Method for a Simple Encryption of Images Based on the Chaotic Map of Bernoulli

International Journal of Computer Science & Information Technology (IJCSIT) Vol 9, No 6, December 2017
DOI:10.5121/ijcsit.2017.9604 43
METHOD FOR A SIMPLE ENCRYPTION OF IMAGES
BASED ON THE CHAOTIC MAP OF BERNOULLI
Luis Alfredo Crisanto Baez, Ricardo Francisco Martinez Gonzalez,
Yesenia Isabel Moreno Pavan and Marcos Alonso Mendez Gamboa
Department of Electric-Electronic Engineering, TecNM - Instituto Tecnologico de
Veracruz, Veracruz, Mexico
ABSTRACT
In this document, we propose a simple algorithm for the encryption of gray-scale images, although the
scheme is perfectly usable in color images. Prior to encryption, the proposed algorithm includes a pair of
permutation processes, inspired by the Bernoulli mapping. The permutation disperses the image
information to hinder the unauthorized recovery of the original image. The image is encrypted using the
XOR function between a sequence generated from the same Bernoulli mapping and the image data,
obtained after two permutation processes. Finally, for the verification of the algorithm, the gray-scale Lena
pattern image was used; calculating histograms for each stage alongside of the encryption process. The
histograms prove dispersion evolution for pattern image during whole algorithm.
KEYWORDS
Algorithm of image encryption; Bernoulli mapping; Permutation.
1. INTRODUCTION
In the last decades, the transmission of information through the internet has increased [1]. This
fact is due to the technological advance in hardware and software, which has caused that
nowadays it is possible to carry out data transfers at high speed and in high definition. To be able
to perform this type of transmissions, a large volume of data must be moved; the fact is
considered trivial by users [2]. However, it presents a latent risk of attacks by third parties; who
want to obtain information about a specific user [3]. Therefore, it is necessary to implement
methods capable of offering certain levels of information security; being encryption the most
common way [4]. Encryption needs to be fast, efficient and easy to implement, to prevent
bottlenecks in the transfer of information [5], and to reduce affectations to the end user.
A good part of the traffic circulating on the network are images [6]; unfortunately, they can be
observed by individuals who can misuse them. The purpose of this algorithm is to use pseudo-
random sequences to control the different processes for encrypting the input image. There are
multiple ways to generate pseudo-randomness, but in [7], the use of chaos is justified for
encryption applications; it is mainly due to the great compatibility between the chaotic properties
with the desired characteristics in the ciphers.

44
As part of the present work, a method is proposed that not only encrypts the image based on a
chaotic one-dimensional mapping but also it includes permutations controlled by the same
mapping. These permutations will not change the statistics of the image; however, they will alter
the image perception, increasing the difficulty to recognize it [8]. To complete the process, the
image is encrypted using a sequence generated by the same mapping. As part of this manuscript;
in the second section, some details of the Bernoulli mapping will be announced. In the next
section, we will detail the used encryption scheme, as well as the logic behind image
permutations. Later, the results obtained with the present work will be reported in the fourth
section. Finally, the conclusions obtained after the realization of the proposal are narrated on the
last section.
2. MATHEMATICAL MODEL OF BERNOULLI
The Bernoulli mapping is also known as the dyadic transformation [9]. This mapping is usually
used in a recursive way, and it can be defined as a piece-wise linear function [10], which is shown
in Eq. 1.
=
2 0 ≤ < 0.5
2 − 10.5 ≤ < 1
(1)
Where xn is the iteration value at the present time, while xn+1 is the iteration value at the next time.
The value range for x is (0, 1). On the other hand, the variable µ represents the feedback factor
and its range is (0, 1).
As it can be seen on the Eq. 1, the domain of the Bernoulli mapping is (0,1); therefore, such
interval is converted to a floating-point representation. Which is, accordingly to [11], the only
way to effectively represent the decimal values that were established in the mapping domain.
On the other hand, when the mapping is used to process files whose coding is in fixed-point
representation, as is the case of the image files, it must be modified to fit within the range of the
file. For this reason, the mapping should have a small adjustment, which is shown on next
{ } = { } ∗ 2 ( 2)
Where {X} is the output set produced by the mapping represented by Eq. 1; {Y} is the set of the
scaled and truncated values obtained from {X}; while bitsis the number of bits that are intended to
represent the binary data.
The function floor truncates the {X} data after being scaled to the range (0, 2bits
); it takes the input
data and removes the decimal part, retaining the entire part. As it was mentioned, the present
proposal aims at image encryption. Typically, in an image file, eight bits are used to represent
pixel depth; so that value will be the number of bits used in this work.
At Figure 1 there is a flow diagram for encrypting data generation. It begins with the definition of
the feedback factor (µ) and initial value x[0] for the mapping; additionally, the total number of
iterations need to encrypt image.
After basic parameters were defined, a loop starts; it iteratively obtains necessary data to encrypt
image. Inside the loop, there is a decision to evaluate if previous outcome iteration is lower or

