# Four-dimensional hybrid chaos system and its application in creating a
secure image transfer environment by cellular automata

R. Parvaz a111<EMAIL_ADDRESS><EMAIL_ADDRESS>, Y.
Khedmatia222Corresponding<EMAIL_ADDRESS><EMAIL_ADDRESS>,
Y. Behroo<EMAIL_ADDRESS>
###### Abstract

One of the most important and practical researches which has been considered
by researchers is creating secure environments for information exchanges. Due
to their structures, chaos systems are efficient tools in the are of data
transferring. In this research, using a mathematical structure such as
composing and transferring, we improve classical chaotic systems by creating a
four-dimensional system. Then we introduce a new encryption algorithm based on
the chaos and cellular automata. The security of the proposed environment
which is evaluated using different types of security tests shows the
efficiency of the proposed algorithm.

aDepartment of Mathematics, University of Mohaghegh Ardabili, 56199-11367
Ardabil, Iran.

Keywords: Cryptography; Image; Chaotic system; Cellular automata.

##  1 Introduction

With the rapid development of social networks, a huge amount of various
information in the form of text, image, audio and video is exchanged through
these networks in any small period of time. Because of their visual features,
digital images have been received more attention among other formats, and so
it is important to present a new method or improve existing methods in order
to transfer images safely. In addition to steganography, cryptography is a
method of secure transfer of information in which the goal is to scramble
pixels of data properly so that it is not detectable [4]. So far, various
cryptographic methods have been proposed based on cellular automata, chaotic
mapping, and so on for example see [8, 16, 24]. Chaos systems are nonlinear
phenomena with random-like behaviors. These maps are very important in
information security transforms with due regard to their special features,
which we will discuss in detail in future sections. Given that researchers
have shown that algorithms that use existing chaotic maps are likely to be
attacked [23], defining appropriate maps is a priority for cryptographic
methods. The dimension of a chaos maps are determined based on the number of
input and output variables. In general, it can be said that high-dimensional
chaotic maps, although having high computational cost, perform better than
one-dimensional maps due to their complex dynamic structure.

Cellular automata is one of the image encryption tools that has been used in
this article due to its fast, easy and high speed process. In [2], Ray et al
use chaos and 3D automation to encode digital images. First, the replacement
operation is performed by a chaotic system, and then the diffusion operation
is performed using a three-dimensional cellular automata. This method has
shown good resistance to most known cryptographic attacks. Wang’s method is an
image encryption system based on chaos theory and cellular automation. This
method uses a logistic map and reversible cellular automata. Pixel values
divided in four-bit units, then in the permutation step of logistic map and
the distribution step, cellular automation is also used. This method of
cryptography, which is one of the symmetric methods, has good security and
shows good resistance against differential attacks [17].

##  2 4D Hybrid Chaos Systems

In the first step of this section, we introduce a novel type of four-
dimensional chaotic system by improving its overall structure, and then in the
second subsection, we evaluate the proposed chaos behavior using the various
tests such as lyapunov exponent and cobweb diagram.

###  2.1 Structure of the Proposed Hybrid System

In this subsection, more details about the proposed new hybrid chaos systems
based on Tent, Sin and Logistic maps are given. The general structure of the
proposed chaos system has been given in Fig. 1. The combination parts in the
proposed system based on Tent, Sin and Logistic maps are shown in Fig. 2. The
mathematics formulae for each of the parts can be written as follows.

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Figure 1:  The structure of the proposed chaotic system.

__

Figure 2:  The structure of the combination parts in the proposed system.

-First combination box:

$\displaystyle
x_{i+1}:=\left\\{\begin{array}[]{ll}\alpha^{x}_{1}f^{x}_{1}\circ
F^{x}_{1}(r,x_{i})+g^{x}_{1}(r,x_{i},y_{i},z_{i},w_{i})\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+h_{1}^{x}(\frac{(\beta^{x}_{1}-r)z_{i}}{2})~{}mod~{}1,~{}~{}when~{}w_{i}<0.5,\\\
\\\ \alpha^{x}_{2}f^{x}_{2}\circ
F^{x}_{2}(r,x_{i})+g^{x}_{2}(r,x_{i},y_{i},z_{i},w_{i})\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+h_{2}^{x}(\frac{(\beta^{x}_{2}-r)(1-z_{i})}{2})~{}mod~{}1,~{}~{}when~{}w_{i}\geq
0.5.\\\ \end{array}\right.$ (2.6)

-Second combination box:

$\displaystyle
y_{i+1}:=\left\\{\begin{array}[]{ll}\alpha^{y}_{1}f^{y}_{1}\circ
F^{y}_{1}(r,y_{i})+g^{y}_{1}(r,x_{i+1},x_{i},y_{i},z_{i},w_{i})\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+h_{1}^{y}(\frac{(\beta^{y}_{1}-r)\zeta_{1}}{2})~{}mod~{}1,~{}~{}when~{}\zeta_{1}<0.5,\\\
\\\ \alpha^{y}_{2}f^{y}_{2}\circ
F^{y}_{2}(r,y_{i})+g^{y}_{2}(r,x_{i+1},x_{i},y_{i},z_{i},w_{i})\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+h_{2}^{y}(\frac{(\beta^{y}_{2}-r)(1-\zeta_{1})}{2})~{}mod~{}1,~{}~{}when~{}\zeta_{1}\geq
0.5.\\\ \end{array}\right.$ (2.12)

-Third combination box:

$\displaystyle
z_{i+1}:=\left\\{\begin{array}[]{ll}\alpha^{z}_{1}f^{z}_{1}\circ
F^{z}_{1}(r,w_{i})+g^{z}_{1}(r,x_{i+1},x_{i},y_{i+1},y_{i},z_{i},w_{i})\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+h_{1}^{z}(\frac{(\beta^{z}_{1}-r)\zeta_{2}}{2})~{}mod~{}1,~{}~{}when~{}\zeta_{2}<0.5,\\\
\\\ \alpha^{z}_{2}f^{z}_{2}\circ
F^{z}_{2}(r,w_{i})+g^{z}_{2}(r,x_{i+1},x_{i},y_{i+1},y_{i},z_{i},w_{i})\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+h_{2}^{z}(\frac{(\beta^{z}_{2}-r)(1-\zeta_{2})}{2})~{}mod~{}1,~{}~{}when~{}\zeta_{2}\geq
0.5.\\\ \end{array}\right.$ (2.18)

-Fourth combination box:

