PixAdapt Image Encryption Algorithm
PixAdapt Image Encryption Algorithm
PixAdapt Image Encryption Algorithm
A R T I C L E I N F O A B S T R A C T
Keywords: Image encryption using genetic approach is a recent and advanced technique which has grabbed attention in
PixAdapt recent years. Currently, most image encryption algorithms (using genetic approach) use a static set of parameters
Hill climb for image encryption without considering the features representative of the image. In this study, an innovative
Genetic algorithm
adaptive image encryption algorithm – PixAdapt is developed. The process of image encryption is being re-
Simulated annealing
UACI
engineered in a way to calculate the fitness of encrypted image using UACI and adapting the respective pa
Chaos rameters using genetic hill climb or simulated annealing. Pseudorandom numbers have been generated using the
linear feedback shift register and chaos-based maps such as the Logistic map, Rossler map, Henon map and Tent
map. PixAdapt algorithm also uses confusion and diffusion process to ensure that plain text image and cipher text
image are completely un-related. The use of metaheuristic search techniques for optimization of image
encryption parameters has been implemented for the first time. The results obtained show that the genetic hill
climb algorithm encrypts the various images giving the most optimal value of UACI. The algorithm has been
tested for fitness improvement, parameter evolution, statistical analysis, and quality of encryption. PixAdapt is
not only unique but has proven the encryption parameter UACI to be an appropriate fitness function to encrypt
an image efficiently.
1. Introduction have been explored and implemented namely: hill climb and simulated
annealing. Hill climb has been implemented as a genetic algorithm while
There are several types of images available varying from application simulated annealing has been used in its primitive essence. Chaotic se
to application [1,2]. Each of these images have unique features in their quences such as Logistic map, Rossler map, Henon map and Tent map
own regard. Images from the medical field and satellite images are much have been used to generate pseudorandom numbers in conjunction with
larger and higher dimensional than trivial JPEG and PNG images [3]. the Linear feedback shift register.
Despite the characteristics of the image, the security of the image while An adaptive system in a more general sense can be defined as a
transferring through media remains vital [4,5]. Image encryption is a system created with the mind set of dealing with and adapting to
process of converting the original image into an unreadable image. changes in the environment while optimizing performance objectives
Tremendous amount of research has been conducted in the usage of [7]. According to this definition, the proposed system does deal with
chaotic sequences for encrypting an image. These sequences are gener changes in its environment which are the new types of images it en
ated using a static set of parameters which do not account for the counters at every pass. As the environment changes i.e., the images
characteristics of the actual image. Currently, very few encryption change, the adaptive system calculates the fitness of the parameters for
methods adapt to the different types of images they are encrypting. The respective image and decides whether the parameters are required to be
lack of an adaptive mechanism sparked curiosity and an effort to re- evolved further [8–11]. The overall process has the objective of main
engineer the process of encrypting an image with the help of a meta taining the performance by creating a solution which lies in the
heuristic search algorithm to find the right pairs of parameters to acceptable fitness range. The proposed mechanism also fits the defini
encrypt an image has been made in this paper. Metaheuristic search tion of an adaptive system as proposed by [12] where they described an
algorithms provide a solution to solve complex optimization problems in adaptive system to satisfy natural selection in accordance with the
a relatively short span of time [6]. In this paper, two such algorithms following conditions of varying entities, having continuity and the
* Corresponding author.
E-mail addresses: rt349@sussex.ac.uk (R. Tuli), hiteshsoneji25@gmail.com (H.N. Soneji), Prathamesh.churi@ieee.org (P. Churi).
https://doi.org/10.1016/j.chaos.2022.112628
Received 11 April 2022; Received in revised form 23 August 2022; Accepted 25 August 2022
Available online 19 September 2022
0960-0779/© 2022 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
R. Tuli et al. Chaos, Solitons and Fractals: the interdisciplinary journal of Nonlinear Science, and Nonequilibrium and Complex Phenomena 164 (2022) 112628
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success of each entity differs from the other. Each of parameters used to substitution process is governed by usage of various maps used in this
generate chaotic sequences have varying values after each pass. These algorithm and confusion-diffusion process is governed by regeneration
values are either derived from each other or they are continually evolved of binary sequences. The process of diffusion is also explained and
depending on the method used. After each pass, a new entity is explored in [14,15].
discovered with the aim of optimizing the fitness value. To be more The sections of the paper are as follows: Section 2 describes the
specific, the proposed mechanisms can also be defined as an adaptive background of this research. Section 3 describes the current trends in
system which changes its behaviour with the aim of accommodating image encryption using non-adaptive and genetic approaches. The
changes in its environment [13]. When the necessary conditions are met Section 4 consists of the methods used during the experimentation and
to adapt the system, the current state is evolved into a new state. This overall proposed image encryption approaches. Section 5 dives deep
definition is to do with the encryption system to be in an idle state until it into the results obtained from the proposed methodology. Section 6
comes across a set of parameters which do not produce the required discusses the results obtained from the respective proposed schemes
fitness. The system will check for the fitness generated and only inter while Section 7 concludes the research.
vene when the fitness is below par.
