Domain Examination of Chaos Logistics Function As A Key Generator in Cryptography

International Journal of Electrical and Computer Engineering (IJECE)
Vol. 8, No. 6, December 2018, pp. 4577 – 4583
ISSN: 2088-8708, DOI: 10.11591/ijece.v8i6.pp4577-4583 4577
Institute of Advanced Engineering and Science
w w w . i a e s j o u r n a l . c o m
Domain Examination of Chaos Logistics Function As A Key
Generator in Cryptography
Alz Danny Wowor1
and Vania Beatrice Liwandouw2
1
Department of Informatics Engineering, Satya Wacana Christian University, Salatiga, Indonesia
2
Department of Computing Science, Radboud University, Nijmegen, The Netherland
Article Info
Article history:
Received September 28, 2017
Revised May 09, 2018
Accepted May 22, 2018
Keyword:
Domain Examination
Logistic Function
Chaos
Cryptography
ABSTRACT
The use of logistics functions as a random number generator in a cryptography algo-
rithm is capable of accommodating the diffusion properties of the Shannon principle.
The problem that occurs is initialization x0 was static and was not affected by changes
in the key, so that the algorithm will generate a random number that is always the same.
This study design three schemes that can providing the flexibility of the input keys in
conducting the examination of the value of the domain logistics function. The results
of each schemes do not show a pattern that is directly proportional or inverse with the
value of x0 and relative error x0 and successfully fulfill the properties of the butterfly
effect. Thus, the existence of logistics functions in generating chaos numbers can be
accommodated based on key inputs. In addition, the resulting random numbers are dis-
tributed evenly over the chaos range, thus reinforcing the algorithm when used as a key
in cryptography.
Copyright c 2018 Institute of Advanced Engineering and Science.
All rights reserved.
Corresponding Author:
Alz Danny Wowor
Affiliation
Faculty of Information Technology, Satya Wacana Christian University,
Jl. Diponegoro 52-60, Salatiga 50711, Indonesia
+62852 0071 0079
alzdanny.wowor@staff.uksw.edu
1. INTRODUCTION
The logistic function f(x) = rx(1 − x) or in the iterative form xi+1 = rxi(1 − xi) was usually used
as a generator in the generation of chaos-based random numbers [1]. This function is able to accommodate the
diffusion properties of Shannon’s principle on cryptography algorithms, since they have a sensitivity to initial
values.
Research [2], [3], [4], uses numbers as static initialization on logistics functions in cryptography algo-
rithms. This means that changes to the key don’t have an effect on random numbers, thus the algorithm will still
acquire a random number that is always the same. Research [5], also uses numbers as input values that directly
fill in by user, nevertheless becomes inefficient since the user needs more input other than the key and plaintext.
This research not only combines algorithms such as [6], [7], [8], but designs a new algorithm to obtain an unique
key. Additionally, the function will generate a random number, if the initialization domain r and x0 as seed is
limited to a certain value, 0 < x0 < 1, and r = 4 .
The value of r is constant, making the strength of the algorithm rests on the value of x0. Moreover
it needs an examination process by using any scheme that can increase x0 complexity space, on the contrary
remain in logistics function domain. This study provides the flexibility of key inputs that are efficient and can
generate different random numbers of logistics functions.
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Institute of Advanced Engineering and Science
w w w . i a e s j o u r n a l . c o m

4578 ISSN: 2088-8708
2. PROPOSED EXAMINATION SCHEME
The examination process of the logistics function domain conducted to a wide variety of inputs that
allow it to be used as a key. Suppose the key k1, k2, . . . , k8 is the result of a conversion of eight ASCII characters.
Key input is set up to eight characters, considering the user’s ability to recall key and also consider the complexity
of the key guessing space when the use of the characters too little. Each key input of less than eight characters,
then do the padding process with character ”§” that equivalent to 167 in ASCII.
This study provides three algorithms that are used for the examination process of initialization value in
domain 0 < x0 < 1, as shown in the general scheme in Figure 1. The ratio of ra is designed as a comparison of
two values to accommodate the domain from initialization. Let ra = pa/qa where qa > pa, for a = 1, 2, 3 and
pa, qa ∈ R.
x0
ASCII encode,
ki, i = 1,2,…, 8.
examination
scheme

