On Linear Complexity of Binary Sequences Generated Using Matrix Recurrence Relation Defined Over Z4

International Journal of Distributed and Parallel Systems (IJDPS) Vol.1, No.2, November 2010
DOI : 10.5121/ijdps.2010.1207 68
On Linear Complexity of Binary Sequences
Generated Using Matrix Recurrence Relation
Defined Over Z4
Ramesh S 1
, K N Haribhat 2
, R Murali 3
1
Research Scholar, Dr MGR University, Chennai, India
Faculty, Department of Electronics & Communication Engineering
Dr. Ambedkar Institute of Technology, Bangalore, 560056, India
Email:rameshs_1@yahoo.co.in
2
Dean Academic and Head, Department of Electronics & Communication Engineering
Nagarjuna College of Engineering & Technology Bangalore, India,
Email:knhari.bhat@gmail.com
3
Professor, Department of Mathematics
Dr. Ambedkar Institute of Technology, Bangalore- 560056, India
Email:dr_muralir@yahoo.co.in
Abstract:
This paper discusses the linear complexity property of binary sequences generated using matrix
recurrence relation defined over Z4. Generally algorithm to generate random number is based on
recursion with seed value/values. In this paper a linear recursion sequence of matrices or vectors over Z4
is generated from which random binary sequence is obtained. It is shown that such sequences have large
linear complexity.
Keywords
Matrix Recurrence Relation, Random Sequence, Linear Complexity, Z4 Sequence, Random numbers
1. INTRODUCTION
Today’s modern cryptography needs to deal with many aspects of security issues.In addition to
providing confidentiality, it has to provide data integrity, authentication and non repudiation
[1].Ever increasing computing power of potential attackers endangers even well-researched
encryption algorithms [ 2].
Cryptography systems can be broadly classified into symmetric-key systems such as DES, AES
and RC4 that use a single key that both the sender and recipient have to encrypt and decrypt
respectively. Public-key or asymmetric systems such as RSA, ElGamal and Elliptic curve
cryptograchy that use two keys, a public key known to everyone and a private key that only the
recipient of messages uses [3], [4, [5]. Symmetric cryptosystem is usually divided into block
ciphers and stream ciphers. Block ciphers operate with a fixed transformation on large blocks of
plain-text data; stream ciphers operate with a time-varying transformation on individual plain-
text digits [1], [5].This classification is not absolute, and any block cipher can be used as a

69
stream cipher by using certain modes of operation. Cipher Feedback mode (CFB), Output
Feedback mode and Counter mode operation on block cipher system can be used to turn a block
cipher into a stream cipher. They can be proven secure under the assumption that the block
cipher is secure [1], [3], [4].
In block cipher data is devided into blocks and encrypt and decrypt block wise, where in a
stream cipher encryption or decryption is bit-by-bit or character by character. Both block
ciphers and stream ciphers are in common use today. Generally in stream encryption system, a
binary message is encrypted by adding bit by bit modulo 2 a binary random sequence called key
sequence. Stream cipher system has the advantage that both encryption and decryption occurs at
real time. Stream ciphers are especially prevalent in business and military applications [2], [3],
[4] [5] [6], [7].
Security of stream cipher system depends on the randomness properties of the sequence called
key sequence. Therefore key sequence generator is very important building block for stream
cipher system. A random bit generator can be used to generate binary random bit sequences with
desirable statistical properties which are important in cryptographic applications.The need for
design of efficient and secure pseudorandom sequence generators remains an ongoing challenge
and an important field in cryptographic research up to the present day.
A method of generation of random binary sequences using matrix recurrence relation defined
over Z4 is discussed in [8].There are standard tests like FIPS-1 and NIST-SP-800-2 revision 1 [9]
test suites to test the randomness properties of binary sequences. It is shown in [8] that
sequences generated using matrix recurrence relation pass these test suites. It is also found that
such sequences exhibit good autocorrelation and cross correlation properties [10].
In this correspondence, we discuss the generation of random sequence defined over Z4 using
matrix recurrence relation and corresponding binary sequence is derived from it. Linear
complexity of the binary sequences so generated is determined using Massey - Berlekamp
algorithm [11] and results are analyzed. It is shown that such sequences exhibit large linear
complexity which is desirable characteristics of random sequences required for key sequences in
stream cipher systems.
Organization of the rest of the paper:
In Section 2 we introduce random sequence generator. Section 3 introduces a method of
generation of random sequence using matrix recurrence relation defined over Z4. Section 4
discusses the results. Section 5 contains concluding remarks.
2. RANDOM SEQUENCE GENERATORS
Random binary sequences are used as running key sequence in stream cipher system. Here
message is in the form of binary sequence is encrypted by adding bit by bit modulo 2 a binary
random sequence called key sequence and decrypted at the receiver using the same random key
sequence generated at the receiver.

