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IJCCCE Vol.15, No.3, 2015
_________________________________________________________________________________
77
New Pseudo-Random Number Generator System
Based on Jacobian Elliptic maps and Standard Map
Asst. Prof. Dr. Luma Fayeq Jalil1 Prof.Dr. Hilal Hadi Saleh2 Ekhlas Abass Albhrany3
1,2 Department of Computer Science, University of Technology, Baghdad
3 Department of Computer Science, Mustansiriyah University, Baghdad, Iraq,
e-mail: dr_lumafaik79@yahoo.com, hhsrq888@yahoo.com akhlas_abas@yahoo.com,
Received: 25/2/2015
Accepted: 11/11/2015
Abstract- Chaotic systems have numerous properties, for example: mixing property,
sensitivity to initial conditions parameters, structural complexity and deterministic
dynamics. These properties were investment in the last decade for cryptographic
applications and developments of pseudorandom number/bit generator. The paper propose
new pseudorandom number (bits) generator (PRNG) based on the Jacobian elliptic chaotic
maps and standard map. The principle of the method consists in generating binary sequence
from elliptic chaotic maps of cn and sn types. These sequence positions is permuted using
standard map. The performance of the generator is studied through conventional statistical
methods and also using the NIST test suite. The results show that the produced sequences
possess high randomness statistical properties and good security level which make it
suitable for cryptographic applications.
Keywords: chaotic function, pseudorandom sequences, Jacobian elliptic maps, Standard
map, NIST test suite.
IJCCCE Vol.15, No.3, 2015
Ekhlas Abass Albhrany et. al.
New Pseudo-Random Number Generator System
Based on Jacobian Elliptic maps and Standard map
78
1. Introduction
The generation of PRNG plays an
essential role in a large number of
applications such as simulations of
numerical, statistical mechanics, gaming
industry, communication or cryptography.
The main advantages of such generators are
the rapidity and the repeatability of the
sequences and require less memory for
algorithm storage. First way to design such
a pseudo-random number generator is
connected to the chaos theory [1]-[3]. That
theory focuses primarily on the description
of these systems that are often very simple
to define, but whose dynamics appears to
be very confused. Indeed, chaotic system
very attractive for pseudo-random number
generators because a slightly change in the
input can cause a large change in the output
(i.e. the extreme sensitivity to the initial
conditions). Moreover, during this last
decade several pseudo-random number
generators have been successfully
developed.
Patidar, 2009 designed a new PRBG
(pseudo random bit generator) based on
running pair of chaotic logistic maps side-
by-side and beginning from random
independent initial conditions [3]. Pareek ,
Patidar , Sud, 2010 proposed a new binary
sequence generator, called Cross- Coupled
Chaotic Random Bit Generator (CCCBG),
which achieve the interesting properties of
a skew tent map. They used two chaotic
maps which are the piecewise linear skew
tent maps and cross-coupled. The CCCBG
generates the binary sequences based on the
comparison between the outputs of the
skew tent and cross coupled chaotic maps
[5]. Francois, Grosges, 2011 proposed an
algorithm for generation of multiple
pseudo-random sequences using a chaotic
function. The algorithm uses permutations.
The permutation positions are computed
and indexed using a chaotic function based
on linear congruencies. These chaotic
permutations are obtained iteratively on this
initial vector to produce two chaotic maps
[6]. Azeem, Adriana, Adrian, 2013 suggest
using tent map to generate pseudorandom
binary sequences. The binary sequences
under investigation are obtained either by
considering all the successive iterations of
the tent map and choosing a threshold equal
to the tent map parameter or by applying a
periodical sampling on the tent map values
and by choosing a threshold equal to 0.5
[7]. Recently, Michael François, David
Defour and Christophe Negre 2014 propose
pseudo-random bit generator based on
combined three logistic maps and generate
a block of 32 random bits at each iteration.
The proposed generator based on the
binary64 double that produced based on the
IEEE 754-2008 standard for floating-point
arithmetic [8].In this paper, we propose a
novel pseudo-random number (bit)
generator (PRNG) based on the Jacobian
elliptic chaotic maps and standard map. It
combines the output of the elliptic chaotic
maps of cn and sn types. The output from
each type is converted to binary sequence.