ra = pa/qa

a = 1,2,3.
max
key = 8
padding character “§”
Y
N
Figure 1. General Scheme of x0 Initialization Value Determination.
2.1. The First Scheme
Given each ki ∈ Z256 for i = 1, 2, . . . , 8 is the decimal base number of ASCII conversion results. To
be able to make changes at every turn of the input, given index value di ∈ Z256 for j = 1, 2, . . . , 8 as a constant
value which is multiplied by the value ki. Scheme-1 is determined by using r1 = p1/q1 where,
p1 = (k1)2
+ k7d7 + k3d3 + k4d4 + k5d1 + k6d6 + k8d8 + k1d5 (1)
q1 =
8
i=1
ki/8 · (k7 + d2)k1d3 + k4d5 + k6d8 + k3d1 + k5d6 + k8d7(k2 + d4) (2)
Determination of p1 value is obtained from the sum of multiplication ki and di which is performed
based on position, it is only for k1 and d5 there is crossing position and at k1 squared. Multiplication of position
difference is also done to acquire the q1 value, however in sum with the average value of each ki. The multi-
plication combination is done as a variation to gain a unique ratio value, taking into account the requirement
q1 > p1.
2.2. Second scheme
Scheme-2 also uses the same decimal number and index value, where each ki, di ∈ Z256 for i, j =
1, 2, . . . , 8. The ratio value is determined by the equation r2 = p2/q2,
p2 = (k1 + k2 + k3 + k4 + k5 + k6)/6 =
6
i=1
ki/6 (3)
q2 = k1d1 + k2d2 + k3d3 + k4d4 + k5d5 + k6d6 + k7d7 + k8d8 =
8
i=1
kidi (4)
The p2 value is obtained using the average of the ﬁrst six values of ki, whereas to earn the q2 value is the sum of
the multiplication of ki and di values according to the order of each value.
2.3. Third scheme
Every ki, dj ∈ Z256 for i, j = 1, 2, . . . , 8. Scheme-3 is obtained based on r3 = p3/q3, where the
determination of the numerator and denominator is given in Equation (5) and Equation (6). The p3 value is the
IJECE Vol. 8, No. 6, December 2018: 4577 – 4583

IJECE ISSN: 2088-8708 4579
average of the multiplication ki with di only at the first value, the third value, and the sixth value with each
index.
p3 = (k1d1 + k2 + k3d3 + k4 + k5 + k6d6)/6 (5)
q3 = 3(k1d1) + 7(k2d2) + 11(k3d3) + 13(k4d4) + 17(k5d5) + 23(k6d6) + 6(k7d7) + 27(k8d8) (6)
The value of q3 is the sum of the ki and di multiplications based on the index with the constants selected by
different increments.
3. RESULT AND DISCUSSION
3.1. Analysis of Examination Process
Domain examination of the logistics function is performed based on the three schemes given in the
previous section. Referring to the general scheme in Figure 1, then testing related variations of inputs that allow
to be used as a key. Each possible key is an ASCII character whose decimal basis is in the range 0 to 255.
The problem that occurs is not all numbers have a character. Consequently the test for the lowest number can’t
start from decimal 0 instead in decimal 32 which is proportional to the space character, whereas testing for the
largest number at decimal 255 is equivalent to the character . In addition to character testing for minimum and
maximum decimals, key tests are also tested with one bit difference, so that it can be seen how sensitive each
scheme is to generating initialization values. Table 1 is the simulation result of each scheme in obtaining the
value of x0.
Table 1. Key Variations Test on Each Scheme.
Test Input Keys Scheme-1 Scheme-2 Scheme-3
1 ZZZZZZ 0.002416239247 0.020477815700 0.082448573324
2 ZZZZZY 0.002417771534 0.020467836257 0.082481509183
3 fti 0.002441029054 0.024268127467 0.082049869712
4 ftj 0.002439355077 0.024284705051 0.082201209246
5 $4LaT1g4 0.006268369980 0.025188594809 0.095909955612
6 $4LaT1g3 0.006272557467 0.025266127990 0.096168154762
7 (space 8 character) 0.018665158371 0.027777777778 0.096054888500
8 ∼∼∼∼∼∼∼∼ 0.003244514742 0.027777777778 0.096054888508
9 ÿÿÿÿÿÿÿÿ 0.001456374283 0.027777777778 0.096054888508
Testing with one bit difference (numbers 1 through 6) shows the changes in values that begin to occur
in the 4th or 5th mantissa of the x0 value in each schema. Accordingly, the process of domain examination for
each scheme succeeds to generate different initialization. This condition corresponds to the need of logistics
functions in obtaining random numbers based on chaos. Tests with the same eight-character input based on the
smallest decimal, the medium decimal, and the largest decimal are given successively in the numbers 7, 8, and 9
in Table 1. The initialization values in scheme-2 and scheme-3 obtain the same value, although the input is very
different. This condition occurs since the determination ratio of r2 and r3 using the average process to get the
numerator, while for the denominator using the addition and multiplication of characters with the index value
with the same position. On the other hand, scheme-1 uses a combination of multiplicity of position difference,
squared and mean process. In that case, scheme-1 keeps generating different initialization values.
3.2. Relative Error Test
Relative error testing [9] was conducted to see whether linear character reduction would yield also
linear results on the xr values with proportional or inversely proportional.
Use ER = |ca − pa| /ca · 100%, the key ÿÿÿÿÿÿÿÿ is chosen as the reference value ca (number 1 in
Table 2), and the approximation value as the key input pa less than eight characters ÿ (iterated from number 2 to
number 8).
Domain Examination of Chaos Logistics Function As A Key Generator in Cryptography (Alz Danny Wowor)