To make the system adaptive, metaheuristic search algorithms such 2. Background
as genetic hill climb, and simulated annealing have been used. When the
system is not producing the correct fitness, the adaptiveness becomes 2.1. Genetic algorithm
active and starts to evolve parameters in accordance with the algo
rithms. This approach has also been extended to encrypt a sequence of A genetic algorithm is an evolutionary model inspired by biological
images which evolves the image solutions and provides a feedback evolution based on the natural selection theory proposed by Darwin
mechanism to either increase or decrease the parameters acting as a [16]. A population of genes for every parameter in the system is created.
negative or positive feedback in the respective cases. These algorithms initially use a genotype obtained from the population
To Sum up, our research seeks following research objectives which for the process which is then evaluated using a fitness function. This
can also be classified as the advantages and novelty proposed through fitness function determines whether the genotype is required to undergo
this algorithm: mutation, or the system has reached convergence.
Objective 1: To develop a novel adaptive image encryption algorithm
- PixAdapt which uses genetic hill climb or simulated annealing 2.1.1. Hill climb
algorithm. This is a simple yet efficient evolutionary algorithm that uses chro
Objective 2: To propose a fitness function with the aim of optimizing mosomes to model possible configurations of parametric sequences of
UACI value of an image using genetic hill climb or simulated annealing. weights and biases [17]. Fig. 1 depicts the process of hill climb. In this
Appropriate experimentation has been designed to verify the same. process, a population of genes is initialized. Genes from the initial
Objective 3: To generate appropriate pseudorandom sequences - population are randomly selected to perform the given task. After the
PixAdapt uses linear feedback shift register and chaos-based maps. A task is completed, the fitness of the system is calculated. If the fitness is
switching mechanism has also been proposed to generate the pseudo within the acceptable range, the algorithm terminates. A gene is only
random sequence. selected for mutation if it has a better fitness score than the previously
Objective 4: To develop a confusion-diffusion process for PixAdapt. highest scoring gene and less than the optimal value. If the fitness is not
Objective 5: To testify the use of metaheuristic search techniques for in the acceptable range, a random gene from the genotype is selected
optimization of PixAdapt algorithm. and mutated. The selected gene is replaced by another gene from the
PixAdapt which is an acronym of the principle “Pixel Encryption respective gene pair initialized at the beginning of the process. The
through genetic adaption”. The algorithm uses genetic approach as well fitness of the newly obtained genotype is then calculated. The process
as it produces correct fitness and adapt to the best value of the param continues until a genotype in the acceptable range is generated.
eters (say, UACI). The aim of the fitness function is to evolve parameters For experimentation purposes, the initial population for analysis has
until they generate an acceptable fitness score hence adaption. The been kept at 100. The fitness of each genotype is calculated by encrypted
proposed method in the near future can prove to be a pioneer to encrypt images with the respective genotype parameters and calculated UACI.
an image using adaptiveness. PixAdapt is Symmetric genetic image The results obtained from each of these have been discussed later in the
encryption algorithm based on substitution-confusion-diffusion. The paper.
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These maps use feedback loops, reliability, and the fractal nature of the proposed algorithm uses four chaotic maps which are: The logistic map,
system to produce such sequences. Most chaos-based encryption systems Tent map, Henon map and Rossler map.
use a single chaotic map for sequence generated while not considering
the effect of the sequences on the respective images [20]. Additionally, • Logistic map
these algorithms usually use the most chaotic range of these maps to
generate pseudo-random numbers used for encryption thus, using a This function has a second degree of polynomial mapping. It is a
static value for all images. Using a static set of parameters for encrypting single variable, discrete-time system which exhibits chaos in selecting
images can lead to inefficient encryption in some cases and this problem suitable “r” values. The mathematical function is as follows:
can be overcome by using more than one pseudo-random sequence and
xn+1 = rxn (1 − xn )
determining the quality of encryption for these parameters. The
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where xn is the current value and xn+1 is the next value. The parameter r Gaussian distribution of mean and sigma equal to 0.1 and 0.2 for the
is known as the reproduction rate. Fig. 4 shows the behaviour of the seed parameter and 0.1 and 0.01 for the r parameter respectively (see
logistic map on changing values of r. According to this figure, the Fig. 5).
function shows chaotic behaviour only when the r value is in the range
of 3.5 to 4. For experimentation, in the Genetic Hill climb algorithm, the • Henon map
genes are generated between 3.6 and 4 while for the simulated
annealing approach, a random value is generated within this range, and The Henon map is a two-dimensional discrete-time dynamical sys
it is continuously modified by adding or subtracting a small value ob tem using two parameters a and b. The Henon map exhibits chaotic
tained from a Gaussian distribution with its mean and sigma values behaviour when the parameter b is kept constant and a is varied in the
equal to 0.1 and 0.01 respectively. The initial seed value for a genetic range of 1 and 1.45. The equation is as follows:
{
algorithm is generated between 0.1 and 1. For the simulated annealing xn+1 = 1 − ax2n + yn
method, the seed value is generated between the range of 0.1 and 1
yn+1 = bxn
while it's continuously modified after each epoch by a random value
from a Gaussian distribution with mean and sigma values equal to 0.1
where xn is the current value and xn + 1 is the next value, yn is the
and 0.2 respectively.
current value and yn + 1 is the next value. Parameters a and b govern the
behaviour of the system. Fig. 8 shows the chaotic performance of the
• Tent map
Henon map when b = 0.3 and a is varied. For experimentation, the
parameters x, y and a are varied in a range of 0 to 1, 0 to 1 and 1 to 1.45
This function uses a single degree of polynomial mapping along with
respectively. In the genetic hill climb approach, genes of size 10 and 100
a parameter “r”. This system can be classified as a discrete-time
for each of the parameters and generated while for the Simulated
dynamical system. Its equation is as follows:
Annealing approach, the initial values chosen at random between the
⎧
⎪
⎪
1 given range and the value for the next iteration is modified by a random
⎨ rxn , xn <
2 value drawn from a Gaussian distribution of mean and sigma equal to
xn+1 =
⎪ 0.1 and 0.2 for the seed parameters (x and y) and 0.1 and 0.01 for the
⎩ r(1 − xn ), xn ≥ 1
⎪
2 parameter a respectively (see Fig. 6).
