Selection of Intelligent Rules for the Evolution of Elementary Cellular
Automata for Image Encryption
NASHAT AL BDOUR
Computer and Communication Engineering,
Tafila Technical University,
Tafila,
JORDAN
Abstract: - The paper is devoted to the search for new approaches to the formation of key arrays for encryption
of color images. Emphasis is placed on using the initial key sequence of the smallest length. In this case, the
key is the initial state of an elementary cellular automaton for implementing evolution based on a given rule.
The use of an evolutionary approach on cellular automata to the formation of large key arrays made it possible
to achieve unpredictable image encryption based on a single rule of an elementary cellular automata. The task
of the research is to search for the rules of elementary cellular automata, which, based on a small initial key bit
sequence, allow one to form a reliable key array of large dimensions for encrypting the bit layers that make up
the image. To solve this problem, an experiment was carried out, on the basis of which the search for the
necessary rules and options for choosing the elements of each bit array was carried out to encrypt the bit layers
of the image. To form each bit key array, different initial conditions were used for elementary cellular
automata. It is shown that for different initial conditions and for the chosen rules, the encryption quality is
preserved. The most reliable encryption is the use of two key arrays formed on the basis of the evolution of one
rule for different initial conditions. As a result of the experiments, the rules were determined (rules 90, 105, 150
and XOR function based on the two previous steps of evolution), which can be used without additional rules.
Each bit layer of the image is encrypted using different subarrays of each generated one key array of the same
dimension. It has been established that the most effective for encryption is the rule 105 and the XOR function
based on the two previous steps of evolution. The resulting histograms of the distribution of brightness for each
color of the encrypted image confirm the high quality of encryption based on the proposed method.
Key-Words: - Encryption, Cellular automata, Key array, Image bit layers, Evolution, Wolfram's rule.
Received: August 28, 2021. Revised: August 29, 2022. Accepted: September 27, 2022. Published: October 25, 2022.
1 Introduction
Modern society is characterized by a high degree of
digitalization, which is being implemented at a rapid
pace in all spheres of human activity. On this basis,
digital information transmission systems have
received great development, which, like tentacles,
have come to every apartment and to every person,
to every desktop. Almost every person who has a
modern smartphone has its own access point, which
allows the user to communicate with anywhere in
the world. Modern smartphones contain high-
performance processors and large amounts of
memory, which allows high-speed transfer of large
amounts of data from point to point.
Images presented in digital form take up
significant amounts of memory. At the same time,
many methods and tools for image compression
have been developed, [1], [2], which allow to reduce
the occupied volumes, but at the same time, in most
methods, part of the information about the image
itself is lost. In most digital systems, images are
represented in raster form.
In most cases, users do not want the images they
represented or send to be viewable by other network
users. Therefore, images are converted or encrypted
so that visual information is not available for
viewing.
There are a large number of image encryption
methods that take into account its digital structure.
The most ideal encrypted image is an image in
which all the colors of the pixels are distributed
evenly and do not make it possible to reveal any
statistical relationships with the original image.
Also, the ideal method for encrypting an image is a
method that is based on using any encryption key
with the smallest possible length.
Existing methods often provide high quality
image encryption. However, these methods cannot
boast of the simplicity and versatility of the method
itself, since they use various additional computing
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tools. In this regard, the search and development of
new methods for encrypting images that are close to
ideal are carried out. Increasingly, there are
publications that describe image encryption methods
implemented using cellular automata (CA)
technologies [3], [4], [5], [6], [7], [8], [9], [10]. In
these works, CA proved to be promising in solving
encryption problems.
2 Problem Statement
One of the main problems in solving the problem of
image encryption is the use of a long key gamma,
which can be generated by the user himself, or can
also generate a pseudo-random number generator
(PRNG) based on the established initial conditions.
Forming a key gamma for encrypting large amounts
of data (images) does not seem realistic. Therefore,
PRNG is most often used. However, PRNG cannot
always form the highest quality sequence of
numbers. In addition, the structure of the generator
may be known to other users. Such shortcomings
force developers to search for new more effective
methods.