Theses sequences are combined to produce
one binary sequence. This sequence is
permuted by using standard map. The
choice of using Jacobian elliptic chaotic
maps and standard map enlarges the
complexity of the system and increases the
difficulty for an attacker to extract sensitive
information from the outputs. Experimental
results and security analysis indicate that,
the elliptic chaotic map is advantageous
IJCCCE Vol.15, No.3, 2015
Ekhlas Abass Albhrany et. al.
New Pseudo-Random Number Generator System
Based on Jacobian Elliptic maps and Standard map
79
from the point of view of large key space
and high level of security.
The produced pseudo-random
sequences have successfully passed the
various statistical tests. The assets of the
generator are: high sensitivity to initial seed
values, high level of randomness and good
throughput.
The paper is structured as follows, the
description of the method as well as the
chaotic functions analysis are given in
Section 2 and 3 and 4. Section 5 presents
the statistical analysis applied on a set of
generated pseudo-random sequences. The
security analysis of the generator is
achieved in Section 6, before conclusions.
2. Jacobian Elliptic Chaotic Map
The core of the generator is the of the cn
and sn type of Jacobian elliptic chaotic
map. Ergodicity and fixed interval of
chaotic orbits are two major properties for
the performance of chaos based
cryptosystems. The families of one-
parameter elliptic chaotic maps of cn and sn
at the interval [0, 1] are defined as the ratio
of Jacobian elliptic functions of cn and sn
types [9] through the following equations:
Obviously, these equations map the
unit interval [0, 1] into itself. The maps
(α , x), w = 1, 2, are (N−1)-nodal
maps, that is, they have (N−1) critical
points in unit interval [0, 1] and they have
only a single period one stable fixed point
or they are ergodic.
As an example of the Jacobian elliptic maps
(1), the following maps can be presented:-
Where x0∈ [0, 1], α∈[0, 4] and k∈[0, 1].
The parameter k (modulus) represents the
parameter of the elliptic functions. Elliptic
chaotic maps are ergodic for certain values
of their parameters.
3. Standard Map
The so-called standard map was introduced
in [10]-[11], and is described by:
where k is the control parameter satisfying
k > 0, and the ith states ai and bi both take
real values in [0,2) for all i. Thestandard
map was discretized in astraightforward
manner [12] by substituting x = aN/2, y =
bN/2, K = kN/2 intoEq. (6), which
maps from [0,2) × [0,2) to N × N. After
discretization, the map becomes
where K is a positive integer. The
properties of this discretized map may not
be as good as the original one, but it can be
implemented in the integer domain, which
reduces the computational complexity and
is more suitable for real-time data
encryption. The standard map is used to
realize data permutation [12].
In the standard map, the pixels at the
corners of a square image have some
IJCCCE Vol.15, No.3, 2015
Ekhlas Abass Albhrany et. al.
New Pseudo-Random Number Generator System
Based on Jacobian Elliptic maps and Standard map
80
special properties. For example, the pixel at
position (0, 0) remains unchanged after any
number of iterations. This is actually a
weakness of the permutation process. And
it can do some help to the attackers
although the permutation process is further
strengthened by a diffusion process. In
order to avoid it, [13] proposed to change
the positions of the pixels at the corners ((0,
0), (0,N-1), (N- 1, 0) and (N- 1,N - 1)). That
is, the normal scan order is changed into a
random one. A random-couple (rx, ry) is
generated (after the iteration of chaotic
map), which represents the position of a
randomly selected pixel in the square
image. Then, the whole image shifts in
horizontal and vertical directions by rx and
ry, respectively as shown in Figure 1.The
normal scan mode by using the random
shift process is changed into a random one,
so it is named a random-scan mode.
The two parameters rx and ry both vary
from 0 to N-1. Thus, the random-scan
process can be combined with the chaotic
permutation process, and the modified
chaotic map becomes
The modified map is still invertible, so
the inverse-permutation process can be
easily realized. The modified chaotic
confusion process has two advantages [13]:
First the random-couple can be
generated under the control of keys, which
enlarges the cryptosystems key space. This
mean that the key space for the random-
couple is 2 × N (N is the width or height of
the matrix).
Second the random-scan process
makes it difficult to break the diffusion key
under known-plaintext attacks. This mean
that the random-scan process confuses the
position of the first pixel, which makes
attackers difficult to get the first pixels
cipher-pixel, and thus increases the
difficulty of breaking the diffusion key.