70
Linear Feedback Shift Registers (LFSRs) are important building blocks for generating key
sequences. Maximum length sequences called m- sequences generated by an n stage Linear
Feedback Shift Registers have very good randomness properties such as long period, ( 2n
-1),
balance, ideal autocorrelation and good statistical properties which are desirable
characteristic[6]. Also LFSR can be easily implemented both in hardware and software.However
m-sequences have low linear complexity. For an m-sequence of length 2n
-1 the linear
complexity is n. In this case only 2n consecutive bits are required to determine the feedback
polynomial of the LFSR and hence the entire sequence. In the practical stream cipher designs, a
large linear complexity of the key stream is obtained by a nonlinear transformation of the LFSR
output sequence or nonlinear feedback shift register.
The use of non linear feedback mechanism to produce pseudorandom sequence is discussed in
[3], [4]. Some of the methods of transformation define three general design categories:
combination generators; filter generators and clock controlled generators [3].
The linearity complexity properties of LSFR’s can be improved by feeding the outputs of several
parallel LFSRs into a non-linear Boolean function to form a combination generator. The various
properties of such a combining function are critical for ensuring the security of the resultant
scheme, like avoiding correlation attacks.
Clock controlled generators [17] serve the function of introducing non-linearity in LFSRs. This
non-linearity is achieved by having LFSR clocked irregularly by being driven by the output of
some other LFSR. Several generators based on this principle have been proposed like stop-and-
go, alternating step generator [12] and the shrinking generator [3], [4], [13], [14][15][16]
Another approach of improving the linearity complexity of sequence generated by LFSR is to
use Filtering Boolean functions [5]. Although, not sufficient to be resistant enough against
several attacks, certain characteristics are supposed to be necessary in stream ciphers with this
structure. These characteristics include: high non-linearity, balance, and algebraic immunity
[17].
Some of the widely used random sequence generators are generator based on recurrence modulo
2 [11],[18], linear feedback shift register generator (LFSR) [4], [20],[21],[22], feedback with
carry shift registers (FCSR) [23],[24], nonlinear combination generators [3],[4], non linear
feedback shift registers (NLFBSR) [3],[4], Marsaglia random number generators [25]-[26], and
elliptic curve based pseudorandom random sequence generator [28] etc. Examples of
cryptographically secure pseudorandom bit generators are RSA pseudorandom bit generator
[29], Micali-Schnorr pseudorandom bit generator [30, 31], Blum-Blum-Shub pseudorandom bit
generator (x2
mod N generator) [32], [33]. Generally in all these schemes a random sequence is
generated using seed key.Generally stream cipher systems are fast and easy to implement both in
hardware and software.
The proposed scheme describes a method of obtaining binary sequences with large linear
complexity.

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3. PROPOSED RANDOM SEQUENCE GENRATOR USING
MATRIX RECURRENCE RELATION
The proposed random sequence generator defined over Z4 is based on Matrix Recurrence
Relation defined by,
Uj= 1ij
1n
0i
i UA −−
−
=
∑ , j ≥ n, arithmetic modulo 4 (1)
Where Ais , i = 0,1,…………..,n-1are k×k matrices called co-efficient matrices , Uis , i =
0,1,……………,n-1are k×k initial matrices called seed matrices over Z4 .With U0,U1,
U2,……………....Un-1 known,Un,Un+1,……….… satisfies the recurrence relation (1), where n is
called order of the recurrence relation.
A Vector recurrence relation can also be defined as follows
Vj = 1ij
1n
0i
iVA −−
−
=
∑ , j ≥ n , arithmetic modulo 4 (2)
Where, as in equation (1) Ai s, i = 0,1,…,n-1 are k×k matrices called co-efficient matrices . Vi s ,
i = 0,1,…,n-1 are k×1 vector called seed values .With V0,V1,V2,…..Vn-1 known,Vn,Vn+1,……
satisfies the recurrence relation (2) .
Based on recurrence relation (1),with n arbitrary initial k×k seed matrices U0,U1,U2, ……..Un-
1,random sequence of k×k matrices U0,U1,U2,…..Uj over Z4 is generated as follows
Un=A0Un-1+A1 Un-2,……..+ An-1 U0 modulo 4, (3)
Un+1=A0Un+A1 Un-1,……..+ An-1 U1, modulo 4 (4)
. . . . . . . . . . . . . .
. . . . . . . . . . . . . .
In general for j ≥ n
Uj=A0Uj-1+ A1Uj-2,……... + An-1 Uj-n , modulo 4 (5)
It can be shown that the sequence generated is strictly periodic if k× k coefficient matrix An-1 is
nonsingular [8].
3.1. General Case of Sequence over Z4
The method of generation of sequences of matrices and sequences of vectors over Z4 is
described below.

