Vigenere followed by multiple bit-level permutations to encrypt a color image

The largest part of the image encryption algorithms operates on the pixel as a central element by implementing diffusion confusion and eventually a permutation. On the other side, a permutation applied at the bit level changes not only the pixel value, but also its location within the image. In this work, we will propose a new technology of medical and color image encryption, based on chaotic permutations acting at the bit level, and a diffusion confusion ensured by an application of the method of Vigenere largely improved and adapted to the subject. Simulations performed on a large number of images of different sizes and formats ensure that our method is not subject to any known attacks.

encryption have flooded the digital world, mostly exploiting number theory and chaos By applying permutations at the pixel level [1 − 2 − 3 − 4 − 5].

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 [10 − 11]. In this work, the new structure of the substitution matrix and its attached replacement function will be described in detail.

1) Our contribution
Our contribution in this work is to start with a deeply improved trick of Vigenere for a suppression of any differential attack and to switch to binary notation to apply multiple permutations to encrypt a color image.

I. the proposed method
Based on chaos [13 − 14 − 15], 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 2 [16 − 17 − 18]. This choice is due to the simplicity of its development and its high sensitivity to the initial parameters.

1) 2D Logistics Map Decryption
It is a two-dimensional chaotic sequence defined by second degree polynomials. This sequence is given by the equation below All the conditions ensure the installation of the chaotic aspect.

2) Chaotic vector design
Our work requires the construction of four chaotic vectors( ), ( ), ( )and ( ) generated from the following algorithm 3) Binary control vector development.
we will construct two control vectors

Axe 2: vIgenere upgrAde
In the first stage, Vigenere's [19 − 20 − 21]technology was greatly modified by integrating the new substitution matrix provided by the new powerful replacement function.

a) Initialization value computation
The initialization value must be calculated in order to change the startup pixel in the future and start the encryption process correctly.
The calculation of this value is described below

2) First Encryption Process
The figure below illustrates the first encryption process improved using only Vigenere This function links the encrypted pixel with the next clear pixel. 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.

1) First round analysis
For a better follow-up of our algorithm, several reference images were tested by this first round, we quote The solid chaining establishes between the encrypted block and the next clear block in the broadcast process, forcing us to start decryption from the last block using the opposite functions. So, the decryption process should follow these steps  Binary conversion  Reciprocal permutation generation  Switching to grayscale  Vigenere reciprocal matrix  Vigenere's reciprocal application

1) Reciprocal permutation generation
Consider ( ) the reciprocal permutation of ( ) , it defines by the following algorithm

AxIs 5: 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 7 = 0,7655412001 , = 3.89541 = 0.865421331, = 0,56120 = 1,3561 = 0,56321 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. This ensures that in the absence of the real encryption key, the original image cannot be restored.

2) Histogram 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.

3) 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 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.

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

i. Position parameter analysis
The values derived from the entropy by applying our approach to over 150 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

c. Correlation analysis
Correlation is a technique that compares two images to estimate the displacement of pixels in one image relative to another reference image. The relevant expression is defined by the following equation

i. Horizontal Correlation
The following figure graphically represents the simulation of 150 color images of the same size, which are selected from an image database of various sizes, formats and related values   1  9  17  25  33  41  49  57  65  73  81  89  97  105  113  121 129 137 145 Figure 5 shows that the vertical correlation values of the encrypted images are close to zero. This ensures high security against correlation attacks.

iii. Diagonal Correlation
Simulations made on 150 of the database gave the diagonal correlation scores are displayed in Figure below

4) Differential analysis
Let be two encrypted images, whose corresponding free-to-air images differ by only one pixel, from ( )and( ), respectively. The mathematical analysis of an image is given by the equation below The study of the 150 revealed the following diagram . 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 fIgure8: 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.

5) 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 Figure below depicts the evaluation of the score for 150 images examined by our approach.  (b) PSNR 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. b) Speed analysis For an evaluation of the execution time, our algorithm is tested on a personal computer "Intel core 5 3337 1.86 8 . We use Matlab as programming software. We measure the encryption and decryption time of the tested images.

II. mAth securIty
The size of the encryption key protects the system from brute force attacks. The construction of the chaotic permutations is difficult to reverse. Moreover, the established chaining protects the algorithm from differential attacks.

III. conclusIon
This new algorithm which starts with an implementation of the greatly improved Vigenere technique ensured by two chaotic substitution matrices and installation of a diffusion ensured also by the two Vigenere matrices, followed by a passage in binary writing for an application of eight chaotic permutations acting at the level of the bits for a better diffusion confusion can face any known attack .