Cost Optimized Design Technique for Pseudo-Random Numbers in Cellular Automata

International Journal of Advanced Information Technology (IJAIT) Vol. 2, No.3, June 2012
DOI : 10.5121/ijait.2012.2302 21
Cost Optimized Design Technique for
Pseudo-Random Numbers in Cellular Automata
Arnab Mitra1, 3
and Anirban Kundu2, 3
1
Adamas Institute of Technology, West Bengal - 700126, India
mitra.arnab@gmail.com
2
Kuang-Chi Institute of Advanced Technology, Shenzhen - 518057, China
anirban.kundu@kuang-chi.org
3
Innovation Research Lab (IRL), West Bengal - 711103, India
anik76in@gmail.com
ABSTRACT
In this research work, we have put an emphasis on the cost effective design approach for high quality
pseudo-random numbers using one dimensional Cellular Automata (CA) over Maximum Length CA. This
work focuses on different complexities e.g., space complexity, time complexity, design complexity and
searching complexity for the generation of pseudo-random numbers in CA. The optimization procedure for
these associated complexities is commonly referred as the cost effective generation approach for pseudo-
random numbers. The mathematical approach for proposed methodology over the existing maximum length
CA emphasizes on better flexibility to fault coverage. The randomness quality of the generated patterns for
the proposed methodology has been verified using Diehard Tests which reflects that the randomness quality
achieved for proposed methodology is equal to the quality of randomness of the patterns generated by the
maximum length cellular automata. The cost effectiveness results a cheap hardware implementation for the
concerned pseudo-random pattern generator. Short version of this paper has been published in [1].
KEYWORDS
Pseudo-Random Number Generator (PRNG), Prohibited Pattern Set (PPS) & Fault Coverage,
Randomness Quality, Diehard Tests, Cellular Automata (CA)
1. INTRODUCTION
Production of random patterns, plays a fundamental role in the field of research work varying
from Computer Science, Mathematics or Statistics to cutting edge VLSI Circuit testing. Random
numbers are also used in cryptographic key generation and game playing. Mathematicians
describe that this random numbers happen in a sequence where the values are homogeneously
distributed over a well defined interval, and it is unfeasible to predict the next values based on its
past or present ones.

22
In probabilistic approach, a random number [2] is defined as a number, chosen as if by chance
from some precise distribution such that selection of a large set of these numbers reproduces the
fundamental distribution. Those numbers are also required to be independent to maintain no
correlations between successive numbers. In the generation process of random numbers over
some specified boundary, it is often essential to normalize the distributions such that each
differential area is equally populated. In order to generate a power-law distribution P(x) from a
uniform distribution P(y), it is P(x)= Cxn
for x є [x0,x1]. Then normalization follows as Equation
(1),
∫ =
+
=
+
1
0
1
0
1
1
1
][
)(
x
x
x
x
n
n
x
CdxxP …….. (1)
From Equation (1), the value of C can be determined as followed in Equation 2.
From Eq We get, 1
0
1
1
1
++
−
+
= nn
xx
n
C …….. (2)
In practice, random number generator requires a specification of an initial number used as the
starting point, which is known as a "seed". Random number generators are classified into several
groups based on the difference in generation procedures of random numbers. Pseudo-random
number and true-random number are most commonly used in scientific works. The classification
of the random number is based upon the selection of seed, that describes how much randomized
way has been adopted for the selection of that seed for generation of random pattern. The quality
of any random number generator is being tested through a statistical test suit named Diehard Tests
[3].
Random numbers or patterns [2], [4] obtained by means of executing a computer program, is
based on implementing a particular recursive algorithm are commonly referred to as pseudo-
random numbers. “Pseudo”, emphasizes that the selection of seed is in deterministic way.
The quality of randomness generated by any random number generator is needed to be verified.
For this purpose, the diehard tests are a battery of statistical tests for measuring the quality of a
random number generator. This statistical test suit was developed by George Marsaglia over
several years and first published in 1995 on a CD-ROM of random numbers [3]. The following
tests are executed as enlisted below to measure the quality of the randomness of any particular
random pattern generator:
i) Birthday Spacings: This name is based on the birthday paradox. Here a pair of random points
on a large interval is chosen such that the spacing between the points should be asymptotically
exponentially distributed.
ii) Overlapping Permutations: In this test, the sequences of five consecutive random numbers
are analyzed to observe overlapping of the permutations; a total 120 possible orderings should
occur with statistically equal probability.
iii) Ranks of Matrices: In this test, some number of bits are selected from some number of