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Case I: Generation of Sequences of Matrices over Z4
In this case the n coefficients Ai s, i = 0, 1……….. n-1are k×k matrices over Z4 which are
arbitrarily chosen with An-1 nonsingular to get strictly periodic sequence. The n initial matrices
Ui s, i = 0, 1… ……..n-1are also k×k matrices over Z4. Random sequence of k× k matrices is
generated using equation (1) by randomly selecting the coefficient matrices and initial matrices
over Z4. Different sequences are generated for different coefficients and initial matrices. The
properties of the sequences generated depend on coefficient matrices, initial seed matrices and n,
order of recurrence relation given in expression (1).
To obtain Binary Sequence from Matrix Sequence over Z4
Coefficient matrices Uis are over Z4, the elements are from the set {0, 1, 2, 3}. Sequence over Z4
can be derived from the sequence of k× k matrices by concatenation of the rows of Ui s .Then a
sequence of N matrices each having k2
elements from Z4 gives rise to a sequence of Nk2
elements
over Z4. By representing the elements 0, 1, 2, and 3 in binary as 00, 01, 10, and 11 respectively
the length of the corresponding binary sequence is 2Nk2
.
Consider k×k seed matrices, each having k2
elements.
The number of possible seed matrices over Z4 for each stage = 4 k**2
(6)
For k=2, number of possible matrices = 44
=256 (7)
For n stages the number of possible seed matrices over Z4 is 4n k**2
. If all the possible states are
in one cycle, then the corresponding length of sequence is 4nk**2
, which is the maximum possible
length. Corresponding maximum possible length of binary sequence is equal to 2(4nk**2
).
However the actual length of the sequence generated is always less than the maximum possible
value and depends on the coefficient matrices and seed matrices.
Case 2: Generation of Sequence of Vectors over Z4
By having k×k coefficient matrices as above and with the initial values from a set of k×1vectors
over Z4 instead of k×k matrices, the sequences of k×1 vectors over Z4 are generated. The number
of possible seed vectors for one stage is 4k
and for n stages it is 4kn
.
To obtain Binary Sequence from Vector Sequence over Z4
The recurrence relation given by equation (2) is used to generate sequence of k×1 vectors {Vi},
over Z4.The seed values V0,V1,V2, Vn-1 are k×1 vectors over Z4.Seed vectors along with known n
coefficient matrices are used to obtain random vector sequence {Vi}. As mentioned earlier with
An-1, a k×k nonsingular coefficient matrix, the sequence generated is strictly periodic. Sequence
over Z4 is generated by concatenating the transpose of vectors in sequence {Vi}.

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Consider vector sequence of length N.The corresponding sequence over Z4 is then of length Nk.
The corresponding binary sequence is of length 2Nk. For a vector sequence of size k×1, with n
number of stages the maximum possible length of the sequence will be equal to 4nk
and the
corresponding maximum possible length of binary sequence is equal to 2(4nk
).
In this case also the actual length of sequences depends on number of stages n, choice of
coefficient matrices and seed values and is less than the maximum possible value.
3.2. Linear Complexity Measure
One of the most useful concepts in the study of key sequences used in stream ciphers is that of
linear complexity. The linear complexity of a binary sequence is defined as the length of the
shortest linear feedback shift register that generates it. If a sequence has small linear complexity,
then the synthesis of a linear equivalent of the sequence generator becomes computationally
feasible. The linear complexity of a finite sequence is determined using Massey –Berlekamp
algorithm [11]. The algorithm is briefly described below.
Algorithm:
INPUT: A binary sequence sn = s0, s1, s2,….., sn−1 of length n.
OUTPUT: Linear complexity L(sn
) of sn, 0 ≤ i ≤ n.
Initialization: C (D) = 1= 0, l= 0, m= −1, B(D) =1, N= 0.
2. While (N < n) do the following:
2.1 Compute the next discrepancy d.
d = (sN+ 1N
L
1i
i
sc −
=
∑ ) mod2,
2.2 If d = 1 then do the following:
T (D) = C (D), C (D) =C (D) + B (D) DN−m
.
If L ≤ N/2 then L =N + 1 − L, m= N, B (D) =T (D).
2.3 N= N + 1.
3. Return (L).
Where C (D) is connection polynomial, L is linear complexity.
It is desirable that random sequence which can be used as key sequence in stream cipher systems
to have a large linear complexity. The necessary (but not sufficient) condition to be secure in
running key stream cipher is to have large linear complexity [11], [34], [35]..
Even for a given order n, for different arbitrarily chosen n coefficient matrices and seed values,
the generated sequence of length m, need not necessarily have same linear complexity. Further

