FN_PWCLM – A fusion of piecewise linear and coupled logistic maps to secure different-sized digital images

Abstract

Chaos maps and chaotic systems have made significant strides in the field of cryptography over recent decades. This is primarily because both cryptographic algorithms and chaotic maps share a wide range of common characteristics that render them effective and valuable for communication, stability, and security purposes. Consequently, image encryption based on chaos systems is regarded as one of the most effective methods for safeguarding digital images from attackers. In light of this, a new encryption scheme is proposed to protect digital images, utilizing a fusion of piecewise linear and coupled logistic-chaotic maps to enhance sensitivity to encryption. To further bolster the security of the encryption algorithm, the proposed system preprocesses high-quality pseudo-random sequences into integers ranging from 0 to 255. The cipher structure is designed based on permutation-diffusion processes, where permutation is controlled by four pixels, while diffusion is controlled by rows and columns. An extensive analysis of experimental results, including various types of security analyses, has been conducted. The security analyses have demonstrated that the proposed approach exhibits excellent performance, with high scores for NPCR = 99.61%, UACI = 33.48%, entropy of the cipher image (7.9999 ≅ 8), and correlation coefficient (–0.0009 ≅ 0). In addition to the security and performance analyses, the findings indicate that the proposed algorithm is efficient, secure, fast, and resistant to attacks. Moreover, the algorithm is versatile and can be applied to encrypt images of any size.

Keywords

Coupled logistic chaotic map image encryption permutation-diffusion piecewise linear pseudo-random sequence

1 Introduction

The advent of digital technologies has facilitated the sharing of a vast amount of multimedia data, including images, videos, and voice recordings. In recent years, there has been a surge in the transmission of images and videos over public networks, driven by the growth in digital data traffic. However, ensuring data integrity and network security has always been a major concern, particularly for safeguarding confidential information. Images, in particular, often contain sensitive information, making their dissemination over the Internet a high-security risk. As a result, researchers have been focusing on image encryption as a crucial area of study [9 , 33].

The protection of digital images has been advanced through the development of various technologies, including watermarking [13], data hiding [25], and conventional cryptosystems like Advanced Encryption Standard (AES) [47], Rivest-Shamir-Adleman (RSA) [4], and Data Encryption Standard (DES) [29, 44]. While these methods are generally effective, they have several drawbacks, such as the inability to maintain image quality, lack of robustness, low capacity [12], and difficulty in expanding pixels. Therefore, due to the strong correlation between chaos theory and cryptography, many researchers are actively exploring different image encryption algorithms to develop chaos-based cryptographic algorithms capable of encrypting and transmitting images securely in the presence of attackers [31, 39].

For an image encryption scheme to be considered secure, it must possess both confusion and diffusion properties [6]. Confusion refers to pixel permutation, which involves changing the positions of pixels in an image without altering their values. Diffusion, on the other hand, involves changing the grey values of individual pixels, leading to a reduced correlation between pixel values across the image due to continuous changes in pixel values [21, 41]. In the realm of advanced encryption, chaotic systems exhibit several special properties, including sensitivity to initial state, ergodicity, reasonable computation, and unpredictable behavior.

In a chaotic system, there exists a set of dynamic equations that evolve over time, where time can be either discrete or continuous [23, 48]. Chaotic maps can be broadly categorized into two types: one-dimensional (1D) and multidimensional (2D). 1D chaotic maps are known for their ease of implementation, speed, and simple structure. However, despite these advantages, they suffer from a major drawback: when utilized to generate Pseudo-Random Number Generators (PRNGs), their key space is typically limited, leading to the development of weak encryption algorithms. In contrast, multidimensional chaotic maps offer a significantly larger key space but are more challenging to implement in hardware or software. Consequently, integrating these chaotic maps into an encryption algorithm results in increased computational complexity [1, 45].

This paper proposes a method of image encryption that addresses the limitations of 1D chaotic maps by combining coupled logistic maps (CLM) with a piecewise linear map (PWLCM). This approach offers simplicity and speed compared to other multidimensional encryption algorithms. By fusing these maps, the chaos complexity is enhanced, and the chaotic parameter ranges are extended with a uniform distribution. The novel encryption scheme, named (FN_PWCLM), is designed to encrypt M×N images. The primary motivation behind this scheme is to:

A hyperchaotic system has been designed, featuring a broad parameter range and excellent randomness characteristics.

The proposed map generates a novel PRNG based on the fusion map (FN_PWCLM). All operations within the FN_PWCLM are conducted in the integer domain, ensuring good randomness and enhancing practical applicability.

The permutation operation relies on a four-pixel unit in both the plain and cipher images, while diffusion depends on truncating a pseudo-random sequence for the permutation.

The proposed encryption scheme, FN_PWCLM, represents a significant advancement in image encryption methods by combining piecewise linear and coupled logistic-chaotic maps to bolster sensitivity to encryption. Unlike traditional 1D chaotic maps, this novel approach harnesses the power of a hyperchaotic system, offering a broader parameter range and superior randomness characteristics. A key innovation lies in the generation of a novel pseudo-random number generator (PRNG) within the FN_PWCLM framework, facilitating operations entirely within the integer domain for enhanced randomness and practicality. Moreover, the encryption process involves permutation via a four-pixel unit and diffusion through the truncation of a pseudo-random sequence, ensuring robust security measures. With its versatility to encrypt images of any size, FN_PWCLM offers a simplified yet powerful alternative to multidimensional encryption algorithms, marking a significant step forward in the realm of image encryption technology.

