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Ziad Alqadi, International Journal of Computer Science and Mobile Computing, Vol.13 Issue.4, April- 2024, pg. 25-40
© 2024, IJCSMC All Rights Reserved, ZAIN Publications, Fridhemsgatan 62, 112 46, Stockholm, Sweden 25
Available Online at www.ijcsmc.com
International Journal of Computer Science and Mobile Computing
A Monthly Journal of Computer Science and Information Technology
ISSN 2320–088X
IMPACT FACTOR: 7.056
IJCSMC, Vol. 13, Issue. 4, April 2024, pg.25 – 40
Analysis of Chaotic Logistic Map
used to Generate Secret Keys
Prof. Ziad Alqadi
Albalqa Applied University, Faculty of Engineering Technology, Jordan-Amman
DOI: https://doi.org/10.47760/ijcsmc.2024.v13i04.004
Abstract:
Chaotic logistic map is a good model which can be used to generate various secret keys used in data cryptography.
In this paper research a detailed analysis of CLM will be introduced. The behavior of CLM will be studied to show
how to select the best values of the CLM parameters (growth rate, initial population and the length of the
population). Various ranges of the population growth and the initial population will be tested. It will be shown how
to use the generated populations to form various secret keys. The sensitivity of CLM will be studied, and it will be
shown that any minor changes in the CLM parameters will lead to change the generated population. The population
generation time will be tested using various in length generated population, and some recommendation will be
provided when dealing with data with big size. It will be shown that using CLM will provide a good security level
when using the CLM parameters as a private key.
Keywords: CLM, population, growth rate, initial population, length, private key, secret key, indices key.
Introduction
Chaotic logistic map (CLM) is a nonlinear difference equation (see equation 1), which maps the maps the population
value at any time step to its value at the next time step [77-84].
CLM is a system, nonlinear and dynamic; a system is just a set of interacting components that form a larger
whole. Nonlinear means that due to feedback or multiplicative effects between the components, the whole becomes
Ziad Alqadi, International Journal of Computer Science and Mobile Computing, Vol.13 Issue.4, April- 2024, pg. 25-40
© 2024, IJCSMC All Rights Reserved, ZAIN Publications, Fridhemsgatan 62, 112 46, Stockholm, Sweden 26
something greater than just adding up the individual parts. Lastly, dynamical means the system changes over time
based on its current state [1-10].
The CLM equation is so simple, but it produces chaos at certain growth rate parameters values, table 1 shows the
populations values for various values of r using x=0.2 as initial population value.
Table 1: Population values using various growth rates (X1=0.2)
Iteration
r=0.5
r=1
r=1.5
r=2
r=2.5
r=3
r=3.5
1
0.2000
0.2000
0.2000
0.2000
0.2000
0.2000
0.2000
2
0.0800
0.1600
0.2400
0.3200
0.4000
0.4800
0.5600
3
0.0368
0.1344
0.2736
0.4352
0.6000
0.7488
0.8624
4
0.0177
0.1163
0.2981
0.4916
0.6000
0.5643
0.4153
5
0.0087
0.1028
0.3139
0.4999
0.6000
0.7376
0.8499
6
0.0043
0.0922
0.3230
0.5000
0.6000
0.5806
0.4465
7
0.0021
0.0837
0.3280
0.5000
0.6000
0.7305
0.8650
8
0.0011
0.0767
0.3306
0.5000
0.6000
0.5906
0.4088
9
0.0005
0.0708
0.3320
0.5000
0.6000
0.7254
0.8459
10
0.0003
0.0658
0.3327
0.5000
0.6000
0.5976
0.4563
11
0.0001
0.0615
0.3330
0.5000
0.6000
0.7214
0.8683
12
0.0001
0.0577
0.3332
0.5000
0.6000
0.6029
0.4002
13
0.0000
0.0544
0.3332
0.5000
0.6000
0.7182
0.8401
14
0.0000
0.0514
0.3333
0.5000
0.6000
0.6072
0.4700
15
0.0000
0.0488
0.3333
0.5000
0.6000
0.7156
0.8719
16
0.0000
0.0464
0.3333
0.5000
0.6000
0.6106
0.3910
17
0.0000
0.0442
0.3333
0.5000
0.6000
0.7133
0.8334
18
0.0000
0.0423
0.3333
0.5000
0.6000
0.6135
0.4859
19
0.0000
0.0405
0.3333
0.5000
0.6000
0.7113
0.8743
20
0.0000
0.0389
0.3333
0.5000
0.6000
0.6160
0.3846
The calculated populations of the CLM can be used to generate various types of secret keys [11-20], which can be
used in data security applications such as message, image and speech cryptography [21-30].