where xn is the current value and xn+1 is the next value. The parameter r • Rossler map
is a real positive constant. The behaviour of the tent map is like that of
the logistic map. The tent map exhibits chaotic behaviour in the range of The Rossler map or attractor is a three-dimensional continuous-time
r values above 1 and shows very chaotic behaviour at r values equal to 2 dynamical system. The system also has three parameters a, b and c
as shown in Fig. 7. For experimentation, the r value is varied in the range which are used to exhibit chaotic behaviour. The system is defined by
of 1 to 2 and the seed pixel value is varied between 0.1 and 1. In the three non-linear ordinary differential equations which are as follows
genetic hill climb approach, genes of size 10 and 100 for each of the (see Fig. 7):
parameters and generated while for the Simulated Annealing approach,
the initial value is chosen at random between the given range and the
value for the next iteration is modified by a random value drawn from a
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Table 1 images. The algorithm used highest value entropy by keeping low error
Details of sample images for image encryption. rate.
Name Size Congruity Abbasi et al. [28] proposed an algorithm which has highest value of
entropy through experimental analysis. The algorithm used lattice map
Image_1 (256, 256) Symmetrical
Image_2 (300,300) Symmetrical function along with GA (genetic algorithm) to obtain the results. The
Image_3 (300, 300) Symmetrical algorithm achieved 2120 key space value which is resistant to brute
Image_4 (512, 512) Symmetrical force attack. Wang [29] focused on sensitivity of the plaintext through
Image_5 (800, 600) Asymmetrical scrambling. The said algorithm did not reveal the computational
Image_6 (512, 512) Symmetrical
Image_7 (492, 600) Asymmetrical
complexity. The algorithm was resistant to brute-force and statistical
Image_8 (594, 670) Asymmetrical attack. Abbasi et al. [28] proposed an algorithm based on chaos
Image_9 (1990, 1342) Asymmetrical sequence and wavelet transform. The algorithm did not promise that the
Image_10 (1026, 1024) Asymmetrical algorithm's time complexity is efficient or not. Wong et al. [30] have
done detailed study on Cryptanalysis theory on image encryption algo
⎧ rithm. It has proven that the encryption scheme is not as secure as Bis
⎪ dx
⎪
⎪
⎪ =− y− z was et al. [31] proposed an algorithm with the help of parallel
⎪ dt
⎪
⎪
⎨ processing capability for multiple bit-planes encryption. The algorithm
dy
= x + ay achieved good encryption speed and is suitable for real time application.
⎪ dt
⎪
⎪
⎪ [32] proposes an algorithm with 1045 key space sensitivity with
⎪
⎪
⎪ dz
⎩ = b + z(x − c) acceptable speed performance 4.7081Mbt/s and resistance to various
dt attacks.
In the year of 2017, Kaur and Kumar [33] proposes encryption al
where x, y and z are coordinates in the three dimensions and, a, b and c
gorithm with beta chaotic map, nonsubsampled contourlet transform,
are parameters that govern the behaviour of the system. Fig. 9 shows the
and genetic algorithm. The algorithm is better at computational speed
chaotic behaviour of the Rossler map for the x dimension and the
and high encryption intensity. [34] proposes improvised parallel Non-
parameter c. For experimentation, parameters a and b were set to 0.2
Dominated Sorting Genetic Algorithm (NSGA-II)-based encryption al
and parameter c was varied. To obtain high chaoticity, parameter c was
gorithm which has less computational complexity. [35] proposes an
varied for values above 9 in the genetic hill climb and simulated
encryption algorithm with 7.99 entropy. The author claimed that the
annealing algorithm approach. For the simulated annealing approach,
algorithm will also be used in maintaining the privacy of the images.
the c value was modified with a value obtained from a Gaussian distri
[36] proposes an algorithm exclusively for medical images which has
bution with mean and sigma values equal to 0.1 and 0.01 respectively.
better decreasing execution time. The algorithm achieved 2128 key space
attack. Algorithm could be enhanced by experimenting better analysis
3. Literature review
on attacks. [37] proposes PWLCM chaotic map to perform bit level
encryption algorithm. SHA-1 algorithm is used to encrypt the key of an
This section acknowledges the work done in image encryption using
algorithm. It is observed that security and efficiency of an algorithm (in
an adaptive approach. To the best of the knowledge, there were very few
just one round) is better and encryption and decryption time is low. [38]
works found which involved an adaptive system to improve the per
proposed an algorithm with the use of RNA code truth table which forms
formance of an encryption algorithm. Most of these algorithms showed
initial population of genetic algorithm. The algorithm claimed to be high
static behaviour and did not take into the account the computation time
resistance of attacks on 256 × 256 image after 30 repetitions. [39] fol
complexity.
lowed the 2D non-linear couple lattice map and basic genetic operations
The initial work on genetic algorithm-based image encryption star
to develop novel image encryption technique. The prosed technique had
ted by focusing the basic operations of genetic algorithm i.e. selection,
better computational complexity and achieved better key space. [40]
crossover and mutation. Wang and Xu [21] proposed an encryption al
proposes the color image encryption system by integrated non-
gorithm which uses logistic map to generate chaotic sequence. However,
dominated sorting algorithm and basic chaotic map is used to tune the
the algorithm lacks in doing extensive security analysis. Das et al. [22]
hyper parameters of 5D chaotic map. The algorithm achieved the better
proposed an encryption algorithm to encrypt multiple images at the
UACI, Entropy values as compared to exiting techniques. An Inter-
same time. The proposed algorithm had less execution time as compared
twinning logistic map is used in [41] for image encryption. The pri
to other genetic encryption algorithms. Ghazvini et al. [23] proposed
mary outcome of the research is that efficiency of Differential evolution
algorithm with genetic map and piecewise linear chaotic map (PWLCM).
is outperformed than genetic algorithm.
