA) Research platform”, Cyber Journals:
Multidisciplinary Journals in Science and Technology, Journal of Selected Areas in Software
Engineering (JSSE), Volume 3, Issue 6, June Edition, 2013, pp 1-15.
[8] K. Salman (2013), “Feedback Shift Registers As Cellular Automata Boundary Conditions”, First
International Conference on Computational Science and Engineering (CSE-2013), pp 11-21,
http://airccj.org/CSCP/vol3/csit3302.pdf
[9] K. Salman (August 2013), “Analysis Of Elementary Cellular Automata Boundary Conditions”,
International Journal of Computer Science & Information Technology (IJCSIT) Vol. 5, No 4, pp. 3551
[10] S.W. Golomb (1986), “Shift Register Sequences”, Aegean Park Press; Revised edition, ISBN-10:
0894120484.
[11] Edward Jack Powley, (2009), “Global Properties of Cellular Automata”, Ph.D., Department of
Computer Science, University of York.

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ANALYSIS OF ELEMENTARY CELLULAR AUTOMATA CHAOTIC RULES BEHAVIOR

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