Optimizing color image security using hybrid cryptographic techniques based on sine and logistic maps

Abstract

This paper proposes an innovative method to enhance the security of color images by integrating hybrid cryptographic techniques, specifically using Sine maps and a double application of the logistic map. The proposed method encrypts a color image in several distinct phases. In the first phase, the image is converted into an M×N matrix. In addition, the Sine maps generate sequences for each color channel: red, green, and blue. A bit-by-bit XOR operation is applied between the matrix and the sequences, producing the first encrypted image. In the second phase, the logistic map is applied to the output from the first phase, following the same sequence-generation procedure until the encrypted image is obtained. The final phase involves reapplying the Logistic map to the image from phase two using modified Logistic map parameters. A comparison with established techniques demonstrates the effectiveness of the proposed strategy in terms of both security and computational efficiency. Experimental results confirm strong performance, with an average correlation coefficient of −0.00014, NPCR = 99.61%, UACI = 33.61%, entropy = 7.9992, and PSNR = 8.60 dB. The average encryption time is 0.445 s for a 512 × 512 color image, demonstrating high robustness and resistance to various attacks, thereby offering valuable insights for practical integration into image-security frameworks.

Introduction

The security of color images is a primary concern in various areas¹, including the secure transmission of visual information and the protection of sensitive graphical data^2,3. Due to the widespread use of digital images, ensuring their protection against unauthorized access, modification, or interception has become a major challenge⁴. Traditional encryption algorithms such as AES or DES are often insufficient for image data because of its large size, high redundancy, and strong pixel correlation. Consequently, many chaos-based cryptographic approaches have been proposed to enhance the confidentiality and robustness of visual information⁵. In this paper, we propose a new approach to enhance the security of color images by exploiting the robust properties of Sine maps. Additionally, we integrate one-dimensional chaotic transformations to further strengthen the encryption process. This approach relies on the joint application of 1D chaotic transformations (Sine and Logistic maps) to generate highly secure pseudo-random sequences^6,7,8.

Sine maps are mathematical tools that exhibit deterministic chaotic behavior^{9,10,11,−12}. Their use in cryptography has shown promise due to their ability to generate robust random sequences, making encryption algorithms resistant to attacks^13,14,,14. This intrinsic property makes Sine maps suitable for designing encryption techniques capable of withstanding cryptanalytic attacks^15,16.

Simultaneously, the dual application of Logistic chaotic maps¹⁷ is an advanced strategy that exploits the chaotic behavior of logistic maps. The use of multiple instances of this map in the encryption process increases the algorithm’s complexity, thereby improving its resistance to intrusion attempts^18,19,20,21. Logistic maps, known for their unpredictable and deterministic nature, provide a solid foundation for encryption methods²².

The integration of Sine maps and Logistic maps allows the advantages of both chaotic approaches to be combined. By using these two types of transformations together, we can create an encryption system that exploits the chaotic properties to generate highly secure pseudo-random sequences^23,24. This approach provides enhanced protection of color images against cryptanalytic attacks and intrusion attempts.

In this article, we detail our approach based on the synergistic combination of these two powerful techniques Sine maps and the double application of Logistic maps in order to optimize the security of color images. We explore the theoretical aspects of these methods by highlighting their respective advantages^25,26, and describe how they are integrated into our proposal for increased security.

Extensive research conducted on applications of Sine and Logistic maps in cryptography²⁷. demonstrate their effectiveness in designing robust encryption schemes. However, exploration of their combined potential particularly for improving the security of color images remains limited²⁸.

In the following sections, we will describe our methodological approach, present the results of our experiments, and discuss the practical and theoretical implications of our method in the context of color image security. This research aims to contribute significantly to the evolution of cryptographic techniques dedicated to the protection of visual information.

Related work

The security of color images is a dynamic area of research, crucial to ensuring the confidentiality and integrity of visual data in various sectors. To reinforce this security, many researchers have studied the field of image encryption in order to design algorithms adapted to the specificities of multimedia data. Most of these algorithms rely on the use of chaotic maps^29,30,31,32, which are characterized by sensitivity to initial conditions, parameter variability, non-periodicity, and pseudo-random value generation.

A range of studies have focused on harnessing chaotic maps for image encryption. For instance, Singh et al.³³, developed a secure communication framework centered on the Logistic map as a principal component. Their analysis underscored the efficacy of this methodology in protecting the privacy of transmitted information. In a parallel effort, Goumidi et al.³⁴ presented an image-encryption technique tailored for satellite imagery, which utilized a modified confusion-diffusion paradigm that incorporated standard, Logistic, and Sine chaotic maps. Their method strategically merged various chaotic maps to boost encryption security through customized confusion and diffusion procedures.

Following a similar thread Zeng et al.³⁵ introduced an image-encryption scheme that employed a composite Logistic–Sine chaotic system. Their findings emphasized this method’s capability in preserving image confidentiality within distributed settings, offering an efficient and sturdy encryption solution by blending the attributes of the Logistic and Sine maps.