4. The Proposed PRNG
The main idea of the proposed PRNG
consists of the following major steps:-
Step 1: the initial condition (x) and control
parameters (α and k) are input to the
Jacobian elliptic chaotic map type cn
equation (3) and sn equation (4). These
numbers are floating point numbers where
the precision is 10−16 for each of x0, α and k,
considered as the keys of the generator.
Step 2: Iterate the Jacobian elliptic maps
100 times and ignore the results, in order to
eliminate the transient effect of chaotic
map.
Step 3: modify the values of x, k and α
using simple XOR operation to increase the
complexity of algorithm.
Step 4: Iterate the Jacobian elliptic maps
one time and convert the floating point
output of each map to binary sequence of
random length. The two sequences are
Figure 1 Scan order in a square image:
(a) Normal scan mode, (b) Random scan mode
IJCCCE Vol.15, No.3, 2015
Ekhlas Abass Albhrany et. al.
New Pseudo-Random Number Generator System
Based on Jacobian Elliptic maps and Standard map
81
combined into one random length binary
sequence.
Step 5: The parameters of standard map
equation (7) are constructed from the half
24-bit of the resulted sequence. These bits
are divided into three 8-bit integer numbers
to produce the parameters of standard map
(r1, r2, k). These 24-bit then is eliminated
from the sequence. The resulted binary
sequence is translate to matrix of 8 x N
where N is an integer number.
Step 6: Diffuse the resulted matrix by using
standard map equation (7).
Step 7: the resulted matrix from the
standard map is transferred column by
column to new binary sequence.
Step 8: repeat from step 4 until the desired
number of bits (numbers) is reached. When
number of bits becomes a chosen number
(in proposed algorithm this number is 500),
the parameters x, k and α are modified
based on last values of cn and sn maps in
order to increase the complexity of detect
the keys.
Step 9: The output of the algorithm can be
either a binary sequence of random length
or a sequence of a random number of
integer numbers.
The flowchart of the proposed algorithm is
presented in Figure 2.
5. Statistical Analysis
A statistical analysis should be
carefully conducted to prove the quality of
the pseudo-random sequences. The quality
of the output sequences produced by any
PRBG must have a high level of
randomness and be completely decorrelated
from each other.
5.1. Randomness evaluation.
This testing is implemented using
statistical tests NIST (National Institute of
Standards and Technology of the U.S.
Government) [14]. The testing is achieved
on sequences produced from nearby or
successive seed values.
Because for very distant seed values,
the chaotic trajectories are very different,
this usually allows obtaining good pseudo-
random sequences.
The testing was realized by generating
a number of m = 1000 different binary
sequences of length 1500. Each sequence is
generated by using different seed values.
The seed values are distributed in the range
[0..1] floating point number. All sequences
generated by the proposed PRNG are
analyzed using the NIST statistical
package. These tests are divided into two
groups based on the sequence length.
The first group consists of tests that can
be evaluated on the sequences that have
length >=100 bits, while the second group
consists of tests that are evaluated on the
sequences have length >=1000, 000.
Therefore we analyzed individual
sequences and concatenate sequences by
using the first group and only the
concatenate sequences using second group.
The results of NIST tests obtained on the
two groups are presented in Table 1 and
Table 2 respectively. The acceptable
proportion should lie above the proportion
previously defined. For individual
sequences, the proportion is:
This mean that the proportion should be
in the confidence interval [0.9994392..
0.9805608]. All the tested sequences
(individual and concatenate) pass
IJCCCE Vol.15, No.3, 2015
Ekhlas Abass Albhrany et. al.
New Pseudo-Random Number Generator System
Based on Jacobian Elliptic maps and Standard map
82
successfully the NIST tests. These results
show clearly the quality of the produced sequences from successive seed values.
Input the secret key x, k and α.
Input N.
Set i=1, j=1.
Convert the resulted floating point of sn
and sn maps into binary sequence. These
sequences are combined in one random
length sequence.
i=500?
Or I=63?
Transfer the resulted binary sequence to matrix
and diffuse it by using standard map.
j <= N?
Transfer the resulted matrix column by column into new
binary sequence which is the output of the generator.
i=i+1, j=j+1.
Iterate the Jacobian elliptic maps one
time using the new values of x, k and α
No
The parameters x, k and α
are modified based on last
values of cn and sn maps.
Set i=1.
End
Iterate the Jacobian elliptic maps cn and sn 100
times to avoid transient effect.
Modify the values of x, k and α using simple
xor operation
No
Yes
Yes
x,k and α → the
parameters of
Jacobian elliptic
map.
i → the counter of
bit number for
modification.
j → the counter of
required number of
bits.