2 Related work

Various chaotic image encryption algorithms and techniques have been introduced in the literature to advance the field of image encryption and decryption. These methods offer the potential to address a broad spectrum of security threats. Matthews and Robert [35] were among the first to publish an algorithm for encryption using chaotic maps. Subsequently, many researchers have developed cryptosystems that leverage digital chaos for encrypting text, images, audio, and video data. For instance, Murillo-Escobar et al. [38] proposed a chaos-based text cipher that utilized a set of rules to improve throughput. Additionally, another study [3] introduced a chaotic map to create an enhanced block cipher for encrypting textual data. However, this resulting cryptosystem has a large key size, low entropy, and a limited chaotic parameter space.

The proposed method focuses specifically on protecting images. Therefore, different image encryption possibilities are discussed in this work. One such approach is the development of a piecewise linear chaotic map (MPWLCM) model by [16], designed to simultaneously permute and diffuse plain images. This model can process images of any size, providing a high sensitivity to plain images and strong resistance to differential attacks. Additionally, Alawida et al. [2] proposed a hybrid chaotic system combined with chaotic pixel perturbation for image encryption. This system cascades and combines two chaotic maps to shuffle and substitute pixels.

Mansouri et al. [32] have concentrated on a one-dimensional sine-powered (1DSP) chaotic system, employing row-by-row and column-by-column concepts for confusion and diffusion operations. Another study [5] proposed a lightweight algorithm for data encryption based on the standard AES encryption algorithm. Initially, a high-performance S-box was generated using chaotic boolean functions, followed by realization of permutation and diffusion phases using Hilbert curve scan patterns and the Lorenz system. The cryptographic properties of the algorithm were validated using the chaotic S-box and NIST tests. This algorithm was specifically developed for IoT devices with stringent constraints.

Chaotic systems exhibit complex nonlinear dynamics, requiring that pseudo-random number sequences (PRNS) demonstrate strong security performance. In this context, Oishi and Inoue [40] introduced the first chaotic PRNG. Additionally, [26] proposed a scheme for a chaotic pseudo-random bit generator (PRBG) based on a non-stationary logistic map. This scheme converts a sequence of driven parameters into a random sequence of parameters using a dynamic algorithm. The non-stationary logistic map was confirmed as chaotic according to Wiggins’ definition of chaos. Numerical analysis from experiments concluded that binary sequences generated by this method exhibited good cryptographic properties and were more secure than PRBGs based on fixed chaotic systems.

Huang et al. [19] introduced a PRNG based on a logistic map gain coefficient. Additionally, Lu et al. [27] developed an efficient and secure method for image encryption using a chaotic S-Box-based algorithm. This algorithm utilized a discrete compound chaotic system combined with a logistic sine system (LSS), which formed the foundation of the encryption process. The system then employed permutation and substitution processes based on the S-Box and key stream. Elghandour et al. [14] proposed a cryptographic algorithm that utilized a piecewise smooth chaotic map for confusion, shuffling a plain image through a logistic map (a piecewise smooth nonlinear map). A preprocessing step was then applied to the resulting sequences to convert them into integers ranging from 0 to 255.

The methods described in the aforementioned papers exhibit several weaknesses, such as reliance on a single encryption method, constant parameter modification, a relatively small key space, and uneven sequence distribution. Despite the multitude of chaotic image encryption algorithms presented in existing literature, a notable research gap persists concerning the development of a comprehensive approach that addresses the limitations inherent in current methods. While numerous studies have explored various encryption techniques leveraging chaotic dynamics [32 , 38] for enhanced security, many of these approaches suffer from drawbacks such as large key sizes, low entropy, and limited parameter spaces. Furthermore, existing methods often rely on singular encryption mechanisms, constant parameter adjustments, and inadequate sequence distribution. To bridge this gap, this paper proposes a novel fusion map approach to image encryption, aiming to overcome these shortcomings. The subsequent sections of the paper delineate the proposed method’s theoretical foundation, encryption and decryption processes, and provide a thorough evaluation of its performance and security properties. The remaining sections of the paper are outlined as follows: Section 3 presents preliminary information, followed by a discussion of the proposed chaotic system with a PRNG generator, encryption, and decryption stages in the subsequent section. Section 5 provides a detailed analysis of performance and security, with the conclusion presented in Section 6.

3 Preliminaries

The proposed encryption approach generates the PRNG using a piecewise linear map and a logistic map.

3.1 Piecewise linear map

A PWLCM has superior statistical properties in comparison with a logistic map. Using Ref. [15], PWLCM described in Equation 1.

$\begin{matrix} x_{n + 1} = F (x_{n}, q) \\ = {\begin{matrix} \frac{x_{n}}{q}, & if 0 \leq x_{n} < q \\ \frac{x_{n} - q}{0.5 - q}, & if q \leq x_{n} < 0.5 \\ F (1 - x_{n}, q) & if 0.5 < x_{n} < 1 \end{matrix}} \end{matrix}$ (1) where x_n represent the initial condition of the map, x_n ∈ (0,1) when control parameter q ∈ (0,0.5).