More recent contributions include the work of Zhou et al.²² who detailed an innovative image-encoding system that exploits a conservative hyperchaotic system in conjunction with closed-loop block diffusion. Their investigation demonstrated that this design enhances visual data security by uniting chaotic encryption with inter-block diffusion to maximize confusion. Finally, Kiran et al.³⁶ proposed a partial image-encryption technique relying on the Sine–Logistic Map (LSM), showcasing its ability to ensure the confidentiality of selected image areas by effectively employing the LSM maps as a cryptographic instrument.

Elazzaby et al.³⁷ present a significant advancement in image cryptography using the bidimensional Arnold Cat Map to rearrange pixel positions based on parameters from the original image. Combined with blur patterns, the multiplicative group $\:Z/nZ$ generated from a 2D logistic–sinusoidal hyperchaotic sequence adds randomness to the statistical properties of the encrypted image. By exploiting the properties of multiplicative groups and chaotic systems, they demonstrate a robust and efficient method for encrypting visual data. In²⁶, an image-encryption method using a 2D sinusoidal–Logistic modulation map and the $\:Z/nZ$ group was proposed to create a highly secure encryption system. By exploiting hyperchaotic properties, it generates a blurred pattern that obscures the original image.

Comparative tests show that this approach achieves high security, optimal complexity, and strong protection against unauthorized access, surpassing existing methods. Es-Sabry et al.³⁸, present a highly efficient color-image-encryption algorithm using Logistic, Sine, and Chebyshev maps along with an intersecting-plane technique in a cube. The algorithm encrypts each color channel (red, green, blue) by first extracting pixel values and performing circular rotations to ensure uniqueness, followed by encryption through intersecting planes and the Arnold Cat Map for confusion. The method demonstrates high performance, reliability, and robustness compared to existing techniques. Another recent work by Es-Sabry et al.³⁹, proposes a method for encrypting 32-bit color images using four one-dimensional chaotic maps (Tent, Logistic, Chebyshev, and Sine) and 16 × 16 matrices for each color channel. The process involves shifting pixel values, encrypting them with the Four-Square cipher, and using the Arnold Cat Map for pixel rearrangement. Evaluations demonstrate the algorithm’s strong performance and security against common attacks.

Recently, several notable studies have advanced chaos-based image encryption, particularly focusing on enhancing security metrics through complex chaotic structures. Alexan and Megalli⁴⁰ developed a method incorporating a memristive hyperchaotic system for multiple image encryption, aiming for high-speed hardware implementation. Similarly, Alexan et al.⁴¹ proposed an efficient scheme based on multiple chaotic systems combined with nonlinear transformations to fortify resistance against statistical and differential attacks.

In another approach, Moysis et al.⁴² introduced a chaotic encryption technique that leverages circular shifts for enhanced efficiency. Addressing the need for high-speed multimedia applications, Alexan et al.⁴³ presented an optimized algorithm featuring an enhanced permutation-diffusion architecture. Furthermore, Alexan et al.⁴⁴ demonstrated a strong multiple image encryption algorithm by combining hyperchaotic systems with Singular Value Decomposition (SVD) and a modified RC5 cipher to achieve strong diffusion.

While the aforementioned studies demonstrate the effectiveness of various chaotic systems, they often rely on single hyperchaotic maps, direct combinations of maps into a single chaotic system, or other complex transformations. The novelty of our proposed method lies in its unique sequential three-phase architecture that synergistically combines distinct cryptographic operations.

Unlike methods that merge chaotic maps into a single formula, our approach first employs a Sine map for a substitution-based XOR encryption. It then significantly enhances security by subjecting the result to a dual application of the Logistic map, utilizing different parameters at each stage to create a deep and complex confusion-diffusion effect. This layered strategy of sequential and distinct chaotic processes offers a more robust defense against cryptanalytic attacks compared to single-stage or blended-chaos systems.

Proposed method

In this section, we detail our proposed algorithm, which combines chaotic techniques based on the Sine map and the Logistic maps for color image encryption. The process begins by applying the Sine map method to the original color image of dimensions $\:M\times\:N$. From this, a key is automatically generated from the calculated variance between the image pixels. Next, the resulting output undergoes two successive application of the Logistic map, using different keys at each step. In other words, the cipher image that emerges after the application of the Sine map must go through a two-step process of the Logistic map. As illustrated in (Fig. 1), the proposed methodology describes the sequential steps involved in transforming the original color image into its final encrypted form.

Fig. 1

Schema of the proposed method.

Logistic map

The Logistic map is a widely utilized chaotic function, holding a primary role in the development of chaotic cryptosystems. A comprehensive understanding of its inherent characteristics and dynamics is essential for its application. This function is acutely susceptible to its initial value, a property critical for cryptography. Its non-linear nature ensures the exhibition of chaotic behavior when the control parameter $\:\alpha\:$ is bounded within the range [$\:3.57,\:4$]. The mathematical definition of the Logistic map is formally given by the following equation⁴⁵:

$$\:{X}_{n+1}=G(\alpha\:,{y}_{n})\:=\:\alpha\:\times\:{y}_{n}(1-{y}_{n})$$

(1)

With α ∈ [0, 4] and $\:{y}_{0}$ $\:\in\:$ [0, 1].