PWLCM has a strong base of confusion and diffusion process which al
Recently, the paper [42] uses quantum genetic algorithm (QGA) and
gorithm resistant to brute force, statistical and differential attack.
compressive sensing (CS) for encryption of images. The algorithm has
Shankar and Eswaran [24] proposed novel approach of using ECC
been found to be resistant to statistical attack and plaintext attack as
(Elliptical curve cryptography) algorithm to generate key used in image
compared to other algorithms. [23] uses Chen's chaotic map and
encryption. The private key of decryption algorithm is generated by
Logistic-Sine map for the process of confusion and diffusion. [43] uses
genetic algorithm. The algorithm lacks in experimenting in security
master slave model for image encryption to improve its computational
analysis and more parameters on statistical analysis.
complexity. [44] uses Keccak algorithm to generate chaotic values.
Over a period of year, perception of sensitivity of the medical images
Genetic operations are managed by Henon map and DNA coding. The
has drawn more attention in the community. To augment this, Pareek
algorithm was found to be resistant to statistical and differential attacks.
and Patidar [25] proposes genetic based image encryption algorithm
[45] invented Pareto-optimal image encryption algorithm which is
which transforms plain image to newly formed image by basic genetic
evaluated against standard parameters such as key sensitivity, entropy,
algorithm operations. The algorithm proven to be resistant to brute-
UACI, NPCR etc. Finally, Chai et al. [46] used color image cryptosystem
force attack, differential attack, plaintext attack and entropy attack
based on improved genetic algorithm and matrix semi-tensor product
prominently. Enayatifar et al. [26] used the DNA sequence and logistic
(STP).
map along with genetic algorithm to encrypt images. However, the al
According to the papers surveyed, none of them have used an
gorithm could be enhanced by proving the computational complexity of
adaptive mechanism to improve the behaviour of encryption system and
the proposed algorithm. Abdullah et al. [27] proposed dynamic
have used constant parameters. Most algorithms did use genetic algo
behaviour wherein enciphered images are constructed using original
rithms but those were for pseudorandom number sequence generation.
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Nearly all surveyed papers used chaos-based sequence generation in one adjusts its parameters to improve itself can also be classified as an
way or another which confirmed the fact that chaos-based image adaptive system. In this paper, two methods of adaptation have been
encryption is the right way to move forward. Another key point explored to improve image encryption. The first method is a genetic
discovered was the difficulties faced to optimize the Unified Average algorithm approach which updates its parameters using the hill climb
Changing Intensity (UACI) value in all the papers. Additionally, the lack algorithm. The second method used is Simulated Annealing which is an
of research in the adaptive encryption field could be bridged by re- improved version of the hill climb algorithm. The goal of each of these
engineering the approach as described in the sections further. adaptive mechanisms is to improve the UACI evaluation parameter to as
close as 33.46 [21,22,24]. Fig. 8 depicts the working of the adaptive
4. Proposed methodology and algorithm image encryption system. The image is fed into the encryption system
which initially encrypts the image using a set of encryption parameters.
4.1. Sample image description The encrypted image is checked for its fitness using the UACI evaluation
parameter. If the fitness is in the acceptable range, the algorithm is
Table 1 shows the sample image sets used during experimentation. terminated. For cases where the fitness is not in the acceptable range, the
Ten images are used to test the proposed algorithm. All the images used parameters are fed into an evolutionary mechanism that makes manip
are of 8-bit grayscale. For the images, an equal number of 5 images each ulations in those parameters. The evolved parameters are then used to
for asymmetrical and symmetrical images have been used. The variation re-encrypt the image. This process continues until near-ideal fitness is
in shapes and sizes helps in validating the viability of the proposed al obtained.
gorithm on different types of images. The image sizes vary from 256 × The purpose of using UACI as a metric of fitness is due to the diffi
256 to 1990 × 1342 thus, covering smaller as well as larger-sized culties obtained while encrypting an image and obtaining an ideal UACI
images. value of 33.46. Other metrics such as entropy and NPCR are not as
sensitive to encryption as UACI and almost all UACI values near 33.46
4.2. Adaptive mechanism of proposed algorithm give us all other metrics in the near-ideal range. For experimentation
purposes, an acceptable range of 33.46 ± 0.01 has been used.