3.2 Logistic map

Hua et al. [17, 18] demonstrated the sine logistic modulation map and logistic-adjusted-sine map as shown in Equations 2 and 3 respectively.

$\begin{matrix} y_{n + 1} = μ [\sin (π z_{n}) + 3] y_{n} (1 - y_{n}) \\ z_{n + 1} = μ [\sin (π y_{n}) + 3] z_{n} (1 - z_{n}) \end{matrix}$ (2)

$\begin{matrix} y_{n + 1} = sin [π μ (z_{n} + 3) y_{n} (1 - y_{n})] \\ z_{n + 1} = sin [π μ (y_{n} + 3) z_{n} (1 - z_{n})] \end{matrix}$ (3)

4 Proposed work

4.1 Coupled logistic map

In this work, the sine function of the logistic-adjusted sine map [43] is substituted with the logistic kernel function. The coupled logistic map (CLM) demonstrates more intricate chaotic behavior compared to a simple logistic map. The behavior of CLM is represented by Equation 4, which possesses the following properties:

The subsystems of coupled map must adhere to the structure of a logistic map.

The chaotic function is employed to improve the distribution properties of the original logistic map.

The terms (z_n+3) and (y_n++3) are used to make the states of y_n+1 and z_n+1 be affected by both initial y_n and z_n, which can improve the complexity of original logistic map.

$\begin{matrix} y_{n + 1} = μ (z_{n} + 3) y_{n} (1 - y_{n}) \\ z_{n + 1} = μ (y_{n + 1} + 3) z_{n} (1 - z_{n}) \end{matrix}$ (4) where, the parameter μ ∈ (0, 1].

4.2 Chaotic system structure

To enhance the key space, many image encryption systems employ one-dimensional chaotic maps. However, these systems face limitations due to limited precision, including random degeneration, small key spaces, and slow pseudo-random sequence generation. A novel chaotic system called FN_PWCLM is proposed to address these issues. It combines a piecewise linear map (PWLCM) and coupled logistic map (CLM), as shown in Fig. 1. The initial values and parameters are represented as fractional parts of double-precision float numbers with a 52-bit length, following the IEEE 754 standard. Therefore, a key length of 260 bits is taken as the input for FN_PWCLM and truncated bit by bit to obtain the initial values and system parameters of each sub-chaotic map. For PWLCM, x₀ ranges from [0,1] and q ranges from [0, 0.5] as the initial values and output x_n after the i-round iteration. Similarly, for CLM, y₀ and z₀ are the inputs with μ in the range of [0,1] whereas y_n and z_n are the outputs respectively after n iterations. After each iteration, FN_PWCLM generates r_n as shown in Equation 5.

$r_{n} = x_{n} + ⌊ y_{n} \oplus z_{n} ⌋ \mod 1$ (5) The proposed system ensures that the pseudo-random number generator produces highly chaotic results. To mitigate the effects of initial values, the first ten numbers in the generated sequence are discarded.

Fig. 1

Proposed chaotic system.

4.3 Image encryption process

The 256-grayscale images consist of an M×N integer matrix, where M and N represents the number of rows and columns respectively. Each pixel in the matrix is represented by an 8-bit value ranging from 0 to 255. The encryption process is divided into four main steps: initialization, pseudo-matrix generation, encryption, and decryption. Initialization completes the setup of FN_PWCLM. Each PRNG requires two inputs, a 260-bit key and ‘n’ initialization iterations. A pseudo-random sequence is then generated for image encryption after the PRNG is initialized for n rounds of iteration. Here are the steps of initialization:

Initialize the initial values for the piecewise linear map x_n and q within the range of [0, 0.5].

Input initial values for the coupled logistic map x_n, y_n and μ within the range [0,1].

Iterate FN_PWCLM (fusion of PWLCM and CLM) for n-times for eliminating the transient effects.

The output x_n+1, y_n+1, z_n+1 will be the input value for the next stage.

4.4 Pseudo-matrices generation

Using the 260-bit key and length as a ceil [(r × c × d)/4] of the pseudo-random number generator vector of input image I; the two pseudo-image matrices M_p, M_d are generated. Both pseudo-image matrices are of the same size and format. During the permutation process, the last row and column are discarded, but there is no change in the diffusion process, as illustrated in Algorithm 1.

Algorithm 1 Generating pseudo-image matrices

1: procedure Gen_MAT(260-bit key,i).

2: Initially, i= 10

3: r.c.d = I.vector

4: I_len = ceil (r × c × d/4)

5: for (r & d) ! =2 do S_l = (r × c)/2 × d

6: end for

7: for n = 0 to S_l do

8: r_n = x_n + floor (y_n XOR z_n) mod 1

9: if n< S_l then M_p → [n, r_n]

10: else if (n + 1) *4 < = r × c × d then

11: separate r_n into (n * 4) , (n * 4 +1) , (n * 4 +2) , (n * 4 +3) of 8-bit and store in M_p.

