Preliminaries

This section describes the fundamental ideas and mathematical tools used in the suggested encryption system, which include the piecewise linear chaotic map, the SHA-256 cryptographic hash function, the permutation and diffusion method, and Shannon entropy as a randomness indicator.

Piecewise Linear Chaotic Map (PWLCM)

The use of chaos in image encryption can be attributed to the features that come with the systems such as sensitivity to changes in the initial conditions, ergodicity, pseudo-randomness, and determinism. Piecewise Linear Chaotic Map (PWLCM) has received much attention in recent times owing to the fact that it is mathematically simple and easy to compute.

The PWLCM is described by Eq (1) as:

(1)

where represents the state variable at the n-th iteration, and is the control parameter. When p lies within this interval, the system exhibits strong chaotic behavior and generates sequences with excellent pseudo-random properties.

Due to these advantages, it is widely used for generating pseudo-random sequences for image encryption by applying the PWLCM. In our proposed method, multiple PWLCM systems with different initial values and control parameters are used for generating independent chaotic sequences for permutation and diffusion operations [41,42].

SHA-256 cryptographic hash function

Cryptographic hash functions play a vital role in enhancing the security of encryption methods by producing dynamic keys, which depend on the plaintext. The Secure Hash Algorithm 256 (SHA-256) produces a 256-bit fixed-size hash value, which depends on the input of any arbitrary size.

The process of hashing can be represented as:

where M represents the input message (or image data), and H denotes the resulting 256-bit hash value.

SHA-256 possesses some important cryptographic properties, such as the avalanche effect, collision resistance, and preimage resistance. These properties guarantee a considerable change in the output hash values even with a small change in the input.

In the proposed scheme, the hash value obtained by using the SHA-256 algorithm on the combined image is used to derive the initial conditions and control parameters of the chaotic systems. This has created a dependency on the plaintext, thus making the scheme more robust against known and chosen plaintext attacks.

Shannon entropy

Shannon entropy is usually used to measure the randomness of encrypted images. A good encryption algorithm should produce encrypted images whose Shannon entropy is close to the maximum possible.

Shannon entropy is defined by Eq (2) as:

(2)

where denotes the probability of occurrence of the pixel intensity .

For an ideal 8-bit image, the maximum entropy is:

An entropy value close to 8 indicates a uniform distribution of pixel intensities, which reflects a high level of randomness in the encrypted image. Therefore, Shannon entropy is an essential tool in measuring the efficacy of image encryption algorithms.

In this research, entropy analysis is used to test the level of randomness in the encrypted image to validate the efficacy of the proposed entropy-controlled diffusion mechanism.

Proposed methodology

The proposed encryption scheme, referred to as ECPD-MIE, is designed for the simultaneous encryption of multiple heterogeneous images, including both grayscale and color images. The framework consists of three major phases: key generation, encryption, and decryption. During the key generation phase, the input images are combined to form a composite image, and a SHA-256 digest is generated. Rather than serving as an independent secret key, the SHA-256 digest acts as a plaintext-dependent perturbation mechanism that dynamically modifies the initial conditions and control parameters of multiple PWLCM systems, thereby enhancing plaintext sensitivity and generating highly random chaotic sequences.

During the encryption phase, grayscale and color images are first processed separately and subsequently combined into composite images. Chaotic row and column permutation operations are then performed using two independent PWLCM-generated sequences. The row and column permutations collectively provide a two-dimensional scrambling effect that significantly reduces the spatial correlation among adjacent pixels and improves the confusion property of the cryptosystem. Furthermore, the SHA-256-based perturbation mechanism dynamically modifies the chaotic parameters for different plaintext images, resulting in distinct permutation patterns for different image sets. After permutation, an entropy-controlled diffusion process is carried out using additional PWLCM-generated chaotic masks. In this stage, adaptive chaotic masks are combined with pixel values through bitwise XOR operations, while the entropy-based weight factor serves as a plaintext-dependent adaptive diffusion parameter. Finally, the encrypted composite image is separated to obtain the corresponding individual cipher images.

The decryption phase is the exact inverse of the encryption procedure. Initially, the received cipher images are recombined to reconstruct the encrypted composite image. Using the same secret chaotic parameters together with the SHA-256-based perturbation mechanism, the chaotic permutation sequences and adaptive diffusion masks are regenerated. The encrypted composite image is then subjected to inverse diffusion and inverse permutation operations to recover the original image arrangement. Finally, the reconstructed composite image is separated to obtain the original grayscale and color images without any loss of information.

A detailed explanation of the individual stages involved in the proposed ECPD-MIE framework is provided below.

Key generation

In the proposed ECPD-MIE system, the secret key parameter values are generated based on a dynamic method dependent on the plaintext. This significantly improves the sensitivity of the key value while ensuring that the keys generated are resilient against cryptanalysis attacks.

1. 1. Suppose there are I numbers of input images to encrypt. The first stage involves combining all the input images to generate an image composite referred to as IGC.
2. 2. The SHA-256 cryptographic hash function is then used on the composite image to obtain a hash value in bits of 256 represented by the hexadecimal sequence:

where is the hexadecimal value of the hash sequence.

1. 3. These hash values are subsequently used to update the initial parameters of five independent PWLCM systems. This process introduces strong randomness and ensures that even a slight variation in the input images produces completely different chaotic sequences.

It is important to note that the SHA-256 digest is not employed as an independent secret key. Instead, it acts as a plaintext-dependent perturbation mechanism that dynamically modifies the initial conditions and control parameters of the PWLCM systems. Consequently, even a slight modification in the composite image produces a substantially different hash digest, resulting in different chaotic sequences for permutation and diffusion. This property enhances plaintext sensitivity and improves resistance against known-plaintext and chosen-plaintext attacks.