74
just by knowing n coefficient matrices and initial values, linear complexity cannot be predicted
.Hence linear complexity can be treated as a random variable. Thus to study the statistical
behavior of the linear complexity, number of sequences are generated and their linear
complexity is computed. From these data we compute Mean, µx Variance, σx
2
, and Standard
deviation, σx of linear complexity. The definition of Mean, Variance, and Standard deviation are
available in [37].The definitions are repeated here.
Mean
Let X1, X2, X3…XN be N linear complexity values of N sequence of same length.The mean value,
µx or expected value of linear complexity is defined as
Mean=µx= ∑
N
1=i
iX
N
1
(8)
Variance and standard deviation, σx
2
Variance of random variable X is a measure of how far the value of random variable deviates
from its mean value. It is defined as
σx
2
=
N
)X(
N
1=i
2
xi∑ µ−
(9)
Very small value of variance indicates that X takes values almost equal to µx. The standard
deviation σx is the positive square root of its variance σx
2
. Standard deviation is a widely used
measure of the variability or dispersion from mean value. It shows how much variation there is
from the "average" (mean or expected) value. A low standard deviation indicates that the values
tend to be very close to the mean, whereas high standard deviation indicates that the values are
spread out over a large range of values, around the mean value.
Through simulation it is seen that by proper choice of size, k of matrix and number of stages, n it
is possible to generate random binary sequences of large linear complexity.
4. SIMULATION RESULTS AND DISCUSSIONS
In general the binary sequence generated will have its linear complexity which depends
on
i) n , the order of recurrence relation
ii) choice of n coefficient matrices and
iii) the n seed value

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4.1. CASE 1: Matrix Sequences and Corresponding Binary Sequences
In the following studies k = 2 is chosen. From equation (6), the number of 2×2 matrices over Z4
is 256.As n increases the linear complexity of the sequence also increases. This is verified by
means of computer simulation. For n=5,10,15,20,25 and 30, the corresponding number of 2×2
coefficient matrices are arbitrarily chosen from a set of 256 , 2×2 matrices. Likewise the n seed
values are also chosen arbitrarily. In each case, 10 different sequences of length m = 5000 bits
are considered. Linear complexities of these 10 sequences are computed using Berlekamp-
Massey algorithm [11].The statistical behavior of linear complexity X is found by computing its
mean µx , variance, σx
2
and standard deviation ,σx after the linear complexities of 10 sequences
are computed. The results are tabulated in Table 1.
Columns of Table 1shows the computed linear complexity of generated sequences using
recurrence relation of order n =5, n=10, 15, 20, 25 and 30 respectively. It is observed from the
Table 1 that the mean linear complexity µx for sequences of length 5000 bits for n = 5 are found
to be 556. Similarly the mean linear complexities µx, for n=10, 15, 20, 25, and 30 are found to be
1875.8, 2495.8, 2493, 2498.3, 2499.3 respectively. The corresponding variance σx
2
is found to be
125.7777, 126.4, 130.4889, 52, 11.3444 and 0.9111 respectively. Likewise the corresponding
standard deviation σx is found to be11.2150, 11.2427, 11.4231, 7.2111, 3.3681 and 0.6749
respectively. From these results it is evident that the linear complexity value is increasing with
the order n of recurrence relation and approaches m/2 for n≥ 15.Correspondingly variance, and
standard deviation decreases .This implies that the sequences generated have mean value of
linear complexity µx almost equal to m/2 with high probability for m=5000 bits and n≥15.
Similarly Table 2 and Table 3, list the linear complexities X , mean µx, variance σx
2
and standard
deviation σx for sequences of length 10000 bits and 20000 bits respectively , for different n =
5,10,15,20, 25 and 30. It is seen from Table 2 and Table 3 that for m= 10000 and 20000 and for
n ≥20 the linear complexity approaches m/2 which is desirable for random sequences [36].
The variation of mean µx, variance σx
2
and standard deviation σx with different m and n are
depicted in Figure.1, Figure.2 and Figure.3.respectively. For different length of sequence m
=5000, 10000 and 20000 it is also seen that the values of variance and standard deviation are
decreasing as n increases.
From the above results it is seen that for n ≥ 15 the mean value of linear complexity µx is almost
equal to m/2 and hence the generated sequences have linear complexity m/2 with high
probability for m=5000 bits. It is also seen that variance σx
2
and standard deviation σx are small.
The same behavior of linear complexity, variance and standard deviation of linear complexity
are observed for m = 10000 and 20000 bits. These are depicted in Figure.1, Figure.2 and
Figure.3 respectively.