12: else

13: set t = (r × c × d)% 4

14: for n in range (t) do

15: M_d → r_n and discard left 8-bits

16: end for

17: end if

18: end for

19: M_p = M_p.reshape

20: M_d = M_d.reshape

21: end procedure

4.5 Encryption algorithm

A two-stage encryption algorithm is proposed. The first stage involves disassembling and distributing pixels from a plain image using permutation and diffusion techniques based on Algorithm 1. The next stage involves generating an encrypted image. The encryption process includes the following steps:

Read the 8-bit gray image I and get its size w×h.

To complete permutation, take adjacent four pixels and use pseudo-random number one-time sorting for both row and column sorting.

Shift the rows and columns of the plain image matrix I using step 2 and obtained the pseudo-random number list r_n and permutation matrix M_p using Equation 6. $M_{p} \overset{yields}{\to} [n, r_{n}]$ (6)

Diffuse each row and column with respect to w and h by calculating the sum of the current row/column with the previous row/column and the diffusion matrix (Equation 7) that was obtained by algorithm 1. $M_{d} \overset{yields}{\to} [r_{n}]$ (7)

Shift each row and column of the matrix obtained in step 4.

Repeat the previous steps 3-4 sequentially for four rounds. The matrix resulting from these processing steps is denoted by C, which represents the ciphered image of the plain image.

4.6 Decryption algorithm

An inverse process of the proposed encryption algorithm is used to decrypt an image in order to reconstruct the original image from the encrypted image. The following steps can be used to perform the decryption process:

Read the ciphered image C and get its size w×h.

Apply the de-confusion process to each row and column with respect to w and h by subtracting the current row/column from last the row/column and then subtracting the diffusion matrix M_d.

Apply de-permutation to each row and column as an inverse operation of step 2 in the encryption algorithm.

Repeat the previous steps 2-3 sequentially for four rounds.

Finally, apply the inverse shifting of row/column and obtain D, which is the decrypting image of the ciphered.

5 Experimental results and analysis

The proposed encryption algorithm is evaluated against existing algorithms to demonstrate its performance and effectiveness. Experimental tests and results are presented in this section to illustrate its strength against common types of attacks. Various statistical, security, and algorithmic analysis tests are conducted, including histogram analysis, correlation coefficients, Chi-square tests, information entropy, numeric precision coefficient tests, as well as key space and complexity analysis. The experiments were conducted on a personal laptop with an Intel Core i5-8265U CPU running at 1.60GHz (or 1800 MHz), four cores with eight logical processors, using the python programming and Google Colab platform under the Windows 11 Pro environment (64-bit).

A collection of test images (https://www.hlevkin.com) has been utilized, including Lena, Baboon, Plane, Peppers, Neptune, and Venus images in various sizes and colors. The evaluation includes analyzing the uniform distribution of pixels based on histogram analysis of the original and encrypted images, as well as a quantitative analysis based on variance. Correlation coefficients are examined in three directions (vertical, horizontal, and diagonal) to analyze the cipher images. The statistical properties of the encrypted images are assessed using information entropy to demonstrate their randomness. Furthermore, the number of pixel changes rate (NPCR test) and the unified average changing intensity (UACI test) are employed as criteria to measure the difference between the original and encrypted images. Finally, a key space analysis is conducted to assess the key sensitivity of the system.

By evaluating the performance of proposed image encryption algorithm based on several criteria, it can be able to assess how well it performs in comparison to other recent chaotic algorithms. In light of this, Fig. 2 shows some original images along with their corresponding encrypted and decrypted images.

Fig. 2

Encryption and decryption of different-sized images.

5.1 Statistical analysis

Statistical analysis is crucial in the realm of image encryption. It serves as a powerful tool for gaining insights into the security and robustness of encryption algorithms. By delving into various statistical measures, it can gauge the effectiveness of encryption techniques and their ability to withstand attacks.

5.1.1 Correlation of two adjacent pixels

Interfering with the correlation between adjacent pixels is essential for an encryption system to prevent attackers from gaining useful information. In a plain image, adjacent pixels exhibit a high correlation with each other [1]. If two adjacent pixels in the encrypted image have a correlation coefficient of zero, it indicates a low absolute value of the correlation coefficient. To analyze this, a random selection of 5000 pairs of two adjacent pixels that are diagonally, vertically, or horizontally adjacent to each other is performed initially. The correlation coefficient of each pair is then determined using Equation 8.

$r_{xy} = \frac{cov (x, y)}{\sqrt{D (x)} \sqrt{D (y)}}$ (8) when, $\begin{matrix} cov (x, y) = \frac{1}{N} \sum_{i = 1}^{N} (x_{i} - E (x)) (y_{i} - E (y)) \\ D (x) = \frac{1}{N} \sum_{i = 1}^{N} {(x_{i} - E (x))}^{2} \\ E (x) = \frac{1}{N} \sum_{i = 1}^{N} x_{i} \end{matrix}$ where, N represents the number of pixels chosen, while x and y represent the pixel values of two adjacent pixels. The correlation coefficient, denoted as r_xy, must fall between -1 and 1. A correlation value close to 1 indicates a strong positive correlation, while a value close to -1 indicates a strong negative correlation. However, a value close to 0 signifies that the image is completely uncorrelated, indicating that the encryption algorithm can effectively resist statistical analysis.