A promising direction in solving such a problem
is the use of CA, on the basis of which PRNGs are
built, which showed a high quality of the generated
bit sequences, [3], [11], [12], [13], [14]. Elementary
CA (ECA) and two-dimensional CA are used, as
well as their states obtained during the formation of
evolution.
Despite the byte structure, bitmaps can be
considered as a sequence of two-dimensional bit
layers that can be encrypted both in parallel and
sequentially. To encrypt one two-dimensional N×M
bit array, it is necessary to use N×M key bits. If a
binary code of length K bits is used to encode the
visual characteristics of one pixel of a raster image,
then K × N × M key bits must be used to encrypt the
entire image. The use of PRNG based on CA
complicates the implementation of the encryption
method.
The paper solves the problem of implementing a
method for encrypting raster images using a key of
small length. To solve this problem, one ECA rule is
used, determined on the basis of experimental
studies.
Since a bitmap image is represented in a
computer system by a sequence of bytes or, more
precisely, a binary sequence, it is easiest to encrypt
images using the streaming encryption method [3],
[4], [5], [11], which consists in applying a bit key
gamma to a bitmap image [3], [4], [5], [6], [7], [8],
[9], [10], [11]. To implement this method, a device
for generating a key gamut is used, which, as a rule,
is a PRNG [12]. To date, a large number of PRNGs
have been developed [12], [13], [14], [15]. which
have different configurations. Almost all existing
PRNGs (mathematical, hardware, etc.) can be
simulated on a PC and implemented in software.
Using such a program model, you can encrypt any
graphic file. In papers [3], [10], [11], [15], methods
using this method are considered. In [11],
experimental studies were carried out and it was
shown that it is enough to encrypt the three most
significant bits of each color byte. In this case, a
pseudo-random bit sequence was used, generated by
a PRNG implemented on a CA with active cells,
[13], [14]. The obtained experimental results
showed a high quality of encryption. However, this
method takes time to form the key gamma and
complete enumeration of all bits that encode the
image.
There are image encryption methods based on
the Fourier transform, [16], [17], [18], and Wavelet
transform, [19]. In recent years, a direction based on
quantum image encryption has been developing
[21], [22], [23], [24]. Papers on these topics describe
complex encryption schemes that do not always
provide high quality encryption. In addition, such
approaches can lead to partial loss of information
about the original image at the decryption stage.
A large number of methods use various methods
based on chaos theory, [25], [26], [27], [28], [29],
[30]. However, these methods require complex
calculations and can lead to partial loss of
information.
There are methods using genetic algorithms,
[31]. [32], [33] [34], [35], DNA calculations, [36],
[37], [38], based on elliptic curves, [39], [40],
Rubik's cube, [41], and artificial neural networks,
[42]. All these methods require special calculations
to be performed to transform the information array
and form a key array, which limits these methods,
since they are not resistant to attacks and cannot
guarantee high reliability.
Image encryption methods using CA
technologies have been greatly developed [3], [4],
[5], [6], [7], [8], [9], [10]. Many of these works use
additional means for encryption. Often an additional
tool is chaos theory , [25], [26], [27], [28], [29],
[30]. There are also works that use separate ECA
rules with additional methods [7], [43]. Thus, in
[43], rules for an ECA with a length of 8 cells were
considered and studied. Rules that give a good result
are defined. However, the limited length of the ECA
does not give full grounds for asserting effective
encryption of color images. In [7], rule 30 is
considered, which does not provide high quality
image encryption. This is proven in paper [10]. In
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this paper, a good encryption result is obtained.
However, this uses multiple rules to encrypt each bit
layer of the image, and also uses different initial
states for each rule. This approach requires the
formation of a large encryption key, which limits
this method. At the same time, paper [10] has
valuable material that shows which ECA rules are
the most effective for encrypting color bitmaps.
3 The Structure of a Bitmap Color
Image
In a computer system, a color image is stored as a
graphic file, consisting of bytes that encode the
structure of the image and the colors for each pixel.