76
Table 1: List of Linear Complexity of Binary Sequences Generated Using Matrix Recurrence Relation
Defined by equation (1), CASE 1 with k = 2
Length of Sequence = 5000 bits
Stage , n 5 10 15 20 25 30
LC of Sequence 1 544 1897 2500 2500 2499 2500
LC of Sequence 2 545 1871 2500 2489 2500 2500
LC of Sequence 3 567 1881 2498 2499 2500 2499
LC of Sequence 4 567 1855 2500 2489 2499 2499
LC of Sequence 5 565 1881 2500 2500 2500 2500
LC of Sequence 6 546 1877 2497 2488 2500 2498
LC of Sequence 7 539 1881 2500 2479 2498 2499
LC of Sequence 8 561 1879 2488 2489 2489 2499
LC of Sequence 9 567 1865 2488 2497 2498 2499
LC of Sequence 10 559 1871 2487 2500 2500 2500
Mean, µx 556 1875.8 2495.8 2493 2498.3 2499.3
Variance, σx
2
125.7777 126.4 130.4889 52 11.3444 0.9111
Standard Deviation, σx 11.2150 11.2427 11.4231 7.21110 3.3681 0.6749
Stage , n 5 10 15 20 25 30
LC of Sequence 1 544 1897 3650 5000 5000 5000
LC of Sequence 2 545 1871 3662 4990 5000 5000
LC of Sequence 3 567 1881 3665 5000 5000 5000
LC of Sequence 4 567 1855 3664 5000 5000 5000
LC of Sequence 5 565 1881 3653 4997 4999 4999
LC of Sequence 6 546 1877 3650 5000 5000 5000
LC of Sequence 7 539 1881 3660 5000 4999 5000
LC of Sequence 8 561 1879 3655 4994 4998 4999
LC of Sequence 9 567 1865 3654 5000 5000 5000
LC of Sequence 10 559 1871 3643 5000 5000 5000
Mean, µx 556 1875.8 3655.6 4998.1 4999.6 4999.8
Variance, σx
2
125.7778 126.4 50.0444 12.1 0.4888 0.1777
Stage , n 5 10 15 20 25 30
LC of Sequence 1 544 1897 3650 9989 9998 10000
LC of Sequence 2 545 1871 3662 9989 9998 10000
LC of Sequence 3 567 1881 3665 9988 9998 9998
LC of Sequence 4 567 1855 3664 9989 9997 9998
LC of Sequence 5 565 1881 3653 10000 10000 10000
LC of Sequence 6 546 1877 3650 9989 9999 10000
LC of Sequence 7 539 1881 3660 9987 9987 9997
LC of Sequence 8 561 1879 3655 9988 9989 9998
LC of Sequence 9 567 1865 3654 9999 9999 10000

