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Image Encryption Through R¨
ossler System,
PRNG S-Box and Recam´
an’s Sequence
Mohamed ElBeltagy ID ,
Wassim Alexan ID ,SMIEEE
and Abdelrahman Elkhamry
Faculty of IET
The German University in Cairo
Cairo, Egypt
wassim.alexan@ieee.org
mohamed.elbeltagy@ieee.org
Mohamed Moustafa
Faculty of Informatics and Computer Science
Administrative Capital,
The German International University in Cairo
Cairo, Egypt
mohamed.dawood@student.giu-uni.de
Hisham H. Hussein
Faculty of Science and Innovation
Universities of Canada
Administrative Capital,
Cairo, Egypt
hisham.hussein@uofcanada.edu.eg
Abstract—This paper proposes a lightweight image
encryption scheme that is based on 3 stages. The first stage
incorporates the use of the R¨
ossler attractor for the R¨
ossler
system, the second stage incorporates the use of a PRNG
S-Box, while the third stage makes use of the Recam´
an’s
sequence. Performance of the proposed encryption scheme
is evaluated using a number of metrics. The computed
values of the metrics indicate a comparable performance to
counterpart schemes from the literature, at a very low cost
of processing time. Such a trait indicates that the proposed
image encryption scheme possesses potential for real-time
image security applications.
Keywords–Cryptography, image encryption, R ¨
ossler sys-
tem, Recam´
an’s sequence, S-Box.
I. INTRODUCTION
The tremendous evolution in digital image processing
and network communications have created a great demand
for real time secure image transmission over the Internet
and through wireless networks [1]. Data security, through
cryptography and steganography [2]–[7], has thus become
a vital means to ensure safe and secure operation and
usage of millions of online applications [8]. Cryptography,
which plays a vital role in information security, has lured
the attention of scientists and engineers, with contribution
in its research and developments ascending in recent
decades [9]–[11]. Global attempts focused lately on
refining the security of image transmission, with novel
cryptosystems proposed including cellular automata,
DNA coding and chaos theory [12]–[14].
Chaos is characterized in pseudo-randomness,
ergodicity and high sensitivity to initial conditions
and parameters. Thus, it is extensively used in image
encryption schemes. Results of such attempts have usually
involved the usage of one or more PRNGs, as well as true
RNGs. The literature on PRNGs incorporate examples
pooling from chaos theory [15], [16], mathematical
sequences [17], electrical circuits [18], quantum physics
[19], as well as many others.
The R¨
ossler system is a third-order continuous-time
system of differential equations with a single quadratic
cross-term and depends on 3 parameters that were
originally introduced by Otto R¨
ossler in the 1970s
[20]. These differential equations create a continuous
time dynamical system that outcomes chaotic dynamics
associated with the fractal properties of the attractor
[21]. The calculated characteristics usually concern the
generation of a single lobe chaotic attractor (spiral-type)
following a period doubling cascade of a limit cycle, or a
more complicate chaotic attractor (screw-type) due to the
presence of homo-clinic orbits [22]. Some properties of
the R¨
ossler system can be concluded from linear methods
such as eigenvectors, however the main features of the
system require non-linear methods such as Poincar´
e
maps and bifurcation diagrams. The original R¨
ossler
paper states the R¨
ossler attractor was designed to operate
likewise the Lorenz attractor, moreover it is also easier to
analyze qualitatively [21].
The Recaman’s sequence is an interesting sequence
of integers that is very simple to define, but the
resulting complexity exhibits how forceful it can be
against cryptanalysis. The authors of [23] made use of
Recaman’s sequence image steganography for 2D images,
where their proposed scheme led to excellent performance
and resiliency against steganalysis.
In cryptography, an S-Box (substitution-box) is a
basic component of symmetric key algorithms which
performs substitution. In block ciphers, they are typically
used to obscure the relationship between the key and the
ciphertext, thus ensuring Shannon’s property of confusion.
The first S-box used on symmetric key algorithms such978-1-6654-8303-2/22/$31.00 ©2022 IEEE
as Advanced Encryption Standard (AES) and the Data
Encryption Standard (DES), but the main problem of such
S-boxes is their statistic behavior. So, in order to produce
a dynamic behavior, PRNG and chaotic systems are used
to construct S-boxes, as in [24] and [25]. The authors of
[26] introduced a novel approach to construct an S-box
based on the R¨
ossler system, where the effectiveness of
the proposed S-box showed is well-exhibited in terms of
being resistant against attacks. Another good example of a
constructed S-box is that in proposed in [27]. The authors
of [27] employ a novel transformation, modular inverse
and permutation to construct their S-box. Performance
evaluation and comparison against set benchmarks from
the literature validate its cryptographic strength.