International Journal of Computer Science &
higher than 0.5. The evaluation result is used to assign which segment of the mathematical
description will be taken for obtaining
stop condition is reached, the loop is broken and total data is scaled using definition shown in Eq.
2.
Figure 1. Flow diagram of the implementation of Eq. 1 and its subsequent adaptation for eight
The graphical tool known as the bifurcation diagram is used
described generator mapping [12].
2.1 Bifurcation diagram
It is a tool that indicates the values that a mapping can take, depending on the feedback factor (
The algorithm for obtaining it is quite simple, and is shown
• Define the range of the feedback factor,
that will be in it; create a vector of feedback factors.
• Select the total number of iterations that will have the sequence to be used to generate the
diagram. The number of iterations must be higher than 300, nevertheless, it is important
to remark that the higher this number the denser diagram will look, but the computational
time will be longer.
• Choose an initial value (
factors and use the mapping described in Eq. 1 to calculate the
feedback factor. This point must be repeated for all values of the vector; therefore, they
will have step {X} sets.
• Scale the {X} sets using Eq. 2, obtaining the same number of
graph the values of the sequences obtained in the corresponding space of the feedback
description will be taken for obtaining x[n]; then loop control variable is increased. When the
he loop is broken and total data is scaled using definition shown in Eq.
Figure 1. Flow diagram of the implementation of Eq. 1 and its subsequent adaptation for eight
resolution files.
The graphical tool known as the bifurcation diagram is used to observe the behavior of the
described generator mapping [12].
It is a tool that indicates the values that a mapping can take, depending on the feedback factor (
The algorithm for obtaining it is quite simple, and is shown below [13]:
Define the range of the feedback factor, µi, and µf, as well as the number of values (
that will be in it; create a vector of feedback factors.
Select the total number of iterations that will have the sequence to be used to generate the
Choose an initial value (x0) arbitrarily. Take the first value of the vector of feedback
factors and use the mapping described in Eq. 1 to calculate the {X} set for the taken
sets.
ing Eq. 2, obtaining the same number of {Y} sets. With the sets,
Information Technology (IJCSIT) Vol 9, No 6, December 2017
45
; then loop control variable is increased. When the
he loop is broken and total data is scaled using definition shown in Eq.
Figure 1. Flow diagram of the implementation of Eq. 1 and its subsequent adaptation for eight-bit
to observe the behavior of the
It is a tool that indicates the values that a mapping can take, depending on the feedback factor (µ).
, as well as the number of values (step)
Select the total number of iterations that will have the sequence to be used to generate the
value of the vector of feedback
set for the taken
sets. With the sets,