4580 ISSN: 2088-8708
Table 2. Relative error test on key variation differences.
Test Input keys Scheme-1 Scheme-2 Scheme-3
Relative Error
Scheme-1 Scheme 2 Scheme-3
1 ÿÿÿÿÿÿÿÿ 0.00145637428 0.027777777778 0.09605488850
2 ÿÿÿÿÿÿÿ 0.00151184278 0.030084945729 0.10267089320 3.81 8.31 6.89
3 ÿÿÿÿÿÿ 0.00237830973 0.032442748091 0.10498487627 63.30 16.79 9.30
4 ÿÿÿÿÿ 0.00247676219 0.032778687034 0.10922797121 70.06 18.00 13.71
5 ÿÿÿÿ 0.00260130457 0.032743277230 0.11525621373 78.62 17.88 19.99
6 ÿÿÿ 0.00272929162 0.032262996942 0.11525043975 87.40 16.15 19.98
7 ÿÿ 0.00287551307 0.031283195241 0.10350303974 97.44 12.62 7.75
8 ÿ 0.00146481070 0.029781420765 0.09849935979 0.58 7.21 2.54
The relative error results in each scheme do not show a proportional or inversely proportional pattern.
So, it will complicate cryptanalyst to see the pattern of input changes on the same character.
3.3. Linear Regression Test
One-to-one correspondence between the input-output is also important so that cryptanalyst difficult to
reconstructing scheme and make prediction of the key used as input. This relationship can be seen through a
linear regression test based on the rate of change on the resulting value of x0.
Figure 2, is the result of each scheme visualized using the Scatter plot. The diagram of each scheme has
no linear relationship, because when the curve matching process is used, the coefficient of determination (R2
)
is close to zero. This test illustrates that any changes to the key characters is done patterned, will not provide an
initial value x0 linearly patterned, either proportional or inversely.
scheme 1
0.000
0.001
0.002
0.003
0.004
0 1 2 3 4 5 6 7 8
y = 9.589E-5x + 0.0018
R² = 0.1502
scheme 2
0.026
0.028
0.030
0.032
0.034
0 1 2 3 4 5 6 7 8
y = 0.0002x + 0.0301
R² = 0.1
scheme 3
0.080
0.091
0.103
0.114
0.125
0 1 2 3 4 5 6 7 8
y = 0.0007x + 0.1026
R² = 0.0567
Figure 2. Diagram Scatter Initialization Value x0 based on Key Pattern Changes.
3.4. Butterfly Effect Test
The butterfly effect test is used to indicate the change in bits in the key input, whether it gives a large
change to the output. Suppose that as a comparator key ZZZZZZ and ZZZZZY are selected which has a dif-
ference of one bit. The result of the two keys with scheme-1 is obtained by a random number of the first 500
iterations, shown in Figure 3.
ZZZZZZ
0
0.25
0.5
0.75
1
i
0 125 250 375 500
ZZZZZY
0
0.25
0.5
0.75
1
i
0 125 250 375 500
xi xi
Figure 3. Results of Random Numbers with Different Inputs 1 Bit for Scheme-1.
Visually, the random number generated is very different although the difference in initialization value
x0 is only 0.0000151538 ≈ 1.538 × 10−6
or in an absolute relative of 0.063%. A minor change in input and a
major change in output proves that the 1st scheme has successfully fulfilled the butterfly effect.