When a system responds to changes in the environment, it can be
called an adaptive system. The method of adaptation varies from 4.3. Proposed PixAdapt algorithm
application to application, but its main objective is to achieve its defined
goal and return to normalcy. A system that monitors its performance and To encrypt an image, two adaptive approaches have been explored
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and implemented. The first approach uses a genetic algorithm based on 4.3.2. Algorithm for chaotic sequence generation
Hill Climb to improve the solution obtained after encrypting an image PixAdapt algorithm uses chaotic values which are generated through
using selection and mutation based on a fitness criterion which is UACI. the above-mentioned pseudorandom sequences. A switching mechanism
Most other image encryption parameters such as entropy, NPCR and has been added into the system as represented in Fig. 9. This switching
correlation do not show much variation irrespective of the method of mechanism gives control to the evolutionary algorithm to activate or
encryption. The second approach uses Simulated Annealing to evolve deactivate the participation of the sequence in generation of the final
the solutions obtained from the cipher image. key at will. For each of the sequences, a single ON or OFF value is also
inserted along with pseudorandom sequence (K1, K2, K3, K4 and K5). A
4.3.1. Algorithm for generating pseudorandom sequences key sequence is generated by XORing all sequences which are active
Algorithms 1 and 2 contain the different methods involved in the finally producing the secret key (K).
generation of pseudorandom sequences, using chaotic methods and Algorithm 3 and Fig. 9 depicts the process of chaotic sequence gen
linear feedback shift register. PixAdapt algorithm uses the following eration used in PixAdapt algorithm.
chaotic methods namely - Logistic map, Tent map, Henon map and
Algorithm 3. Chaotic sequence generation.
Rossler map.
Algorithm 1. Pseudorandom chaotic sequences.
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Fig. 10, and Algorithm 4 depict the whole process confusion and generated using the following ranges.
diffusion process. Logistic map
r: 3.6 – 4.0
Algorithm 4. Confusion and diffusion. Seed value: 0.01 – 1
Rossler map
c: 9, 10, 13 or 18
Tent map
Seed value: 0.01 – 1
r: 1 – 2
Henon map
x seed value: 0.1 – 1
y seed value: 0.1 – 1
a: 1- 1.4
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Table 2
Fitness analysis (UACI).
Image name UACI value (without an adaptive mechanism) UACI value (with Genetic Hill Climb) UACI value (with simulated annealing)
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Image_3 and Image_10. The initial value of Image_3 is around 20 while 5.2.3. Tent map
initial value of Image_10 is observed between 26.0 and 28.0. According to Fig. 17, for the simulated annealing approach,
Image_10 draws an attention to the optimal value of UACI. Using
5.2.2. Rossler map simulated annealing approach, when the map was ON both the seed
According to Fig. 16, in the genetic hill climb approach, c values value and r value, UACI value observed was low and when the map was
around 13 did not work well and in fact, when the Rossler map was off, OFF, it was in the acceptable range.
the UACI value was in the acceptable range. For most images, Rossler
map did not contribute to increasing fitness. Using the simulated 5.2.4. Henon map
annealing approach, for Image_3 and Image_10 achieves optimal value Henon map using genetic hill climb approach gives better results in
of UACI when the map was ON. case of 5 images (see Fig. 18). While for Image_5, it can be deduced that
when the map was ON, the fitness value was low. Henon map using
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Fig. 18. Henon map parameters x seed, y seed and a value parameter.
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Table 3 ∑M ( ∑M )( ∑M )
image entropy readings.
1
i=1 xi − M j=1 xj yi − M1 j=1 yj
rxy = √̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
( ) ̅√̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
( ̅
Image Entropy ∑M 1
∑M 2 ∑M 1
∑M ) 2
name i=1 x i − M j=1 x j i=1 y i − M j=1 y j
Original Encrypted image
Image
Non- Genetic Simulated
where xi and yi are pairs of horizontal, vertical, and diagonal ith adjacent
adaptive algorithm annealing
coordinates and M represents the overall number of adjacent pixel pairs
Image_1 7.360105 7.996818 7.996992 7.997248
under consideration. For our analysis, 10,000-pixel pairs adjacent to
Image_2 6.889566 7.997899 7.998016 7.997651
Image_3 4.961201 7.997850 7.997777 7.998098
each other in the horizontal, diagonal and verticals directions are chosen
Image_4 7.558386 7.999226 7.999277 7.999323 at random, and their correlation coefficients have been calculated.
Image_5 4.931948 7.999658 7.999618 7.999557 Fig. 19 depicts the distribution of the correlation coefficients before
Image_6 7.052831 7.999248 7.999314 7.999311 and after encryption. The correlation distribution of the original images
Image_7 6.559510 7.999307 7.999323 7.999279
is high and concentrated in a single region. After encryption, the dis
Image_8 7.452617 7.999585 7.999521 7.999580
Image_9 7.535319 7.999932 7.999932 7.999931 tribution of the image is scattered and the correlation coefficients are
Image_10 7.314666 7.999833 7.999818 7.999838 highly dispersed thus, the image after encryption has a very low degree
of correlation in comparison to the original image.
In Table 4, the individual correlation coefficient of the test images in
simulated annealing approach creates a positive impact when the map is the vertical, diagonal, and horizontal directions have been shown. The
ON. correlation coefficient of the original image is very close to unity thus,
According to results obtained from the above analysis, it can be showing high correlation. After encryption, using the genetic hill climb
concluded that a single pseudo-random number does not always give the as well as the simulated annealing in all three directions show very low
correct pseudorandom sequence to encrypt the image with. Logistic map correlation to zero correlation. Thus, after encrypting an image using the
parameters had the most to contribute to but for the other images, the proposed algorithm, the cipher image pixels are uncorrelated to each
tent and Henon map sequences worked very well. For the most part, the other in the vertical, diagonal, and horizontal directions exhibiting high
switching mechanism provides another level of adaptation to the resistance against statistical attacks.