$\displaystyle
w_{i+1}:=\left\\{\begin{array}[]{ll}\alpha^{w}_{1}f^{w}_{1}\circ
F^{w}_{1}(r,z_{i})+g^{w}_{1}(r,x_{i+1},x_{i},y_{i+1},y_{i},z_{i+1},z_{i},w_{i})\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+h_{1}^{w}(\frac{(\beta^{w}_{1}-r)\zeta_{3}}{2})~{}mod~{}1,~{}~{}when~{}\zeta_{3}<0.5,\\\
\\\ \alpha^{w}_{2}f^{w}_{2}\circ
F^{w}_{2}(r,z_{i})+g^{w}_{2}(r,x_{i+1},x_{i},y_{i+1},y_{i},z_{i+1},z_{i},w_{i})\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+h_{2}^{w}(\frac{(\beta^{w}_{2}-r)(1-\zeta_{3})}{2})~{}mod~{}1,~{}~{}when~{}\zeta_{3}\geq
0.5,\\\ \end{array}\right.$ (2.24)

where $a_{1}:=\\{r,x_{i},y_{i},z_{i},w_{i}\\},$
$a_{2}:=\\{r,x_{i},x_{i+1},y_{i},z_{i},w_{i}\\}$,
$a_{3}:=\\{r,x_{i},x_{i+1},y_{i},y_{i+1},z_{i},w_{i}\\}$,
$a_{4}:=\\{r,x_{i},x_{i+1},y_{i},y_{i+1},z_{i},z_{i+1},w_{i}\\}$, and
$\xi_{\tau}=\tau_{i}$ or $\tau_{i+1},$ for $\tau=x,y,z.$
$F_{\varsigma}^{\tau},$ for $\tau=x,y,z,w,\varsigma=1,2$, can be considered as
Sin or Logistic maps. $\alpha_{\varsigma}^{\tau},\beta_{\varsigma}^{\tau},$
for $\tau=x,y,z,w,\varsigma=1,2,$ are arbitrary number in R, and
$g_{\varsigma}^{\tau},h_{\varsigma}^{\tau}$ are considered as arbitrary
sufficiently smooth functions. In the proposed system, the best feature of the
different chaos maps as Tent, Sin and Logistic maps have been improved by
using Composition and transfer operator. In the following, the basic
properties of the hybrid system have been studied. In order to study hybrid
system, the following cases have been considered.

Case i:
$\\{\alpha_{1}^{x},\alpha_{2}^{x},\alpha_{1}^{y},\alpha_{2}^{y},\alpha_{1}^{z},\alpha_{2}^{z},\alpha_{1}^{w},\alpha_{2}^{w}\\}$
$=\\{1,16,10,20,10,20,10,20\\}$,
$\\{\beta_{1}^{x},\beta_{2}^{x},\beta_{1}^{y},\beta_{2}^{y},$
$\beta_{1}^{z},\beta_{2}^{z},\beta_{1}^{w},\beta_{2}^{w}\\}$
$=\\{6,2,50,30,50,30,50,30\\}$, $\\{\xi_{x},\xi_{y},\xi_{z}\\}$
$=\\{x_{i},y_{i},z_{i}\\}$, $f_{1}^{x}(p)=\cosh(p)$, $f_{2}^{x}(p)=\cot(p)$,
$f_{1}^{y}(p)=f_{1}^{z}(p)=f_{1}^{w}(p)=p$,
$f_{2}^{y}(p)=f_{2}^{w}(p)=\sin(\pi p)$, $f_{2}^{z}(p)=\exp(\pi p),$
$g_{1}^{x}(a_{1})=15\tanh(rx_{i}+z_{i})+\sin((w_{i})+12\cos(rx_{i})$,
$g_{2}^{x}(a_{1})=-7ry_{i}+\exp(1+2w_{i})+z_{i}+7\log(\pi rx_{i})$,
$g_{1}^{y}(a_{2})=2\tan(rx_{i}+y_{i}+2z_{i}+w_{i}),$
$g_{2}^{y}(a_{2})=z_{i}+w_{i}+14\exp(20rx_{i})$,
$g_{1}^{z}(a_{3})=2\tan(rx_{i}+y_{i})+w_{i}+z_{i},$
$g_{2}^{z}(a_{3})=14\exp(20rx_{i}+w_{i})+\sin(z_{i})$,
$g_{1}^{w}(a_{4})=2\tan(rx_{i}+y_{i}+z_{i})$
$+w_{i},g_{2}^{w}(a_{4})=14\exp(20rx_{i}+w_{i})$
$+z_{i},h_{1}^{x}(p)=\sin(2p)$ ,$h_{2}^{x}(p)=4p,h_{2}^{y}(p)=\cot(p)$,
$h_{1}^{y}(p)=h_{1}^{z}(p)=h_{1}^{w}(p)=\exp(2p)$, $h_{2}^{z}(p)$
$=h_{2}^{w}(p)=\cot(4p)$.

Case ii:
$\\{\alpha_{1}^{x},\alpha_{2}^{x},\alpha_{1}^{y},\alpha_{2}^{y},\alpha_{1}^{z},\alpha_{2}^{z},\alpha_{1}^{w},\alpha_{2}^{w}\\}$
$=\\{7,12,14,14,3,15,15,10\\}$,
$\\{\beta_{1}^{x},\beta_{2}^{x},\beta_{1}^{y},\beta_{2}^{y},$
$\beta_{1}^{z},\beta_{2}^{z},\beta_{1}^{w},\beta_{2}^{w}\\}=$
$\\{69,28,68,36,33,2,5,7\\}$, $\\{\xi_{x},\xi_{y},\xi_{z}\\}=$
$\\{x_{i+1},y_{i+1},z_{i+1}\\}$, $f_{1}^{x}(p)=\cos(p)$,
$f_{2}^{w}(p)=\cos(\sin(\pi p))$, $f_{2}^{x}(p)=f_{2}^{y}(p)=p$,
$f_{1}^{z}(p)=\sin(p)$, $f_{1}^{w}(p)=\sin(\pi p)$, $f_{2}^{z}(p)=\exp(\pi
p),$
$g_{1}^{x}(a_{1})=15\tan(rw_{i}+x_{i}+2z_{i})+\sin(w_{i})+12\cos(rx_{i})$,
$g_{2}^{x}(a_{1})=7\sin(ry_{i}+w_{i})-7ry_{i}+x_{i}+2w_{i}+z_{i}-1$,
$g_{1}^{y}(a_{2})=14\cos(20rx_{i}+x_{i+1})$,
$g_{1}^{z}(a_{3})=\tan(x_{i+1}+y_{i+1})+w_{i}$
$z_{i}+2(rx_{i}+y_{i})$,$g_{2}^{z}(a_{3})$
$=14(rx_{i}+w_{i})+y_{i+1}+\sin(z_{i})$,
$g_{1}^{w}(a_{4})=14\cos(20rx_{i}+x_{i+1})$ $+\log(x_{i}+w_{i})$,
$g_{2}^{w}(a_{4})=14\sinh(x_{i}+rx_{i+1}+w_{i})$
$+z_{i}+\sin(z_{i+1}+y_{i+1})$,
$h_{1}^{x}(p)=\cos(20p),h_{1}^{x}(p)=\sin(2p),$ $h_{1}^{y}(p)=\exp(2p)$,
$h_{2}^{x}(p)=h_{2}^{y}(p)=4p$, $h_{1}^{z}(p)=\cosh(2p),$
$h_{1}^{w}(p)=\exp(4p)$, $h_{2}^{z}(p)=h_{2}^{w}(p)=\coth(p)$.

Also, in the case i, $\\{F_{1}^{x},F_{1}^{z},F_{2}^{z},F_{2}^{w}\\}$ are
considered as Logistic map, and $\\{F_{2}^{x},F_{1}^{y},F_{2}^{y},$
$F_{2}^{z},F_{1}^{w}\\}$ are considered as Sin map. For case ii,
$\\{F_{1}^{x},F_{2}^{y},F_{1}^{z},$ $F_{2}^{z},F_{1}^{w},F_{2}^{w}\\}$ and
$\\{F_{2}^{x},F_{1}^{y}\\}$ are considered as Logistic and Sin maps,
respectively.