Color depth is encoded by the number of bits
allocated for each pixel (4, 8, 16, 24, etc. bits). The
more bits per pixel color code, the more color
gradations can be displayed by one pixel of the
image and the more realistic the visual picture can
be presented.
The paper [14] presents the structure of a color
image, which displays it as a sequence of bit layers.
Each bit layer is a matrix (two-dimensional array),
each element of which represents the bit value of the
corresponding bit, which occupies a position in the
binary code corresponding to the binary layer
number. Each i-th bit layer contains the i-th bits of
the codes of all pixels. The number of bit layers is
equal to the number of bits in the code of each
image pixel.
Since each bit layer contains only logical "0" and
logical "1", they are considered as two-dimensional
cellular automata or as arrays formed as ECA
evolution in one of four directions.
4 Color Image Encryption
Methodology
The image encryption method uses the
representation of an image as a matrix of codes that
encode the color and brightness of each pixel. The
pixel code consists of three parts (3 bytes). The first
high byte of the code encodes the blue color and its
shades, the second and third code bytes encode the
green and red colors and their shades, respectively.
A color image can be represented as a sequence
consisting of a sequence of 24 bit layers forming an
image matrix in depth.
The paper [10] describes the encryption
methodology for color images based on the
technology of cellular automata with different rules.
Each bit layer is encrypted separately with a key
array of the same dimension. Key arrays are formed
based on the use of ECA evolution construction
technology. The structure of the encryption method
on Fig. 1 is shown.
Fig. 1: Structure of the color image encryption
method
Each binary layer of the image is encrypted using
one of the key layers formed according to the
selected ECA rule. The initial state of the ECA is set
and the evolution is formed. The resulting evolution
is commensurate with the dimension of the bit layer
of the image and is a key array. Each key array as a
key gamma is superimposed on each binary bit layer
of the image. As a result, we get encrypted binary
layers, which form an array of binary layers Еi of
the encrypted image. The binary encrypted array E
is formed by applying the XOR operation
.
In this case, the key layers K are formed on the
basis of different rules and are used for encryption
for different binary layers of the initial image. If the
same rules are often used, then an offset on key
layers is applied for different initial layers. To
implement this approach, evolutions form two-
dimensional arrays of a larger dimension than the
dimension of the bit layer of the image.
Using the same initial conditions to form key
layers with different rules requires a large number of
rules with different forms of evolution. In this case,
there is a need to select the initial conditions, as well
as the rules for each layer of the initial image.
Different initial conditions (initial keys) for each
rule improve the picture of encryption. However, for
images of large dimensions, it is necessary to use
key bit sequences of large length. As the number of
rules increases, the number of large length
encryption keys increases.
In this method, encrypted images contain some
geometric shapes that are inherent in the ECA
evolution forms for some rules. Therefore, image
encryption requires careful selection and placement
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of rules before encryption, which partially limits the
method.
5 Experimental Selection of One Rule
for Image Encryption
The main task of the experiment is to analyze the
ECA evolutions for different rules and the rules
selection that can be used to form a key array. In
this case, only one rule should be used and a
combination of several rules should not be used,
since this circumstance complicates the encryption
system. The rules were selected both visually by the
user and possible rules without visual analysis that
could be effective.
To search for optimal rules, the following rules
were studied: 30, 45, 51, 90, 105, 111, 150, and 184.
These rules were selected from the analysis of
evolutions described earlier in various works [44],
as well as visual analysis of these evolutions. On
fig. 2 is shows the evolutions for these rules. A
visual analysis was carried out for the predictability
of states at each step of the iteration.
Fig. 2: ECA evolutions for rules: 30, 45, 51, 90,
105, 111, 150 and 184.
Visual analysis of the obtained evolutions
showed that the most suitable rules are the rules: 90,
105 and 150. Rules 111 and 184 can only be used in
addition to other rules and cannot be used as
separate rules. Evolutions show constant state
changes throughout the field of the formed two-
dimensional array. If rules 90 and 150 form
evolutions with triangular shapes, then rule 105 is
the most attractive. However, encryption was
carried out for all the described rules.