In this paper, we propose an image encryption scheme
that is based on 3 stages. The first stage incorporates the
use of R¨
ossler system, while the second stage incorporates
the use of S-Box and the third stage incorporates the
use of Recam´
an’s sequence. This paper is organized as
follows. Section II briefly presents the R¨
ossler system,
followed by a PRNG S-Box and the Recam´
an’s sequence
used for the proposed image encryption scheme. Section
III outlines the numerical results of the computations and
testing and provides appropriate commentary on them.
Section IV finally draws the conclusions of the paper and
suggests a future work that can be further pursued.
II. THE PROP OSED IMAGE ENCRYPTION SCHEME
The proposed image encryption scheme is composed
of three stages. The first stage makes use of the R¨
ossler
system, while second stages makes use of a PRNG
S-Box, and in the third stage the Recaman’s sequence
was employed. The next few sections introduce each of
those three concepts.
A. R¨
ossler attractor
The R¨
ossler system is an infamous prototype of a
continuous dynamical system defined by the following set
of 3 nonlinear differential equations:
˙x= (y+z)
˙y=x+ay
˙z=b+z(xc)
(1)
where a,band care non-negative parameters. This well-
known system approaches chaos through a period doubling
bifurcation route. In the proposed encryption scheme, the
employed parameter values are a= 0.1,b= 0.01 and
c= 14 resulting in Fig. 1. Listing the computed x, y and z
values in succession and plotting them against the iteration
number yields a plot for the R¨
ossler attractor points as
shown in Fig. 2.
Fig. 1: R¨
ossler attractor 3D graphical representation.
Fig. 2: R¨
ossler attractor 2D graphical representation.
B. S-Box
A substitution-box is a pivotal constituent of modern-
day block ciphers that helps in the generation of a
muddled ciphertext for the specified plaintext. Through
the incorporation of S-box, a nonlinear mapping among
the input and output data is established to create confusion
[28]. The security of data relies on the substitution process.
Substitution is a nonlinear transformation which performs
confusion of bits. It provides the cryptosystem with
the confusion property described by Shannon [29]. In
Fig. 3: The 2D shape of Recam´
an’s sequence.
general, an S-box takes m input bits and transforms them
into noutput bits. This is called an mn S-box and is
often implemented as a lookup table. These S-boxes
are carefully selected to resist and obstruct linear and
differential cryptanalysis. Through the incorporation of an
S-box, a nonlinear mapping among the input and output
data is established to create confusion [30], [31].
For the proposed encryption scheme, we randomly
generate an S-box of dimensions 16 ×16 utilizing
Wolfram Mathematica®. Table IV displays its values.
C. Recam´
an’s Sequence
In order to generate the Recaman’s sequence, one can
let a1= 1 and follow the mathematical expression shown
in (2) to generate its elements an.
an=(an−1−n, if an−1−n > 0
an−1+n, otherwise,(2)
where nis the position of the element in the
sequence. This delivers the first few elements as
1,3,6,2,7,13,20,12,21,11,22,10,23,9, ...
Fig. 3 is the 2D graphical representation for the first
200 iterations which are used in our proposed encryption
scheme to generate a key of randoms bits.
D. Image Encryption and Decryption Processes
The proposed image encryption scheme is implemented
as follows.
First, an image of appropriate dimensions is chosen, and
then the image’s pixels are converted into a 1D stream of
bytes. Lastly, these bytes are converted into a bit stream d.
Second, the mean intensity of the image pixels
is calculated. The resulting value is a rather small
number, which we multiply by a magnifying factor fM.
resulting value of the multiplication shall be denoted by µ.
Next, we cyclically shift dto the right by µplaces and
the resulting bit stream, now denoted dµ, is then XORed
with kCA .kCA is the first key, a bit stream of the same
length as dand dµ, that is made up of a repetition of the
first NCA bits resulting from the binary representation of
the first 250 R¨
ossler numbers in R¨
ossler attractor. Let us
denote the resulting bit stream as C1. This concludes the
first step of encryption.
Next, the randomly generated S-Box is used for
substituting the decimal representation for each 8 bits
from the bitstream acquired after the first step as in
Table IV. The randomly generated S-box is mainly
used to provide 256 as total from 16 rows 16 columns
randomly distributed numbers starting from 0 to 255,
RandomSample function from Wolfram Mathematica®
was used in order to satisfy this condition. Next,
we change those resulted Decimal representations to
a bitstream C2. At this point, we take the xand
ycoordinates of each of the points of the Recam´
an’s
sequence equations and flatten them into a single 1D array.
