Connection of two conventional enhanced to encrypt color images

This document introduces a new cryptosystem mixing two improvement standards generally used for text encryption, in order to give birth a new color image encryption algorithm capable of dealing with known attacks. Firstly, two substitution matrixes attached to a strong replacement function will be generated for advanced Vigenere technique application. At the end of this first round, the output vector is subdivided into size blocks according to the used chaotic map, for acting a single enhanced Hill circuit insured by a large inversible matrix. A detailed description of such a large involutive matrix constructed using Kronecker products will be given. accompanied by a dynamic translation vector to eliminate any linearity. A solid chaining is established between the encrypted block and the next clear block to avoid any differential attack. Simulations carried out on a large volume of images of different sizes and formats ensure that our approach is not exposed to any known attacks. Article Highlights This new algorithm is the mixture result of two deeply improved classical systems which we mention the most important changes made.  New size substitution matrix (256,256) Construction  Strong replacement functions definition  Improved Vigenere technology application  Kronecker products Theoretical reminder  Kronecker product Application  Large invertible matrix design  Single enhanced Hill revolution application Notation ⎩ ⎪ ⎨ ⎪ ⎧ G = Z tZ ring A(j: ): Line number j of matrix A A(: j): colunm number j of matrix A ⨁: Xor operator ⨂: Kronecker product E(x): The whole part of the real number x Key word: Vigenere grid; Involutive matrix; Chaotic map; Broadcast function; genetic operator Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 19 April 2021 doi:10.20944/preprints202104.0507.v1 © 2021 by the author(s). Distributed under a Creative Commons CC BY license. 2


I. IntroductIon
The rapid development of chaos theory in mathematics provides researchers with opportunities to further improve some classic encryption systems. In front of this great security focus, many techniques for color image encryption have flooded the digital world, mostly exploiting number theory and chaos [1,2]. Others are attempting to update their policies by improving some classical techniques, such as Hill [3 − 4], Cesar, Vignere [5 − 6], Feistel [7 − 8].

1) Vigenere's classical technique
This technology is based on static ( )matrix defined by the following algorithm Even though Vigenere's matrix was known, the encryption was able to withstand several centuries. But, Babagh's cryptanalysis is not efficient in not knowing the size of the encryption key. Several attempts to improve Vigenere's technique have invaded the digital world we quote [9 − 10]. In this work, the new structure of the substitution matrix and its attached replacement function will be described in detail.

2) Hill's classical technique
Hill's method is a classic symmetric encryption system that uses a linear transformation provided by an invertible matrix in a carefully selected ring [11 − 12]. Unfortunately, this technique is still exposed to known attacks. Since then, over time, several improvements have been made to the system [13 − 14]. The improvement of the classic Hill technique tracked in this document is to prove the construction of a large invertible matrix whose size will be randomly calculated based on the chaotic graph used. Some advanced techniques in the same axis should be taken into consideration [15 − 16].

3) Our contribution
Our contribution in this article is to couple two conventional deeply improved systems to better adapt to color image encryption. These improvements, made on the two classic systems are

II. the proposed method
Based on chaos, our new technology is divided into three main stages The image below shows this technology steps This new cryptosystem is based on the following axes

Axe 1: chAotIc sequences development
In order to build a new algorithm using a single-encryption key, we will use the 2d logistics map. This choice is due to the simplicity of its development and its high sensitivity to the initial parameters.

2) Chaotic used vector design
Our work requires the construction of three chaotic vectors( ), ( )and ( ) , with a coefficient in ( ), and two ( ) , ( )binary vector will be regarded as the control vector. This construct is seen by the following algorithm

3) Global permutation Design
Sorting the ( )vector in a large decreasing order generates the ( ) global permutation This method is illustrated by the following algorithm

III. InstAll the new encryptIon method
Based on chaos, our new technology is based on two encryption processes each using a deep improvement of a classic system.

Axe 2: vIgenere upgrAde
In the first stage, Vigenere's technology was greatly modified by integrating the new substitution matrix provided by the new powerful replacement function.

2) Initialization Value Design
The( 1) initialization value must be recalculated to change the value of the starting pixel. Ultimately, the ( ) value is provided by the next algorithm This initialization value is set only to modify the value of the seed pixel and start the encryption process. By implementing chaotic mapping in the calculation of the initialization value, the problem of uniform image color (black, white ,... ) can be solved.

3) Vigenere's advanced methods
This new technology requires the establishment of two ( ) and ( ) substitution matrices through the process described by the following steps  permutation ( ) obtained by descending ordering the first 256 of the sequence ( )  permutation ( ) obtained by increasing the ordering the first 256 of the sequence ( ), with the following restrictions  We noticed that this new expression is firmly attached to the ( ) decision vector, and uses two general substitution matrices to contain powerful substitution functions.

2) First Encryption Process
With a powerful broadcast function, this first encryption process will protect the system from differential attacks and increase the time complexity of the attack. The figure below illustrates the first encryption process improved using only Vigenere This schema is translated by the algorithm below We note that this first step uses only substitutions, which ensures an extreme speed in the execution. The output vector ′( ′ , ′ , … … . . , ′ ), will undergo a second encryption attempt.

Axe 3: ApplyIng hIll upgrAde
Before starting the second encryption stage, we will introduce in detail some mathematical knowledge useful for constructing inversible matrix and, install a hill lap.