###  2.2 Chaotic Behavior Analysis

In this subsection some important tests for the proposed chaos system are
discussed. One of the important value in the study of the behavior of the
chaos system is Lyapunov exponent or Lyapunov characteristic exponent. An
$n$-dimensional chaos systems in general have n values for Lyapunov exponent.
There are many methods for calculating this value [21, 11, 3]. The method
based on QR algorithm has been used for obtained Lyaponov exponent in Fig. 3
for the case i. More details about this method can be found in [3]. The
positive or negative values of the resulting values are related to the
structure of a chaos system. This relation had been studied in many papers. In
[14], the relation has been given as follows “In an n-dimensional dynamical
system we have n Lyapunov exponents. Each $\lambda_{k}$ represents the
divergence of k-volume. The sign of the Lyapunov exponents indicates the
behavior of nearby trajectories. A negative exponent indicates that
neighboring trajectories converge to the same trajectory A positive exponent
indicates that neighboring trajectories diverge [14]”. Also the following
theorem has been given in [7] for this value.

######  Theorem 2.1.

If at least one of the average Lyapunov exponents is positive, then the system
is chaotic; if the average Lyapunov exponent is negative, then the orbit is
periodic and when the average Lyapunov exponent is zero, a bifurcation occurs.

The results in Fig. 3 show that all four value of Lyapunov exponent in the
proposed systems are positive. By using above studies, we can say that the
proposed system in the all neighboring trajectories diverge. In order to
compare proposed system, the Lyapunov exponent has been compared with 4D
Chaotic Laser System [9] in Fig. 3. It is observed that the proposed system
has better chaos behavior than Chaotic Laser System. Another tool for study
chaotic behavior is bifurcation analysis. In the Fig.s 4-5, the results for
the bifurcation analysis of the case i and ii of the proposed system have been
shown. The chaotic attractors can be studied by this figure. The attractor for
$r\in(0,1.2]$ is given in the vertical line at the chosen $r$. Also, the
cobweb plot (or Verhulst diagram) for case i have been given in the Fig. 6. By
using this results, it is observed that for the given values, the resulting
sequences of the proposed system has chaotic behavior.

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Figure 3:  Lyapunov exponent values for: (a) Chaotic system in [6], (a) Case
i, (b) Case ii.

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Figure 4:  Bifurcation diagram results for the case i.

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Figure 5:  Bifurcation diagram results for the case ii.

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Figure 6:  Cobweb plots for the case i.

Distribution is another important factor in evaluating chaotic system. One of
the reasons for the weakness of chaos systems against the statistical attack
is nonuniform distribution. The histogram plots of the proposed system for the
case i are given in the Fig. 7. Also, the distribution patterns of the case ii
are shown in Fig. 8. By using these results, it can be seen that the generated
sequence of the proposed maps have flat distributions.

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Figure 7:  Histogram plots of the case i for r=0.5.

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Figure 8:  Distribution patterns of the case ii for $r=0.4$.

######  Remark 2.2.

In the continuation of this paper outputs of the proposed four-dimensional
chaos system with initial values $\gamma=(x_{0},y_{0},z_{0},w_{0})$ and $r$ is
shown by using the following notation

${}^{\gamma}\Psi^{n}_{r}:=\left[{\begin{array}[]{*{20}{c}}{\begin{array}[]{*{20}{c}}{x_{1}}\\\
{y_{1}}\\\ {z_{1}}\\\ {w_{1}}\\\
\end{array}}&{\begin{array}[]{*{20}{c}}{x_{2}}\\\ {y_{2}}\\\ {z_{2}}\\\
{w_{2}}\\\ \end{array}}&{\begin{array}[]{*{20}{c}}{\cdots}\\\ {\cdots}\\\
{\cdots}\\\ {\cdots}\\\ \end{array}}&{\begin{array}[]{*{20}{c}}{x_{n}}\\\
{y_{n}}\\\ {z_{n}}\\\ {w_{n}}\\\ \end{array}}\\\ \end{array}}\right].$

Also ${}_{\tau}^{\gamma}\Psi^{n}_{r}$ denotes ${}^{\gamma}\Psi^{n}_{r}$, which
the decimal part of the numbers is cut from the $\tau$-th decimal number to
the next digits.

##  3 One and three dimensional shift Functions

In this section, we introduce two types of functions named
$\operatorname{ULTS1D}$ and $\operatorname{S3D}$ to intermix pixels of images.
The former is upper and lower triangular shift function which is suitable for
grayscale images and the latter is suitable for RGB images. Note that applying
these functions on images does not change the original image size and only the
pixel locations are moved. In the continuation of this section,
$n_{1},n_{2},n_{3},n_{4}$ are integers and $n,m$ are natural numbers which $m$
is divisible by 3 and $\frac{m}{3}$ is perfect square. $A,B$ and $C$ are
matrices of size $n\times n$ and $P$ is a permutation of size $m$. The inputs
of $\operatorname{ULTS1D}$ are $A$, $n_{1}$ and the output matrix
$\operatorname{ULTS1D}(A,n_{1})$ is expressed as follows.

As seen in Figure 9, the pixel shuffling by $\operatorname{ULTS1D}$ starts
from $A(1,1)$ entry and continues along the first row. Entries are shifted in
the number of $n_{1}$ units. After reaching $A(1,n)$, it changes direction to
the secondary diameter until it reaches $A(n,1)$. Upon reaching $A(n,1)$,
entries keep on moving across the last row and last column, respectively. In
$A(2,n)$, the route again change to diagonal, and this time along the sub-
secondary diagonal. In this shift, there are some mutations of different
lengths over secondary diameters that the first and shortest jump is from
$A(n-1,3)$ to $A(n-1,1)$. After first jump, we move straight from bottom to
top towards $A(2,1)$ and after entering this entry, we move towards
$A(2,n-3)$. After a unit of diagonal movement, we enter the third row to
traverse $A(3,2)$. Now we enter $A(4,2)$ and similarly continue the
displacement of entries. Note that in this process, we do not enter a
situation that we have already passed through, and as mentioned before, we
make jumps wherever necessary.

Figure 9:  The path of one dimentional upper and lower triangular shift
function ($\operatorname{ULTS1D}$).

Now, by applying composition of functions and the shift function defined
above, we will introduce $\operatorname{FCS1D}$ and $\operatorname{SCS1D}$
functions. The outputs $B=\operatorname{FCS1D}(A,n_{1},n_{2},n_{3},n_{4})$ and
$C=\operatorname{SCS1D}(A,n_{1},n_{2},n_{3},n_{4})$ are obtained as follow.

$B=\operatorname{ULTS1D}(A,n_{1})$;

$B=\operatorname{ULTS1D}(B^{\prime},n_{2})$;

$B=\operatorname{ULTS1D}\big{(}\operatorname{fliplr}(B),n_{3}\big{)}$

$B=\operatorname{ULTS1D}\big{(}\operatorname{flipud}(B),n_{4}\big{)}$;

$C=\operatorname{ULTS1D}\big{(}\operatorname{flipud}(A),n_{1}\big{)}$;

$C=\operatorname{ULTS1D}\big{(}\operatorname{fliplr}(C),n_{2}\big{)}$;

$C=\operatorname{ULTS1D}(C^{\prime},n_{3})$;

$C=\operatorname{ULTS1D}(C,n_{4})$;

which $B^{\prime}$ is transpose matrix of $B$. For better understanding,
examples for one-dimensional shift functions are given in Fig 10.

Figure 10:  Examples for 1D shift functions.