23
random numbers to form a matrix over {0, 1}. Thereafter the rank is counted based on the
determinant value of the matrices.
iv) Monkey tests: This name is based on the infinite monkey theorem. Here the sequences of
some number of bits are assumed as "words". Thereafter the count is completed over the
overlapping words in a stream. The number of "words" that does not appear should follow any
known distribution.
v) Count the 1’s: In this test, the 1 bit in each of either successive or chosen bytes is counted and
then the counted value is converted into "letters" and finally following the counting of the
occurrences of five-letter "words".
vi) Parking Lot Test: In this test, randomly placed unit circles in a 100 x 100 square are tested to
find whether any of the circles, overlaps an existing one. After an iteration of 12000 tries, the
number of successfully "parked" circles should follow a certain normal distribution.
vii) Minimum Distance Test: In this test, the minimum distance between the pairs of randomly
placed 8,000 points in a 10,000 x 10,000 square is measured. The square of this distance is
exponentially distributed with a certain mean.
viii) Random Spheres Test: In this test, randomly chosen 4000 points with the center a sphere on
each point is placed in a cube of edge 1000 and the radius is calculated. The smallest sphere's
volume should be exponentially distributed with a certain mean.
ix) The Squeeze Test: In this test a multiplication is performed with 231
by random floats on [0,
1) until 1 is achieved. The iteration of this multiplication occurs around 100000 times. The
number of floats needed to reach 1 should follow a certain distribution.
x) Overlapping Sums Test: In this test a long sequence of random floats on [0, 1) is generated.
Thereafter an addition of sequences of 100 consecutive floats is performed. The sums should be
normally distributed with characteristic mean and sigma.
xi) Runs Test: In this test a long sequence of random floats on [0, 1) is generated to perform the
ascending and descending runs tests. The counts should follow a certain distribution.
xii) The Craps Test: In this test the games of craps is played for 200000 times to obtain the wins
and the number of throws per game. Each count should follow a certain distribution.
Cellular Automata [5, 6] is used to represent a dynamic mathematical model. A Cellular
Automaton (pl. cellular automata, in short CA) is a discrete model studied in computability
theory, mathematics, physics, complexity science, theoretical biology and microstructure
modeling. It consists of a regular grid of cells, each in one of a finite number of states, such as "1"
for ON and "0" for OFF. The framework might be in higher degree of dimensions. For each cell, a
set of cells called its neighborhood (usually including the cell itself) is defined relative to the
specified cell. For example, the neighborhood of a cell might be defined as the set of cells a
distance of 2 or less from the cell. A primary state (time t=0) is selected by assigning a state for
each cell. A new generation is created (advancing t by 1), according to some predetermined
mathematical function that determines the new state of each cell in terms of the current state of
the cell and the states of the cells in its neighborhood.

24
The simplest CA is one-dimensional with only two neighbors defined at the adjacent cells on
either side of it. This combination forms a neighborhood of 3 cells, with 23
=8 possible patterns
for this neighborhood. There are then 28
=256 possible rules. These 256 CAs are generally
referred to by their Wolfram code. Wolfram code is a standard naming convention invented by
Wolfram; it gives each rule a number from 0 to 255. A number of papers have analyzed and
compared these 256 CAs, either individually or collectively.
CA has a significant role for computation as it is easy to implement through hardware
components. Following Figure 1 shows a rough outline of CA implementation and Figure 2
describes the hardware implementation of CA through flip-flop.
Figure 1. Implementation of 3-cell Null Boundary CA
Figure 2. Hardware implementation of CA
Rest of the paper is organized as follows: Section 2 briefs about the related work; Section 3
describes proposed work; Experimental observations and result analysis are shown in Section 4
and Conclusion is reflected in Section 5.
2. RELATED WORK
Several methods have been implemented to generate good quality random numbers. Some novel
efforts have been established to generate good quality random numbers [5], [7-19]. Some
important efforts have also been made to generate pseudo-random numbers using CA. It has been
already established that the pseudo-random numbers generated using maximum length CA
exhibits a high degree of randomness.
The most common way to generate pseudo-random number is to use a combination of
“randomize” and “rand” functions. Random patterns can be achieved based on the following
recursive PRNG Equation 3.

