FEEDBACK SHIFT REGISTERS AS CELLULAR AUTOMATA BOUNDARY CONDITIONS

Sundarapandian et al. (Eds) : CSE, CICS, DBDM, AIFL, SCOM - 2013
pp. 11–21, 2013. © CS & IT-CSCP 2013 DOI : 10.5121/csit.2013.3302
FEEDBACK SHIFT REGISTERS AS CELLULAR
AUTOMATA BOUNDARY CONDITIONS
K. Salman
Middle Tennessee State University
Murfreesboro, TN 37132, USA
ksalman@mtsu.edu
ABSTRACT
We present a new design for random number generation. The outputs of linear feedback shift
registers (LFSRs) act as continuous inputs to the two boundaries of a one-dimensional (1-D)
Elementary Cellular Automata (ECA). The results show superior randomness features and the
output string has passed the Diehard statistical battery of tests. The design is good candidate
for parallel random number generation, has strong correlation immunity and it is inherently
amenable for VLSI implementation.
KEYWORDS
Linear Feedback Shift Registers Cellular Automata Boundary Conditions Diehard
1. INTRODUCTION
Both LFSRs and CAs have been used extensively in a wide area of applications, particularly
random number generation for Mont Carlo simulation, communications, cryptography and
network security [1-8]. LFSRs, albeit simple in structure and design were proven to have
comparatively weak statistical featured when utilized in the production of pseudo random
numbers [2]. The weakness can be attributed to the linearity of the exclusive-or function used in
the feedback network.. Additionally, non-linear feedback shift registers have their problems as
well [1]. On the other hand, a uniform 1-D CA, where one rule is implemented throughout the
spatiotemporal evolution of the CA, have shown unique and useful characteristics, and have been
suggested by [3,4] and others for use in random number generation. A notable impediment
however, is the input to the boundaries of the CA, where it is confined to a limited span. For
example, a necessary condition for an unbounded (bi-infinite) 1-D ECA to produce a pseudo
random contiguous string of length x ¥Î is to have a span of 2x + cells long. Hence, a
relatively long string of output will render the CA overly unsuitable. However, a shorter span
K ¥Î implies a constant span length and therefore a fixed and limited number of cells. Hence,
inputs are needed to feed the two extremities of the ECA. One approach attempted to solve this
problem is to make the ECA evolve in a continuous loop (referred to as autonomous or periodic),
in which case the peripheral cells (i.e. the last and the first extreme cells) are made adjacent to

12 Computer Science & Information Technology (CS & IT)
each other, as depicted in figure 1. An alternative technique used earlier in the literature is to feed
the peripheral cells with fixed inputs. Figure 2 depicts the various boundaries from (2)GF used.
t
k
ct
k 1
c +
t
k 1
c −
t
K 1
c −
t
K 2
c −
t
K 1
c −
t
0
ct
1
ct
0
c
Figure 1, ECA periodic boundary configuration
1
t
K
c 1
t
k
c t
k
c 1
t
k
c 1
t
c 0
t
c2
t
K
c
0 1k K
Figure 2. Common one dimensional cellular automaton fixed boundary conditions.
All these methods running under chaotic rule 30 on uniform one-dimensional ECAs have
produced much shorter periods than the LFSR and drastically failed the well-established Diehard
battery of tests [7]. This paper reports the findings of a new method whereby a pair of
uncorrelated LFSRs are used to generate the two boundary conditions. With this design the output
string of the center bit of the ECA evolving for time steps 2= K
T , where K is the span length,
has passed the Diehard battery of tests and produced attractive parallelism and correlation
properties. This paper is arranged such that the theoretical analysis and the proposed approach are
included in the section called Preliminaries, while the results section discusses the improvement
in the performance of the ECA. The conclusion finalizes the outcome of the paper.