Fig. 2 visually presents the chaotic features of this system. The bifurcation diagram highlights the chaos zone in the interval [$\:3.57,\:4$], an observation supported by the Lyapunov exponent diagram. More specifically, Lyapunov exponent values are negative for $\:\alpha\:\:<\:3.57$ and become positive for $\:\alpha\:\:>\:3.57$. In addition, the values recorded on the diagram are in the range [$\:0,\:1$].

To formalize the sequences generated with the logistic maps for the different channels (red, green and blue) in matrices that match size of original image, the following formula is used:

$$\:G{\prime\:}(\alpha\:,{y}_{0})=E(255\times\:G(\alpha\:,{y}_{n}\left)\right)\:\:$$

(2)

Fig. 2

Logistic map: (L1) Bifurcation diagram (L2) Lyapunov exponent diagram.

Sine map

The Sine map incorporates the sine function into security algorithms to produce unpredictable elements (Fig. 3). This approach exploits the chaotic characteristics of the sine function to introduce randomness, thus strengthening data protection in cryptographic systems. It is a non-linear function exhibiting a chaotic attitude same to the Logistic maps, and it’s defined by⁴⁶:

$$S_{{n + 1}} = T(r,x) = r \times x_{n} \times \sin \left( {\pi \times x_{n} } \right)$$

(3)

with $\:r\:\in\:\:[0,\:4]$ and $\:{x}_{0}$ ∈ $\:[0,\:1]$.

Fig. 3

Sine map: (S1) Bifurcation diagram. (S2) Lyapunov exponent diagram.

Proposed encryption process

The suggested system relies on chaotic sine and logistic maps to encrypt a color image, the high-efficiency image encryption algorithm. The proposed detailed Method procedure is described as follows:

→Input: a color image I of size $\:M\times\:N$ with 3 channels (R, G, B), where each value is in the interval [$\:\text{0,255}$].

S1. Sine-map sequence generation:

• A chaotic sequence $\:X$ is first generated using the sine map.

• A normalization function is applied to scale the values between 0 and 1:

$$X^{\prime } = normalize\left( X \right)$$

(4)

Then, the sequence is further transformed follows:

$$\:{\text{Y}}^{{\prime\:}}=\text{a}\text{b}\text{s}\left(\text{f}\text{l}\text{o}\text{o}\text{r}\left(\text{sin}\left({\text{X}}^{{\prime\:}}{\uppi\:}\right)\times\:256\right)\right)$$

(5)

S2. Encryption using the Sine map:

The original image $\:\text{I}$ is encrypted using the Sine Map through the XOR operation for each pixel:

$$\:{\text{C}}_{\text{S}}(\text{i},\text{j})=\text{I}(\text{i},\text{j})\:\oplus\:{\text{X}}^{{\prime\:}}(\text{i},\text{j})$$

(6)

→$\:{\text{C}}_{\text{S}}$: Image encrypt with sine map.

S3. Generation of New Chaotic Sequences with the Logistic Map (First Application).

Two new chaotic sequences $\:X^{{\prime\:}{\prime\:}}$ and $\:Y^{{\prime\:}{\prime\:}}$ are generated using the Logistic Map defined by:

$$\:{X}^{{\prime\:}{\prime\:}}\left(n+1\right)={\upalpha\:}\times\:{\text{X}}^{{\prime\:}}\left(\text{n}\right)\left(1-{\text{X}}^{{\prime\:}}\left(\text{n}\right)\right){Y}^{{\prime\:}{\prime\:}}\left(n+1\right)={\upalpha\:}\times\:\text{Y}{\prime\:}\left(\text{n}\right)(1-{\text{X}}^{{\prime\:}}\left(\text{n}\right))$$

(7)

where $\:\alpha\:\:\in\:\:\left(\text{3.57,4}\right]$.

S4. First Application of Logistic Map:

• The encryption of $\:{\text{C}}_{\text{S}}$ (Sine map encryption) with the Logistic Map is performed as follows:

$$\:{\text{C}}_{\text{R}1}(\text{i},\text{j})=\left\{\begin{array}{c}{\text{C}}_{\text{s}}\left(\text{i},\text{j}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:if\:j\:mod\:2=0\\\:{\text{C}}_{\text{s}}\left(\left(\text{i}+\text{k}\right)\text{m}\text{o}\text{d}\:256,\text{j}\right),\:\:if\:not\end{array}\right.$$

(8)

where$\:\:k$ is a shift parameter used to rearrange the pixel indices and enhance diffusion.