25
)(211 ModNPXPX nn +=+ .........(3)
Here P1, P2 are prime numbers, N is range for random numbers. Xn is calculated recursively using
the base value X0. X0 is termed as seed and it is a prime number.
If X0 (seed) is same all time or selected in any deterministic way, then it yields pseudo-random
number [2].
It has been observed that in the generation process of pseudo-random pattern using Maximum
Length CA, only a large cycle is responsible for yielding the pseudo random patterns. If a single
cycle with larger numbers of states is used to generate the random patterns, the associated cost
would be increased. All the cost associated with the generation process of random numbers; i.e.,
time, design and searching costs are having higher values as the complexity is directly
proportional to the number of cell sizes used in the cycle. So, it is convenient to reduce the cycle
without affecting the randomness quality of the generated random patterns for reducing these
associated costs.
For the generation process of high quality of pseudo-random numbers, several works have been
established with the uses of Cellular Automata. The research on Maximum Length CA reveals
that with the increased number of cells, the resulting maximum length cycle includes the
maximum degree of randomness. In Maximum Length CA, prohibited pattern set (PPS) is
excluded from the cycle for achieving higher degree of randomness. In Maximum Length CA, the
generation procedure of random patterns is complex and costly. In this methodology, it is
mandatory to keep track of PPS in maximum length cycle and to exclude the portion in resulting
maximum length cycle. Thus it makes the generation procedure complex. The associated
complexity costs i.e., time complexity, design complexity and searching complexity are high
along with this procedure. The quality of randomness in generated integer pattern is quite high
[1]. Therefore, an alternative easy to implement generation methodology of random numbers
would be much more beneficial which deal with the flaws of Maximum Length CA. In our
proposed methodology, we have proposed a system that will be dealing with the flaws of the
Maximum Length CA. Thus the proposed methodology should be more cost effective in terms of
design complexity, time complexity and searching complexity though the generated random
integer patterns are having the quality of randomness, is at least equal to the quality of
randomness achieved by Maximum Length CA. The detailed discussion of the proposed
methodology is following in Section 3.
3. PROPOSED WORK
A cost effective and simpler design methodology always should be beneficial for the generation
of random integer patterns. The cost efficiency refers to the space, time, searching and design
complexity of any involved algorithm. In the cost optimization generation methodology of the
universal random pattern, a one dimensional Equal Length Cellular Automata, is proposed over
the existing Maximum Length Cellular Automata for random pattern generation.
In the proposed methodology, a new approach has been proposed to achieve high degree of
randomization in terms of better cost optimization with respect to all the concerning complexities
mentioned earlier. The resulting methodology should also be very much convenient in hardware

26
implementation. The quality of randomness of the generated pattern by the proposed
methodology is compared through Diehard Tests with Maximum Length CA random number
generator. The conclusion is made about the quality of randomness of any data set based on the
number of Diehard Tests passed. Several data set are analyzed through Diehard, is described in
Table 4 to compare for randomness quality of different random number generators.
In the proposed PRNG, we propose an equal length one dimensional CA over the maximum
length CA. In our work, we propose the decomposition of the larger cycle of maximum length
CA, into more relevant sub-cycles such that our concerning complexities cost can be reduced as
well as the fault coverage can be more flexible than of previously established maximum length
CA. The proposed PRNG system is described in Figure 3 [1].
Figure 3. Flowchart of proposed system
A new mathematical approach has been proposed to achieve the same amount of randomization in
less cost with respect to various complexities and hardware implementation, according to the
flowchart of the proposed system as mentioned in Figure 3.
The resulting larger cycle of maximum length CA is divided into two or more equal sub-cycles
instead of taking the full cycle of maximum length. These sub-cycles are capable to generate
random patterns as equivalent to the randomness quality achieved from maximum length CA. The
following Algorithm 1 has been used for the decomposition of an n-cell maximum length CA [1].