already evolving solutions to check for the presence and absence of a few
pseudorandom sequences. 5.3.3. Contrast
Contrast of an image is the overall variation in intensity of pixels
5.3. Statistical analysis with respect to their neighbours. Images after encryption should exhibit
a high degree of contrast in comparison to the original image. Contrast is
It is essential for a cryptosystem to be resistant against statistical calculated using the following mathematical expression:
attacks. Parameters such as (i) entropy, (ii) correlation, (iii) contrast,
∑levels− 1
(iv) histogram analysis, and (v) chi-square test provide statistical in C= Pi,j × (i − j)2
sights into the behaviour of the plain and cipher image. The results
i,j=0
obtained are as follows: where P is grey-level co-occurrence matrix containing the intensity
values which are calculated from the original image, and i and j are the
5.3.1. Entropy row and column values for the respective GLCM matrix. The total
The algorithm's robustness is a crucial factor. For any encryption number of levels is 256.
algorithm to be successful, the measure of entropy should be as high as Table 5 shows the contrast values obtained before and after
possible (ideally ~7.999 for 8-bit images). Image entropy is calculated encryption in the vertical, horizontal, and diagonal directions. Accord
using the following: ing to the results, before encrypting the image, the contrast of the images
∑ ( )
1 is significantly lesser than after encryption. For both encryption
H(m) = pi log2
pi schemes, the contrast is amplified significantly thus, there is very low
predictability of the plain image from the cipher image and high security
where pi denotes the likelihood of the ith pixel value. is demonstrated.
Table 3 shows the entropy values obtained from the original image,
non-adaptive (static parameter) image encryption, genetic hill climb 5.3.4. Histogram analysis
image encryption and simulated Annealing image encryption. Accord For a cryptosystem to be resistant against statistical attacks, the
ing to the results, it can be observed that the cipher image obtained from histogram of cipher image must be uniformly distributed.
the three methods of encryption has a much higher value of entropy than According to Fig. 20, the histogram prior to encryption depicts the
the original image. The cipher images have entropy values in the ideal features present in the image and has a regular shape. After encryption,
range thus, the encryption algorithm with static parameters and the the image has no similarity with respect to the original image and this
adaptive image encryption system demonstrates high resistance to en argument is solidified by looking at the flattened histogram obtained
tropy attack. after encryption. Thus, the proposed cryptosystem is resistant to histo
gram statistical attack.
5.3.2. Correlation
Correlation can be defined as a method of determining the similarity 5.3.5. Chi square test
between the plain image and the encrypted image. Correlation is Statistical features can be further analysed by using the chi square
calculated in the vertical, horizontal, and diagonal directions. Plain test. The mathematical formula is as follows:
images have a high degree of correlation in all directions while a suc
∑
N v− 1
(oi − ei )2
cessfully encrypted image should have near zero correlation. The cor x2exp =
relation coefficient is defined as follows: i=0
ei
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R. Tuli et al. Chaos, Solitons and Fractals: the interdisciplinary journal of Nonlinear Science, and Nonequilibrium and Complex Phenomena 164 (2022) 112628
Fig. 19. Correlation coefficients of the plain and encrypted image in the horizontal, vertical and diagonal.
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R. Tuli et al. Chaos, Solitons and Fractals: the interdisciplinary journal of Nonlinear Science, and Nonequilibrium and Complex Phenomena 164 (2022) 112628
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R. Tuli et al. Chaos, Solitons and Fractals: the interdisciplinary journal of Nonlinear Science, and Nonequilibrium and Complex Phenomena 164 (2022) 112628
Table 4
Plain image and cipher image correlation coefficient in the vertical, diagonal, and horizontal direction.
Image name Original image Genetic hill climb Simulated annealing
Image_1 0.936756 0.914034 0.966708 9.7e-05 0.000809 − 0.001513 − 0.000741 0.003105 − 0.003962
Image_2 0.845865 0.753027 0.852888 − 0.008226 − 0.002724 0.005994 − 0.004255 − 0.001084 − 0.000428
Image_3 0.442394 0.440309 0.440271 0.003212 0.001237 − 0.000249 − 0.002066 0.003409 − 0.000793
Image_4 0.978764 0.969448 0.978875 − 0.000439 − 0.00223 0.000922 0.002257 0.000504 0.00059
Image_5 0.940822 0.919419 0.941767 0.001529 − 0.000761 2.5e-05 0.002624 0.000459 − 0.001285
Image_6 0.983025 0.973168 0.989951 − 0.001511 0.00029 − 0.002639 -4e-06 − 0.002383 0.003106
Image_7 0.99759 0.995939 0.998192 0.001126 − 0.000423 0.002554 0.002422 − 0.001185 0.000181
Image_8 0.882058 0.834933 0.933329 0.001117 0.003009 0.000799 0.001033 0.000127 − 0.002202
Image_9 0.968638 0.936685 0.962609 6.8e-05 − 9.4e-05 − 6.5e-05 0.250232 0.25383 0.255514
Image_10 0.887737 0.826111 0.89895 0.001656 − 0.000524 − 0.001162 0.00178 0.000171 − 0.000249
Table 5
Horizontal, vertical, and diagonal contrast values for plain and cipher images.