S5. Generation of New Chaotic Sequences (Second Application of Logistic Map):

• A second set of logistic sequences $\:X^{{\prime\:}{\prime\:}{\prime\:}}\:$and $\:Y^{{\prime\:}{\prime\:}{\prime\:}}\:$is generated from the previous ones as follows:

$$\:{X}^{{\prime\:}{\prime\:}{\prime\:}}\left(n+1\right)={\alpha\:}^{{\prime\:}}\times\:{\text{X}}^{{\prime\:}{\prime\:}}\left(\text{n}\right)\left(1-{\text{X}}^{{\prime\:}{\prime\:}}\left(\text{n}\right)\right){Y}^{{\prime\:}{\prime\:}{\prime\:}}\left(n+1\right)={\alpha\:}^{{\prime\:}}\times\:\text{Y}{\prime\:}{\prime\:}\left(\text{n}\right)(1-{\text{X}}^{{\prime\:}{\prime\:}}\left(\text{n}\right))$$

(9)

where $\:\alpha\:{\prime\:}$ represents a new control parameter.

S6. Second Application of Logistic Map Encryption:

• The previously encrypted image $\:{\text{C}}_{\text{R}1}$ is re-encrypted using the second logistic map application:

$$\:{C}_{R2}(i,j)=\left\{\begin{array}{c}{\text{C}}_{\text{R}1}\left(\text{i},\text{j}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:if\:j\:mod\:2=0\\\:{\text{C}}_{\text{R}1}\left(\left(\text{i}+{\text{k}}^{{\prime\:}}\right)\text{m}\text{o}\text{d}\:256,\text{j}\right),\:\:if\:not\end{array}\right.$$

(10)

where $\:k{\prime\:}$ is a new shift parameter.

S7. Reconstruction of the Final Encrypted Image:

• Finally, the three-color channels (R, G, B) of the encrypted image $\:{\text{C}}_{\text{R}2}$​ are recombined to form the final ciphered image:

$$\:{\text{C}}_{\text{F}\text{I}\text{N}\text{A}\text{L}\text{E}}=\text{s}\text{t}\text{a}\text{c}\text{k}({\text{C}}_{\text{R}2}\text{}\text{}\left[:,:,0\right],{\text{C}}_{\text{R}2}\text{}\left[:,:,1\right],{\text{C}}_{\text{R}2}\text{}\left[:,:,2\right],\:axis=-1)$$

(11)

→output

• The final encrypted image $\:{\mathbf{C}}_{\mathbf{F}\mathbf{I}\mathbf{N}\mathbf{A}\mathbf{L}\mathbf{E}}$.

Following this analysis, we present an explanation of the proposed workflow procedure, illustrated by a diagram. This graphical representation, designed as a didactic tool, facilitates the understanding of the encryption process. The visual format, known for its pedagogical effectiveness, enhances the clarity and accessibility of our cryptographic innovation, as shown in (Fig. 4).

Experimental results

Statistical analysis

The purpose of this section is to present a comprehensive analysis of the proposed primitive cryptographic scheme in terms of its security and efficiency against well-known attack methods reported in the literature. Statistical techniques are applied to the cryptographic data to evaluate its resistance to various attacks. This process involves examining specific features or characteristics of the encrypted and decrypted data in order to identify potential weaknesses or information leakage. To achieve this, five different types of security analyses were conducted on encrypted images, and all operations were implemented using the Python programming language.

Fig. 4

Flowchart of the proposed method.

Computational complexity analysis

The computational efficiency of the proposed encryption algorithm is analyzed in terms of its theoretical complexity. For a color image of size $\:M\times\:N$ with three channels (R, G, and B), each pixel undergoes a fixed number of arithmetic and logical operation in both the confusion and diffusion phases, while the chaotic sequences are generated once the algorithm can be expressed as:

$$\:O\left(3\:\times\:\:M\:\times\:N\right)\approx\:O(M\:\times\:N)\:$$

(12)

This linear complexity indicates that the algorithm scales efficiently with the image size, ensuring a good balance between computational cost and encryption strength.

Execution time analysis

Execution time in-depth analysis was performed for a variety of image sizes, in order to dispel any doubts about the computational cost of the proposed hybrid cryptographic method. The main points that will be discussed in this section are performance according to the algorithm’s execution time and its comparison to similar cryptographic methods.

Experimental setup

• Processor: Intel(R) Core (TM) i7-4600U CPU @ 2.10 GHz (2.69 GHz).

• Programming Environment: Python 3.12, with libraries including NumPy and Pillow for optimized calculations.

• RAM:8Go.

• Dataset: A dataset consisting of multiple-sized 32-bit color images was used. The encryption technique put forward was applied to every image in sequence, with execution times measured by Python’s time module. Each operation was independently timed and repeated across scenarios for consistency and measurement of true performance metrics.

Table 1 The measured execution times for the algorithm on different image sizes.

The results are summarized in ‎Table 1. It can be observed that the execution time is approximately proportional to the number of pixels in the image. For medium-sized images (512 × 512), the execution time is as low as 0.445 s, demonstrating the suitability of the proposed method for applications that prioritize security over real-time performance. For larger images (1024 × 1024), the execution time remains reasonable at 1.747 s.