27
Algorithm 1: Cycle Decomposition
Input: CA size (n), PPS Set
Output: m-length cycles excluding PPS
Step 1: Start
Step 2: Initialize the number of n-cell CA to generate random patterns using n-cell CA
Step 3: Decompose the cell number (n) into two equal numbers (m) such that n=2*m
Step 4: Check each PPS whether it belongs to a single smaller cycle CA
Step 5: Repeat Step 3 and Step 4 until each PPS belongs to separate smaller cycles
Step 6: Allow m-length cycles of n-cell CA after excluding all the PPS containing cycles
Step 7: Stop
In the proposed methodology, the primary concern is to reduce the PPS. Therefore it has been
taken care of that the occurrence of every PPS must be completed in some of the smaller sub-
cycles, such that by eliminating all those smaller cycles, the remaining prohibited patterns free
cycles can be allowed to generate random patterns. Thus, this methodology implies a better cost
effectiveness approach. The proposed methodology thus simplifies the design complexity and
empowers the searching complexity. The terminology design complexity refers to the
implementation procedure for generation of random pattern and empowering searching
complexity means the zero overhead for keeping track for PPS for random pattern generation. In
comparison with an n-cell maximum length CA, more number of smaller cycles instead of one
maximum length cycle should be used.
The maximum length CA cycle has been decomposed into smaller equal length cycles using
Algorithm 1. The Algorithm 1 explains the decomposition procedure of maximum length CA.
Let us assume, for the n-length CA, the total number of states= 2n
.
We can generate this same amount of CA cells using Equation 4.
We have 2n
= 2*(2m
); i.e., at least 4 numbers of equal length cycles.
Thus ‘m’ is always less than ‘n’.
Let one maximum length CA is divided into two sub-cycles.
Then, 2n
= (2m1
+ 2m2
)
If m1= m2
Then it becomes 2n
= 2 (2m1
)
Upon illustrating this model, let us assume the case where maximum length CA, n=8.
So, 28
=27
+27
(i.e., two equal length of CA cycles)
=26
+26
+26
+26
(i.e., four equal length of CA cycles)
=25
+25
+25
+25
+25
+25
+25
+25
(i.e., eight equal length of CA cycles)
=24
+24
+24
+24
+24
+24
+24
+24
+24
+24
+24
+24
+24
+24
+24
+24
(i.e., sixteen equal length of CA
cycles)
The PPS is excluded from the cycle as per the procedure for generating the maximum random
pattern in maximum length CA. In our procedure, the PPS can be totally removed as we are
having more number of cycles for generation of random sequences. In proposed methodology, the
cycles producing PPS can be removed from the generation of pseudo random numbers.

28
According to the proposed methodology, a CA size of n=24 might be decomposed into equal
length smaller cycles instead of one maximum length circle. In this case it can be divided into 16
smaller cycles of length 20 or more. Let us assume that there exist nine number prohibited
patterns as PPS. So in worst case, assuming every single prohibited pattern occurs in a single
cycle, there exist (16 - 10=6) number of 20 CA cycle to generate random patterns. The pattern
generation on this scenario is followed as in Figure 4. Figure 4(a) shows one maximum length
cycle with prohibited patters and on the hand Figure 4(b) shows 16 equal length smaller cycles
where some the cycles contains prohibited set only. The following Figure 4 is based on Null
Boundary 3 cell CA where the rules <90, 90, 150> have been applied. The PPS is denoted as
{PS0, PS1……PS9}.
Figure 4(a).
Figure 4(b).
Figure 4. Figure. 4(a). Maximum length CA Cycle for n=24 and Figure 4(b). Proposed equal length CA of
smaller cycle size