1) Math reminder
Several new knowledges with mathematical bases must be clarified

a. related information
In this section, we will introduce the concept of diastereomeric matrices and Kronecker products used to construct large invertible matrices.

i. Involutive matrix a) Definition
A is an involutive matrix if and only if we have

=
In other words 15 = b. Building the ( ) matrix Assume that the size of ( ) matrix is (2 , 2 ), and is defined by block as follows

= ( , )
We got Since ( ) matrix is given randomly, other matrices can be selected by the following formula Under these conditions, the obtained matrix ( ) is involute. This choice is not unique, and it increases the complexity of matrix ( ) reconstruction.

2) Kronecker matrix product
Matrix theory is widely used in cryptography. Due to its difficulties, classical computing techniques are rarely used. For example, it is difficult to reverse the large matrix using conventional methods. At present, the Kronecker product and the tensor product are not only connected with physics or biology, but also take a safe path in cryptography. We are going to take advantage of the Kronecker product to increase the size of the involutive matrix.

3) ( ′) Vector Adaptation
In order to facilitate the implementation of the second encryption stage, the vector ( ′) must be cut into blocks uniform, we are going to calculate two constants

4) Switching to a size matrix
The following figure illustrates the transaction from vector to size matrix( , ) used in the Hill circuit

2) New Hill matrix Development
In our algorithm, in order to overcome this problem, we will introduce in detail a new simple method of constructing large invertible matrix based on involute matrix.

3) New encryption matrix construction
This second encryption process is illustrated by the following figure

AxIs6: exAmples And sImulAtIons
In order to measure the performance of our encryption system, we randomly select a large number of reference images, and then use our method to test them

1) Brutal assaults
They consist in reconstructing the encryption keys in a random manner.

a) Key-space analysis
The chaotic sequence used in our method ensures strong sensitivity to initial conditions and can protect it from any brutal attacks. The secret key to our system consists of = 0,7655412001 , = 3.89231541, = 0.865421331, = 0,563215 = 1,3561 = 0,563215 If we use single-precision real numbers to operate, the total size of the key will greatly exceed ≈ ≫ , which is enough to avoid any brutal attacks.

b) Secret key's sensitivity Analysis
Our encryption key has a high sensitivity, which means that a small degradation of a single parameter used will automatically cause a large difference from the original image. The image below illustrates this confirmation

Figure11: Encryption key sensitivity
This ensures that in the absence of the real encryption key, the original image cannot be restored.

a) Hystogram analysis
all images tested by our algorithm have a uniformly distributed histogram. This reflects that the entropy of the encrypted images is around 8, which makes the system immune to histogram attacks. The table1 shows that the horizontal correlation values of the encrypted images are close to zero. This ensures high security against correlation attacks.

2) Statistics Attack Security a) Entropy Analysis
Entropy is the measure of the disorder diffused by a source without memory. The entropy expression is determined by the equation below The entropy values on the 150 arbitrarily chosen from a large database of images of different sizes and formats, tested by our method are represented graphically by the following figure   Fugure12: Entropy of 150 images All the entropy values of the images tested by our algorithm are close to 8, which confirms the uniformity of the histogram. This proves that the method is far from a statistical attack.

a) Entropy statistical analysis
We will study the uniformity of the distribution of entropy released by the test.

(a) Position parameter analysis
The values derived from the entropy by applying our approach to over 150 images in our image database, constitute a statistical series with position, dispersion and concentration parameters have been recalculated to verify the safety of our approach.
The purpose of this analysis is to show that the distribution follows a reduced central normal distribution. So

Position Parameters
The moustache box of the entropy is illustrated in the diagram in Figure below       . These values are largely sufficient to affirm that our crypto system is protected from known differential attacks The study of the 150 selected images revealed the following diagram fIgure19: UACI of 150 images All detected values are inside the confidence interval [33; 34 33,35]. These values are largely sufficient to affirm that our crypto system is protected from known differential attacks.

a) Avalanche effect
The avalanche effect is a required property in virtually all cryptographic hash functions and block coding algorithms. It causes progressively more important changes as the data is propagating in the structure of the algorithm. This constant determines the avalanche impact of the cryptographic structure in place. It is approximated by the equation below 32 = ∑ ℎ ∑ * 100 The signal-to-peak noise ratio, often abbreviated , is an engineering term for the ratio between a signal's maximum possible power and the power of distorted noise that affects the precision of its display. The mathematical analysis of an image is given by the next equation

= 20 √
For color images, the definition of is the same except that the is the sum of all square value changes. In the alternative, for color images, the image is transcoded into a separate color space and the is displayed for each channel in that color space.

a) Speed analysis
For an evaluation of the execution time, our algorithm is tested on a personal computer "Intel core i5 3337 U 1.86 Gz CPU 8 GB ram. We use Matlab as programming software. We measure the encryption and decryption time of the tested images.

4) Math Security
The first round eliminates any correlations and protects the system from differential attacks. Not knowing the size of the encryption matrix will increase the complexity of the attack. Without knowing the encryption key, it is difficult to replicate the structure of this matrix.

Iv. conclusIon
The first encryption stage is provided by the deep enhancement of Vigenere's classic system, which can protect our new algorithm from any differential attack, while the second stage uses dynamic affine transformation, which consists of a large invertible matrix of uncertain size Provide without knowing the secret encryption keys, are difficult to construct from Kronecker's products, making the system resistant to any known attacks. The system only uses the replacement, which greatly reduces the execution time.

Conflict of Interest
I am the sole author of this article, and there are no private or public organizations or laboratories to fund my research, thus avoiding any expected conflicts.
This document does not contain any research or experiments conducted on animals.