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Figure 1: Plot of Mean Linear Complexity, µx
v/s Number of Stages,n
Case 1 Matrix sequence with k=2
Figure 2: Plot of Variance of Linear Complexity, σx
2
Figure 3: Plot of Standard Deviation of Linear
Complexity, σx v/s Number of Stages,n
0
2000
4000
6000
8000
10000
12000
5 10 15 20 25 30
AverageLinearComplexiy
Number of Stages
Plot of Average Linear
Complexity
v/s Number of Stages
5000 bits
10000 bits
20000 bits
0
20
40
60
80
100
120
140
5 10 15 20 25 30Variance
Number of Stage , n
Plot of Variance of Linear
Complexity
v/s Number os Stages
5000 bits
10000 bits
20000 bits
0
2
4
6
8
10
12
5 1015202530
StandardDeviation
Number of Stages , n
Plot of Standard Deviation v/s
Number of Stages
5000 bits
10000 bits
20000 bits
LC of Sequence 10 559 1871 3643 9989 10000 10000
Mean, µx 556 1875.8 3655.6 9990.7 9996.5 9999.1
Variance, σx
2
125.7778 126.4 50.0444 22.0111 21.1666 1.4333

78
4.2. CASE 2: Vector Sequences and Corresponding Binary Sequences.
For this case also k= 2 is chosen.
The simulation study is carried out with seed values of 16 possible 2×1 vectors over Z4 instead
of matrices. For n=5,10,15,20,25 and 30, the corresponding number of 2×2 coefficient matrices
are arbitrarily chosen from a set of 256 possible 2×2 matrices over Z4.
In each case, 10 different sequences of length m = 5000 bits are generated after binary
transformation of sequences over Z4 as discussed in section 3.1. Linear complexities of these 10
sequences are computed using Berlekamp- Massey algorithm [11].The statistical behavior of
linear complexity is found by computing its mean value, variance and standard deviation.The
results are tabulated in Table 4.Columns of Table 4 shows the computed linear complexity of 10
sequences generated using recurrence relation of order n =5, 10, 15, 20, 25 and 30 respectively.
The corresponding mean linear complexity µx, variance σx
2
and corresponding standard deviation
σx are also listed. From these results it is evident that the linear complexity value is increasing
with the order n of recurrence relation and approaches m/2 for n≥ 15.Correspondingly variance
and standard deviation decreases .This implies that the sequence generated have linear
complexity almost equal to m/2 with high probability for m=5000 bits and n≥15.
Similarly in Table 5 linear complexities X, mean value of linear complexity µx, variance σx
2
and
corresponding standard deviation σx for sequences of length 10000 bits for n = 5, 10,15,20,25
and 30 are listed.
Likewise in Table 6 linear complexities X, mean valueof linear complexity µx, variance σx
2
and
corresponding standard deviation σx for sequences of length 20000 bits for n = 5, 10,15,20,25
and 30 are also listed. It is seen from Table 5 and Table 6 that for m= 10000 and 20000 and for n
≥20 the linear complexity approaches m/2.
The variation of mean µx, variance σx
2
and standard deviation σx with different m and n are
depicted in Figure.4, Figure.5 and Figure.6.respectively for different length of sequence m
=5000, 10000 and 20000. It is seen that the values of variance and standard deviation are
decreasing as n increases.
From the above results it is seen that for n ≥ 15 the mean value of linear complexity µx is almost
equal to m/2 and hence the generated sequences have linear complexity m/2 with high
probability for m=5000 bits. It is also seen that variance σx
2
and standard deviation σx are small.
The same behavior of linear complexity, variance and standard deviation of linear complexity
are observed for m = 10000 for n ≥ 20 and for m = 20000.For n≥30, linear complexity nearly
m/2.
Table 4: List of Linear Complexity of Binary Sequences Generated Using Vector Recurrence Relation
Stage , n 5 10 15 20 25 30
LC of Sequence 1 251 789 2497 2500 2500 2500
LC of Sequence 2 254 765 2500 2500 2500 2500
LC of Sequence 3 245 778 2475 2489 2499 2499