IJECE ISSN: 2088-8708 4581
The initial value difference of x0 is 0.0487% in scheme-2. But the rate of change occurs very signif-
icantly that appears in the Scatter diagram in Figure 4. So, the scheme-2 has also fulfilled the butterfly effect
test.
ZZZZZZ
0
0.25
0.5
0.75
1
i
0 125 250 375 500
ZZZZZY
0
0.25
0.5
0.75
1
i
0 125 250 375 500
xixi
Significant changes also occur at random values with scheme-3, as shown in Figure 5, although the
difference is 0.0399% at input value x0. So, the scheme-3 also meets the properties of the butterfly effect.
ZZZZZZ
0
0.25
0.5
0.75
1
i
0 125 250 375 500
ZZZZZY
0
0.25
0.5
0.75
1
i
0 125 250 375 500
xixi
The three schemes have succeeded in fulfilling the properties of the butterfly effect, thus the existence
of the logistics function in generating chaos numbers can be accommodated based on key inputs. Each scheme
can be used as a complement to a cryptography algorithm to meet the diffusion properties of Shannon’s principle.
3.5. Analysis Of The Algorithm Ability
Each scheme is tested in correlation [10], to detect the connectedness of the random number generated
based on the input. While MAPE [11], it is used to find out how massive the difference of random value
of key change. Used three variations of the input [12], the first is the same character input, a second input
alphabetic characters that revolve around the 26-character alphabet. While the last test, used alphabet, symbols,
and numbers.
Calculation of correlation in Table 3 show that there are two values on scheme-1 and scheme-3 which
correlation is negative, besides the rest is positive. Cryptographically, the negative value is not too influential,
hence it seen how close the value to zero indicating the unrelated two random numbers are generated. In the
context of the relations, this same analogy can be used to test the difference of two random numbers generated.
Table 3. 1 Bit differences test with key variations.
Test Input keys
Correlation value MAPE
Scheme-1 Scheme 2 Scheme-3 Scheme-1 Scheme-2 Scheme-3
1 ZZZZZZ - ZZZZZY 0.1739823 0.04831424 −0.0685288 23.9966 3486.022 13.192
2 fti - ftj 0.0220125 0.05469541 0.00900384 612.172 210.2993 46.6942
3 $4LaT1g4 - $4LaT1g3 −0.011209 0.02911591 0.00971799 64.9543 38.1335 380.158
Overall the correlation value generated by each scheme is within the range of 0.00 - 2.99. Based on
[13], the interval shows the strength of a very weak relationship. This condition provides information that 1 bit
key input difference, can generate different random numbers on each scheme.

4582 ISSN: 2088-8708
In addition to the correlation and MAPE analysis, we also tested the distribution of random number
data using box-plot diagrams.

-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Figure 6. Box Plot Random Number of Each Scheme Output with Various Input Variations
Based on Figure 6, each box has almost the same size in which the upper and lower whisker lines
vary slightly, but the maximum value is always close to one and the minimum is near zero. The distribution of
data in the chaos range will strengthen the cryptography algorithm if used as a key, this condition will certainly
complicate cryptanalyst to be able to search for infinitely many numbers although limited.
4. CONCLUSION
Each designed scheme is capable of providing key input flexibility that can execute domain logistics
function values. A 1 bit difference in the key character affects every random number generation, so each key
will generate a different random number sequence. Under the key input conditions of the same eight characters,
the 1st scheme is better at generating different initialization values than the scheme-2 and schema-3. In addition,
the one character reduction of the eight identical characters in the key input does not show a proportional or
reverse pattern with the initial x0 values and the relative error x0. The resulting random numbers distributed
evenly over the chaos range will amplify the algorithm when used as a key in cryptography. This condition
will certainly complicate cryptanalyst to be able to search for infinitely many numbers although limited. The
three schemes have succeeded in fulfilling the nature of the butterfly effect, thus the existence of the logistics
function in generating chaos numbers can be accommodated based on key inputs. Each scheme can be used as a
complement to a cryptography algorithm to satisfy the diffusion properties of the Shannon principle.
REFERENCES
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[3] Munir, R., 2011, Enkripsi Selektif Citra Digital dengan Stream Cipher Berbasiskan pada Fungsi Chaotik Logistic Map,
Seminar Nasional dan ExpoTeknik Elektro, Universitas Achmad Dahlan.
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IJECE ISSN: 2088-8708 4583
ACKNOWLEDGMENT
Thank you to Satya Wacana Christian University Research and Community Service Center (BP3M) for
the research funding support through the Internal Obligatory Research scheme in the 2016 ﬁscal year.
BIOGRAPHIES OF AUTHORS
Alz Danny Wowor is currently a lecturer at the Faculty of Information Technology, Satya Wacana
Christian University in Salatiga, Indonesia. He received bachelor and master degree in mathemat-
ics and informatics from Satya Wacana Christian University, in 2005 and 2011 respectively. His
researches are in ﬁelds of Primitive Cryptography, Symmetric Cryptography: Block Cipher and
Pseudorandom.
Vania Beatrice Liwandouw is a Master of Cyber Security student at Faculty of Science, Radboud
University, Nijmegen, The Netherland. She received her Bachelor degree in engineering informatics
at the Faculty of Information Technology, Satya Wacana Christian University, Salatiga, Indonesia.
Her research interests are in the Design and Implementation of Symmetric Cryptographic Algo-
rithms.

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