The CLM can be useful to generate the following kinds of secret key [31-40]:
- Generation 1D key of fractional values:
Here the calculated population values will be used as a key [41-50], the length of the key can be controlled by the
user. The following example shows how to generate a 10 elements key with fractional values:
Ziad Alqadi, International Journal of Computer Science and Mobile Computing, Vol.13 Issue.4, April- 2024, pg. 25-40
© 2024, IJCSMC All Rights Reserved, ZAIN Publications, Fridhemsgatan 62, 112 46, Stockholm, Sweden 27
- Generation 2D key of fractional values:
Here the calculated population values will be used as a key [51-60], the length (number of rows and number of
columns) of the key can be controlled by the user. The following example shows how to generate a 10 elements key
with fractional values:
- Generation 1D key of integer values:
Here the calculated population values will be used as a key [61-70], the length of the key can be controlled by the
user. The following example shows how to generate a 10 elements key with fractional values:
- Generation 2D key of integer values:
Here the calculated population values will be used as a key, the length (number of rows and number of columns) of
the key can be controlled by the user. The following example shows how to generate a 10 elements key with
fractional values:
Ziad Alqadi, International Journal of Computer Science and Mobile Computing, Vol.13 Issue.4, April- 2024, pg. 25-40
© 2024, IJCSMC All Rights Reserved, ZAIN Publications, Fridhemsgatan 62, 112 46, Stockholm, Sweden 28
- Generation 1D indices key:
Here the calculated population values will be used as a data set to generate the secret indices key [71-80], the
generated population will be sorted to get the indices key, this key will contain a set of unrepeated integer values,
the first value will point to the minimum element in the data set, the second value will point to the second minimum
and so on, the length of the key can be controlled by the user. The following example shows how to generate a 10
elements key with fractional values:
- Generation 2D indices key:
The generated population can be used to generate a 2D indices key by selecting the values of the rows and columns,
the generated population can by sorted row or columns wise to form a 2D matrix of indices keys, the following
example shows how to generate a set of indices keys, each row of the matrix forms an indices key:
Ziad Alqadi, International Journal of Computer Science and Mobile Computing, Vol.13 Issue.4, April- 2024, pg. 25-40
© 2024, IJCSMC All Rights Reserved, ZAIN Publications, Fridhemsgatan 62, 112 46, Stockholm, Sweden 29
Analysis of CLM populations
To maximize the benefits of using CLM to generate secret keys, the following must be taken in considerations:
a) CLM population’s behaviors.
b) Key generation time.
c) Required memory space.
d) Key sensitivity.
a) CLM populations behaviors
CLM was implemented varying the value of the population growth r, figure 1 shows the obtained populations:
Here we can see how the population changes over the time, when using different values of r. The line representing a
growth rate of 1 is quickly drops to zero. The population dies out. The line representing a growth rate of 2.0
(remember, the replacement rate) and it stays steady at a population level of 0.5 after 4 iterations. The growth rates
of 2.5 keeps the population value stable) 0.6) after 4 iterations. The growth rates of 2.9 keeps the population stables
with repeated values 0.7 and 06. Based on the previous behaviors of the populations, the values of r must be greater
than 3. Changing the initial value of the population will not enhance the generated values of the population (see
figure 2).
Figure 1: Generated populations when varying r (r<3) with initial population=0.1
Ziad Alqadi, International Journal of Computer Science and Mobile Computing, Vol.13 Issue.4, April- 2024, pg. 25-40
© 2024, IJCSMC All Rights Reserved, ZAIN Publications, Fridhemsgatan 62, 112 46, Stockholm, Sweden 30
Figure 2: Generated populations when varying r (r<3) with initial population=0.9
An attractor is the value, or set of values, that the system settles toward over time. When the growth rate parameter
is set to 1 or less, the system has a fixed-point attractor at population level 0. In other words, the population value is
drawn toward 0 over time as the model iterates. When the growth rate parameter is set to 3.5 9 (see figure 3), the
system oscillates between four values. This attractor is called a limit cycle.
But when we adjust the growth rate parameter beyond 3.5, we see the onset of chaos. A chaotic system has a strange
attractor, around which the system oscillates forever, never repeating itself or settling into a steady state of
behavior. It never hits the same point twice and its structure has a fractal form, meaning the same patterns exist at
every scale no matter how much you zoom into it.