46
factor used for its generation; producing a sweep of all the output values for the mapping
for each µ used.
Figure 2. Bifurcation diagrams for {X} set (right) and {Y} (left)
The main difference of the {X} and {Y} sets diagrams is their intervals. The diagram for the {X}
set is in (0, 1), while the one for the {Y} set is in (0, 2bits
). According to some investigations, the
truncation process degenerates the pseudo-random behavior of the mapping; which could be
manifesting in the form of the lines present in the bifurcation diagram of the {Y} set. Thanks to
the bifurcation diagram, the presence of such detail can be observed.
3. DESCRIPTION OF THE PROPOSED ALGORITHM
The proposed algorithm considers loading a gray-scale image (img), it has a single layer image
and extending it into a vector (bitplane) that has a length obtained to multiply the length (m) and
width (n) of the loaded image. This vector is permuted following a criterion established in Eq. 3.
This permutation was inspired by the Bernoulli mapping, and it can be seen below:
!"# $%&' = (
!"# $%&)' ≤
*∙
)
!"# $%&)', *∙ >
*∙
)
(3)
Where pbitplane is the vector obtained after the permutation criterion. bitplane was previously
mentioned that are the values of the img image in a vector form. l is a consecutive number that
must cover all the cells of the bitplane vector. And finally, m and n are the dimensions of the
loaded image.
According to the flow diagram in Figure 3, the permutation process is the second step in the
proposed algorithm. Following the same diagram, it is observed that the next step is the
transposition of pbitplane. The transposition process starts with the reconstruction in two
dimensions from the pbitplane vector; for such reason, the vector is chunk in n pieces, where each
piece is a row in the reconstructed matrix.

With the information in the form of a matrix, called
using function available in Matlab for such purpose. The function exchanges the matrix values
from horizontal to vertical and viceversa, obtaining the transposed matrix,
Figure 3. Flow diagram for the algorithm proposed in the present
After transposition, the matrix
process of permutation is concluded following the mathematical definition described in Eq. 3,
whereby pbitplane2 is obtained. To explain the permuta
representation was created that can be seen in Figure 4.
The permutation process seeks to distribute image information contained in the matrix, in such a
way that the information of each pixel is as separate as poss
This process will make it difficult to reconstruct the original image by not presenting a reference
point of where the information came from.
Although the process of permutation produces a dispersion of the image
to have an adequate level of security. A disadvantage of any permutation process is that the
permuted matrix statistics remains the same as the original matrix [15]. For this reason, the last
process is required; it is the data enc
With the information in the form of a matrix, called imgp1, next step is to transpose the matri
from horizontal to vertical and viceversa, obtaining the transposed matrix, imgt1.
Figure 3. Flow diagram for the algorithm proposed in the present manuscript
After transposition, the matrix imgt1 is again transformed into a vector, called
is obtained. To explain the permutation process described above, a graphic
representation was created that can be seen in Figure 4.
way that the information of each pixel is as separate as possible from its original position [14].
point of where the information came from.
Although the process of permutation produces a dispersion of the image data, this is not enough
process is required; it is the data encryption in itself, which is performed twice.
47
, next step is to transpose the matrix
manuscript
is again transformed into a vector, called bitplane2. The
tion process described above, a graphic
ible from its original position [14].
data, this is not enough

Figure 4. Graphical description of the permutation process.
To begin the encryption process, a sequence is generated using the Bernoulli mapping. As a
parameter for the mapping, a feedback factor greater tha
because according to the diagram in Figure 2, it guarantees that the data obtained have a pseudo
random behavior. By using a sequence similar to the random, the encrypted data will have a good
level of security; preventing that someone wants to break into them [16].
The encryption process is simple, it starts by defining the feedback parameter and initial
condition; and then, to generate the sequence using Eq. 1. It is important to take into account that
the length of the generated sequence must be equal to the vectorized image to be encrypted. After
obtaining it, the generated sequence is scaled using Eq. 2, with a representation of eight bits. Such
length was chosen because the image data have it the same length. Each
generated from Eq. 2 is input along with the data from the
function [17].
It should be noted that the encryption process takes only one of the data from each source at a
time, and it outputs a single data of eight bits at a time. The process continues until the data of
both input vectors, pbitplane2
4. RESULTS
The first result presented is that corresponding to the histograms
the different stages of the proposed algorithm. As can be seen in Figure 5, the permutation
processes do not alter the data, only its location; so their respective histograms are the same.
According to the presented histogr
with the eight-bit range that was established for the algorithm. With respect to the data incidence,
its maximum value is 2500; and this incidence is reached for a value close to 160.
The encrypted image histogram changes significantly, and this is because the encryption process
takes data and replaces it with a new one; and not only exchanges it as in the case of the
Figure 4. Graphical description of the permutation process.
parameter for the mapping, a feedback factor greater than 0.75 is used; such value is agreed
because according to the diagram in Figure 2, it guarantees that the data obtained have a pseudo
ing that someone wants to break into them [16].
he generated sequence must be equal to the vectorized image to be encrypted. After
length was chosen because the image data have it the same length. Each of the sequence data
generated from Eq. 2 is input along with the data from the pbitplane2 vector into a XORbitwise
single data of eight bits at a time. The process continues until the data of
vector and the pseudo-random sequence vector, are exhausted [18].
The first result presented is that corresponding to the histograms obtained from the output data of
According to the presented histograms, the data starts at 0 and ends at 255; which is consistent
bit range that was established for the algorithm. With respect to the data incidence,
its maximum value is 2500; and this incidence is reached for a value close to 160.
ypted image histogram changes significantly, and this is because the encryption process
48
n 0.75 is used; such value is agreed
because according to the diagram in Figure 2, it guarantees that the data obtained have a pseudo-
he generated sequence must be equal to the vectorized image to be encrypted. After
of the sequence data
vector into a XORbitwise
single data of eight bits at a time. The process continues until the data of
random sequence vector, are exhausted [18].
obtained from the output data of
ams, the data starts at 0 and ends at 255; which is consistent
bit range that was established for the algorithm. With respect to the data incidence,
ypted image histogram changes significantly, and this is because the encryption process