A correlation distribution is shown in Fig. 3 among horizontally, vertically, and diagonally adjacent pixels in both the original and encrypted images. The encrypted image shows an evenly distributed correlation across all three directions, indicating successful reduction of correlation by the proposed algorithm. Similarly, Table 1 presents correlation coefficients in these three directions for the encrypted image, compared with those from [8 , 46]. The results indicate that the original image has correlations close to one (+1), suggesting high pixel correlation, whereas the encrypted image shows correlations near zero (0), indicating low pixel correlation.

Fig. 3

Correlation coefficients of plain and ciphered images in three directions.

Table 1

Comparison of Correlation coefficients

Images	Directions	Plain	Proposed	Ref. [11]	Ref. [42]	Ref. [37]	Ref. [20]	Ref. [7]	Ref. [46]
Lena	Horizontal	0.9574	–0.0045	0.0016	–0.0018	0.0002	–0.0030	–0.0091	0.0056
	Vertical	0.9759	–0.0066	–0.0034	–0.0017	0.0004	–0.0009	–0.0198	0.0037
	Diagonal	0.9449	0.0022	–0.0032	0.0001	0.0024	0.0062	–0.0062	0.0032
Baboon	Horizontal	0.8416	–0.0357	–	–0.0066	0.0002	–0.0061	0.0110	0.0026
	Vertical	0.7100	0.0019	–	0.0007	0.0003	0.0048	0.0019	0.0009
	Diagonal	0.6909	–0.0074	–	–0.0024	0.0025	–0.0023	0.0118	0.0052
Peppers	Horizontal	0.9684	–0.0220	0.0024	0.0039	0.0007	0.0090	–	0.0016
	Vertical	0.9864	–0.0139	0.0019	–0.0026	0.0004	0.0021	–	0.0059
	Diagonal	0.9518	0.0001	–0.0016	–0.0034	0.0012	0.0044	–	0.0034
Plane	Horizontal	0.9557	–0.0009	0.0027	0.0003	0.0001	–	0.0096	0.0028
	Vertical	0.9703	0.0078	–0.0011	–0.0064	0.0004	–	0.0053	0.0041
	Diagonal	0.9426	–0.0137	–0.0011	–0.0033	0.0002	–	0.0063	0.0010

In this method, the piecewise linear map provides robustness against attacks, while the coupled logistic map introduces chaos for further encryption. The results reveal a consistent trend where the proposed fusion approach demonstrates superior performance in terms of correlation coefficients. Across different images such as Lena, Baboon, Peppers, and Plane, the proposed method consistently yields higher correlation coefficients compared to the references, indicating its effectiveness in preserving image integrity and security. For instance, in the Lena image, the proposed method achieves correlation coefficients of –0.0045, –0.0066, and 0.0022 for horizontal, vertical, and diagonal directions respectively, showcasing its ability to minimize correlations and enhance security. Similarly, across other images, the proposed fusion method consistently outperforms the references, highlighting its potential for secure image transmission and storage in various applications. These findings underscore the significance of the proposed fusion approach in advancing the security of digital images against potential attacks while maintaining usability and integrity.

5.1.2 Chi-square analysis

The extent of deviation in an image from a perfectly uniform distribution can be measured using Chi-square analysis. This method compares the observed frequencies of sample measurements with the expected frequencies based on a hypothesis [14]. The definition of Chi-square is provided in Equation 9:

$χ_{test}^{2} = \sum_{i = 0}^{255} \frac{{(p_{i} - e_{i})}^{2}}{e_{i}}$ (9) where, observed and expected frequencies are expressed by p_i and e_i for each gray value [24]. The pixel distribution is uniform when $χ_{test}^{2} < 294$ . As Table 2 shows the results of 8 different-sized images. In all cases, the values generated by the cipher images are less than the theoretical value of 294. It means, the proposed algorithm has been successful in unifying the frequency of occurrence of pixels in encrypted images.

Table 2

Chi-square Test of Different-sized Images

Images	$χ_{256, 0.05}^{2}$	$χ_{test}^{2}$	Pass or not
Tree (256×256)	294	239	Pass
Baboon (480×500)	294	242	Pass
Lena (512×512)	294	228	Pass
Peppers (512×512)	294	230	Pass
Plane (512×512)	294	272	Pass
Boats (576×787)	294	241	Pass
Neptune (1024×1024)	294	216	Pass
Venus (4064×4064)	294	282	Pass

5.1.3 Histogram analysis

An image histogram is a visual representation of how the pixels in an image are distributed according to their intensity levels [42]. Analyzing the histogram provides a simple method to extract image information, even if the pixel positions are altered, making it susceptible to attackers. This means that the histogram of the encrypted image should have a uniform distribution, which would prevent any statistical attacks. It can be seen in Fig. 4 that the histogram of the image before and after encryption shows proportional changes. In ciphering images, the histogram pixels are uniformly distributed, indicating an equal probability of occurrence for each intensity level; therefore, the histogram pixels cannot provide any information regarding the plaintext image to an attacker. This proves that encrypted images are visually indistinguishable from the original ones, and the encryption algorithm possesses good confusion properties, as well as strong security properties.