29
Further a CA size of n=64 might be decomposed into equal length smaller cycles instead of one
maximum length cycle. In this case it can be decomposed it into 128 smaller cycles of length 57
or more. Let us assume that there exist hundred number prohibited patterns as PPS. So in worst
case, every single prohibited pattern will be in a single cycle, there exist (128 -100=28) number
of cycles of length 57 to generate random patterns. The pattern generation is followed as in
Figure 5. Figure 5(a) shows one maximum length cycle with prohibited patters and Figure 5(b)
shows 100 equal length smaller cycles contains prohibited pattern only. Figure 5 is based on Null
Boundary 3 cell CA where the rules <90, 90, 150> have been applied. The PPS is denoted as
{PS0, PS1……PS99}. Thus higher numbers of PPS are discarded from random pattern generating
equal length CA cycles.
Figure 5(a).
Figure 5(b).
Figure 5. Figure. 5(a). Maximum length CA Cycle for n=64 and Figure 5(b). Proposed equal length CA of
smaller cycle size

30
In terms of fault coverage, the maximum length CA follows a procedure to handle the PPS if it
exists in the larger cycle, which is responsible for generating random pattern. The fault coverage
mechanism is as followed in following section.
In practice, the maximum length CA based random number generator produces the random
patterns of integer using the cycle which might contain some of the prohibited patterns. In this
scenario, the PPS are excluded form that maximum length CA cycle. Assume that there exist
some n-numbers of PPS in the maximum length cycle. Let the PPS are {PS0,…,PSn }. For the
exclusion purpose of these PPS from the cycle, the minimum length of arc (Arcmin) between the
prohibited pattern PS1 and PSn should be measured so that we can utilize the remaining cycle arc
i.e. effective arc (Earc) for random number generation, which is typically free from PPS. This
scenario can be efficiently visualized in Figure 6.
Figure 6. Typical Cycle structure of an n-cell Maximum Length CA
The Figure 6 is based on the facts that there exists total n-number of prohibited patterns with the
following set: PPS={PS1,…..,PSn}. Figure 6 also explain the procedure to measure Arcmin and Earc
from an n-cell maximum length CA for fault coverage in random pattern generation.
Definition 1: The minimum length of arc (Arcmin) is the minimum distance between the first and
last prohibited pattern in an n-cell maximum length CA cycle.
Definition 2: The effective arc (Earc) is the remaining arc length of an n-cell maximum length CA
cycle which excludes Arcmin from the corresponding CA cycle of states and it is responsible for
generating pseudo random patterns of integers.
In our proposed methodology, we have discarded all the cycles containing any prohibited pattern.
Some equal length cycles containing PPS are being discarded, yet we have enough cycles for
generation of random patterns. The following Table-1 shows the comparison of fault coverage
between maximum length CA and our proposed CA for CA cell size n=9. Let us assume that there
exists 9 numbers of prohibited patterns. This procedure is described in Figure 7.

31
Figure 7. Typical cycle structure for maximum length CA cycle for n=9
4. EXPERIMENTAL OBSERVATIONS AND RESULT ANALYSIS
The PPS containing arc in random pattern generating cycle is excluded from the cycle as per the
procedure for generating the maximum random pattern in maximum length CA. On the other
hand, the PPS containing cycles are totally removed to generate the random sequences. There is
no overhead to calculate Earc and Arcmin in proposed equal length CA methodology. The following
Table 1 compares the procedures of maximum length CA and equal length CA based random
pattern generators.
Table 1: Comparison of fault coverage procedures
Maximum length CA Equal Length CA
PPS 9 9
Earc 503 N/A
Arcmin 9 N/A
Table 1 reflects the advantages of our proposed methodology over maximum length CA for fault
coverage in random pattern generation for CA size n=9. In proposed equal length CA, the cycles
containing any prohibited pattern is excluded from generating random patterns.
The p-value analysis, generated for each sample data set by Diehard battery series test, helps to
decide whether the test data set passes or fails the diehard test. Diehard returns the p-value, which
should be uniform [0, 1) if the input file contains truly independent random bits. Those p-values
are obtained by p=F(x), where F is the assumed distribution of the sample random variable ‘x’,
which is often normal. The value p<0.025 or p>0.975 means the RNG has “failed the test at the
0.05 level”. Table 2 indicates the different tests of Diehard Test Suit.