N→ the required
number of bits.
Figure 2 The Flowchart of Proposed PRNG.
IJCCCE Vol.15, No.3, 2015
Ekhlas Abass Albhrany et. al.
New Pseudo-Random Number Generator System
Based on Jacobian Elliptic maps and Standard map
83
5.2. Correlation evaluation
Correlation evaluation is to check the
correlation between the produced pseudo-
random number sequences. This can be
done in two different ways.
Firstly, the correlation between
generated sequences is analyzed globally by
computing the Pearson's correlation
coefficient of each pair of sequences [15].
Secondly correlation based directly on
the bits of sequences is analyzed. The
Hamming distance between two binary
sequences (of the same length M) is the
number of places where they differ, i.e., the
number of positions where one has a 0 and
the other a 1 [16].
5.2.1. Pearson’s correlation coefficient
Consider a pair of sequences given by:
S1 = [x1, . . . ,xN] and S2 = [y1, . . . , yN].
Therefore, the corresponding correlation
coefficient is[15]:
Where the mean values of S1 and S2 are:
and
Two uncorrelated sequences are
characterized by CS1,S2 =0. The closer the
value of CS1,S2 is to ±1, the stronger the
correlation between the two sequences. In
the case of two independent sequences, the
value of CS1,S2 is equal to 0. Correlation
coefficients are computed for each pair of
sequences and the distribution of their
values is presented by a histogram.
The correlation between each pair of
the 1000 produced sequences is computed
using Pearson's correlation coefficient. The
results of the coefficient are represented in
the histograms shown in Figure 3. The
histogram shows that the computed
coefficients are very close to 0. This means
that around 99.9% of the coefficients
belong to [-0.08, 0.08] and the correlation
between the produced sequences is very
small.
5.2.2. Hamming distance
Given two binary sequences S =
[s0,……, sM-1 ] and S׳ = [s׳0,…… s׳M-1 ] of
same length (M), the Hamming distance is
the number of positions where they differ.
The distance is given as [16]:
In the case of truly random binary
sequences, such distance is typically around
M/2, which gives a proportion (i.e. d(S, S׳) /
M) of about 0.50. For each pair of produced
sequences, this proportion is determined
and all values are represented through a
histogram. The interest of both approaches
is to check the correlation for generated
sequences mainly from nearby or
successive seed values.
The bits of 1000 produced pseudo-
random sequences are analyzed using
Hamming distance. All resulted values are
represented through a histogram shown in
Figure 4. The distributions show that all the
proportions are around 50% and 99.3% of
the coefficients are belonging to [0.465,
0.535]. This testing provides another
IJCCCE Vol.15, No.3, 2015
Ekhlas Abass Albhrany et. al.
New Pseudo-Random Number Generator System
Based on Jacobian Elliptic maps and Standard map
84
indication about the decorrelation between
the generated sequences.
Test name
η of
Individual
Final Result
p-value of concatenate
Final Result
Frequency
0.994
Success
0.040903443
Success
Block Frequency
0.988
Success
0.553430376
Success
Runs
0.991
Success
0.880462635
Success
Longest Run
0.993
Success
0.231078877
Success
Rank
0.993
Success
0.823833254
Success
FFT
0.989
Success
0.514489087
Success
Non-Overlapping
0.984
Success
0.398411794
Success
Serial (1)
0.989
Success
0.684240449
Success
Serial (2)
0.993
Success
0.999648182
Success
Cumulative Sums
0.995
Success
0.063395343
Success
Test name
p-value of
Concatenate
Final
Result
Overlapping
0.957408045388764
Success
Universal
0.745350098931327
Success
Linear Complexity
0.402768848028256
Success
Approximate Entropy
0.110486457868039
Success
Random Excursions
(8 p-values)
0.807101754304452
0.485159643724902
0.94699932242908
0.719851271910543
0.150518027509838
0.286566773029429
0.539296753686001
0.850627304185789
Success
Success
Success
Success
Success
Success
Success
Success
Random E-Variant
(18 values)
0.349048077620093
0.490537427022143
0.351985540956662
0.253294182514715
0.268472090963728
0.1927496064346 1
0.467725747869967
0.843110263994956
0.759058537369923
0.080102266851582
0.128301667190791
0.759139341678901
0.832576263792955
0.704772436102199
0.866081601939654
Success
Success
Success
Success
Success
Success
Success
Success
Success
Success
Success
Success
Success
Success
Success
Table 1Results of first group of the NIST tests on the 1000 generated sequences for individual
and concatenate sequences. The ratio η of p-value concerns individual sequences while the p-
value concerns the concatenate sequences.