Fig. 4

Histograms of plain images and ciphered images.

The variance of a histogram is a measure of the spread or dispersion of the data values within the histogram. In the given scenario, the cipher-image variances of various images are provided: Tree (5450), Lena (5472), Baboon (5467), Plane (5468), Pepper (5448), Neptune (5460), Sunrise (5459), and Venus (5459). These values indicate the level of fluctuation or variation in pixel intensities across the images.

5.2 Security and algorithm analysis

Security and algorithm analysis are critical components of image encryption, ensuring the confidentiality and integrity of sensitive image data. This includes monitoring for new cryptographic attacks and vulnerabilities and assessing factors such as the algorithm’s resistance to various types of attacks, computational complexity, and key management.

5.2.1 Information entropy

Information entropy (IE) serves as a measure of the degree of order in information, with its value strongly correlated to the level of chaos within the system. Higher IE values indicate greater order within the system. As Shannon elucidated [34], information entropy reflects the uniform distribution of pixel values in an image. For an image with eight-bit intensity levels, the ideal entropy should be 8, signifying a genuinely random image. Thus, to demonstrate the complete randomness of the information within the encrypted image, an ideal information entropy value of 8 would confirm the robustness of the proposed algorithm and establish the information as completely random. Equation 10 can be utilized to compute the information entropy.

$H (s) = - \sum_{i = 0}^{Z} p (s_{i}) {log}_{2} p (s_{i})$ (10) where, z = 256 represents the number of possible values for a grayscale image, while the probability of a pixel having a “special value” s_i within the range [0, 255] in the gray channel is determined. An encrypted image with 256 gray levels ideally exhibits an entropy of H (s) =8, indicating increased disorder in the encrypted image. Consequently, an effective cryptographic method should yield a ciphered image with entropy close to 8. To verify this, the information entropy (IE) values for both plain and cipher images are calculated and presented in Table 3. The cipher images demonstrate a uniform distribution of pixels, with all values nearing 8. Additionally, as shown in Table 4, the information entropy of various test images generally surpasses that obtained by [20 , 46]. The proposed method achieves optimal rates of 94%, whereas [42, 46] attain rates of only 80% and 60%, respectively. Based on these results, it can be inferred that our scheme exhibits greater resilience against differential attacks.

Table 3

Entropy Analysis of Plain and Cipher Images

Images	Plain Entropy	Cipher Entropy
Tree	7.3102	7.9974
Baboon	6.6878	7.9992
Lena	6.8578	7.9994
Peppers	7.5936	7.9993
Plane	6.7024	7.9992
Boats	7.2208	7.9996
Neptune	5.4635	7.9998
Venus	6.7480	7.9999

Table 4

Comparison of entropy analysis

Methods	Baboon	Lena	Pepper
[20]	7.9969	7.9973	7.9974
[37]	7.9965	7.9989	7.9975
[42]	7.9992	7.9992	7.9993
[36]	7.8546	7.8993	7.9973
[Proposed]	7.9992	7.9994	7.9993

The Table 4 presents a comparison of entropy analysis results for different image encryption methods applied to three distinct images: Baboon, Lena, and Pepper. Entropy is a measure of randomness or uncertainty in the information content of an image, with higher entropy values indicating greater randomness and thus potentially stronger encryption. Among the methods compared, method [42] consistently demonstrates the highest entropy values across all three images, suggesting potentially stronger encryption in terms of randomness. The proposed method also shows competitive entropy values, particularly matching or exceeding those of existing methods for Lena and Pepper images, indicating promising performance in terms of randomness preservation during encryption.

5.2.2 NPCR and UACI tests

In assessing the resistance of images to differential attacks in encryption, two primary measures are commonly employed: the Number of Changing Pixels Rate (NPCR) and the Unified Average Changed Intensity (UACI). During a differential attack, the attacker seeks to establish relationships between pixels in pairs of plain images, potentially affecting the differences between plain and ciphered images. This implies that even a minor alteration in the initial image, such as a single-bit change, should result in a complete transformation of the ciphered image. Consequently, UACI quantifies the average intensity of discrepancies between paired ciphered images, while NPCR scrutinizes the behavior of individual pixels between such pairs [43]. These measures serve to demonstrate the robustness of any algorithm against potential differential attacks.

Mathematically, NPCR, denoted as N(C¹, C²), and UACI, denoted as u(C¹, C²), between two sets of ciphertext images corresponding to plaintext images with slight differences as shown in Equations 11, 12:

$N (C^{1}, C^{2}) = \sum_{i, j} \frac{D (i, j)}{T} \times 100 %$ (11) when $\begin{matrix} D (i, j) = {\begin{matrix} 0, & if C^{1} (i, j) = C^{2} (i, j) \\ 1, & if C^{1} (i, j) \neq C^{2} (i, j) \end{matrix} \end{matrix}$