Computer Science & Information Technology (CS & IT) 13
2. PRELIMINARIES
For the purpose of this paper we will restrict our attention towards one dimensional cellular
automaton. The cells are arranged on a linear finite lattice, with a symmetrical neighborhood of
three cells and radius 1=r . Each cell takes its value from the set {0,1,..., }=G p and we let
2=p . All cells are updated synchronously and the cells are restricted to local neighborhood
interaction with no global communication. The ECA will evolve according to one uniform
neighborhood transition function, which is a local function (rule) 2 1
: r
f G G+
® where the
ECA evolves after certain number of time steps T. Out of a total of
2 1+rp
p rules we use rule 30 as
suggested by Wolfram and adopt his numbering scheme [3,4]. It follows that a 1-D ECA is a
linear register of , Î ¥K K , memory cells. Each cell is represented by
t
k
c , where [1 : ]k K=
and [1, )= ¥t , that describes the content of memory location k at time evolution stept . Since
2=p the cell takes one of two states from (2)GF . This implies the applicability of Boolean
algebra to the design over (2)GF . A minimum Boolean representation of chaotic Rule30 in terms
of the relative neighborhood cells is:
1
1 1( )t t t t
k k k kc c c c+
+ -= Å + or 1
1 1 1( ) mod 2t t t t t t
i k k k k kc c c c c c+
+ - -= + + + × , where
2 2k K£ £ - , as depicted in figure 3.
1
t
K
c 2
t
K
c 1
t
k
c t
k
c 1
t
k
c 1
t
c 0
t
c
1
1
t
K
c 1
2
t
K
c 1
1
t
k
c 1t
k
c 1
1
t
k
c 1
1
t
c 1
0
t
c
Time step
t
Time step
t+1
Figure 3. Illustration of Rule 30 operating on the present state of neighborhood of time step t to
produce the next state cell of time step t+1.

Furthermore, since the ECA is actually a finite state machine then the present state of the
neighborhood of cell
t
k
c , 1 1
( , , )t t t
k k k
c c c+ -
at time step t and the next state
1t
k
c +
at time step t+1,
can be analyzed by the state transition table and the state diagram depicted in figure 4.
1 1+ −−t t
k kc c
1 1+ −−t t
k kc c
1 1+ −−t t
k kc c1 1+ −−t t
k kc c1 1+ −−t t
k kc c
t
kc
Figure 4, State machine analysis of Rule 30.
It can be seen from above that in order to evolve from the present time step t to the next time step
t+1, each cell at lattice location k would require the present state of itself
t
k
c as well as the
present state of the other two cells in its neighborhood 1+
t
k
c and 1−
t
k
c . Therefore, if the ECA is
allowed to expand freely, leftwise and rightwise the total number of cells at each time step t+1,
say K ∈N would need 2K + cells at time step t. For example to produce the string
1101110T = from the evolution of the unbounded ECA of span length WÎ ¥ by the
concatenation of the center cell would require a span of length 2 1TW= + , i.e. 13 cells, as can
be seen in figure 5. Hence, if the ECA is unbounded then for a string of T-bits would require the
evolution of the ECA of span 2 1T + as illustrated in figure 5. This condition will eventually lead
to an unpractical span of the ECA. Hence, it is imperative that the ECA has to be bounded. The
open literature is rich with research on fixing the size of the ECA and provides data for the
extreme cells of the bounded ECA. Figure 2 gives a brief account of some common fixed
boundary conditions. Figure 6 categorizes the boundary conditions to include the new boundary
condition proposed in this paper using LFSR as a new source for boundary conditions. The fixed
boundary conditions are already illustrated in figure 1. The miscellaneous category includes
either some ad hoc permutations of the fixed boundaries or some fixed sequence of inputs. The
autonomous category, commonly referred to as periodic, make the extreme cells of the ECA
adjacent, as illustrated in figure 2. The resultant ECA becomes circular as depicted in figure 8,
and with time evolution it can be visualized as a cylinder. The expression for the extreme left and
right cells at time step t+1 are, respectively
1
1 0 1 2
( )t t t t
K K K
c c c c+
- - -
= Å + and
1
0 1 1 0
( )t t t t
K
c c c c+
-
= Å + .
The published results of these different types of boundary conditions produced poor results when
used as a source of generating random numbers. In this paper we are proposing a new source for
the boundaries. We have used the well established LFSR as the source of inputs to the extreme
cells of the fixed 1-D ECA, as shown in figure 9. A LFSR of span N memory cells can be
described by the following simple recurrence equation,
1
0 0 0 1 1 1 1
t t t t t
i i N NL a L a L a L a L+
- -Å Å Å Å= ××× ××× where (2)Îia GF .
The choice of ia are exactly the coefficients of a primitive polynomial of degreeN . The extreme
cells of the new design at time step t+1 can now be described by