Table 2Results of second group of the NIST tests on the 1000 generated sequences for
concatenate sequences. The p-value concerns the concatenate sequences.
IJCCCE Vol.15, No.3, 2015
Ekhlas Abass Albhrany et. al.
New Pseudo-Random Number Generator System
Based on Jacobian Elliptic maps and Standard map
85
6. Security analyses
When a new PRNG is proposed, it
should always be accompanied by some
security analyses. All the critical points of
the cryptosystem and cryptographic
requirements should be taken into account
when the analysis is done [1]. The analyzed
points are: the size of the key space, the key
sensitivity, randomness quality of the
outputs and two basic attacks are evaluated:
brute-force attack, differential attack
.
6.1. Key space
Crucial part of each cryptosystem is the
key. Keeping in mind the end goal to make
brute-force attacks infeasible, PRNG ought
to have a large key space. The size of key
space that is smaller than 2128 is not secure
sufficiently. Here, the key space is
constructed form the parameters of
Jacobian elliptic maps cn and sn (initial
value x0 and control parameters k and α)
types which are floating point numbers,
where x0[0, 1], α[0, 4] and k[0, 1].If
the precision is 10−16 for each of x0, α and k,
the size of key space for initial conditions
and control parameters is 2160((1016)3).
In addition the parameters of the
standard map (permutation parameter k and
the random scan keys [rx,ry]) which are
integer number, where k, rx and ry
[0..255]. If each parameter has 256 possible
keys (28), the total number of keys is
(28)3=224. So the total space of keys is
2160+224.
6.2. Key sensitivity.
Sensitivity analysis is the investigation
of how the instability in the yield of a
model can be distributed to various sources
of instability in the model input [17]. An
essential factor for the pseudo-random
generation is the sensitivity on the key. In
other words, a small changing in the
starting seeds should cause a large change
in the pseudo-random sequences.
Figure 3 Histogram of Pearson's correlation
coefficient values on interval [-0.10, 0.10] for the
1000 sequences
0.726128320880061
0.592148885856078
0.843894535051766
Success
Success
Success
IJCCCE Vol.15, No.3, 2015
Ekhlas Abass Albhrany et. al.
New Pseudo-Random Number Generator System
Based on Jacobian Elliptic maps and Standard map
86
Figure 4 Histogram of Hamming distance on
interval [0:435; 0:565] for the 1000 sequences
This means that a small difference on seed
values, the output sequences should be
completely uncorrelated.
Actually in the test of correlation, the key
sensitivity was already tested due to the
successive seed values. To ensure the
sensitivity of the key, additional analyses
have been done using Pearson's correlation
coefficients and Hamming distance. Four
large pseudo-random sequences of size N
=500,000 bits S1, S2, S3, S4 produced from
slightly different initial seeds are
considered.
A sequence S1 is produced by using
the seed values X0=0.2344587645985498,
K0=0.5123678943210314,
α0=2.8231406754308769.
A sequence S2 is produced by using the
seed values X1=X0+8×10-16, K1=K0+8× 10-
16, α1=α0+8×10-16.
A sequence S3 is produced by using the
seed values X2=X1+8×10-16, K2=K1+8× 10-
16, α2=α1+8×10-16.
A sequence S4 is produced by using the
seed values X3=X2+8×10-16, K3=K2+8× 10-
16, α3=α2+8×10-16.
The analysis is done using the linear
correlation coefficient of Pearson and the
Hamming distance between the four
pseudo-random sequences produced with
slightly different seed. The results show
that, the sequences are highly correlated
from each other as shown in Table 3.
6.3 Attacks of proposed PRNG
Any new PRNG must be analyzed against
attacks to check if the generator cannot be
Tests
S1/S2
S1/S3
S1/S4
S2/S3
S2/S4
S3/S4
Pearson Corr.
Coef.
-0.00089
-0.00104
-0.00163
0.00166
0.001881
0.001594
Hamming
Distance
0.500012
0.50062
0.500818
0.499150
0.499050
0.499200
Table 3 Correlation Coefficients between four pseudorandom sequences produced
with slightly different seeds
IJCCCE Vol.15, No.3, 2015
Ekhlas Abass Albhrany et. al.