Next, we list plot those values into 2D, as shown in Fig. 3.
Examining the plot in Fig. 3, we change those integer
values to bits. This newly obtained bitstream of length
NLwould make up the seed of our Recam´
an’s sequence
based key. We repeat those NLbits until they are of the
same length as dand C1, thus forming the second key.
Let us denote it kL.
Next, we XOR kLwith C2obtaining C3. This
concludes the third step of encryption.
Finally, C3is reshaped back into an image of the same
dimensions as those of the plain image, obtaining the
encrypted image.
The decryption process is implemented in a reverse
manner as to that of the encryption process.
III. NUMERICAL RES ULTS A ND PERFORMANCE
EVALUATI ON
his section outlines the numerical results of the
proposed image encryption scheme. Performance is
evaluated and compared to counterpart algorithms
found in the literature. The proposed scheme is
implemented using the computer algebra system Wolfram
Mathematica®on a machine running Windows 10
Enterprise. The PC is equipped with a 2.3 GHz 8-Core
Intel®CoreTM i7 processor and 32 GB of 2400 MHz
DDR4 of memory. The utilized keys are assigned the
following values: NCA = 250, NL= 200 and fM= 106.
Four images that are commonly used in image processing
applications/experimentation are utilized in this section.
These are Lena, Mandrill, Peppers and House, all of
dimensions 256 ×256.
The graphical representation denoted by Table II
is a histogram representing the pixels distributions
characteristics of sample images. As observable, the plain
images (prior to encryption) and decrypted images’ pixels
are non-uniformly distributed on the histogram. In contrast
to the histogram pixels distribution for the encrypted
image which show a uniform pattern all along with
the histogram. Noting that statistical analyses, attempts
of breach and attacks do not yield any cryptanalytic
results in comparison to those carried out on images of
non-uniform distribution. Thus, this observation yields to
that no information could be distinguished or determined
from any of the characteristics of the encrypted images.
Fig. 4 shows the correlation coefficient diagrams of the
plain and encrypted Lena image. It is clearly seen that the
horizontal, vertical and diagonal correlation coefficients
of the adjacent pixels for the plain image are linear.
However, on inspecting the plots generated from the
encrypted image, it is clear that the plots are uniform and
have a scatter-like distribution. This signifies a resistance
of the proposed scheme to statistical analyses or attacks.
A time-size complexity metric is utilized to assess
the efficiency of the proposed image encryption scheme
in order to identify whether the scheme is adequate for
real-time applications. Table I displays the processing
time required for encryption, decryption, and their
summation for certain different standardized square
image dimensions such as {128,256,512,1024,2048}.
Furthermore, Table I shows that for an image dimensions
of 128 ×128, a decryption time of less than a single
second is enough to successfully decrypt the image.
In turn, this means that the proposed image encryption
scheme is appropriate for real-time image exchange
among handheld devices. This also translates into better
resource management and optimization concerning the
power consumed during image processing on the devices.
Moreover, Table I shows that the amount of time required
for the encryption-decryption process increases with
increases in the dimensions of the image. This behavior
is exhibited with a certain rate that starts to converge into
a slower rate beyond an image dimensions of 1024×1024.
Table III lists the computed values of MSE and PSNR
of our proposed scheme, as well as those of 2 of its
counterparts from the literature, specifically [32] and
[33]. A larger value of the MSE signifies an improved
level of security. Our proposed scheme is shown to
outperform the MSE values of [33], while it achieves
a lower performance than that achieved in [32]. Since
the PSNR as a metric is inversely proportional to the
TABLE I: Processing time for various dimensions of the
Lena image.
Image Time [s]
dimensions Encryption Decryption Total
128 ×128 5.39173 0.268098 5.659828
256 ×256 6.41808 1.336811 7.754891
512 ×512 7.70297 3.989731 11.692701
1024 ×1024 17.86431 15.879460 33.74377
2048 ×2048 52.73229 63.00331 115.7356
MSE, the comparison among those 3 schemes in terms of
PSNR still holds the same significance as aforementioned.
Information entropy is employed to measure the
randomness of the distribution of gray pixel values of
an encrypted image. Theoretically, the entropy value of
a randomly encrypted image is 8because a gray scale
image has 256 symbols and the data of the pixel has
28possible combinations. The entropy values of various
encrypted images are shown in Table V. As can be seen,
each of the values is a little over 7.999 which reveals
that the proposed encryption scheme randomizes the
distribution of the pixels of the plain image, making
it impossible for an attacker to gain any information
about the plain image. Moreover, Table V provides a
comparison among the achieved information entropy
values with those achieved by counterpart schemes from
the literature [32]–[34].