$u (C^{1}, C^{2}) = \sum_{i, j} \frac{∣ C^{1} (i, j) - C^{2} (i, j) ∣}{L . T} \times 100 %$ (12) In this context, the difference function D (i, j) determines whether two pixels from the image grid (i, j) of (C¹, C²) are equal or not. The parameter T signifies the total number of pixels in the ciphertext, while L denotes the maximum pixel value supported by the cipher-image format. In this study, Table 5 provides the NPCR and UACI test results. NPCR exhibits an average value of 99.605%, while UACI demonstrates an average value of 33.453%. These results indicate that proposed algorithm consistently withstands differential attacks across various image sizes, with both average values satisfy the threshold of 0.001. Following this, Table 6 presents comparisons of UACI and NPCR between the proposed algorithm and the methods referenced in [2 , 46]. The Table 6 presents a quantitative comparison of NPCR (Number of Pixels Change Rate) and UACI (Unified Average Changing Intensity) values for various image encryption methods across three different images: Baboon, Lena, and Pepper. NPCR measures the percentage of pixels that change their values between the original and encrypted images, while UACI assesses the average changing intensity between corresponding pixels in the original and encrypted images. Overall, the proposed method shows competitive NPCR and UACI values across all three images, indicating effective preservation of image content and quality during encryption.

Table 5

NPCR(% ) and UACI(% ) of Different-sized Images

Images	NPCR	UACI
Tree	99.60%	33.48%
Baboon	99.60%	33.46%
Lena	99.61%	33.43%
Peppers	99.61%	33.44%
Plane	99.60%	33.46%
Boats	99.61%	33.44%
Neptune	99.61%	33.47%
Venus	99.60%	33.45%

Table 6

Comparison of NPCR(% ) and UACI(%)

	Baboon		Lena		Pepper
Methods	NPCR	UACI	NPCR	UACI	NPCR	UACI
[46]	99.59	33.52	99.60	33.50	99.61	33.52
[2]	99.61	33.52	99.62	33.50	99.61	33.39
[20]	99.62	33.46	99.63	33.55	99.61	33.49
[37]	99.60	33.46	99.62	33.44	99.61	33.46
[42]	99.61	33.55	99.60	33.43	99.60	33.51
[36]	99.57	33.35	99.69	33.41	99.58	33.41
[Proposed]	99.60	33.46	99.61	33.43	99.61	33.44

5.2.3 Key space

For any reliable ciphering algorithm to withstand exhaustive attacks, it requires a sufficiently large key space. Within any encryption scheme, there should exist a maximum size for the key space, ensuring that brute-force attacks remain impractical [24]. Therefore, in proposed approach, pseudo-random number sequences were generated using a fusion of PWLCM and CLM maps. To instantiate a chaotic system, a 260-bit key is necessary, resulting in a key space of 2⁵²⁰, significantly surpassing 2¹²⁴. As illustrated in Table 7, the key space of proposed algorithm surpasses that of recent algorithms, enhancing its resilience against brute-force attacks and establishing it as a more robust encryption method.

Table 7
Comparison of Key space

Methods Key-Space

[42] 2¹⁹²

[37] 2¹²⁸

[20] 2²¹⁰

[28] 2¹²⁴

[2] 2³¹²

Proposed 2⁵²⁰

Methods	Key-Space
[42]	2¹⁹²
[37]	2¹²⁸
[20]	2²¹⁰
[28]	2¹²⁴
[2]	2³¹²
Proposed	2⁵²⁰

5.2.4 Robustness analysis

An effective encryption algorithm should be capable of withstanding partial data loss during the transmission of encrypted images. Even if the encrypted image is affected by noise or experiences data loss, it should still be possible to produce a visually recognizable decrypted image. This resilience indicates the robustness of the encryption algorithm. To assess the algorithm’s performance against noise attacks, various levels of Gaussian and Salt-and-pepper noise can be introduced to the ciphertext image. By decrypting the image under different noise conditions, its peak signal-to-noise ratio (PSNR) can be calculated. A higher PSNR signifies a stronger resistance to noise, indicating better anti-noise capabilities of the encryption method.

The magnitude of the Mean Square Error (MSE) plays a crucial role in the process of calculating Peak Signal-to-Noise Ratio (PSNR). The calculation method is given in Equations 13, 14.

$PSNR = 20 \times lg \frac{255}{\sqrt{MSE}}$ (13)

$MSE = \frac{\sum_{i = 1}^{W} \sum_{j = 1}^{H} {(P (i, j) - C (i, j))}^{2}}{W \times H}$ (14) where, W × H represents the dimensions of the plaintext image. P (i, j) and C (i, j) denote the pixel size before and after encryption, respectively. The algorithm’s performance against noise attacks at levels of 0.005%, 0.05%, and 0.2% is illustrated in Fig. 5. Additionally, Table 8 presents the comparision of PSNR values of the ciphertext images.

Fig. 5

Salt-and-Pepper noise attack with (a), (b), (c); Gaussian noise attack with (a), (b), (c).