32
Table 2: Diehard Tests
No. Name of the test
1 Birthday Spacings
2 Overlapping Permutations
3 Ranks of 31x31 and 32x32 matrices
4 Ranks of 6x8 Matrices
5 The Bitstream Test
6 Monkey Tests OPSO,OQSO,DNA
7 Count the 1`s in a Stream of Bytes
8 Count the 1`s in Speci_c Bytes
9 Parking Lot Test
10 Minimum Distance Test
11 The 3DSpheres Test
12 The Sqeeze Test
13 Overlapping Sums Test
14 Runs Test
15 The Craps Test
The comparison result in Table 3 shows the degree of randomness achieved by different random
number generators in Diehard Tests.
Table 3: Performance result through Diehard for different n-values in maximum length CA-RNG
Diehard Test
Number
Max-length CA
n=8
Max-length CA
n=10
Max-length CA
n=23
Max-length CA
n=64
1 Fail Fail Pass Pass
5 Fail Fail Fail Fail
6 Fail Fail Fail Pass
Total Number of
Diehard Test
Passes
0 0 10 14

33
Diehard Tests
0
5
10
15
n=8 n=10 n=23 n=64
No. of Cells in Max Length CA
No.ofDiehard
TestsPassed
Randomness Quality
Figure 8. Randomness Quality for Maximum Length CA for different cell sizes (n)
Table 3 shows the degree of randomness achieved in Maximum Length CA for different cell sizes
in terms of the total number of Diehard Tests passes. Figure 8 reflects this degree of randomness
graphically.
Table 4: Performance result through Diehard for different CA random number generators
Diehard Test Number Max-length CA Equal length CA
n=23 n=64 n=23 n=64
1 Pass Pass Pass Pass
5 Fail Fail Fail Fail
6 Fail Pass Fail Pass
Total Number of Diehard
Test Passes
10 14 10 14

34
Figure 9. Diehard Performance Graph
The results obtained from Table 4 and Figure 9 ensures that the proposed random number
generator, posses the maximum degree of randomization with a reference to the number of
Diehard Test passes. This result is similar to the result achieved for maximum length CA based
random number generator. The various complexities of these two CA based methodologies are
reflected in Table 4.
Table 5: Complexity Comparison between Maximum Length CA and Proposed Methodology
Name of the
Complexity
Comparison Result
Space Same
Time Slightly Improved
Design Improved
Searching Improved
Table 5 signifies that the space complexity is same for both procedures as total length of an n-cell
CA is same for both the cases, but there are some changes in other complexities. Other
complexities have been improved in case of our proposed methodology. The proposed
methodology is allowed only to generate random patterns from smaller cycles that exclude PPS.
The PPS exclusion feature from the main cycle, improves the design and searching complexities.
Results based on Table 4, Table 5 and Figure 8, it can be concluded that our proposed
methodology can be used as a better source of random patterns as it posses the maximum degree
of randomness with reference to all other established RNGs and it is less costly to implement.