In order to shuffle pixels of matrices $A,B$ and $C$ by swapping their parts
based on $P$, we define three-dimensional shift function which is shown by
$\operatorname{S3D}$. To get the output matrices of
$\operatorname{S3D}(A,B,C,P)$, first, we block the matrices $A,B$ and $C$ to
obtain the block matrices
$\tilde{A},\tilde{B},\tilde{C}\in(\mathbb{R}^{p\times
p})^{\frac{n}{p}\times\frac{n}{p}}$, which $p=\frac{n}{\sqrt{\frac{m}{3}}}$.
Then, we convert these block matrices to block vectors
$V\tilde{A},V\tilde{B},V\tilde{C}\in(\mathbb{R}^{p\times
p})^{1\times\frac{n^{2}}{p^{2}}}$. Next, we consider the block vector
$\tilde{V}=[V\tilde{A},V\tilde{B},V\tilde{C}]\in(\mathbb{R}^{p\times
p})^{1\times\frac{3n^{2}}{p^{2}}}$. Now is the time to change the location of
blocks of $\tilde{V}$ based on $P$ to get block vector
$\tilde{W}\in(\mathbb{R}^{p\times p})^{1\times\frac{3n^{2}}{p^{2}}}$ as
follow.

for j=1:$\operatorname{size}$($\tilde{V}$,2) do

$\tilde{W}\\{1,j\\}=\tilde{V}\big{\\{}1,P(1,j)\big{\\}}$;

end for

Output matrices of $\operatorname{S3D}(A,B,C,P)$ can be obtained by dividing
cell vector $\tilde{W}$ into three equal parts and deform them into matrices.
In Fig. 11, the steps of blocking and relocating the blocks under a specific
permutation are given.

Figure 11:  Example for the 3D shift function for $m=12$.  Figure 12:  Example
for the 3D shift function for $m=48$.

Also Fig 12 is the step of relocating blocks of block matrices for $m=48$. All
the above operations are invertible and inverses of $\operatorname{FCS1D}$ and
$\operatorname{SCS1D}$ and $\operatorname{S3D}$ are shown by
$\operatorname{InvFCS1D}$ and $\operatorname{InvSCS1D}$ and
$\operatorname{InvS3D}$, respectively.

##  4 Cellular Automata

The study of cellular automata(CA) dates back to Von.Neumonn [10]. CA is a
model to describe a dynamic system composed of discrete cells where each cell
can assume either the value $0$ or $1$. These cells create a lattice which
changes in discrete time according local rules. Generally, a CA can be defined
as $\\{C,S,V,F\\}$, where $C$ is the cell space and $S$ is the discrete state
sets, like $\\{0,1\\}$. $V$ determines cellular neighborhoods and $F$ is
transfer function[5, 19, 20]. CA are categorized in different dimensions [12]
and the process of running one dimensional CA with local rule $30$ and triple
neighborhoods are shown in Fig. 13. At this fig, 1 and 0 are shown with black
and white squares, respectively, and $[0101110]$ is considered as the first
input of CA. Eight possible states for $[0101110]$ with triple neighborhoods
are shown in the second row. Third row shows the next state of cells. This
process up to three steps, is shown in the last row of Fig. 13. Cellular
automat ,which works based on simple logic computations, with pseudorandom
hash behavior [13], is highly parallel and distributed systems that can
simulate complicated behaviors [19]. One of the ways to create a strong cipher
system is to use cellular automata with the large number of rules.

In our proposed algorithm, one-dimensional CA with triple neighborhoods is
used like Fig. 13, which the rule number for each input is obtained based on
proposed chaotic system.

__

Figure 13:  (a) Cellular automata with rule number $30$, (b) Cellular automata
process.

######  Remark 4.1.

The outputs of the decryption by reversible cellular automata is shown by

$\displaystyle[\textbf{x}^{\prime},\textbf{y}^{\prime}]:=\Phi(\textbf{x},\textbf{y},rule,rep),$

where x and y are inputs in the $t-1$ and $t$ times, respectively. ”$rule$” is
the rule number after ”$rep$” repetitions.

In the proposed algorithm, the matrix $M_{n\times n}$ is encrypted based on
reversible cellular automata by following algorithm.

$R=\big{[}{}_{2}^{\gamma}\Psi^{n/2}_{r}(1,:),{}_{2}^{\gamma}\Psi^{n/2}_{r}(2,:),{}_{2}^{\gamma}\Psi^{n/2}_{r}(3,:),{}_{2}^{\gamma}\Psi^{n/2}_{r}(4,:)\big{]}$;

for $i=1:n$ do

if $i==1$ then

$T=\operatorname{diag}(M,i)$;

$X=T(1:n/2)$;

$Y=T(n/2+1:n)$;

else if $i!=1$ then

$X=\operatorname{diag}(M,i)$;

$Y=\operatorname{diag}(M,-i)$;

end if

$[\textbf{x},\textbf{y}]=\Phi(\textbf{x},\textbf{y},rule=R(i),1);$

end for

##  5 Proposed encryption algorithm

The details of the encryption algorithm are given in this section. The
proposed algorithm includes two parts. In the first part a sensitive algorithm
for generating key space is introduced and in the second part, an algorithm is
proposed for image encryption.

###  5.1 Key space generation algorithm

In the process of encryption and decryption methods, the important part that
affects the output of the algorithm is the key space. In this section, an
algorithm to generate an efficient key space is introduced. One of the most
important features of the proposed key space is the sensitivity to slight
changes in input images, which is evaluated in the section related to
simulation results. First, we design the proposed algorithm for image as
follows.

Algorithm 1: Key space generation algorithm for $I\in R^{n\times m}$.

Input: $I,r$;

$r_{0}=4\operatorname{rand}()$;

$r_{1}:=\lfloor r_{0}\times 10^{3}\rfloor$;

$I_{1}=\operatorname{circshift}\big{(}\operatorname{reshape}(I^{\prime},1,[]),r_{1}\big{)}$;

$\gamma_{0}:=I_{1}(1:4)$;

$q:=^{\gamma_{0}}\Psi^{1}_{r}$;

for $i=7:3:nm$ do

$\gamma_{1}:=\Big{(}\operatorname{mean}\big{(}q(2:4)\big{)},q(1),I_{1}(i-2),I_{1}(i-1),I_{1}(i)\Big{)}$;

$q=^{\gamma_{1}}\Psi^{1}_{r}$;

end for

output: $r_{0},q$;

This part of the algorithm is shown by the following notation according to the
output and input values:

$\displaystyle(r,r_{0},q)=\operatorname{key}(I,r).$

For a color image $I_{n\times m\times 3}$, consider $I_{1}\in R^{n\times 3m}$
as follows.

$\displaystyle I_{1}\big{(}1:n,(i-1)m+1:im\big{)}:=I(:,:,i),~{}~{}i=1,2,3.$

Then define

$\displaystyle I_{2}=\operatorname{circshift}\big{(}I_{1},\lfloor r_{0}\times
10^{3}\rfloor\big{)},$

where $r_{0}:=4\operatorname{rand}()$. By using Algorithm 1, obtain

$\displaystyle(r,r_{i},q_{i})=\operatorname{key}\Big{(}I_{2}\big{(}1:n,(i-1)m+1:im\big{)},r\Big{)},~{}~{}i=1,2,3.$

In the last step of key generation for color image, consider key space as

$\displaystyle(r_{0},q)=\Big{(}\operatorname{mean}\big{(}\\{r_{i}\\}^{3}_{i=1}\big{)},\operatorname{mean}\big{(}\\{q_{i}\\}^{3}_{i=1}\big{)}\Big{)}.$

__

Figure 14:  The structure of the proposed key generation algorithm.