The first step of the experiment was to use only
one key array, which, using the XOR function,
encrypted all the bit layers into which the images
were divided. Lighter colors barely changed the
color characteristics of the corresponding pixel. The
encryption results at this stage did not show high
quality. You can see the outlines of differences in
brightness and colors, which can be used to restore
the original image.
At the second stage of the experiment, one rule
was used, but each bit layer of the image was
encrypted with the received key array, the beginning
of which was shifted for each bit layer (Fig. 3). The
initial key array was formed with a larger dimension
along one coordinate. The value of the generated
array is determined by the number of shifts of the
initial key array and the number of pixels by which
the shift is performed.
Fig. 3: Selection of key arrays at the second stage of
the experiment
In many cases (for many rules), this approach
gives a sufficiently high quality of encryption. It
depends on the used initial states of the initial CA,
which is used to form the initial keys. However, the
outlines of the original image are often traced on the
obtained images (Fig. 4).
Fig. 4: Results of image encryption obtained at the
second stage of the experiment
To improve the quality of encryption, key arrays
obtained as a result of using the XOR function were
used at two adjacent previous steps of evolution
(Fig. 5). In this case, the initial key bit sequence is
doubled, but the number of rules does not change.
Changes start from the third line of evolution and
then form unpredictable states if the previous ones
are not known to the user.
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Fig. 5: Formation of the next step of evolution based
on the two previous steps
To form the state of the cell, the XOR function
of six arguments was used. The arguments are the
states of the six cells in the previous steps. These
cells are shown in Fig. 4 and are indicated by the
beginnings of outgoing arrows. The state of the cell
at the time (t + 2) is determined by the following
formula.
An example of CA evolution for this variant is
shown in Fig. 6.
Fig. 6: An example of CA evolution based on the
two previous steps of evolution
On fig. 4 is shows the result of image encryption,
which also does not give the desired quality. In
some places of the encrypted image, the outlines of
the original image are visible, which, after a detailed
analysis, can lead to a complete restoration of the
original image.
For more reliable encryption, several key arrays
were used, which were obtained using the same rule,
but with different initial states of the ECA. In this
case, an additional shift in each key array was used
to encrypt individual bit layers of a color image. The
results of such encryption in Fig. 7 are shown. To be
sure, are used a test image (presented first), which
contains different solid color zones.
Fig. 7: Results of image encryption based on one
rule and several key arrays
Encryption results (Fig. 7) showed high quality
for rules 90, 105, 150 and XOR functions based on
two evolution steps for both test images. The high
quality of encryption for these rules is confirmed by
the obtained histograms of the distribution of
brightness for each color (Fig. 8). On fig. 8 color
distribution histograms are presented for the initial
image (leftmost) and encrypted images for each
rule.
Fig. 8: Color distribution histograms for each
encryption rule
The first row describes the distribution for red,
the second for green, and the third for blue. The
resulting histograms show a uniform distribution of
colors over the entire field of the encrypted image.
Such a distribution does not allow the adversary to
determine and restore the original image.
Histograms are shown for the second image shown
in Fig. 7.
If the images obtained using rules 90 and 150
contain many triangular shapes, then the encrypted
images obtained using the remaining two rules do
not have any geometric shapes that could prompt the
opponent to determine the structure of the key array.
The analysis of the obtained images made it possible
to assert that the rule 105 and the XOR function are
the most effective based on the two previous steps
in the ECA evolution.
6 Conclusion
The paper considers the process of experimental
search for the rules of elementary cellular automata,
with the help of which a key array is formed for
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encrypting color images. As a result of the
experiment, rules were defined that give high
quality encryption (90, 105, 150 and XOR function
based on the two previous steps of evolution). It has
been proven that to encrypt an image, it is sufficient
to use only one of these rules, which gives two key
arrays based on different initial conditions. The use
of these rules made it possible to reduce the length
of the initial key, as well as to simplify the image
encryption scheme. These rules can also be used
together. At the same time, the quality of encryption
remains high.
In further research, the author plans to use
cellular automata that implement other paradigms
that give unpredictable evolutions. Research will
also focus on the use of two-dimensional cellular
automata.
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