79
LC of Sequence 4 254 754 2498 2489 2499 2499
LC of Sequence 5 257 775 2500 2478 2489 2498
LC of Sequence 6 251 779 2500 2500 2500 2500
LC of Sequence 7 255 765 2500 2500 2500 2500
LC of Sequence 8 243 770 2475 2489 2500 2498
LC of Sequence 9 254 764 2497 2489 2499 2498
LC of Sequence 10 255 775 2500 2500 2500 2500
Mean, µx 251.9 771.4 2494.2 2493.4 2498.6 2499.2
Variance, σx
2
20.7666 97.6 103.955 59.1555 11.6 0.8444
Stage , n 5 10 15 20 25 30
LC of Sequence 1 251 789 2680 4989 5000 5000
LC of Sequence 2 254 765 2670 4988 5000 5000
LC of Sequence 3 245 778 2680 4979 4999 4999
LC of Sequence 4 254 754 2689 4998 4999 5000
LC of Sequence 5 257 775 2686 4993 5000 5000
LC of Sequence 6 251 779 2690 4985 4997 4998
LC of Sequence 7 255 765 2697 4999 5000 5000
LC of Sequence 8 243 770 2680 4983 5000 5000
LC of Sequence 9 254 764 2697 4989 5000 5000
LC of Sequence 10 255 775 2690 4989 4999 4998
Mean, µx 251.9 771.4 2685.9 4989.2 4999.4 4999.5
Variance, σx
2
20.7666 97.6 71.8777 38.8444 0.93333 0.7222
Stage , n 5 10 15 20 25 30
LC of Sequence 1 251 789 3650 4989 6600 7560
LC of Sequence 2 254 765 3662 4988 6600 7568
LC of Sequence 3 245 778 3665 4979 6612 7569
LC of Sequence 4 254 754 3664 4999 6604 7565
LC of Sequence 5 257 775 3653 4993 6602 7565
LC of Sequence 6 251 779 3650 4985 6603 7564
LC of Sequence 7 255 765 3660 4999 6603 7560
LC of Sequence 8 243 770 3655 4983 6603 7564
LC of Sequence 9 254 764 3654 4989 6602 7562
LC of Sequence 10 255 775 3643 4989 6601 7560
Mean, µx 251.9 771.4 3655.6 4989.3 6603 7563.7
Variance, σx
2
20.7666 97.6 50.0444 40.9 11.7777 10.4555

80
Figure 4:Plot of Mean Linear Complexity, µx
Case I Matrix sequence with k=2
Figure 5: Plot of Variance of Linear
Complexity, σx
2
Figure 6: Plot of Standard Deviation of Linear
Complexity, σx
2
The statistical results µx, σx
2
based on 10 trials in each case are used to get the bound on
probability of linear complexity X lying within the range µx-25 ≤ X ≤ µx+25 using Chebychev
inequality [20]. First Chebychev inequality as defined in [38] is given below.
0
1000
2000
3000
4000
5000
6000
7000
8000
5 15 25
AverageLinearComplexity
Number of Stages , n
Plot of Average Linear Complexity
5000 bits
10000
bits
20000
bits 0
20
40
60
80
100
120
5 101520 2530
VarianceofLinearComplexity
Number Stages , n
Plot of Variance of Linear
Complexity
5000 bits
10000 bits
20000 bits
0
2
4
6
8
10
12
5 1015202530
StandardDeviation
Number of stafges , n
Plot of Standard Deviation
5000 bits
10000 bits
20000 bits

81
P ( | X- µx| ≥ δ ) ≤ σx
2
/ δ2
; δ > 0 (10)
Where P ( | X- µx| ≥ δ ) is the probability that X lies outside the range (µx - δ ) ≤ X ≤ (µx + δ).
This probability is always less than or equal to σx
2
/ δ2
.
Chebychev inequality is generally expressed as in equation (8). It can also be interpreted as,
P ( | X- µx| ≤ δ ) ≥ 1- σx
2
/ δ2
; δ> σx (11)
where P ( | X- µx| ≤ δ ) is the probability that X lies inside the range (µx - δ ) ≤ X ≤ (µx + δ
).This probability is always greater than or equal to 1- σx
2
/ δ2
Equation.11 is used to observe the change in probability bound with increase in n from 15.The
probability bound is computed for n = 15,20,25 and 30 using Table 1 to table 6. In all these cases
δ is taken as 25.The results are tabulated in Table 7, 8, 9, 10, 11 and 12, corresponding to matrix
and vector recurrence relations.
Table 7: Statistics computed for Matrix Sequences
Length of the sequence: 5000
Number of stage, n 15 20 25 30
Corresponding mean, µx 2495.8 2493 2498.3 2499.3
Corresponding variance , σx
2
130.4889 52 11.3444 0.9111
P ( | X- µx| ≤ 25 ) ≥ 1- σx
2
/ 625 0.7912 0.9168 0.9818 0.9985
Corresponding mean, µx 3655.6 4998.1 4999.6 4999.8
2
50.04444 12.1 0.488889 0.177778
P ( | X- µx| ≤ 25 ) ≥ 1- σx
2
/ 625 0.9199 0.9806 0.9992 0.9997
2
50.0444 22.0111 21.1666 1.4333
P ( | X- µx| ≤ 25 ) ≥ 1- σx
2
/ 625 0.9193 0.9648 0.9661 0.9977
Table 10: Statistics Computed for Vector Sequences
2
103.955 59.1555 11.6 0.8444
P ( | X- µx| ≤ 25 ) ≥ 1- σx
2
/ 625 0.8336 0.9053 0.9814 0.9986