###  5.2 Encryption and decryption algorithm

The steps of the proposed encryption algorithm for a grayscale image $I\in
R^{n\times m}$ based on the new proposed chaos system are written as follows.

step 1. By using key space generation algorithm obtain:

$\displaystyle(r,r_{0},q)=\operatorname{key}(I,r),$

where $q=(x,y,z,w)$.

step 2. Define $\textbf{v}_{1}\in R^{1\times n}$, $\textbf{v}_{2}\in
R^{1\times 8m}$ and $M\in R^{n\times m}$as

$\displaystyle\textbf{v}_{1}:=\lfloor\chi^{n}_{r}\times 10^{3}\rfloor,$
$\displaystyle\textbf{v}_{2}:=\lfloor\chi^{8m}_{r_{0}}\times 10^{3}\rfloor,$
$\displaystyle
M:=mod\Big{(}\big{\lfloor}\operatorname{reshape}\big{(}\chi^{nm}_{\operatorname{mean}([r,r_{0}])}\times
10^{3},[n,m]\big{)}\big{\rfloor},255\Big{)},$

where

$\displaystyle\chi^{j}_{r}=\big{[}^{q}\Psi^{\frac{j}{4}}_{r}(1,:),~{}~{}^{q}\Psi^{\frac{j}{4}}_{r}(2,:),~{}~{}^{q}\Psi^{\frac{j}{4}}_{r}(3,:),~{}~{}^{q}\Psi^{\frac{j}{4}}_{r}(4,:)\big{]},~{}~{}j=n,8m,nm.$

step 3. Transfer $I$ to binary image $I_{1}\in R^{n\times 8m}$, and shift as

for i=1:n do

$I_{1}(i,1:8m)=\operatorname{circshift}\Big{(}\big{(}I_{1}(i,1:8m)\big{)}^{\prime},\textbf{v}_{1}(i)\Big{)}^{\prime}$;

end for

for i=1:4 do

$R\big{(}(i-1)n+1:2n,1:2m\big{)}=I_{1}\big{(}1:n,2(i-1)m+1:2im\big{)}$;

end for

for i=1:2m do

$SR(1:4n,i)=\operatorname{circshift}\big{(}R(1:4n,i),\textbf{v}_{2}(i)\big{)}$;

end for

for i=1:4 do

$I_{2}\big{(}1:n,2(i-1)m+1:2im\big{)}=SR\big{(}(i-1)n+1:in,1:2m\big{)}$;

end for

step 4. Consider most and least significant bits in two separate matrices
$I_{2}^{m},I_{2}^{l}\in R^{n\times 4m}$, respectively. Then define $I_{3}\in
R^{s\times 4m}$ with $s=n/2-mod(n/2,3)$ by

$\displaystyle I_{3}=I^{m}_{2}(1:2:2s,:).$

step 5. By using proposed algorithm based on cellular automata in Section 4,
obtain $I_{4}$ for input values $I_{3},q$ and $r$.

step 6. By inverting steps 3 and 4 on $I_{4}$ obtain $I_{5}\in R^{n\times m}$
and shift $I_{5}$ as

$\displaystyle
I_{6}=\operatorname{FCS1D}\big{(}I_{5},\textbf{v}_{1}(10),-\textbf{v}_{1}(20),\textbf{v}_{1}(30),-\textbf{v}_{1}(40)\big{)}.$

step 7. Obtain final encryption image $EI$ by using bitwise XOR operation as

$\displaystyle EI=I_{6}\oplus M\oplus M_{s},$

where $M_{s}:=\operatorname{circshift}\Big{(}M,\big{[}\lfloor
q(1)\rfloor,\lfloor q(0)\rfloor\big{]}\times 10^{2}\Big{)}$.

Now, we develop the above proposed algorithm for color image $I\in R^{n\times
m\times 3}$. In the first step, by using key space algorithm for color image
obtain $(r,r_{0},q)$. Next by $\operatorname{S3D}$ shift function obtain
$I_{1}$ as

$\displaystyle\big{(}I_{1}(:,:,1),I_{1}(:,:,2),I_{1}(:,:,3)\big{)}=\operatorname{S3D}\big{(}I(:,:,1),I(:,:,2),I(:,:,3),p\big{)},$

where $p=\operatorname{sort}(^{q}\Psi^{nm}_{r})$. Finally, each layer of
$I_{1}$ is encrypted using the encryption algorithm that described above.
Decryption processes are achieved simply by following reverse steps of the
above algorithms.

__

Figure 15: The structure of the proposed encryption algorithm.

##  6 Simulation results

In this section, we review the results of the algorithm and to check the
security of the proposed algorithm, we perform different types of security
tests.

###  6.1 Key space analysis

In order to show the sensitivity of the key generation algorithm to the input
values, Table 1 shows the output results of the algorithm for different
inputs. According to the results, it can be seen that small changes in the
input of the algorithm have caused changes in the key space. To withstand the
brute force attacks the key space must be large enough. According to [1], this
size must be greater than $2^{100}$. If the precision computing is considered
as $10^{-15}$, then the key space size is $10^{90}=2^{90log_{2}10}\approx
2^{298}$, and this is large enough to resist the attack.

Table 1:  Key space generated by the proposed algorithm for various inputs and $r=0.7$. | Keys
---|---
Image |  | $r_{0}$ | $x$ | $y$ | $z$ | $w$
lena | Original | `constant(0.7)` | 0.6038 | 0.2828 | 0.7086 | 0.4590
$(256\times 256\times 3)$ | One bit changed | `constant(0.7)` | 0.5495 | 0.3590 | 0.9294 | 0.3930
| Original | `random` | 0.6600 | 0.7191 | 0.3064 | 0.8206
| One bit changed | `random` | 0.4336 | 0.4298 | 0.2942 | 0.4056
| First time run | `random` | 0.2499 | 0.4357 | 0.5456 | 0.5322
| Second time run | `random` | 0.4351 | 0.2396 | 0.6711 | 0.5193
lena | Original | `constant(0.7)` | 0.7753 | 0.3087 | 0.5087 | 0.1720
$(256\times 256)$ | One bit changed | `constant(0.7)` | 0.6800 | 0.6547 | 0.2556 | 0.3093
| Original | `random` | 0.8145 | 0.4966 | 0.3931 | 0.0714
| One bit changed | `random` | 0.0451 | 0.4445 | 0.0295 | 0.7943
pirate | Original | `constant(0.7)` | 0.3329 | 0.9476 | 0.2881 | 0.5595
$(256\times 256)$ | First pixel changed | `constant(0.7)` | 0.8755 | 0.1108 | 0.6811 | 0.4796
| Last pixel changed | `constant(0.7)` | 0.3318 | 0.6714 | 0.1125 | 0.6366
| Original | `random` | 0.8654 | 0.2863 | 0.0157 | 0.5612
| First pixel changed | `random` | 0.7584 | 0.9722 | 0.1689 | 0.9224
| Last pixel changed | `random` | 0.7952 | 0.4099 | 0.1525 | 0.0080

###  6.2 Statistical analysis

Statistical test consists of three basic tests, correlation values,
information entropy and histogram analysis. The correlation values for an
image are calculated using the following formula

$\displaystyle C_{x,y}=\frac{E(x-\bar{x})(y-\bar{y})}{\sigma_{x}\sigma_{y}},$

where $E,\bar{x},\bar{y},\sigma_{x}$ and $\sigma_{y}$ represent expectation,
mean values and standard deviation, respectively. If this number is close to
zero, the encrypted image will not have significant information from the
original image. The results of this value for original and encrypted image are
give in Tables 2 and 3. By comparing values of encrypted and original images,
it is observed that the results for the encrypted images are close to the
ideal value, i.e., zero. Also the correlation distributions for the original
and encrypted images are shown in Figures 16-18. By using these figures, it
can be seen that encrypted images have more uniform distribution.