Table 2 A comparison to other approaches in terms of execution time (seconds).

‎Table 2 compares the suggested scheme’s execution time to alternative techniques for encrypting 512 × 512 pixel color images. Even under different testing conditions, our proposed algorithm performs better than others, which are optimized under similar conditions. The findings obtained show that the encryption process may be completed at remarkable speeds using the method we have provided, making it ideal for real-time applications.

Histogram

To evaluate the intensity distribution in both the original and encrypted images, the histogram is commonly used as the most effective and widely recognized tool for representing pixel value distributions^50,51. Based on this principle, we performed a comparative histogram analysis on eight different original images, each with distinct content, and their corresponding encrypted counterparts generated using our proposed method. Looking at the details of the study and referring to (Figs. 5, 6, 7, 8 and 9), they found that the histogram, expressed in terms of pixels of the encrypted images, resembles a uniform assignment. However, when comparing the original pictures histograms, there was a big difference in one image, while the other had much higher values in a different region. These substantial differences between the histograms of the original and encrypted images confirm that the proposed encryption algorithm effectively ensures security and provides strong resistance against statistical attacks.

Fig. 5

Histograms of the encoded and original female image.

Fig. 6

Histograms of the encoded and original Baboon image.

Fig. 7

Histograms of the encoded and original Tapi image.

Fig. 8

Histograms of the encoded and original peppers image.

Fig. 9

Histograms of the encoded and original Barbara image.

Correlation analysis between neighboring pixels

Unencrypted images are typically characterized by a strong statistical correlation between adjacent pixels. A robust cryptographic system for visual data must, therefore, be capable of effectively disrupting these correlations to guarantee sufficient protection against various statistical analysis attacks. To quantify the effectiveness of the proposed method in achieving this decorrelation, we selected a random sample consisting of 30,000 pairs of adjacent pixels from the original image and their corresponding positions in the encrypted image. Subsequently, we computed the correlation coefficients across three primary directions horizontal, vertical, and diagonal using the following formula^52,53:

$$\:{Cor}_{xy}=\frac{Cov\left(x,y\right)}{\sqrt{Lx}\sqrt{Ly}}$$

(13)

$$\:Cov\left(x,\:y\right)=\frac{1}{M}{\sum\:}_{i}^{M}({x}_{i}-{S}_{i})({y}_{i}-{S}_{y})$$

(14)

$$L_{x} = \frac{1}{M}\sum\; _{i}^{M} (x_{i} - S_{i} )^{2}$$

(15)

$$\:{S}_{x}=\frac{1}{M}{\sum\:}_{i}^{M}{x}_{i}$$

(16)

Table 3 Coefficient of correlation between neighboring pixels in the original and encrypted pictures.

Table 3 represents the calculated values of the correlation coefficient. It is obvious that the correlation coefficients display values close to 1, indicates that these pixels are strongly correlated with those of original images. Conversely, the encrypted image has a correlation coefficient that is approaching to 0, suggesting the absence of correlation between pixels.

Fig. 10

Horizontal correlation distributions for the Female image. (h1-h3): Red, Green, and Blue channels of the original image. (h4-h6): Corresponding channels of the encrypted image.

Fig. 11

Vertical correlation distributions for the Female image. (v1-v3): Red, Green, and Blue channels of the original image. (v4-v6): Corresponding channels of the encrypted image.

Fig. 12

Diagonal correlation distributions for the Female image. (d1-d3): Red, Green, and Blue channels of the original image. (d4-d6): Corresponding channels of the encrypted image.

We therefore conclude that, as shown in Figs. 10, 11, and 12 our method demonstrates a strong ability to break the dependency between neighboring pixels.

Table 4 The correlation coefficient comparison of adjacent pixels with other methods.

Table 4 compares the correlation coefficients of adjacent pixels for several encryption methods. Lower values particularly those near zero indicate stronger resistance to statistical attacks. The proposed method achieves the lowest average coefficient (-0.00014), outperforming all referenced schemes and confirming excellent decorrelation. While L. Moysis et al.⁴² reported a strong result with 0.00023, our method goes further by producing negative correlations in all directions, including Horizontal (-0.00006) and Vertical (-0.0024). Such negative values are rare and demonstrate a highly effective disruption of pixel relationships. This, combined with the near-zero average, highlights the strong scrambling capability of our three-phase architecture and confirms its superiority over recent approaches, including Alexan et al.⁴⁰.