permutation. This fact drastically alters the data statistics, which is refle
presented in figure 5 for the encrypted image
The histogram of the encrypted image should be ideally uniform [15]; in other words, completely
flat, with all its equal incidences for all values from 0 to 255. In practice, this is extremely
difficult to achieve, so the result presented, where the data a
the same incidence, can be taken as a good result, due its entropy is 7.9963. The entropy was
calculated using the Matlab function, which calculates it with next expression,
. = − ∑0 ∙
where s is the entropy value, and
Figure 5. Histograms of the output data of the different stages of the proposed algorithm
For the present work, it was decided to start with the histograms obtained
delivered after each permutation and encrypted images, they are found in Figure 6.
The input image for the proof was the gray
distorted but the image of Lena is still
process of transposition; using it, the image is rotated and then the second permutation process is
performed; resulting in a much less recognizable image. Finally, the encryption alters either the
statistics of the image or aspect, which at first glance seems like noise.
permutation. This fact drastically alters the data statistics, which is reflected in the histogram
presented in figure 5 for the encrypted image
difficult to achieve, so the result presented, where the data are much more dispersed and almost
calculated using the Matlab function, which calculates it with next expression,
0 ∙ 1) 2 (4)
is the entropy value, and p is the incidence of every pixel in 256 bins of gray
For the present work, it was decided to start with the histograms obtained from the input, those
The input image for the proof was the gray-scale Lena. After the first permutation, the image is
distorted but the image of Lena is still recognizable. Between the two permutations there is a
istics of the image or aspect, which at first glance seems like noise.
49
cted in the histogram
re much more dispersed and almost
is the incidence of every pixel in 256 bins of gray-scale.
from the input, those
scale Lena. After the first permutation, the image is
recognizable. Between the two permutations there is a

Figure 6. Input image (top left), obtained after the first permutation (top right), second permutation (bottom
left) and the final cipher image (bottom right).
5. CONCLUSIONS
With the images of Figure 6 and the histograms previously presented, it is possible to conclude
that the Bernoulli mapping can serve as a basis for pixel permutation processes; thus, to generate
pseudo-random sequences, that can be used for image encrypting.
Additionally, the present algorithm is simple not only to perform but to understand in its
operation; so it can be taken as a frame of reference for future developers of image encryption
schemes. So, a double objective is achieved, the realization of a simp
encryption and a guide for future work in this area.
left) and the final cipher image (bottom right).
the images of Figure 6 and the histograms previously presented, it is possible to conclude
random sequences, that can be used for image encrypting.
dditionally, the present algorithm is simple not only to perform but to understand in its
schemes. So, a double objective is achieved, the realization of a simple method for image
encryption and a guide for future work in this area.
50
the images of Figure 6 and the histograms previously presented, it is possible to conclude
dditionally, the present algorithm is simple not only to perform but to understand in its
le method for image

51
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