Table 2: Correlation coefficient and Information entropy values in the plain and cipher grayscale images. | Correlation coefficient | Information |  |
---|---|---|---|---
| Horizontal | Vertical | Diagonal | Diagonal | entropy |  |
|  |  | (lower left to top right) | (lower right to top left) |  |  |
plain images |  |  |  |  |  |  |
5.1.12 | 0.95649 | 0.97408 | 0.93627 | 0.93893 | 6.7057 |  |
5.2.08 | 0.93707 | 0.89264 | 0.85427 | 0.85572 | 7.201 |  |
boat($512\times 512$) | 0.93812 | 0.97131 | 0.92585 | 0.92216 | 7.1914 |  |
lena($256\times 256$) | 0.92578 | 0.95926 | 0.92577 | 0.90374 | 7.4429 |  |
lena($512\times 512$) | 0.20131 | 0.21056 | 0.19815 | 0.19016 | 7.4455 |  |
encrypted images |  |  |  |  |  |  |
5.1.12 | $-0.00047766$ | $0.00061399$ | $0.001538$ | $-0.0001677$ | 7.9973 |  |
5.2.08 | $-0.0003788$ | 0.00013229 | $-0.00082042$ | $0.00027628$ | 7.9993 |  |
boat($512\times 512$) | 0.00093687 | 0.00057866 | 0.00038615 | $-0.00084275$ | 7.9993 |  |
lena($256\times 256$) | $-0.0021188$ | $-0.0025797$ | $-0.0015225$ | 0.00026687 | 7.9972 |  |
lena($512\times 512$) | 0.00039856 | 0.00015902 | $-0.00023439$ | $0.00069882$ | 7.9993 |  |
Table 3: Correlation coefficient values in the plain and cipher grayscale images. |  | Correlation coefficient | Information |  |
---|---|---|---|---|---
|  | Horizontal | Vertical | Diagonal | Diagonal | entropy |  |
|  |  |  | (lower left to top right) | (lower right to top left) |  |  |
plain images |  |  |  |  |  |  |  |
| R | 0.94928 | 0.95616 | 0.91631 | 0.91764 | 6.2499 |  |
4.1.02 | G | 0.93077 | 0.95338 | 0.8968 | 0.90017 | 5.9642 |  |
| B | 0.91784 | 0.94421 | 0.88698 | 0.88898 | 5.9309 |  |
| R | 0.97865 | 0.9879 | 0.96875 | 0.96841 | 7.2549 |  |
4.1.04 | G | 0.96598 | 0.98201 | 0.95147 | 0.95072 | 7.2704 |  |
| B | 0.95231 | 0.97178 | 0.93069 | 0.93057 | 6.7825 |  |
| R | 0.92307 | 0.86596 | 0.85187 | 0.85434 | 7.7067 |  |
4.2.03 | G | 0.86548 | 0.76501 | 0.72493 | 0.7348 | 7.4744 |  |
| B | 0.90734 | 0.88089 | 0.84244 | 0.83986 | 7.7522 |  |
| R | 0.97527 | 0.98531 | 0.97339 | 0.96484 | 5.0465 |  |
lena | G | 0.96662 | 0.98017 | 0.96303 | 0.95357 | 5.4576 |  |
($512\times 512\times 3$) | B | 0.93339 | 0.95579 | 0.92643 | 0.91863 | 4.8001 |  |
encrypted images |  |  |  |  |  |  |  |
| R | 0.0011619 | 0.0016396 | 0.00053731 | 0.00082608 | 7.9973 |  |
4.1.02 | G | 0.0007757 | $-0.00021986$ | 0.00069429 | 0.00088685 | 7.9971 |  |
| B | 0.00029618 | 0.0005109 | 0.00098397 | $-0.0015792$ | 7.9972 |  |
| R | 0.00036129 | $-0.00047313$ | -0.0015582 | 0.0026443 | 7.9973 |  |
4.1.04 | G | 0.0029232 | 0.0014998 | 0.0010652 | 0.001758 | 7.9972 |  |
| B | 0.0031419 | 0.00095978 | 0.0019824 | $-0.001847$ | 7.9973 |  |
| R | $-0.00057646$ | 0.00020068 | 0.0010341 | $-0.00034744$ | 7.9993 |  |
4.2.03 | G | 0.00053615 | $-0.00029491$ | 0.00061376 | $-0.00040553$ | 7.9993 |  |
| B | $-0.00080725$ | $-0.0017738$ | $-0.0002713$ | 0.0009324 | 7.9993 |  |
| R | $-2.2351e-06$ | 0.00022357 | $-0.00074232$ | $-0.00055947$ | 7.9993 |  |
lena | G | 0.0002728 | $-3.7743e-05$ | 0.0012758 | $-0.00036377$ | 7.9993 |  |
($512\times 512\times 3$) | B | 0.00031603 | $0.00067082$ | $-0.00071107$ | 0.0012129 | 7.9993 |  |

__

Figure 16:  Correlation of neighborhood pixels at different directions before
(a-c) and after encryption (d-f) of lena ($256\times 256$).

__

Figure 17:  Correlation of neighborhood pixels at different directions for
4.1.04.

__

Figure 18:  Correlation of neighborhood pixels at different directions for
encrypted 4.1.04

The uncertainty of information for an image is evaluated by using Shannon
information entropy. This value is calculated by

$\displaystyle H=-\sum_{i}p(i)\log\big{(}p(i)\big{)},$

where $p(i)$ is the probability of occurrence of each pixel. The ideal value
for this test is 8. If the results of the algorithm are close to this number,
it will be difficult to obtain valid information from the image. The results
of this test for different images with different sizes are given in Tables 2
and 3. The results in these tables show that the information entropy of
encrypted image by proposed algorithm are close to the ideal value. The
distribution characteristics of pixel values is evaluated by histogram.
Histograms of plaintext and encrypted images are given in Figure 19. Also the
intensity histogram is shown in Figure 20. Using these results, it can be seen
that the histograms of the cipher image are uniformly distributed in
comparison with the original image. Therefore, according to the results the
proposed algorithm has a good ability to resist statistical attacks.

###  6.3 Differential attack test

One of the most important attacks evaluated on cryptographic algorithms is
differential attacks. This type of attack is studied by two tests, UACR test
(number of pixels change rate) and NPCI test (unified average changing
intensity). These values are calculated by

$\displaystyle NPCR=\frac{\sum_{i,j}D_{i,j}}{n\times m}\times 100\%,$
$\displaystyle UACI=\frac{\sum_{i,j}|C^{1}_{i,j}-C^{2}_{i,j}|}{n\times
m}\times 100\%,$

where $C^{1}$ and $C^{2}$ are two encrypted images such that their plain
images differ only by one pixel. Also if $C^{1}_{i,j}\neq C^{2}_{i,j}$ then
$D_{i,j}=1$ otherwise $D_{i,j}=0$. The ideal values for these tests are
$100\%$ and $33.\overline{33}\%$, respectively. The results are shown in
Tables 4-6. The best study on these values is discussed in [22], which
critical intervals for NPCR and UACI are considered. In Tables 4 and 5, the
results of the proposed algorithm for different images are compared to the
critical values. These results indicate the success of the proposed method in
these tests. Also in Table 6, the proposed method are compared to other
studies.