Image name Original image Genetic hill climb Simulated annealing
Image_1 254.039614 345.894487 134.060218 10,898.589246 10,888.302284 10,913.363189 10,851.312914 10,810.927213 10,885.171768
Image_2 907.680569 1454.248991 865.94194 11,037.591828 10,978.066957 10,883.186132 10,984.660814 10,952.946723 10,946.485095
Image_3 1540.183266 1547.189125 1546.264705 10,890.990691 10,913.429391 10,930.336154 10,964.742798 10,905.218051 10,953.53262
Image_4 122.666811 176.441297 121.98689 10,925.40336 10,944.926479 10,910.346307 10,875.98856 10,894.944895 10,894.370861
Image_5 718.630033 978.933667 707.220831 10,910.639439 10,935.793089 10,927.328274 10,900.421233 10,924.496117 10,943.913765
Image_6 130.736213 206.838002 77.355163 10,923.918619 10,904.579256 10,935.780157 10,922.493594 10,948.272169 10,888.418018
Image_7 34.25758 57.695769 25.726665 10,948.818031 10,966.063085 10,933.311349 10,894.412743 10,933.073599 10,917.516756
Image_8 935.513768 1308.701815 528.775022 10,907.32826 10,886.679989 10,910.832138 10,899.459805 10,909.019908 10,934.546124
Image_9 146.138616 295.044101 174.229581 10,931.009161 10,932.918384 10,932.399051 7394.536379 7359.426842 7342.757700
Image_10 479.069466 742.119379 433.447717 10,901.4611 10,925.377448 10,932.471862 10,876.090467 10,893.788095 10,898.309924
following equation: According to Table 7, the NPCR value after encryption is best for
genetic hill climb. All the values obtained using genetic hill climb are
M×N
ei = above the ideal threshold. For simulated annealing, few values (image_1
256
and image_3) have NPCR values less than the ideal threshold. The non-
The ideal chi-square value of a histogram to be resistant to histo adaptive method performs the worst and has several images with less-
grams attacks with a significance level of 0.05 should be, x2th(255, 0.05) than-ideal threshold. Thus, genetic hill climb approach gave the ideal
= 293. Thus, according to Table 6, all chi-square values for the genetic NPCR values.
hill climb algorithm and other than two values from the simulated
annealing method are less than the ideal value thus, showing resistance 5.3.8. Key space
to histogram attacks. For the encrypted image_9 and image_10 using Key space can be defined as the total number of possible keys which
simulated annealing, the chi-square value is greater than ideal thus, it is can be produced to be used in the encryption method. To create an
not resistant to histogram attacks. According to the chi-square results, encryption system which is resistant to brute force attacks, the crypto
genetic hill climb outperformed simulated annealing for image system should have a key space which is greater than 2100 [47] The
encryption. proposed algorithm consists of five pseudorandom sequence generators
namely, (i) Logistic map (r and xseed), (ii) Tent map (r and xseed), (iii)
5.3.6. Quality of encryption Henon map (xseed, yseed and a), (iv) Rossler map (c) and (v) Linear
Each of the images encrypted have been ciphered using adapting feedback shift register (lfsrseed). For the first four methods, a precision of
parameters to optimize the UACI value. While UACI does play an 10− 16 can be used for each parameter. For the proposed encryption
important role in determining the encryption, the parameters such as scheme, an image is encrypted only if a minimum of three sequences are
NPCR, key space and key sensitivity give us additional insights into the used.
success or failure of the encrypted image. The minimum key space can be summarized as:
( ( ) )
5.3.7. NPCR Key space = 4 × 1016 + 28 = 2220
Number of pixels change rate is a parameter used to determine the The maximum key space can be summarized as:
resistance of cipher images to differential attacks. The difference be ( ( ) )
tween the plain image and the ciphered image in percentage is defined Key space = 8 × 1016 + 28 = 2432
as the NPCR. The ideal value for NPCR is above 99.60 %. It is defined as
Since the minimum key space is greater than 2100, the proposed
follows:
encryption scheme is resistant to brute force attacks.
∑
D(i, j)
NPCR = × 100%
H×W 5.3.9. Key sensitivity
{ Even a small change in the keys of the encryption system should lead
D(i, j) =
0, C1 ∕
= C2 to generation of a new chaotic sequence dissimilar from the original one.
1, C1 = C2 In the proposed algorithm, four dynamical systems in their chaotic states
are used to generate the chaotic sequences used for encryption. Even a
where D(i, j) is a bipolar array and, C1 and C2 are cipher and original
small change in the parameters in the chaotic range of parameters for
images.
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R. Tuli et al. Chaos, Solitons and Fractals: the interdisciplinary journal of Nonlinear Science, and Nonequilibrium and Complex Phenomena 164 (2022) 112628
Fig. 20. Analysis of histograms (original image, original image histogram, cipher image, cipher image histogram).
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R. Tuli et al. Chaos, Solitons and Fractals: the interdisciplinary journal of Nonlinear Science, and Nonequilibrium and Complex Phenomena 164 (2022) 112628
Table 6 correlation values for both techniques (Genetic Hill Climb & Simulated
Chi-square test results for genetic hill climb algorithm and simulated annealing Annealing) showed similar results. While for metrics such as Contrast,
ciphered images with 0.05 significance level. chi-square, and NPCR, the Genetic Hill Climb algorithm displayed
Image name Genetic algorithm Chi-square Simulated annealing Chi-square slightly better results in comparison to Simulated Annealing. Both the
Image_1 229.664062 278.765625
adaptive approaches showed significantly better results than the static
Image_2 237.458689 293.897435 parameters in terms of statistical, histogram and quality of encryption
Image_3 216.894586 237.658119 analysis.