35
5. CONCLUSION
The Diehard Test results show the quality of randomness achieved from the various samples of
random data sets. The number of passes shows the quality of randomness achieved by particular
method. Table 4 shows the randomness achieved from the random data sets produced through
Equal Length CA is having the maximum randomness as it passes the maximum number of
Diehard Tests which is same as for the maximum length CA. Hence this methodology is suitable
for generating random sequences as the cost associated is much cheaper in terms of time
complexity and hardware implementation. The Equal Length CA uses only the sub cycle which
consists of lesser number of states than of maximum length CA. Thus it completes its full cycle in
lesser time and requires lesser hardware implementation cost. The time complexity of this
proposed methodology maintains a linear time complexity with increased number of cells in the
CA. It is also convenient that the Equal Length CA random number generator is more flexible as
it excludes the prohibited pattern set (PPS).
REFERENCES
[1] Arnab Mitra, Anirban Kundu; (2012) Cost optimized Approach to Random Numbers in Cellular
Automata; The Second International Conference on Computer Science, Engineering & Applications
(ICCSEA); India.
[2] S. Wolfram; Wolfram Mathematica Tutorial Collection: Random Number
Generation.http://www.wolfram.com/learningcenter/tutorialcollection/RandomNumberGeneration/
RandomnumberGeneration.pdf
[3] Robert G. Brown. dieharder: A Random Number Test Suite, 2006a. http://www.phy.
duke.edu/~rgb/General/dieharder.php. C program archive dieharder, version 1.4.24.
[4] Dirk Eddelbuettel; Random: An R package for true random
numbers;http://dirk.eddelbuettel.com/bio/papers.html
[5] http://www.random.org/
[6] S. Wolfram; (1986) Theory and Application of Cellular Automata; World Scientific.
[7] Sukanta Das, Biplab K. Sikdar, P. Pal Chaudhuri; (2004) Characterization of Reachable/
Nonreachable Cellular Automata States; International Conference on Cellular Automata for Research
and Industry; ACRI; Netherlands.
[8] Sukanta Das, Anirban Kundu, Biplab K. Sikdar, P. Pal Chaudhuri; (2005) Design of Nonlinear CA
Based TPG Without Prohibited Pattern Set In Linear Time; JOURNAL OF ELECTRICAL TESTING:
Theory and Applications.
[9] Biplab K. Sikdar, Sukanta Das, Samir Roy, Niloy Ganguly, Debesh K. Das; (2005) Cellular Automata
Based Test Structures with Logic Folding; VLSI Design; India.
[10] Sukanta Das, Haffizur Rahaman and Biplab K Sikdar; (2005) Cost Optimal Design of NonlinearCA
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efficient design of non-linear CA based PRPG for VLSI circuit testing; ASP-DAC; Japan.
[12] Sukanta Das, Anirban Kundu, Biplab K. Sikdar; (2004) Nonlinear CA Based Design of Test Set
Generator Targeting Pseudo-Random Pattern Resistant Faults; Asian Test Symposium; Taiwan.
[13] Sukanta Das, Biplab K. Sikdar, Parimal Pal Chaudhuri; (2004) Nonlinear CA Based Scalable Design
of On-Chip TPG for Multiple Cores; Asian Test Symposium; Taiwan.
[14] Sukanta Das, Anirban Kundu, Subhayan Sen, Biplab K. Sikdar, Primal PalChaudhuri; (2003) Non-
Linear Celluar Automata Based PRPG Design (Without Prohibited Pattern Set) In Linear Time
Complexity; Asian Test Symposium; Chaina.
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BIST (UBIST) Structure; VLSI Design; India.

36
[16] Niloy Ganguly, Anindyasundar Nandi, Sukanta Das, Biplab K. Sikdar, Parimal Pal Chaudhuri; (2002)
An Evolutionary Strategy To Design An On-Chip Test Pattern Generator Without
Prohibited Pattern Set (PPS); Asian Test Symposium; Guam.
[17] Peter D. Hortensius,Werner Pries,Howard C. Card; (1989) Cellular Automata based Pseudorandom
Number generators for Built-In Self-Test; IEEE Transactions on Computer-Aided Design; Vl. 8 No. 8.
[18] Paul H. Bardell;( 1990) Analysis of Cellular Automata Used as Pseudorandom pattern Generators;
International Test Conference.
[19] D. de la Guia Martinez, A. Peinado Doinguez; ( 1999) Pseudorandom number generation based on
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Technology.
Authors:
Mr. Arnab Mitra
Mr. Mitra is presently working as an Assistant Professor in the Department of CSE with
Adamas Institute of Technology-W.B. (India). He is pursuing his research work as a
Research Assistant at Innovation Research Lab, under guidance of Dr. Anirban Kundu.
His broad research interests are Artificial Intelligence and applications of Cellular
Automata.
Dr. Anirban Kundu
Dr. Kundu is a Post Doctoral Research Fellow at Kuang-Chi Institute of Advanced
Technology, P.R. China. He has obtained his Ph.D. from Jadavpur University (India).
In India he is associated with Netaji Subhash Engineering College as an Assistant
Professor & Head of the Department (IT). He is also associated with Innovation
Research Lab-W.B. (India) as an Advisor & Consultant. His research interests include
Search Engine Oriented Indexing, Ranking, Prediction, Web Page Classification with
essence of Cellular Automata. He is also interested in Multi-Agent based system
design, Fuzzy controlled systems and Cloud Computing. He has authored more than 40 journal and
conference papers.

Cost Optimized Design Technique for Pseudo-Random Numbers in Cellular Automata

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