__

Figure 19:  (a)-(d) the original image and its histogram, (e)-(f) the output
image of the proposed algorithm and its histogram

__

Figure 20: (a)-(c) 3D histogram representation of original image and (d)-(f) encrypted image for 4.1.04.  Table 4: Numerical results of the UACI and NPCR for different images ($256\times 256$). |  UACI critical values [22] |  |  NPCR critical values [22]
---|---|---|---
|  |  |  u${}^{*-}_{0.05}$=33.2824 |  u${}^{*-}_{0.01}$=33.2255 |  u${}^{*-}_{0.001}$=33.1594 |  |  N${}^{*}_{0.05}=$ |  N${}^{*}_{0.01}=$ |  N${}^{*+}_{0.001}=$
Image |  | UACI |  u${}^{*+}_{0.05}$=33.6447 |  u${}^{*+}_{0.01}$=33.7016 |  u${}^{*+}_{0.001}$=33.7677 | NPCR | 99.5693 | 99.5527 | 99.5341
5.1.12 |  | 33.4570 | Pass | Pass | Pass | 99.6127 | Pass | Pass | Pass
lena |  | 33.4931 | Pass | Pass | Pass | 99.6112 | Pass | Pass | Pass
| R | 33.4931 | Pass | Pass | Pass | 99.6213 | Pass | Pass | Pass
4.1.02 | G | 33.4763 | Pass | Pass | Pass | 99.6000 | Pass | Pass | Pass
| B | 33.4421 | Pass | Pass | Pass | 99.6147 | Pass | Pass | Pass
| R | 33.4882 | Pass | Pass | Pass | 99.6191 | Pass | Pass | Pass
4.1.04 | G | 33.4581 | Pass | Pass | Pass | 99.6032 | Pass | Pass | Pass
| B | 33.5000 | Pass | Pass | Pass | 99.6001 | Pass | Pass | Pass
Table 5: Numerical results of the UACI and NPCR for different images ($512\times 512$). |  UACI critical values [22] |  |  NPCR critical values[22]
---|---|---|---
|  |  |  u${}^{*-}_{0.05}$=33.3730 |  u${}^{*-}_{0.01}$=33.3445 |  u${}^{*-}_{0.001}$=33.3115 |  |  N${}^{*}_{0.05}=$ |  N${}^{*}_{0.01}=$ |  N${}^{*+}_{0.001}=$
Image |  | UACI |  u${}^{*+}_{0.05}$=33.5541 |  u${}^{*+}_{0.01}$=33.5826 |  u${}^{*+}_{0.001}$=33.6156 | NPCR | 99.5893 | 99.5810 | 99.5717
5.2.08 |  | 33.4571 | Pass | Pass | Pass | 99.6115 | Pass | Pass | Pass
boat |  | 33.4750 | Pass | Pass | Pass | 99.6137 | Pass | Pass | Pass
lena |  | 33.4683 | Pass | Pass | Pass | 99.6046 | Pass | Pass | Pass
| R | 33.4546 | Pass | Pass | Pass | 99.6215 | Pass | Pass | Pass
4.2.03 | G | 33.4681 | Pass | Pass | Pass | 99.6064 | Pass | Pass | Pass
| B | 33.4502 | Pass | Pass | Pass | 99.6110 | Pass | Pass | Pass
| R | 33.4641 | Pass | Pass | Pass | 99.6112 | Pass | Pass | Pass
lena | G | 33.4541 | Pass | Pass | Pass | 99.6080 | Pass | Pass | Pass
| B | 33.4543 | Pass | Pass | Pass | 99.6101 | Pass | Pass | Pass
Table 6: Correlation coefficient and Information entropy values in the plain and cipher grayscale images. | NPCR | UACI |
---|---|---|---
image | Propose |  Ref.[15] |  Ref.[18] |  Ref.[6] |  | Propose |  Ref.[15] |  Ref.[18] |  Ref.[6]
5.1.09 | 99.6117 | 99.6078 | 99.6016 | 99.6064 |  | 33.4490 | 33.4563 | 33.4700 | 33.4456
5.1.10 | 99.6164 | 99.6098 | 99.6191 | 99.6154 |  | 33.4733 | 33.4510 | 33.4826 | 33.4946
5.1.11 | 99.6122 | 99.6077 | 99.6042 | 99.6244 |  | 33.4654 | 33.4832 | 33.5648 | 33.5541
5.1.14 | 99.6118 | 99.6129 | 99.6199 | 99.6364 |  | 33.5111 | 33.4848 | 33.4725 | 33.4655
7.1.01 | 99.6161 | 99.6040 | 99.6053 | 99.5992 |  | 33.4682 | 33.4779 | 33.4820 | 33.5037
7.1.02 | 99.6093 | 99.6016 | 99.6080 | 99.6075 |  | 33.4730 | 33.4172 | 33.4357 | 33.4237
7.1.09 | 99.6152 | 99.6061 | 99.6112 | 99.6162 |  | 33.4591 | 33.4814 | 33.4596 | 33.4177
7.1.10 | 99.6110 | 99.6052 | 99.6106 | 99.6045 |  | 33.4817 | 33.4852 | 33.4538 | 33.4344

###  6.4 Noise and data loss attack

In the transmission of images over the network and through physical channels,
part of the data is naturally or intentionally lost due to noise and cropped
attacks. Therefore, efficient cryptographic schemes are capable of retrieving
the original images even after these types of attacks. In order to test the
anti-noise ability of the proposed method, the decrypted images after the
noise attack with different intensities $0.1$ for gray and $0.2$ for color
images are given in Fig.21. The strength of the proposed method against
cropped attacks with varying degrees of data loss is shown in Fig.22.

__

Figure 21: Salt $\&$ pepper noise attack results with different noise density
and extracted secret images: (a), (d) decrypted images with noise intensity
0.1 and 0.2, respectively.

__

Figure 22: Results of extracting the secret image after slight change in the
key: (a) almost 0.17 data loss, (b) almost 0.07 data loss, (c) almost 0.35
data loss, (d) almost 0.23 data loss, (e)-(h) decrypted images.

##  7 Conclusions

Create a secure environment for transferring images a burgeoning subject
branch. To study this topic, in the first step, by the development and
improvement of the chaos system, a new chaos system is introduced. The
structure of the new system is studied with different tests and the results
show the efficiency of the new system. In the next step, the proposed system
is used to create a secure image transfer environment in the form of an
encryption algorithm. The results of studying the proposed algorithm by
various security tests show that the proposed algorithm is efficient and safe.

Funding This work does not receive any funding.

## Compliance with ethical standards

Conflict of interests The authors declare that they have no conflict of
interests.
Human participants and animals This paper does not include human participants
and animals.