The primary objective of this specific analysis is to determine the quantitative relationship linking each pixel value in the source image to its counterpart in the ciphered output. This computation relies on the following established formulas:

$$\:CC=\frac{{\sum\:}_{i,j}^{M,N}({C}_{i,j}-\:\overline{C})({C{\prime\:}}_{i,j}-\:\overline{C{\prime\:}})}{\sqrt{{\sum\:}_{i,j}^{M,N}{({C}_{i,j}-\:\overline{C})}^{2}}\sqrt{\sum\:_{i,j}^{M,N}({{C}^{{\prime\:}}}_{i,j}-\:\overline{{C}^{{\prime\:}}}\:{)}^{2}}}$$

(17)

With

$$\:\overline{C}=\frac{1}{M\:\times\:N}{\sum\:}_{i,j}^{M,N}{C}_{i,j}$$

(18)

$$\:\:\:\overline{\:C^{{\prime\:}}}=\frac{1}{M\:\times\:N}{\sum\:}_{i,j}^{M,N}{C^{{\prime\:}}}_{i,j}$$

(19)

where C is the simple image and $\:\overline{C}$ s its meaning. Similarly, C’ is the encrypted image, and $\:\overline{{C}^{{\prime\:}}}$ is its meaning. The dimensions of the matrices C and C’ are denoted by N is the length and M is the width, respectively.

Table 5 Comparing the correlation coefficient between the encrypted image and the original image using different techniques.

Table 5 demonstrates that our methodology performs exceptionally in comparison to the outcomes derived from the methodologies outlined in³⁷, which introduce an innovative image encryption algorithm using multiple chaotic maps with the intersection plane method. Moreover, our approach goes beyond the technique described in¹³, which depends on the coupling of a multiplicative group and the chaos theory in the image ciphers, as well as the method explained in⁵⁷ And⁵⁸. The value obtained is responsible for this superiority, which shows a remarkable proximity to 0.

Differential attacks

The cryptographic system is resistant to differential attacks, which are similar to was cantered attacks on a single image selected with the objective of identifying the secret key to be protected⁵¹. A robust cryptographic scheme, whether employing symmetric or asymmetric techniques, must demonstrate an exceptionally high sensitivity to minor alterations in the input image. This mandates that even a minute single-bit modification within the plaintext image should lead to substantial changes in the resulting cipher image. To quantitatively assess this crucial sensitivity characteristic of the cryptosystem, especially concerning the source image, two standard metrics are employed: the Number of Pixels Changing Rate (NPCR) and the Unified Average Change Intensity (UACI)⁵⁹. We conducted this evaluation by introducing a minute single-pixel change in the source images (Peppers, Baboon, and Bird). The calculated NPCR and UACI values, derived using Eqs. 20 and 21 respectively, consistently yielded results exceeding 99.63 and 33.61% (detailed results are available in Tables 6 and 7). Furthermore, our approach demonstrates a superior level of efficiency compared to most existing and recent methods, with only a few specific schemes exhibiting marginally higher performance.

Table 6 The result of the algorithm’s sensitivity to a single pixel change in the original image.

Table 7 Result for NPCR and UACI of the suggested method.

$$\:NPCR=\frac{1}{M\:\times\:N}\sum\:_{i=1}^{M}\sum\:_{j=1}^{N}D\left(i,j\right)\times\:100\text{\%}$$

(20)

$$UACI = \frac{1}{{M~ \times N}}\mathop \sum \limits_{{i = 1}}^{M} \mathop \sum \limits_{{j = 1}}^{N} \frac{{\left\| {C_{1} \left( {i,j} \right) - C_{2} \left( {i,j} \right)} \right\|}}{{255}} \times 100\%$$

(21)

With

$$\:D(i,j)=\left\{\begin{array}{c}0,\:\:if\:{C}_{1}\left(i,j\right)={C}_{2}\left(i,j\right)\\\:1,\:\:if\:{C}_{1}\left(i,j\right)\ne\:{C}_{2}\left(i,j\right)\end{array}\right.$$

(22)

where $\:{C}_{1}$ and$\:{\:C}_{2}$ denote two different levels of image encryption. $\:M,\:N$ make up the total number of image pixels. According to the optimal holding values, the NPCR threshold denoted as $\:{N}_{\alpha\:}^{*}$,while the UACI should remain within the designated critical range ($\:{U}_{\alpha\:}^{*-},{U}_{\alpha\:}^{*+})$.This can be expressed through the following equations:

$$N_{\alpha }^{*} ~ = \frac{{Q - \Phi ^{{ - 1}} \left( \alpha \right)\sqrt {Q/H} }}{{Q + 1}}$$

(23)

$$\left\{ {\begin{array}{*{20}c} {U_{\alpha }^{{* - }} = \mu _{u} - \Phi ^{{ - 1}} \left( {\frac{\alpha }{2}} \right)\beta _{u} } \\ {U_{\alpha }^{{* + }} = \mu _{u} - \Phi ^{{ - 1}} \left( {\frac{\alpha }{2}} \right)\beta _{{u^{'} }} } \\ \end{array} } \right.$$

(24)

Sensitivity key

Sensitivity of the ideal image to the private key is a crucial property of an encryption technique. Any minimal change in this key must yield an entirely different image. As such, we thoroughly tested the key sensitivity for the new image encryption technique. The aim of this study was to determine whether a change of the algorithm’s chaotic maps by 10^− 15 in the algorithm’s parameter would result in a successful decryption process (see Tables 8, 9).

