Development of Standard Hill Technology for Image Encryption over a 256-element Body

This technique, discovered by HILL [1],[2] in 1929, was only applicable to the text. It is based on two main steps. The first step is to divide the message to be encrypted into n character (natural number) blocks, and the second step is in a carefully selected ring as usuallyZ/26ZorZ/256Z) The difficulty of constructing a large invertible matrix prompts researcher to only Use a matrix with n ≥ 4. Equation 1 fully describes this standard technique

With(C_i)is the clear block, $\left(C_i^{\text{'}}\right)$is the encrypted block, and (K) is the encryption key. Each (C_i) block is translated to an element of a well selected (G_t) ring. In such a case, the encryption matrix (K) is assigned coefficients in the same ring (G_t). Due to the high degree of linearity, this technique is always exposed to selected plain text and known statistical attacks. On the other hand, the high correlation between adjacent pixels and diagonal pixels of the image makes this technique unsuitable for image encryption. Finally, there is no chain in the encryption system, so this method is vulnerable to differential attacks. The decryption operation is described by next equation.

Several successive developments in the methodology have taken place over time, but all using a reference ring such as (G₂₅₆) or(G₂₆) which significantly reduces the number of invertible matrixes and increases the risk of brutal attacks.

A first improvement [3-4-5] consists in modifying at each iteration; the encryption matrix by a secret permutation (h) and fixed in the ring (G₂₅₆) , on the rows or on the columns. This improvement is given by the equation 3

where h (K_i−1) is the transform of the matrix (K_i−1)by fixed permutation (h). Other improvements accompany the static encryption matrix of a translation vector(T) , and this to overcome the problem of uniform blocks [6−7] and null blocks, still others modify the translation vector at each iteration by a linear transformation provided by a fixed matrix (Q) of size (n, n), not necessarily inversible. This method is described by equation 4.

These improvements overcome statistical attacks and selected text attacks, but due to the lack of clear links between the original pixels, encrypted pixels, and encryption keys, they are still vulnerable to differential attacks. However, unless there is a strong correlation between the adjacent pixels of the image and the diagonal, all these methods are still powerless. Recently, the algorithm based on the classic chaotic method has exploded, and the chaotic suite [8−9] has been created to increase the key space, thereby protecting the method from brutal attacks. Unfortunately, due to the difficulty of calculating the inverse of higher-order matrices, these methods still use general non-invertible matrices with n>4 [10],[11], which poses complexity problems. So far, all encryption algorithms have considered the pixel values of the image in ring G₂₅₆, and the number of invertible matrices of size (n, n) with coefficient G₂₅₆ is given by the following formula [12], [13].

Faced with the great difficulty of inverting in large-size matrices, the researchers were content to handle matrices of sizes generally less than five, in the classic ring (G₂₅₆) To correct this anomaly, our contribution provides a convincing solution by treating all pixel values as elements of one of the constructed subjects. Working in the body greatly increases the number of invertible matrices and provides protection. To facilitate algebraic calculations, two chaos tables will be generated.

Moreover, taking advantage of the properties of the involuntary matrices, a new technique for constructing invertible matrices of random size will be determined.

Our algorithm uses two of the most famous and widely used chaotic maps in cryptography.

The most important step is to create an entity with 256 elements, which will replace the classical (G₂₅₆) in the calculation.

(2) F₂₅₆ Elements Representation

Any element of the F₂₅₆ body can be represented in five different forms:

a) Polynomial Writing

We know that

$$F_{256}=\left\{h\left(x\right)\in \frac{F\left[x\right]}{d}°h\le 7\right\}.$$

(9)

Consequently, any element can be written in the form of a polynomial of degree at most equal to 7 with (G₂) components. For example:

$$h\left(x\right)= x^6+x^4+x^3+x^2+1 , q\left(x\right)=x^2+x$$

b) Vector Writing

Any element of the body F₂₅₆ can be represented as a size vector (1, 8) with a coefficient in (G₂)

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c) Binary Writing

By simple conversion from vector writing to binary writing, any element of such a subject can be written in binary form

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d) Integers Writing

All body elements are displayed in 8 bits, consequently their value is located between 0 and 255. So, we have

$$F_{256}=\{\mathbf{0},\mathbf{1},\mathbf{2},\mathbf{3},\dots ,\mathbf{255}\}$$

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This notation can be extended to the coefficient matrix in F₂₅₆. For example:

$$M=\left(\begin{array}{cc}{x}^2+x+1& x\\ x^3+x^2& x^6+x^4+1\end{array}\right)\Rightarrow M=\left(\begin{array}{cc}7& 2\\ 12& 81\end{array}\right)$$

e) Discrete Exponential Writing

Note that, (F₂₅₆,, ⊕;⊗) is a finite body, consequently, the set $\left(F_{256}^{*},\otimes \right)$is a cyclic group. As a result, it is generated by a single element g(x) closely related to the constructor polynomial p(x). This can be illustrated by the following formula:

$$F_{256}^{*}=\left\{h\left(x\right)\in F\left[x\right] that d °h\le 7 ,and h\left(x\right)\ne 0\right\}=F_{256}-\left\{0\right\}$$

$$\{\begin{array}{c}\forall h\in F_{256}^{*}\exists !i\in \left[0 254\right] : h\left(x\right)=g{\left(x\right)}^i mod \left(p\left(x\right)\right), \\ by convention g{\left(x\right)}^{255}=0,\end{array}$$