1
1 2 0 1( )t t t t
K K Kc c L c+
- - -= Å + and 1
0 0 0 1( )t t t t
c R c c+
= Å +
0000001000000
00001110000
001100100
1101111
00100
111
0
Figure 5, Simple time evolution of an unbounded 1-D ECA under GF(2).
T
T
c
1T
T
c −
2T
T
c −
3T
T
c − 3
3
T
T
c −
+
3
2
T
T
c −
+
3
1
T
T
c −
+
3
1
T
T
c −
−
3
2
T
T
c −
−
3
3
T
T
c −
−
2
2
T
T
c −
−
2
1
T
T
c −
−
2
1
T
T
c −
+
2
2
T
T
c −
+
1
1
T
T
c −
−
1
1
T
T
c −
+
Figure 6, Illustration of time evolution of a bi-infinite 1-D ECA.
Figure 7, Categorization of a fixed span 1-D ECA boundary condition sources.

1
t
K
c
2
t
K
c 1
t
c
0
t
c
1
t
k
c
t
k
c
1
t
k
c
Figure 8, Another illustration of the autonomous (periodic) boundary conditions.
3. RESULTS
In order to test the statistical properties of the new proposed design, we developed a suite of
programs emulating all known types of boundary conditions for wide range of spans for both the
ECA and the LFSRs used as boundaries. We will include snapshots of results obtained for
representative runs on the Diehard battery of tests [8], which has been adopted in this paper due
to its well established stringent requirements on the statistical randomness of the output string.
Due to the restrictions imposed by this test the ECA span K has to be at least 27-bit long evolving
for a minimum of
K
2 time steps. Table 1 shows the results of running the diehard tests on the
periodic boundary conditions for spans 32, 33, up to 512. The ECA running in the periodic
boundary mode has not been able to pass all the tests even for a span of 256-bits. The results of
running the diehard tests on the fixed boundary conditions have totally failed and therefore not
worth reporting here. The results of the diehard tests on the ECA using two LFSRs of span 3-bit
each as the boundary conditions for various increasing spans of the ECA did not show any
significant change and therefore it is not reported. It is clear that such boundary conditions will
give slightly better results than fixed boundary conditions but do not show improvement over the
periodic boundary condition. However, when the LFSR span increased to 15-bits for both
registers some improvement were noticeable as shown by the results reported in table 2. Excellent
results were obtained when the span of the LFSRs were increased to match the span of the ECA.
The ECA has passed all tests with extremely superior p-values, as shown in table 3.
4. CONCLUSIONS
The string of contiguous stream data collected from the evolution of the 1-D ECA for the center
cell of various boundary conditions were tested by the 15 Diehard battery of tests. The various
fixed boundary conditions failed the diehard tests almost completely and were considered
unworthy reporting. The autonomous boundary conditions have shown far better statistical
properties than the fixed boundary conditions. However, it still falls far below the minimum
requirements of the diehard tests for reliable considerations in producing dependable random
numbers even for long spans of the ECA (512-bit). When the boundaries were fed from LFSRs
results did not improve significantly until the span of the LFSRs were comparable to that of the
ECA. The results steadily improved up to the upper bound when the two spans were comparable.