New Pseudo-Random Number Generator System
Based on Jacobian Elliptic maps and Standard map
87
broken. Here, the resistance of the
generator against two basic attacks as the
brute-force attack and differential attack is
analyzed.
a) Brute-force attack.
The size of the key space must be large
enough to prevent a brute-force attack [18].
This attack consists in checking
systematically all possible keys until the
correct key is found. In the worst case, all
the combinations are tested, that
necessitates trying all the key space.
When it is not possible to detect any
weakness in the algorithm, such an attack
might be utilized that would make the task
easier. To resist this kind of attack, the size
of the key space must be large. It is
generally accepted that a key space of size
larger than 2128 is computationally secure
against such attack. In the proposed PRNG,
the size of the key space
is around 2184 [1], which clearly allows
resisting the brute force-attack.
b) Differential Attacks
This attack is similar to the chosen-
plaintext attack; its principle is studying
how differences in an input can affect the
resultant difference at the output in an
attempt to derive the key [18].Trying to
make a slight change on the input pair,
attacker observes the change of the
produced sequences. Such technique of
cryptanalysis was introduced by Biham and
Shamir [19]. Given two inputs In1 and In2 to
the generator and the corresponding outputs
Out1 and Out2, there are two methods to
find the differences between the two
outputs.
Firstly, the difference can be computed
between the two pseudo-random sequences
relatively to the bits or blocks of bits by
subtraction method. This can be done by
∆in =│ In1-In2│ and ∆out = │Out1 –
Out2│, respectively.
Secondly, the difference between the two
output sequences can be computed by
∆in = In1 In2
and
∆out = Out1 Out2
Differential probability is then used to
measure the diffusion aspect on the initial
conditions. In proposed PRNG, we iterate
the jacobain maps 100 times before the
beginning of the generation. In addition, the
results of the analyses showed that even
with a slight difference on the seeds, the
produced outputs are almost uncorrelated
from each other. So, the proposed PRNG is
designed to avoid this kind of cryptanalysis.
7
8 7. Comparative Results of proposed
PRNG
The proposed CRNG is compared with
the PRNG proposed by et al [8]. The
deference between the proposed PRNG and
the M. François PRNG is made on numbers
of factors which include key space, number
of bits in each iteration, the result of NIST
test and the result of correlation
coefficients. Table 4 shows the result of
comparison.
The key space of proposed PRNG is
larger than that of M. François PRNG. It is
by and large acknowledged that a key space
IJCCCE Vol.15, No.3, 2015
Ekhlas Abass Albhrany et. al.
New Pseudo-Random Number Generator System
Based on Jacobian Elliptic maps and Standard map
88
of size bigger than 2128 is computationally
secure against brute-force attack [8].
The number of bits in each iteration of
proposed PRNG is randomly in the rang
[64..80] bits in each iteration. This means
that the proposed PRNG is faster than M.
François PRNG.
8. Conclusions
In this paper, a novel pseudo-random
number generator based on the Jacobian
elliptic chaotic map types sn,cn and
standard map. The initial condition x0 and
the control parameters k and α is the input
to the Jacobian elliptic chaotic map to
produce binary sequence. Chaotic standard
map is used to diffuse the binary sequence.
Such a generator has shown its ability to
produce a very large number of pseudo-
random sequences which can be useful in
several cryptographic applications because
it has many properties which are the
adaptive size of the key space, the
sensitivity to the initial inputs (keys), the
quality of pseudo-random sequences, the
security level against several attacks.
It can be used as pseudo random bit
generator or as pseudo random number
generator.
The quality of the output sequences
randomness is evaluated through NIST tests
which are 15 tests. Table 4 shows that the
output sequence of proposed PRNG is
random with large values for the tests than
that for M. François PRNG.
References
[1] G. Alvarez, S. Li, “Some Basic Cryptographic
Requirements for Chaos-Based Cryptosystems”,
International Journal of Bifurcation and Chaos,
World Scientific, Vol. 16, 2006, pp 2129-2151.
[2] F. Zheng, X. Tian, J. Song and X. Li, “Pseudo-
random sequence generator based on the
generalized Henonmap”, The Journal of China
Universities of Posts and Telecommunications,
vol. 15, no. 3, 2008, pp. 64–68.