(9)

where i is called the exponential notation of the polynomial h (x).

$$We note that ; Exp\left(i\right)=h\left(x\right)$$

We confirm that the change of the generator g(x) will lead to a fundamental change in the sign of the exponent, which will result in serious distortion of the entire encryption system.

f) Discrete logarithm Writing

Function (Exp) is bijective, and its inverse function is the function defined by the following formula (Log):

$$Al3 \{\begin{array}{c}for i=0 to 254\\ Log\left(Exp\left(i\right)\right)=i\\ Next i\\ by convention Log\left(0\right)=255\end{array}$$

This rating will greatly facilitate algebraic calculations

After extraction of the three color channels (RGB) and their conversion into vectors (Vr),(Vg),(Vb), a cohabitation is carried to form the vector X(x₁,x₂,… …..,x_3nm). To apply Hill’s new method, the vector (X) must be cut into blocks of the size of (r_h) calculated from the chaotic map and the original image size.

So, we can conduct as:

In order to implement the new technology, we need to cut the image vector (X) into large and small blocks (2r_h). This operation follows the following formula:

The vector (X) must be imputed by (s) pixels by the following method:

We noticed that this decomposition is completely controlled by the decision vector (CR)

In parallel, convert two chaotic vectors (KR) and (KL) into matrices (MR) and (ML) of size (t, 2r_h) following Fig. 2. After adjusting the size of the image vector, convert the latter to a matrix (MC) of size (t, 2r_h) as shown in Fig. 3.

Fig. 2. Converting two chaotic vectors.

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Fig. 3. Image vector decomposition.

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First, the(VI) initialization vector of size (1, 2r_h)) must be recalculated to change the value of the starting block. Ultimately, the (VI) value is provided by the next algorithm

To surpass the uniform image problem (Black,White …) the setup value (VI) will be combined with the chaotic vector (TT) specified by the following algorithm.

The value calculated from the clear image and the chaotic map, will only be used to change the value of the start pixel and restart the encryption process.

According to our technical steps, it will be easier to construct a large invertible matrix based on involute blocks and non-empty eigenvalue matrices

In our example simulation we took as encryption key

The high sensitivity of the encryption keys used in our system indicates that a very slight degradation of the encryption key automatically leads to an image that is so different from the original image. This confirmation can be viewed below the scheme in the next figure:

Fig. 6. Secret key’s sensitivity.

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We note that a 10⁻¹⁵ change in a single encryption parameter of this technology is incapable of restoring the clear image by the same decryption process.

Our design has given a new opportunity to survive and to partner with the strongest members in the hope of rebuilding a new population more adapted to intruder aggression. To do that, we randomly selected an image and studied the strength of the original populations and the new generation, with the following results:

Let be two encrypted images, whose corresponding free-to-air images differ by only one bit, from (C₁)and(C₂), respectively. The expressions of these two statistical constants (NPCR)and (UACI) are given by equations below

The UACI mathematical analysis

a) Signal-To-Peak Noise Ratio (PSNR)

i. MSE

Mean Square Error (MSE): This is the cumulative square deviation between the original image and other noisy images. When the MSE level decreases, the error also decreases. This constant measure the distance between the pixels of the clear image and the encrypted image. Calculated by the next equation.

$$MSE={\displaystyle \sum }_{i,j}{\left(P\left(i,j\right)-C(i,j\right)}^2,$$

(33)

where (P(i, j)): pixel of the clear image and (C(i, j)): pixel of the cypher image.

ii. PSNR

Since many signals have a large dynamic range, PSNR is usually expressed on a logarithmic decibel scale. The next equation gives the PSNR mathematical analysis of the image:

$$PSNR=20Lo{g}_{10}\left(\frac{I_{max}}{\sqrt{MSE}}\right).$$

(34)

Table 6. Differential parameters..

b) Avalanche effect

Our algorithm uses a strong link between encrypted pixels and pixels with clear policies. As a result, as data propagates through the structure of the algorithm, gradual changes become increasingly important. The avalanche effect is the number of bits that have been changed if a single bit in the original image is changed. The mathematical expression of this avalanche effect is given by

$$AE=\left(\frac{{{\displaystyle \sum }}_{i}bit change}{{{\displaystyle \sum }}_{i}bit total}\right)*100.$$

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Table 7. Avalanche effect.

c) Performance time

In our technique, the encryption and decryption times are very similar and vary in the interval [0.05, 0.1].

Table 8. Performance time.

d) Speed analysis

To approve and document the quality of our methodology in a timely fashion. And finally, thanks to these properties, we have selected the “Lena” grayscale image with three different sizes (256×256) (512×512) and (1024×1024). The results are presented in Table 9.

Table 9. Execution time (in second).

We compared the results with two classic algorithms, AES and DES, and can determine that the execution time is reasonable. The test was conducted on other images of different sizes, and we obtained an acceptable score. This is due to the low algorithm complexity of the algorithm implemented in our strategy.