It can be concluded that the new approach can produce random numbers even at modest size of
the ECA (i.e. 27-bit). More in depth study of the results show that the new approach produced
superior p-values than the best of the autonomous results. Further assertion of the diehard results
are also apparent from visual inspection of the spatiotemporal output as can be seen from figure
10. It is easy to expect that the fixed boundary conditions cause a ECA running under Rule 30,
which is in group III (i.e. the chaotic class) to evolve into Group I or II (i.e. point attractors or
limit cycles with extremely small periods), according to Wolfram’s ECA classification [4-5].
Therefore, such boundary conditions preclude these ECAs from achieving strong random number
generators. The autonomous (periodic) boundary conditions, on the other hand gave better results
which is indicative of better distribution during ECA evolution. However, the periods of this type
were far lower than the maximum length obtainable from LFSRs. The proposed design have an
added favorable feature when considering the initial seeds. It is clear that all the possible
K
2 K-
tuples can be used as seeds including the all 0’s and all 1’s that usually yield quiescent states.
This is not possible with any other known boundary conditions including the autonomous type.
All the tests were performed using a single one as the initial seed. This is admittedly not the case
in a practical situation. Some patterns were observed during the initial evolution of the ECA but
did not persist. Although these initial patterns did not negatively impact the diehard tests it was
found that avoiding the use of trinomials for the LFSRs and replace them with primitive
polynomials of better distribution of the coefficients managed to remove these patterns. One
salient feature of the design is the almost total destruction of the cross-correlation between
different cells as shown in figure 11(a). This strong correlation is an inherent feature of LFSRs
that can be observed as maximum and constant between any two cells of the LFSR and as linear
patterns on the diagonal ridge between the outputs of the LFSR cells, figure 11(b). An immediate
consequence is the ability to use the ECA as a parallel source of pseudo random numbers that can
be considered a strong candidate for parallel data compaction (signature analysis) in VLSI testing
[8]. This is justified since the structure as depicted in figure 9 presents a simple memory-based
and inherently parallel design that is amenable to large scale integration. Inspection of rule 30
reveals that the function is surjective. Since reversibility implies bijection, it follows that the
proposed system is not clear cut reversible. Hence analytical techniques may not be available to
adequately and inversely describe the spatiotemporal data evolution in at most polynomial time.
For a LFSR of spanN , there are
N
(2 1)− N-tuple words as seeds. The two LFSRs are
uncorrelated and running independently and synchronously, hence the effective input
computational complexity from these registers to the ECA would be N 2
(2 1)− . The 1-D ECA of
span K can be initialized with a total of
K
2 K-tuple words as initial seeds. There are a total of
3
2
2
rules, which is the rule space of a1-D ECA. Thus the computational asymptotic complexity of the
system is
Ο
3
N 2 2 K
((2 1) 2 2 )− ⋅ ⋅ ≈ Ο 3K
(2 ) for K N, as compared to 2N for the LFSR and Ο K
(2 ) for a 1-D
ECA with autonomous boundary conditions.

1
t
K
c 2
t
K
c 1
t
k
c t
K
c 1
t
k
c 1
t
c 0
t
c
0
t
L 1
t
L t
j
L 2
t
N
L 1
t
N
L
t
j
R 0
t
R1
t
R
1
0
t
L 1
0
t
R
1
1
t
c1
1
t
k
c 1t
k
c
1
1
t
k
c1
1
t
K
c 1
2
t
K
c 1
0
t
c
2
t
N
R1
t
N
R
0 1j N
0 1k K
Figure 9, Block Diagram Representation of the Proposed ECA System, reversing the order of
indexing, such that the most significant cell is the extreme left hand cell and vice versa for the
extreme right hand cell which becomes the least significant cell.
Table 1, Diehard tests results for 1-D ECA of variable spans and with autonomous boundaries.
S32 S33 S64 S128 S256 S512
T_1 0.4913 0.6089 0.4871 0.5683 0.5976 0.7166
T_2 1 1 1 1 1 1
T_3 0.759 0.7895 0.5035 0.4525 0.643 1
T_4 1 1 1 1 1 1
T_5 1 1 0.4973 0.5195 0.5777 0.7068
T_6 1 1 0.804 0.7769 0.4856 0.7756
T_7 1 1 0.999 1 1 1
T_8 1 1 1 1 1 1
T_9 1 1 0.6235 1 0.431 0.3777
T_10 1 1 0.4376 0.6587 0.489 0.5068
T_11 1 1 0.5549 0.5016 0.457 0.4066
T_12 1 1 1 0.019 1 1
T_13 0.3985 0.4106 0.337 1 0.2194 0.375
T_14 1 1 1 0.3576 1 1
T_15 1 0.8809 1 1 0.8697 0.8524
3 pass
12 fail
4 pass
11 fail
8 pass
7 fail
8 pass
7 fail
9 pass
6 fail
8 pass
7 fail
P_VALUES
Summary