0 2 4 6 8 10 12 14 16 18 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Elem ent
Value
X=0.1
r=3
r=3.1
r=3.4
r=3.7
r=3.9
Figure 3: Generated populations when varying r (r>=3) with initial population=0.1
Ziad Alqadi, International Journal of Computer Science and Mobile Computing, Vol.13 Issue.4, April- 2024, pg. 25-40
© 2024, IJCSMC All Rights Reserved, ZAIN Publications, Fridhemsgatan 62, 112 46, Stockholm, Sweden 31
Growth rate with value above 3.0 the possible population values fork into two discrete paths (see figure 4). At
growth rate 3.2, the system essentially oscillates exclusively between two population values: one around 0.5 and the
other around 0.8. In other words, at that growth rate, applying the logistic equation to one of these values yields the
other.
Just after growth rate 3.4, the diagram bifurcates again into four paths. When the growth rate parameter is set to 3.5,
the system oscillates over four population values. Just after growth rate 3.5, it bifurcates again into eight paths. Here,
the system oscillates over eight population values.
Beyond a growth rate of 3.6, however, the bifurcations ramp up until the system is capable of eventually landing
on any population value. This is known as the period-doubling path to chaos. As you adjust the growth rate
parameter upwards, the logistic map will oscillate between two then four then eight then 16 then 32 (and on and on)
population values. These are periods, just like the period of a pendulum.
By the time we reach growth rate 3.9, it has bifurcated so many times that the system now jumps, seemingly
randomly, between all population values. We only say seemingly randomly because it is definitely not random.
Rather, this model follows very simple deterministic rules yet produces apparent randomness. This is chaos:
deterministic and periodic.
2.8 3 3.2 3.4 3.6 3.8 4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Growth Rate r
The Attractor
Figure 4: Bifurcation diagram of the population
Taking the CLM behaviors in consideration the best values to be selected to generate a population to be used to
generate a secret key must be between 3.5 and 4.
The initial value of the population x must be within the range 0 to 1 (0<x<1), for these values the populations will
oscillate between two values as shown in figure 5
Ziad Alqadi, International Journal of Computer Science and Mobile Computing, Vol.13 Issue.4, April- 2024, pg. 25-40
© 2024, IJCSMC All Rights Reserved, ZAIN Publications, Fridhemsgatan 62, 112 46, Stockholm, Sweden 32
010 20 30
0.2
0.4
0.6
0.8
1x=0.3
010 20 30
0.2
0.4
0.6
0.8
1x=0.5
010 20 30
0.2
0.4
0.6
0.8
1x=0.7
Papulation
Value
010 20 30
0.2
0.4
0.6
0.8
1x=0.9
Figure 5: Required values of the initial population
For initial values greater than 2 or less than 0 the population value will have a minus infinite value after 8 to 10
iterations as shown in figure 6:
Figure 6: Infinite populations
It can be hard to tell if certain time series are chaotic or just random when you don’t fully understand their
underlying dynamics. Take these two as an example (see figure 7):
Both of the lines seem to jump around randomly. The blue line does depict random data, but the red line comes from
our logistic model when the growth rate is set to 3.99. This is deterministic chaos, but it’s hard to differentiate it
from randomness.
Ziad Alqadi, International Journal of Computer Science and Mobile Computing, Vol.13 Issue.4, April- 2024, pg. 25-40
© 2024, IJCSMC All Rights Reserved, ZAIN Publications, Fridhemsgatan 62, 112 46, Stockholm, Sweden 33
For key generation it is better to use CLM and use the generated population to form the secret key because of the
following reasons:
- The generated population using fixed values of r and x remain stable; these values will generate the same
population.
- Using CLM we have to save the values of r, x and the population length, thus we need 24 bytes to store these values
to be used as a private key.
- Random values are changing from time to time, so here we have to store all random numbers, and the private key
here will need 8 multiplied by the key length byte.
0 5 10 15 20 25 30 35 40 45 50
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Generation
Papulation
r=3.99
Chaos
Random
Figure 7: Random VS chaos values
b) Key generation time
Despite the simplicity of using CLM to generate a secret key, the key generation requires a key generation time
(KGT),which must be added to the encryption-decryption time, thus the key generation process will affect the
efficiency of the crypto system, increasing the KGT will negatively affect the speed of data cryptography.
The KGT will mainly depend on the length of the key.
To show the required time needed to generate a population, the following CLM was execute varying the population
length, and table 2 shows the obtained results:
Ziad Alqadi, International Journal of Computer Science and Mobile Computing, Vol.13 Issue.4, April- 2024, pg. 25-40
© 2024, IJCSMC All Rights Reserved, ZAIN Publications, Fridhemsgatan 62, 112 46, Stockholm, Sweden 34
Table 2: KGT results
Length
KGT(second)
Length
KGT(second)
100
0.000001
15000
0.167000
200
0.000001
20000
0.250000
400
0.003000
25000
0.379000
500
0.004000
30000
0.528000
1000
0.017000
40000
0.976000
2000
0.052000
50000
1.549000
2500
0.054000
100000
10.599000
4000
0.060000
200000
57.864000
5000
0.064000
400000
331.892000
10000
0.101000
1000000
2311.878000
As we can see from table 2 the KGT will rapidly increase when increasing the population size (see figure 8), thus it
is recommended to use a population size with size less than 5000.