[3] V. Patidar and K.K. Sud, “A pseudo random bit
generator based on chaotic logistic map and its
statistical testing”, Informatics, 33, 2009, pp.
441–452.
[4] N. Pareek, V. Patidar, and K. Sud, “A Random
Bit Generator Using Chaotic Maps”,
International Journal of Network Security,
vol.10, No.1, 2010, pp. 32-38.
[5] C. E. Shannon. “Communication theory of
secrecy systems”. Bell System Technical
Journal, July, 1948, p.623.
key
space
number
of bits at
each
iteration
Freq.
Block
Freq.
Runs
Longest
Run
Rank
FFT
Non-
Over-
lapping
Serial
(1)
Serial
(2)
Cumu-
lative
Sums
proposed
PRNG
2160+224
Randoml
y in the
range
[64..80]
bits.
0.994
0.988
0.991
0.993
0.993
0.989
0.984
0.989
0.993
0.957
M.
François
PRNG
2147
32 bits.
0.991
0.991
0.989
0.989
0.988
0.987
0.993
0.989
0.990
0.991
Table 4 Comparison between the proposed PRNG and M. Francois PRNG
IJCCCE Vol.15, No.3, 2015
Ekhlas Abass Albhrany et. al.
New Pseudo-Random Number Generator System
Based on Jacobian Elliptic maps and Standard map
89
[6] M. Francois, T. Grosges, D. Barchiesi and R.
Erra, “A New Pseudo-Random Number
Generator Based on Two Chaotic Maps”,
INFORMATICA, vol. 24, no. 2, 2013, pp. 181–
197 Vilnius University.
[7] Ilyas, A. Vlad and A. Luca, “Statistical Analysis
of Pseudo Random Binary SequencesGenerated
By Using TentMap”, U.P.B. Sci. Bull., Series A,
vol. 75, no. 3, 2013.
[8] M. François, D. Defour and C. Negre, “A Fast
Chaos-Based Pseudo-RandomBit Generator
Using Binary64 Floating-Point Arithmetic”,
Informatica vol. 38, 2014, pp. 115–124.
[9] M. A. Jafarizadeh and S.Behnia, “Hierarchy of
one- and many-parameter families of elliptic
chaotic maps of cn and sn types”, Physics
Letters, vol. 310, April 2003, pp. 168–176.
[10] E. A. Jackson, Perspectives in nonlinear
dynamics, Cambridge University Press, vol. 1,
Reprint Edition, 1991.
[11] F. Rannou, “Numerical study of discrete plane
area-preserving map”, Astron & Astrophys: vol.
31, 1974, pp. 289–301.
[12] J. Fridrich, “Symmetric ciphers based on two-
dimensional chaotic maps”, International Journal
Bifurcation Chaos, vol. 8, no. 6, June 1998,
pp.1259-1284.
[13] J. Soto and L. Bassham, “Randomness Testing of
the Advanced Encryption Standard Finalist
Candidates”, Computer Security Division
National Institute of Standards and Technology,
March 2000.
[14] Rukhin, J. Soto, J. Nechvatal, M. Smid, E.
Barker, S. Leigh, M. Levenson, M. Vangel, D.
Banks, A. Heckert, J. Dray and S. Vo, “A
Statistical Test Suite for Random and
Pseudorandom Number Generators for
Cryptographic Applications”, NIST Special
Publication Revision 1a,vol. 800, April 2010, pp.
131.
[15] V. Patidar, N. K. Pareek, G. Purohit and K. K.
Sud, “A robust and secure chaotic standard map
based pseudorandom permutation-substitution
scheme for image encryption”, Optics
Communications, vol. 284, no. 19, September
2011, pp. 4331- 4339.
[16] W. Janke, “Pseudo random numbers: generation
and quality checks”, Quantum Simulations of
Complex Many-Body Systems, vol. 10, 2002,
pp. 447–458.
[17] J.C. Ascough, , T.R. Green, L. Ma and L.R.
Ahjua, “Key Criteria and Selection of Sensitivity
Analysis Methods Applied to Natural Resource
Models”, Research Get, pp. 2463 - 2469.
[18] B. Schneier. Applied Cryptography: Protocols,
Algorithms, and Source Code in C, 2nd Edition,
John Wiley & Sons; 1996.
[19] E. Biham and A. Shamir, “Differential
cryptanalysis of DES-like cryptosystems,”
Journal of Cryptology, vol. 4, no. 1,January
1991, pp. 3–72.