Any PRNG can be easily tested for randomness using
the test devised by the National Institute of Standards
and Technology (NIST). A good PRNG should satisfy its
randomness criteria by a number of tests that comprise
the NIST analysis suite. Specifically, the probability, or
p−value of each of the tests should be greater than 0.1for
any bitstream to be regarded as random. Table VI shows
the results of the NIST analysis as run on an encrypted
Lena image. It is clear that the values for all the tests
are indeed larger than 0.1, deeming the success of our
proposed image encryption scheme at passing the NIST
analysis.
IV. CONCLUSIONS AND FUTURE WOR KS
In this paper, we proposed an image encryption scheme
that is based on 3 stages. The first stage incorporated
the use of the R¨
ossler System, while the second stage
incorporated the use of a PRNG S-Box and the final
stage incorporated the use of the Recam´
an’s sequence.
Performance evaluation of the proposed scheme was
carried out utilizing a number of appropriate metrics
and analyses. Those included visual inspection of both
plain and encrypted images, a histogram analysis, a
cross-correlation analysis, entropy values, MSE and
TABLE II: Numerical results of the achieved values for various metrics.
Image data Plain image/histogram Encrypted image/histogram Decrypted image/histogram
Lena
d= 256 ×256
Mandrill
d= 256 ×256
Peppers
d= 256 ×256
House
d= 256 ×256
TABLE III: A comparison of MSE and PSNR values among the proposed scheme and its counterparts from the literature.
Image Proposed Scheme [32] [33]
MSE PSNR [dB] MSE PSNR [dB] MSE PSNR [dB]
Lena 8893.04 8.6403 10869.73 7.7677 4859.03 11.3
Mandrill 8286.99 8.94683 10930.33 7.7447 7274.44 9.55
Peppers 10064.2 8.10303 N/A N/A 6399.05 10.10
House 8361.94 8.90773 N/A N/A N/A N/A
TABLE IV: S-Box values generated from Wolfram Mathematica®.
102 216 26 199 187 45 252 245 204 154 125 19 238 215 208 43
6 198 195 11 67 223 20 255 7 1 211 162 14 236 145 9
107 170 147 246 196 232 109 133 33 34 179 212 234 197 27 190
82 206 99 18 75 172 12 63 167 203 160 122 78 94 79 51
184 235 37 243 150 143 40 244 10 137 50 186 247 68 185 100
210 169 61 123 253 76 180 16 159 142 21 88 38 237 81 129
71 230 175 217 35 65 202 90 29 136 177 121 80 115 95 140
127 85 110 93 153 225 124 62 209 231 224 54 146 4 157 161
58 86 72 138 250 201 222 116 104 165 47 5 2 39 249 84
170 83 0 174 87 58 172 189 29 135 86 105 223 156 143 132
48 200 112 23 105 164 148 181 0 73 32 56 44 131 178 36
60 92 218 113 254 103 241 108 98 52 117 101 28 220 25 46
242 151 13 168 219 59 213 17 87 158 182 192 171 126 155 227
134 141 42 41 193 106 83 31 166 128 91 176 111 114 74 248
132 144 69 228 57 240 119 207 77 139 174 70 221 189 97 214
226 251 188 53 30 183 15 55 229 22 89 49 156 120 149 194
TABLE V: Entropy values for encrypted images.
Image Proposed [32] [34] [33]
Lena 7.9991 7.9990 7.9978 7.9968
Mandrill 7.9990 7.9991 7.9993 N/A
Peppers 7.9991 N/A N/A N/A
House 7.9989 N/A N/A N/A
TABLE VI: NIST analysis on an encrypted image of Lena.
Test name p-value Remarks
Frequency 0.521667 Success
Block Frequency 0.779001 Success
Run (m= 50162)0.455298 Success
Long runs of ones 0.011365 Success
Rank 0.177465 Success
Spectral FFT 0.683215 Success
No overlapping 0.332454 Success
Overlapping 0.412563 Success
Universal 0.987111 Success
Linear complexity 0.566321 Success
Serial 0.089741 Success
Approx. Entropy 0.521547 Success
Cumulative sum forward 0.987411 Success
Cumulative sum reverse 0.321577 Success
PSNR values. A comparison with counterpart schemes
from the literature was carried out and the proposed
scheme exhibited comparable security performance.
Finally, the processing time was computed and was
shown to be rather low, showcasing the appropriateness
of the proposed scheme for secure image exchange
between handheld devices. A future work that could be
further pursued would be the construction of a secure
S-box, instead of relying on a PRNG S-box generated
via Wolfram Mathematica®.
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