The outcomes of the decryption using the correct key and the modified key are shown in Figs. 13 and 14. These findings demonstrate that a small adjustment to the key produced a noteworthy change of more than 99.63%.

Table 8 The responsiveness of the algorithm to a single pixel change in the original image.

Table 9 The percentage change rate (%).

Fig. 13

Visual demonstration of key robustness: (s1) Original Image. (s2) Encrypted Image. (s3) Successful Decryption using the correct secret key. (s4) Failed Decryption using a slightly incorrect key.

Fig. 14

Visual demonstration of key robustness: (k1) Original Image. (k2) Encrypted Image. (k3) Successful Decryption using the correct secret key. (k4) Failed Decryption using a slightly incorrect key.

Entropy analysis

In image cryptography, entropy is used to evaluate the randomness or unpredictability of pixel values within an image. A high entropy value indicates that the image contains a large amount of random information, making it more difficult to compress or predict. The confusion and diffusion processes employed during encryption increase entropy, thereby enhancing the security of the encrypted image and protecting sensitive visual data from malicious attacks⁴⁶. To calculate entropy, the following equation is used:

$$\:H\left(x\right)=\sum\:_{i=0}^{{2}^{n}-1}\text{P}\left({x}_{i}\right){\text{log}}_{2}[P\left({x}_{i}\right)]$$

(25)

where $\:P\left({x}_{i}\right)$ represents the probability of each $\:{x}_{i}$ occurring.

In this study, the information entropy is assessed independently for each color channel (Red, Green, and Blue). Given that each pixel in a grayscale (or color channel) image can adopt 256 distinct intensity levels, the inherent number of bits per pixel is $\:n=8$. The theoretical maximum value for entropy, $\:H\left(x\right)$, is reached when the source data exhibits complete statistical unpredictability. To successfully simulate a truly random data source and effectively preempt any potential prediction of the generated values, an encryption algorithm of high quality must aim to achieve this maximum entropy threshold.

For each color channel (red, green, and blue), the results of our method’s entropy calculation for the original and encrypted image are shown in Tables 10 and 11 using various images. Additional techniques are also employed for comparison.

The results obtained in Table 11 are close to the optimal value of 8. This suggests that our method can effectively withstand entropy attacks, thus exceeding the performance of the methods of El Azzaby et al.³⁷, Chen et al.⁶⁰ and Es-sabry et al.³⁹

Table 10 The entropy of the secret and original images.

Table 11 The entropy comparison with other methods.

Key space analysis

A crucial aspect in assessing the robustness of an encryption algorithm is the key space size, which defines the total number of possible key combinations. A sufficiently large key space guarantees strong resistance against brute-force attacks, as testing all potential keys becomes computationally impractical.

In the proposed encryption algorithm, six independent parameters are employed as secret keys:$\:\:r\:$and $\:x$₀ for the Sine map, $\:\alpha\:$ and $\:y$₀ for the first Logistic map application, and $\:\beta\:$ and $\:z$₀ for the second Logistic map application. Each parameter is represented using the IEEE 754 double-precision floating-point standard, which provides an accuracy of approximately 10^− 15. This precision implies that each parameter can take around 10¹⁵ distinct possible values within its defined range. Hence, the total key space can be estimated as:

$$KeySpace = (10^{{15}} )^{6} = ~10^{{90}} \approx 2^{{299}}$$

(26)

This exceptionally large key space makes exhaustive key search attacks computationally infeasible, ensuring a high degree of confidentiality and robustness. Therefore, the proposed encryption algorithm demonstrates excellent resistance to brute-force attacks and meets modern cryptographic security standards.

MSE and PSNR analysis

The Mean Squared Error (MSE) and the Peak Signal-to-Noise Ratio (PSNR) are two widely utilized measures employed for evaluating image quality, particularly within the context of image encryption performance. The MSE quantifies the average of the squared intensity differences between the pixel values of the original image and those of the encrypted (or decrypted) image. In the case of robust encryption, a high MSE value is expected, as it signifies a substantial disparity between the original and ciphered versions. Conversely, the PSNR serves as a metric to gauge the fidelity of image reconstruction, achieved by comparing the encrypted (or decrypted) output against the initial source image. It is measured in decibels (dB) and is based on the MSE. A low PSNR usually indicates better data protection. These two values are calculated between an encrypted image and the original image using the following equations:

$$\:MSE=\frac{1}{M\times\:N}\sum\:_{i=1}^{M}\sum\:_{j=1}^{N}[I\left(i,j\right)-K(i,j){]}^{2}$$

(27)

$$\:PSNR=10\:{\text{log}}_{10}\left(\frac{{L}^{2}}{MSE}\right)$$

(28)

With:

$\:M$ and $\:N$ are the image size.

$\:I\:(i,\:j)$ the value of the pixel at the position $\:(i,\:j)$ in the original picture.

$\:K\:(i,\:j)$ the value of the pixel at the position (i, j) in the encrypted image.

$\:L$ is the largest possible value of a single pixel in the image.

Table 12 PSNR and MSE values for each of the encrypted and unencrypted images.