Digital data (with size greater than 5000) to be encrypted-decrypted must be divided into blocks, block size must be
less than 5000 elements, this will save the generation time and will increase the speed of the crypto system.
Figure 8: KGT vs population size
c) Chaos sensitivity
The generated populations of the CLM are very sensitive to the selected values of the growth rate r, the initial
population x and the length of the population, the generated population will change applying the following:
- Any minor changes in x.
- Any minor changes in r.
Ziad Alqadi, International Journal of Computer Science and Mobile Computing, Vol.13 Issue.4, April- 2024, pg. 25-40
© 2024, IJCSMC All Rights Reserved, ZAIN Publications, Fridhemsgatan 62, 112 46, Stockholm, Sweden 35
- Any minor changes in both r and x.
- Any minor changed in the length of population.
A CLM with r=3.99 and varying x was used to generate a population with 12 elements, tables 3 and 4 show the
obtained populations:
Table 3: Obtained populations when varying x by small changes
Population index
X=0.1
X=0.11
X=0.111
1
0.3940
0.3906
0.3937
2
0.9527
0.9498
0.9524
3
0.1798
0.1904
0.1807
4
0.5884
0.6150
0.5908
5
0.9663
0.9448
0.9646
6
0.1298
0.2083
0.1363
7
0.4508
0.6579
0.4697
8
0.9879
0.8980
0.9938
9
0.0479
0.3654
0.0245
10
0.1819
0.9252
0.0953
11
0.5938
0.2763
0.3440
12
0.9624
0.7978
0.9003
Table 4: Obtained populations using bigger x
Population index
X=0.2
X=0.3
X=0.4
1
0.8379
0.9576
0.9975
2
0.5419
0.1620
0.0100
3
0.9905
0.5417
0.0393
4
0.0376
0.9906
0.1507
5
0.1444
0.0373
0.5106
6
0.4930
0.1432
0.9971
7
0.9973
0.4894
0.0117
8
0.0107
0.9971
0.0462
9
0.0423
0.0117
0.1759
10
0.1617
0.0462
0.5784
11
0.5408
0.1758
0.9730
12
0.9909
0.5781
0.1050
It is shown from tables 3 and 4 that any minor changes in the initial population x will lead to change the population,
thus the generated secret keys obtained from the population will also change. The values of the population length, x
and are unique and the can be used as a private key. The same things we can see when changing the population
growth r, changing r will also change the generated population as shown in figure 9.
Ziad Alqadi, International Journal of Computer Science and Mobile Computing, Vol.13 Issue.4, April- 2024, pg. 25-40
© 2024, IJCSMC All Rights Reserved, ZAIN Publications, Fridhemsgatan 62, 112 46, Stockholm, Sweden 36
0 5 10 15
0.2
0.4
0.6
0.8
1r=3.7
Papulation
Value
0 5 10 15
0.2
0.4
0.6
0.8
1r=3.71
Papulation
Value
0 5 10 15
0.2
0.4
0.6
0.8
1r=3.69
Papulation
Value
0 5 10 15
0
0.5
1r=3.8
Papulation
Value
Figure 9: Changing r will change the generated population
d) Security issues
Using the CLM parameters as a private key will provide a good key space, if one CLM is used to generate the
populations required to form the secret key, then the private key will contain the values of three parameters, each
parameter will have a double type data value, thus the private key length will equal 192 bits, the key space will
calculated using equation 2:
The entropy of the key space equal 192, this entropy (>128) [83] will make the key strong enough to resist hacking
attempts.
Conclusion
The CLM was analyzed, it was shown that the process of population generation is so simple, these population can be
easily used to form the required for data cryptography process. It was shown that the best value of the growth rate
must be between 3 and 4, while the best of the initial population must be between 0 and 1, using these values we can
generate a population with various contents and lengths. The CLM parameters can be used as a PK, and the
generated secret keys will be very sensitive to the selected values of the private key.
Ziad Alqadi, International Journal of Computer Science and Mobile Computing, Vol.13 Issue.4, April- 2024, pg. 25-40
© 2024, IJCSMC All Rights Reserved, ZAIN Publications, Fridhemsgatan 62, 112 46, Stockholm, Sweden 37
The main disadvantage of CLM is the required population generation time, increasing the population length will
rapidly increase the generation time, in this case we can use a population with smaller length, this population can be
used to process a data block, here the data to be processed must be divided into block to save the population
generation time and to increase the speed of data cryptography.
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