Image_4 275.691406 219.435546 The experimentation conducted was to test the performance of the
Image_5 254.260266 294.988799
adaptive mechanism with the aim of optimizing fitness. PixAdapt was
Image_6 248.544921 225.015625
Image_7 276.844752 245.379011 tested for parameter evolution, fitness optimization, statistical analysis,
Image_8 264.498069 231.903474 and quality of encryption using non-adaptive, simulated annealing, and
Image_9 272.774805 305.953887 genetic hill climb methods. Genetic hill climb performed better than all
Image_10 254.779239 319.307992 other methods therefore, the system also extended its adaptiveness by
activating and deactivating a few pseudo-random sequences based on
image thus, creating a dependency on the original image as well. The
Table 7 addition of a switching mechanism proved to increase the overall
NPCR values for ciphered non adaptive, genetic hill climb and simulated adaptiveness of the algorithm. This mechanism improved the search
annealing approaches. space of the metaheuristic and genetic algorithms by adding more detail
Image name NPCR
into the behaviour of the cipher image. Without the switching mecha
nism, the non-adaptive image encryption sequence did not produce re
Non-adaptive Genetic algorithm Simulated annealing
sults in the acceptable range thus, cementing its place in the adaptive
Image_1 99.600220 99.644470 99.597168 mechanism.
Image_2 99.658889 99.637777 99.618889
During experimentation, all the parameters were adapted to opti
Image_3 99.595556 99.602222 99.578889
Image_4 99.624634 99.615478 99.603653
mize the solutions and it can be concluded that using a single set of
Image_5 99.587708 99.603750 99.606458 parameters to encrypt an image is not ideal. Through this paper, it was
Image_6 96.174240 99.621200 99.614334 also discovered that using more than a single pseudo-random sequence
Image_7 99.608062 99.621612 99.603997 may be needed to generate a key that will produce ideal image
Image_8 99.618323 99.609779 99.613548
encryption results.
Image_9 99.607950 99.606714 99.609785
Image_10 99.589197 99.606805 99.613468 In the proposed image encryption scheme, there are several param
eters used to generate chaotic maps and a loop which is used to reach
convergence (fitness in the acceptable range). These factors contribute
these dynamical system leads to a large change in the value. It was found to high time complexity for the overall algorithm and can be overcome
that a change of 10− 16 in the original parameters caused a completely using approaches such as parallel processing [48], map reduce [49] or
new chaotic sequence to be formed thus, chaoticity adds to the sensi GPUs [50]. Additionally, the permutation process can be split into
tivity of the sequence generated. several parts and the ideal range of values could be used to encrypt the
image [51].
6. Discussion Fig. 21 shows the skeleton workflow of an encryption scheme using
parallel processing. The chaotic sequence and the flattened original
According to the results obtained, it can be concluded that the in image will be partitioned into smaller parts such that each of the sub-
clusion of an adaptive mechanism did improve the quality of encryption. partitions will be handed by a separate processor or a GPU. Individual
Using an adaptive system almost certainly guaranteed better encryption chaotic sequence will be used to encrypt the partitioned image as shown
results in comparison to using a non-adaptive system. Two metaheuristic in the figure. The encrypted partitions will then be combined to obtain
approaches for parameter evolution were discussed which were the the final image. Additionally, for the algorithm proposed in this paper,
genetic hill climb and simulated annealing algorithm. Both these algo each set of parameters from the initial population can be used to
rithms showed improved results in comparison to the non-adaptive generate a chaotic sequence which will be handled by a single processor
encryption system. Simulated annealing, genetic hill climb and non- or process in n processes/processors. These chaotic sequences will
adaptive image encryption showed similar performance in terms of encrypt the entire image and they will be checked against the fitness
entropy, correlation, and contrast. The UACI value obtained from non- condition. If the fitness of the newly encrypted image from a single
adaptive image encryption was below par. Genetic hill climb algo process is better than the current benchmark, the new sequence will be
rithm outperformed simulated annealing algorithm in terms of NPCR used to check the fitness with the next set of parameters until conver
and chi-square test. gence is achieved.
As described in the introductory section, PixAdapt fits all definitions
of an adaptive system. Both these approaches show adaptive behaviour 7. Conclusion and future scope
with the main aim to optimize the quality of encryption and display
resistance against statistical, brute-force, plain-text and cipher-text at Most image encryption algorithms use only a static set of image
tacks. UACI was used as a parameter to measure the fitness of the encryption parameters to encrypt an image and do not consider the
encrypted image. The results obtained also reaffirmed the fact that using features present in the image. In this paper, PixAdapt – an adaptive
UACI as a fitness parameter was appropriate. The entropy, NPCR, cor image encryption process was proposed and implemented. PixAdapt
relation, contrast, and histogram analysis revealed that using UACI as a uses an evolutionary algorithm to encrypt an image by evolving pa
fitness parameter did optimize the other parameters as well. These rameters to achieve optimal fitness in an accepted range of values. The
image evaluation parameters showed near ideal results for adaptive evolutionary algorithm or metaheuristic algorithm activates itself to
image encryption as well adaptive sequence encryption approaches. evolve the parameters when the optimal fitness is not obtained. Through
Further, the cryptosystem showed a very high key space and key this research, it was observed that UACI is an appropriate parameter to
sensitivity which proved that the overall adaptive mechanism worked be used as a fitness function. A novel approach towards activating and
well for all kinds of images. deactivating the pseudorandom sequences was implemented in this
According to Tables 2, 3, 4 and Fig. 20, the UACI, entropy, paper. The addition of this process demonstrated better results than non-
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R. Tuli et al. Chaos, Solitons and Fractals: the interdisciplinary journal of Nonlinear Science, and Nonequilibrium and Complex Phenomena 164 (2022) 112628
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