The calculated MSE and PSNR values for the original and encrypted images across all three color channels (blue, green, and red) are documented in Table 12. The data presented robustly substantiates the high efficacy and cryptographic strength of our proposed algorithm, particularly in creating a significant distinction between the plaintext and ciphertext images.

Fig. 15

Key visual results of encrypted images under noise attack: (a) Original encrypted image, (b) Gaussian noise, (c) Salt & Pepper noise.

Noise attack analysis

To further experiment with the strength of the proposed encryption scheme, additional experiments were conducted in noisy conditions. Two sources of commonly encountered noises, the Gaussian noise and Salt & Pepper noise, were imposed on the encrypted image. Real-world interference that may be encountered during image acquisition, transmission, or storage was simulated using Gaussian noise with variance 0.02 and Salt & Pepper noise with density 0.02.

Figure 15 clearly shows that both types of noise-affected encrypted images suffer severe distortion and are visually indistinguishable from the original image. This is to be expected and thus validates that the proposed algorithm continues to possess its encryption strength notwithstanding external interference from noise. These results demonstrate the strength and integrity of the encryption process, establishing their suitability in secure image transmission over noisy communication links.

NIST randomness test analysis

To comprehensively evaluate the robustness and statistical quality of the proposed image encryption algorithm, we employed the National Institute of Standards and Technology (NIST) statistical test suite (SP 800–22). This suite is internationally recognized for assessing the randomness and cryptographic strength of binary sequences generated by encryption systems. The NIST test suite produces P-values, which serve as critical indicators of randomness. A test is considered passed if its P-value exceeds 0.01, corresponding to a 99% confidence level that the evaluated data exhibits no detectable deviation from ideal randomness. During our assessment, the encrypted image was first converted into a binary sequence and then subjected to all 15 standard NIST tests, including Frequency, Runs, Rank, and Linear Complexity analyses.

Table 13 NIST test results for the encrypted image.

The obtained results, summarized in Table 13, confirm that all P-values surpass the 0.01 threshold. This indicates that the encrypted data demonstrates excellent random statistical characteristics, comparable to those of a truly random sequence. These outcomes provide strong evidence of the high randomness and robustness of the proposed encryption scheme, confirming its resilience against statistical and probabilistic attacks.

Overall, the successful completion of the NIST tests validates the effectiveness and reliability of the encryption process, making it highly suitable for secure image transmission and data protection applications.

Future work

Although the obtained results have shown the robustness of the proposed encryption method, some improvement is still necessary to optimize its applicability in practical contexts. Therefore, the following avenues will be followed within this paper:

• Experimental evaluation in real-life conditions: Implementation of the method under different hardware conditions will enable in-depth analysis of its performance when necessary to operate under constraints given by limited computing capacities, or network fluctuations. These tests will allow checking the resistance of the encryption under real conditions of use.

• Performance optimization: Although our approach is competitive from the computational perspective, the use of optimization methods, such as parallelization or GPU, will be studied to improve the efficiency of encryption to enable real-time implementation.

• Securing and Managing Chaotic Keys: It is about developing a proper key management mechanism in order to enhance the security of the system. The work will consider the integration of efficient cryptographic protocols, such as Diffie-Hellman or elliptic curves, along with digital signatures for key integrity.

• Post-quantum attack resilience: Evaluating the resistance of the proposed scheme against quantum attacks is a priority research area because of the emergence of quantum computing. Concretely, exploring post-quantum algorithms will be considered to let encryption live for a longer time, especially those based on code-based cryptography.

Specifically, the incorporation of these enhancements is expected to significantly augment the robustness and operational efficiency of the proposed scheme. This increased capability enables its practical deployment in security-critical domains, such as the protection of sensitive industrial and medical communication channels.

Conclusion

In this scientific article, a new encryption method is proposed, consists in combining two 1D chaotic maps to create a fast and secure encryption technique, namely the logistic map and sine map, with the addition of an additional layer of security thanks to the dual application of logistic maps. This is to ensure an adequate response to the confusion criteria within the cryptosystem. We reinforce this method during the diffusion phase by modifying the keys used during the encryption process. A rigorous procedure guarantees a high level of encryption and successfully removes any resemblance or binding to the original image. There is judicious use of a wide range of evaluation criteria, including careful entropy analyses, accurate histogram analyses, a careful correlation coefficient that includes neighboring pixels and plain and encrypted images, and a careful examination of differential attacks NPCR, UACI, MSE, and PSNR. Every criterion is carefully and thoroughly examined. As a result, it is evident that the suggested method offers a practical way to encrypt images, improving both the security of such images and the resilience of image processing security systems.

Nevertheless, the system presents some limitations. Its strong sensitivity to initial conditions, while beneficial for security, can make it vulnerable to computational rounding errors or parameter mismatches. Moreover, the method’s computational cost may increase for large or real-time image datasets. Future work will focus on optimizing computational efficiency and exploring higher-dimensional or hybrid chaotic models to further enhance encryption strength.