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# Recent results and future of the NA61/SHINE strong interactions
program††thanks: Presented at the XXXV Mazurian Lakes Conference on Physics,
Piaski, Poland, September 3-9, 2017

Bartosz Łysakowski for the NA61/SHINE Collaboration  Institute of Physics,
University of Silesia, Poland

###### Abstract

NA61/SHINE is a fixed target experiment at the CERN Super-Proton-Synchrotron.
The main goals of the experiment are to discover the critical point of
strongly interacting matter and to study the properties of the onset of
deconfinement. In order to reach these goals the collaboration studies hadron
production properties in nucleus-nucleus, proton-proton and proton-nucleus
interactions.
In this talk, recent results on particle production in p+p interactions, as
well as Be+Be and Ar+Sc collisions in the SPS energy range, are reviewed. The
results are compared with available world data. The future of the NA61/SHINE
scientific program is also presented.

##  1 Introduction

The NA61/SHINE experiment performs a unique two-dimensional scan of the phase
diagram of strongly interacting matter. The main goals are the study of the
properties of the onset of deconfinement by measurements of hadron production
and the search for the critical point of strongly interacting matter by
measuring event-by-event fluctuations. Measurements are performed in a wide
beam momentum range (from 13 up to
150/158$A\,\mbox{Ge\kern-1.00006ptV}\\!/\kern-0.70007ptc$) and for various
systems (p+p, p+Pb, Be+Be, Ar+Sc, Xe+La and Pb+Pb). The program is motivated
by the discovery of the onset of deconfinement in Pb+Pb collisions at
30$A\,\mbox{Ge\kern-1.00006ptV}\\!/\kern-0.70007ptc$ by the NA49 experiment[1,
2].

NA61/SHINE is a fixed target experiment at the CERN SPS [3]. The detection
system is based on eight Time Projection Chambers (TPC) providing acceptance
in the full forward hemisphere, down to $p_{\text{T}}$ = 0. The TPCs allow for
tracking, momentum and charge reconstruction as well as measurement of mean
energy loss per unit path length. Time of Flight (ToF) walls provide
additional particle identification by measuring particles mass. The Projectile
Spectator Detector (PSD), a zero-degree calorimeter, allows selecting
collision centrality based on the measurement of forward energy. The full
detector system of the NA61/SHINE experiment is presented in Fig. 1.

Figure 1: NA61/SHINE detector system

##  2 Onset of deconfinement

One of the main purposes of NA61/SHINE is to study the phase transition
between hadron gas (HG) and quark-gluon plasma (QGP). The measurements
concentrate on observables for which the Statistical Model of the Early Stage
(SMES) predicts characteristic signals [4].

Mean multiplicities of all pions, $\langle\pi\rangle$, normalized to the
average number of wounded nucleons, $\langle$W$\rangle$, are shown in Fig. 2.
The results were compared with world data from other experiments [5, 6, 7]. At
higher SPS energies the slope of the energy dependence is larger for the heavy
systems (Pb+Pb, Ar+Sc) than for the light ones (p+p, Be+Be). SMES predicts an
increase of the slope in the quark-gluon plasma due to a larger number of
degrees of freedom. All A+A data come from central collisions.

Figure 2: Energy dependence of the ratio
$\langle\pi\rangle$/$\langle$W$\rangle$ of pion multiplicity to the number of
wounded nucleons.

Fig. 3 presents the multiplicity ratio of charged kaons to pions at mid-
rapidity. Fig. 4 shows the energy dependence of the inverse slope parameter of
transverse mass spectra of charged kaons. NA61/SHINE results on p+p
interactions [8] and Be+Be collisions were compared with results from central
Pb+Pb collisions from NA49 [1, 2] and other experiments [9, 10, 11, 12, 13,
14].

Figure 3: Energy dependence of the $\textup{K}^{+}$ (left) and
$\textup{K}^{-}$ (right) multiplicity divided by the corresponding charged
$\pi$ multiplicity at mid-rapidity.

A plateau visible in the energy dependence of the inverse slope parameter in
Fig. 4, and peaks seen in Fig. 3 (left panel) for Pb+Pb and Au+Au collisions
in the SPS energy range, were predicted by the SMES model as signatures of the
onset of deconfinement. In the SPS energy range the NA61/SHINE results on p+p
interactions exhibit a qualitatively similar energy dependence (step) for the
inverse slope parameter, and a step instead of a peak in the kaon-to-pion
ratio. Thus some properties of hadron production previously attributed to the
onset of deconfinement in heavy-ion collisions are present also in p+p
interactions. Surprisingly, while the inverse slope parameter in Be+Be
collisions lies slightly above that in p+p interactions, the values of the
charged-kaon-to-pion ratio are very close in Be+Be and p+p reactions.

Figure 4: Energy dependence of the inverse slope parameter of the transverse
mass spectra of $\textup{K}^{+}$ (left) and $\textup{K}^{-}$ (right).

##  3 Search for the critical point

NA61/SHINE uses the strongly intensive measures $\Sigma[P_{T},N]$ and
$\Delta[P_{T},N]$ to study transverse momentum and multiplicity fluctuations
[15]. Within the Wounded Nucleon Model, they depend neither on the number of
wounded nucleons (W) nor on fluctuations of W. Moreover, in the Grand
Canonical Ensemble, they do not depend on volume and volume fluctuations.
$\Sigma[P_{T},N]$ and $\Delta[P_{T},N]$ have two reference values, namely,
they are equal to zero in case of no fluctuations and one in case of
independent particle production. Fig. 5 shows deviations of $\Sigma[P_{T},N]$
and $\Delta[P_{T},N]$ from unity (independent particle model) that are
smoothly growing with energy, which may be due to the increasing azimuthal
acceptance. So far, there are no prominent structures in the NA61/SHINE data
which could be related to the critical point (see the expected signal from the
critical point in Fig. 5, upper panel).

Figure 5: Upper panel: a hill expected in an energy and system size scan for a
fluctuation observable. Lower left panel: $\Delta[P_{T},N]$ Lower right panel:
$\Sigma[P_{T},N]$.Both lower panels are for negatively charged hadrons for 5%
of the most central collisions.

##  4 NA61/SHINE detector upgrade

This year NA61/SHINE started a pilot program of open charm measurements with
the new Small Acceptance Vertex Detector (SAVD).

The construction of a high-resolution vertex detector was mostly motivated by
the importance and the possibility of the first direct measurements of open
charm meson production in heavy-ion collisions at SPS energies. In the first
step a limited-acceptance version of the detector, named SAVD, was built. This
device covers 35% of the acceptance of the final version, and is foreseen to
operate beyond 2020. The constructed SAVD is based on MIMOSA-26AHR sensors
developed in IPHC Strasbourg.

A detector upgrade is planned during the long shutdown LS2 at CERN during the
years 2019–2020: the readout speed will be increased to 1 kHz and the Large
Acceptance Vertex Detector will be constructed.

The upgraded detector will allow a high-statistics beam momentum scan with
Pb+Pb collisions for precise measurements of D-meson production and multi-
strange hyperon production in 2021–2024.

##  5 Summary

This contribution briefly discusses recent results from the NA61/SHINE energy
and system size scan performed to study the onset of deconfinement and search
for the critical point of strongly interacting matter. Results on produced
particle multiplicity and fluctuations were presented.

The measured fluctuations show no indication of a critical point. Still, such
features may be revealed in future results on Xe+La and Pb+Pb collisions.

In addition, the extension of the NA61/SHINE program, such as the detector
system upgrade and plans to measure precisely open charm in 2021–2024, were
discussed in this contribution.

## Acknowledgments

This work was partially supported by the National Science Centre, Poland
(grant 2015/18/M/ST2/00125).

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