Table 2, Diehard tests results for 1-D ECA of variable spans and with two LFSRs as boundaries
of span15-bit each.
S27 S28 S29 S30 S64 S128 S256
T_1 0.8893 0.5869 0.6834 0.005 0.2502 0.5655 0.5638
T_2 1 1 1 1 1 1 1
T_3 0.402 0.2845 0.52 0.485 0.681 0.0795 0.144
T_4 0.997 1 1 1 1 1 1
T_5 1 1 1 1 0.4418 0.5753 0.4226
T_6 1 1 1 1 0.8211 0.6235 0.7855
T_7 1 1 1 1 0.8848 0.992 1
T_8 1 1 1 1 1 1 1
T_9 1 1 1 1 0.4273 0.0373 0.5168
T_10 1 1 1 1 0.2837 0.00035 0.0049
T_11 1 1 1 1 0.2291 0.1859 0.0362
T_12 1 1 1 1 1 1 1
T_13 1 0.0537 0.6473 0.4943 1 0.1131 0.0442
T_14 1 1 1 1 1 1 1
T_15 1 1 1 1 1 1 0.9056
3 pass
12 fail
3 pass
12 fail
3 pass
12 fail
3 pass
12 fail
8 pass
7 fail
9 pass
6 fail
10 pass
5 fail
P_VALUES
Summary
Table 3, Diehard tests results for ECA of variable spans and with 2LFSRs for boundaries of same
as the ECA spans.
S27 S28 S32 S64 S128
T_1 0.242 0.43 0.3046 0.2398 0.2695
T_2 0.0744 0.4376 0.1128 0.5284 0.2123
T_3 0.8442 0.6365 0.3417 0.3317 0.5543
T_4 0.4688 0.47 0.4323 0.2713 0.0628
T_5 0.52235 0.4697 0.5166 0.4421 0.5454
T_6 0.4755 0.32 0.5486 0.5584 0.4654
T_7 0.6092 0.485 0.3151 0.4642 0.6849
T_8 0.5581 0.5083 0.4601 0.5009 0.6135
T_9 0.2253 0.6181 0.6947 0.5722 0.5413
T_10 0.8818 0.2469 0.9452 0.728 0.0897
T_11 0.7111 0.3404 0.1944 0.7524 0.5147
T_12 0.456 0.423 0.9646 0.9847 0.1522
T_13 0.3026 0.1387 0.2413 0.1063 0.3202
T_14 0.2085 0.6276 0.1753 0.3521 0.4801
T_15 0.343 0.5539 0.7578 0.4428 0.4845
15 pass
0 fail
15 pass
0 fail
15 pass
0 fail
15 pass
0 fail
15 pass
0 fail
P_VALUES
Summary

Figure 10, Spatiotemporal output of 1-D ECA span 31-bit with two LFSRs as boundary inputs
source of the same span. One hundred time steps is shown.
0
0.2
0.4
0.6
0.8
1
1.2
1 3 5 7 9 11 13 15 17 19 21 23 25 27
(a) (b)
Figure 11, Spatiotemporal images of ECA28, LFSR28 and correlation properties.

REFERENCES
[1] SIEGENTHALER, T. : ‘Correlation Immunity of Nonlinear Combining Functions for Cryptographic
Applications’, IEEE Transactions on Information Theory, Vol. IT-30, No. 5, September 1984, pp.
776-780.
[2] GUSTAVSON, F. G.: ‘Analysis of the Berlekamp-Massey Linear Feedback Shift-Register Synthesis
Algorithm.’ IBM J. Res. Dev. 20, Number 3, pp. 204-212, 1976.
[3] WOLFRAM, S.: ‘A New Kind of Science’. Champaign, IL: Wolfram Media, 2002.
[4] WOLFRAM, S.: ‘Random Sequence Generation by Cellular Automata’, Advances in Applied
Mathematics. Volume 7, Issue 2, June 1986, Pages 123-169.
[5] SEREDYNSKI, FRANCISZEK, BOUVRY PASCAL, and ZOMAYA, ALBERT Y.: ‘Cellular
automata computations and secret key cryptography’, Parallel Computing, Vol. 30, 2004, pp. 753-
766.
[6] LLACHIINSKI, Andrew: ‘Cellular Automata: A Discrete Universe’, World Scientific, 2001, pp. 94.
[7] HORTENSIUS, P.D., McLEOD, and CARD, H.C.:‘Parallel Random Number Generation for VLSI
Systems Using Cellular Automata’, IEEE Transactions on Computers, Vol. 38, Issue 10, October
1989, pp. 1466-1473.
[8] ‘The Marsaglia Random Number CDROM including the Diehard Battery of Tests of Randomness’,
Florida State University, http://i.cs.hku.hk/~diehard/

FEEDBACK SHIFT REGISTERS AS CELLULAR AUTOMATA BOUNDARY CONDITIONS

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