Table 8

Comparison of PSNR(dB) against Noise

Noise	Intensity	Proposed	Ref. [10]
Salt-and-Pepper	0.005	44.03	32.27
	0.05	29.78	22.42
	0.2	19.27	16.19
Gaussian	0.005	20.23	14.08
	0.05	14.04	11.76
	0.2	13.25	10.59

5.2.5 Complexity analysis

The computational complexity of the proposed algorithm primarily involves the computations related to permutation diffusion. In both the diffusion and permutation processes, M × N operations are needed for matrix generation, along with 2 × M × N operations for random sequence generation. Consequently, the computational cost of the suggested algorithm is θ (M × N). Similarly, the encryption and decryption times for images of various sizes are detailed in Table 9, and compared with [37] for specific images in Table 10. These comparisons indicate a significant enhancement in encryption and decryption times in our study. Therefore, based on these findings, it can be concluded that our algorithm is well-suited for practical applications.

Table 9
Encryption and Decryption Time (unit:sec) of Different-sized Images

Images Encryption Decryption

Tree (256×256) 0.0399 0.0304

Baboon (480×500) 0.1229 0.1680

Lena (512×512) 0.1420 0.1384

Peppers (512×512) 0.2462 0.1346

Plane (512×512) 0.1334 0.1441

Boats (576×787) 0.2453 0.2422

Neptune (1024×1024) 0.5225 0.7265

Venus (4064×4064) 11.2985 11.2690

Images	Encryption	Decryption
Tree (256×256)	0.0399	0.0304
Baboon (480×500)	0.1229	0.1680
Lena (512×512)	0.1420	0.1384
Peppers (512×512)	0.2462	0.1346
Plane (512×512)	0.1334	0.1441
Boats (576×787)	0.2453	0.2422
Neptune (1024×1024)	0.5225	0.7265
Venus (4064×4064)	11.2985	11.2690

Table 10

Comparison of Time Consumption (ET: Encryption Time, DT: Decryption Time)

	Proposed		[37]
Images	ET	DT	ET	DT
Lena	0.1420	0.1384	0.1838	0.4595
Tree	0.0399	0.0304	0.1149	0.2875
Plane	0.1334	0.14441	0.1151	0.2992

While the paper presents a promising image encryption algorithm utilizing the FN_PWCLM map but there is a limitation within this scheme. In network transmission, it is customary to disclose the hash value of an image to ensure its integrity. Nevertheless, this disclosure of the hash value provides attackers with an opportunity for exploitation. With access to this publicly available hash value, attackers can orchestrate an effective differential attack, leading to the execution of a chosen-plaintext attack (CPA). Consequently, this vulnerability poses a security risk to the image encryption scheme, as attackers may gain unauthorized access to the encrypted image data through differential attacks facilitated by the disclosure of hash values.

In summary, the comparison of correlation coefficients across different images and directions indicates that the proposed method maintains correlation coefficients close to or even exceeding those of reference methods, suggesting strong preservation of relationships between pixels and thus retaining image integrity. The analysis of entropy demonstrates that the proposed method achieves comparable or even higher entropy values compared to existing methods for all three images, indicating effective randomization of image data and robust encryption. Additionally, the comparison of NPCR and UACI values showcases that the proposed method consistently achieves high NPCR percentages and low UACI percentages across all images, indicating minimal alteration of pixel values and intensity during encryption, which are crucial for maintaining image quality and security. The analysis reveals that the proposed method achieves a significantly larger key space of 520 bits compared to the key spaces of other reference methods, which range from 124 to 312 bits. A larger key space indicates a higher level of complexity and variability in encryption keys, which enhances the security of the encryption algorithm and makes it more resistant to brute-force attacks. Therefore, the substantial increase in key space size with the proposed method provides strong empirical evidence of its effectiveness in bolstering security and safeguarding against potential cryptographic attacks.

6 Conclusion

This paper introduces a novel image encryption algorithm utilizing the FN_PWCLM map, which combines piecewise linear and coupled logistic maps, for encrypting grayscale images. The cryptosystem built upon FN_PWCLM can handle images of any size, generating chaotic sequences essential for digital image encryption. Extensive testing has validated the efficancy of FN_PWCLM as a suitable encryption method for digital images. Security and experimental analyses were conducted to demonstrate the effectiveness. These analyses confirm the adequacy of the key space in resisting brute-force attacks. Statistical analysis reveals the method capability to better protect images against statistical attacks. Additionally, the dynamic performance of the new map exhibits uniform distribution and high generation efficiency, offering significant value for practical applications.

Future work will focus on enhancing efficiency, such as simultaneously employing confusion and diffusion. This approach can reduce the complexity of video protection and improve its effectiveness.

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FN_PWCLM – A fusion of piecewise linear and coupled logistic maps to secure different-sized digital images

Abstract

Keywords

1 Introduction

2 Related work

3 Preliminaries

3.1 Piecewise linear map

4.1 Coupled logistic map

4.4 Pseudo-matrices generation

4.5 Encryption algorithm

5 Experimental results and analysis

5.1.1 Correlation of two adjacent pixels

5.2.1 Information entropy

Table 7 Comparison of Key space Methods Key-Space [42] 2192 [37] 2128 [20] 2210 [28] 2124 [2] 2312 Proposed 2520

References

Table 7
Comparison of Key space

Methods Key-Space

[42] 2¹⁹²

[37] 2¹²⁸

[20] 2²¹⁰

[28] 2¹²⁴

[2] 2³¹²

Proposed 2⁵²⁰