82
2
71.8777 38.8444 0.93333 0.7222
P ( | X- µx| ≤ 25 ) ≥ 1- σx
2
/ 625 0.8849 0.9378 0.9985 0.9988
5. CONCLUSIONS
Use of matrix recurrence relation (1) and (2) defined over Z4 for the generation of random binary
sequences derived from sequences over Z4, results in random sequences with large linear
complexity determined using Massey- Berlekamp algorithm. To study the statistical properties
of the linear complexity, mean µx, variance σx
2
and standard deviation σx are determined by
considering randomly chosen sequences of different lengths. Six cases for n=5, 10,15,20,25 and
30 are considered.In each case sequences of length 5000, 10000 and 20000 bits is considered.
For each combination of n and m, by randomly chosing different initial matrices and coefficient
matrices 10 sequnces are generated and their linear complexity property is investigated.
It is seen from the Tables 7 to12, that for n ≥ 15 the probability that the linear complexity X
differ by mean value µx by 25 always increases with n and m. For m = 20000 and n = 30
probability that linear complexity X is within (10000 ± 25) is always greater than or equal to
0.9977 in case of random binary sequence derived from recursion relation (1). Similarly it is
greater than 0.9832 for random binary sequence derived from recursion relation (2).
The proposed method of generation of sequences over Z4 is linear; the corresponding binary
sequence with binary conversion 0,1,2,3 to 00,01,10,11 respectively turns out to be nonlinear.
Hence the linear complexity of the sequence is increased compared to m- sequence. Algorithm is
simple with modulo4 arithmetic. It is possible to implement both in hardware and
software.There are large choices for number of stages,n initial content ( Ui’s or Vi’s) of different
size and the different coefficient matrices (Ai’s) which can be chosen to generate large number
of sequences.
REFERENCES
Corresponding mean, µx 3655.6 4989.3 6603 7563.7
2
50.0444 40.9 11.7777 10.4555
P ( | X- µx| ≤ 25 ) ≥ 1- σx
2
/ 625 0.9199 0.9358 0.9811 0.9832
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Authors
Ramesh S received the B.E degree in Electronics &
Communication Engineering from Gulbarga University,
Karnataka, India in 1990, and M.Tech Degree in Industrial
Electronics from Visvesvaraya Technological University,
Belgaum, India in 2001 and currently working towards Ph.D
Degree at Dr MGR University, Chennai, India. He is working as
Faculty in the Department of Electronics & Communication
Engineering, Dr Ambedkar Institute of Technology, Bangalore,
India. His research areas include Analog Communication, Digital
Communication, Cryptography and VLSI Design.
Dr K N Haribhat received the B.E Degree with honors from
Mysore University in 1966, M.Tech and Ph.D in Electronics &
Communication Engineering from Indian Institute of Technology,
Kanpur, in 1973 and 1986, respectively. He is currently working as
Dean Academic and Head, Department of Electronics &
Communication Engineering at Nagarjuna College of Engineering
& Technology, Bangalore, India. He was with Karnataka Regional
Engineering College, Suratkal, India (Currently known as NIT-K)
for more than 30 years. His research areas include Analog
Communication, Digital Communication and Cryptography. He
has authored more than 25 papers in National/international
Conferences and Journals.He has coauthored two books on
communication.
Dr Murali R received the M.Sc Degree in Mathematics from
Bangalore University, Karnataka, India in 1990, and Ph.D Degree
in Mathematics from Bangalore University, India in 1999 and
currently working as Professor in the Department of Mathematics,
at Dr Ambedkar Institute of Technology, Bangalore, India. His
research areas include Graph Theory-Hamiltonian graphs. He has
authored more than 12 papers in National/international Journals.
[37] Vijay K Rohatgi, A. K. Md.Ehsanes Saleh, “An Introduction to Probability and Statistics”, John
Wiley & Sons( ASIA) Pte Ltd,2nd
Edition
[38] John G. Proakis “Digital Communication” , McGraw Hill Higher Education, 4th Edition,
December 1, 2000

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