The updated parameters of PWLCM-1 are calculated by Eq (3) as:

(3)

Similarly, the updated parameters of PWLCM-2 are obtained by Eq (4) as:

(4)

The updated parameters of PWLCM-3 are calculated by Eq (5) as:

(5)

The updated parameters of PWLCM-4 are obtained by Eq (6) as:

(6)

Finally, the updated parameters of PWLCM-5 are computed by Eq (7) as:

(7)

where , , , , and represent the original initial conditions and control parameters of PWLCM-1 to PWLCM-5, respectively. , , , , and denote the updated key parameters. The operator represents the ceiling function.

Parameters obtained by the above method are used as input parameters in PWLCM systems. The obtained parameters ensure the generation of good quality pseudo-random sequences which determine the permutation and diffusion operations in the encryption scheme.

Encryption

The encryption process for the proposed ECPD-MIE scheme involves adaptive image concatenation, chaos-based permutation, entropy-driven masking, and cross bit plane diffusion. The complete encryption process is shown in Fig 1 below.

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Fig 1. Block diagram for ECPD-MIE encryption scheme.

https://doi.org/10.1371/journal.pone.0357571.g001

1. Input image processing and composite formation.

Consider a set of I input images consisting of P grayscale and Q color images.

All P grayscale images are resized (if required) and horizontally concatenated to form:

where and represent the height and width of the concatenated grayscale image, respectively.

Each color image is decomposed into its RGB components:

The components are concatenated as:

Then, the composite color image is formed as:

where and denote the dimensions of the concatenated color image.

Finally, the unified composite image is constructed as:

where M and N denote the dimensions of the unified composite image.

During the composite-image formation stage, the parameters maxH, maxW, , , , and are maintained as auxiliary reconstruction metadata. These parameters record the dimensions of the grayscale and color image groups as well as the original dimensions of each individual image. The reconstruction metadata is subsequently used during decryption to accurately separate the recovered composite image and restore the original image set.

2. Chaotic row permutation.

Using PWLCM-1, generate a chaotic sequence:

The sequence is discretized as:

where denotes the floor operation.

The row permutation is performed as

The sequence v is generated from PWLCM-1 using SHA-256-perturbed chaotic parameters, thereby providing plaintext-dependent row permutation patterns.

3. Chaotic column permutation.

Using PWLCM-2, generate

The sequence is discretized as

The column permutation is performed as

Although the discretization process may generate repeated shift values, the row and column permutation sequences are generated independently using PWLCM-1 and PWLCM-2, respectively. The successive application of row-wise and column-wise circular shifts provides a strong two-dimensional scrambling effect that substantially rearranges pixel positions over the composite image. Moreover, the SHA-256-based perturbation mechanism dynamically modifies the initial conditions and control parameters of the PWLCM systems for different plaintext images, resulting in distinct permutation patterns for different image sets. Consequently, this design reduces the likelihood of persistent fixed-point structures or short-cycle artifacts that may arise when M or N are not coprime to the sequence period.

4. Generation of diffusion sequences.

Using PWLCM-3, PWLCM-4, and PWLCM-5, three chaotic diffusion sequences are generated:

The chaotic matrices are transformed into 8-bit diffusion sequences as follows:

These diffusion sequences are plaintext-dependent since the initial conditions and control parameters of the PWLCM systems are dynamically perturbed using the SHA-256 digest of the composite image.

5. Entropy-controlled weight computation.

The Shannon entropy of the permuted image is computed as

where P(k) denotes the probability of occurrence of the gray level k.

The entropy-based weight factor is then obtained as

The weight factor WF serves as a plaintext-dependent adaptive diffusion parameter that introduces image-dependent variability into the diffusion process. It is not treated as an independent secret-key component; rather, it complements the chaotic diffusion sequences generated by the PWLCM systems.

6. Chaotic masking.

The entropy-based weight factor is incorporated into the chaotic diffusion sequences using the XOR operation:

thereby producing three adaptive diffusion masks.

7. Cross-bit-plane diffusion.

The final cipher image is obtained by combining the three adaptive chaotic masks using XOR operations:

for all and .

Since the XOR operation satisfies both the commutative and associative properties, the ordering of the adaptive chaotic masks does not affect the final ciphertext. Therefore, the sequence is adopted for implementation consistency and to maintain a direct correspondence with the chaotic sequences generated by PWLCM-3, PWLCM-4, and PWLCM-5, respectively.

The security of the diffusion stage primarily originates from the plaintext-dependent chaotic masks generated by PWLCM-3, PWLCM-4, and PWLCM-5. These adaptive chaotic masks are incorporated into the diffusion process through XOR operations, while the entropy-based weight factor serves as a plaintext-dependent adaptive diffusion parameter that introduces additional image-dependent variability. Consequently, the security contribution of the diffusion stage is mainly derived from the randomness and plaintext sensitivity of the chaotic masks rather than from the ordering of the XOR operations or the entropy-based weight factor alone.

8. Cipher image separation.

The final encrypted composite image is divided to obtain individual cipher images.

To ensure accurate reconstruction during decryption, the auxiliary reconstruction metadata consisting of maxH, maxW, , , , and is maintained together with the encryption parameters. These parameters preserve the partition information of the grayscale and color image groups as well as the original dimensions of all images.

The grayscale portion is horizontally segmented into P grayscale cipher images, while the color portion is separated into CR, CG, and CB components and subsequently partitioned and recombined to reconstruct Q color cipher images.

Thus, the encrypted image set is given by

where the associated reconstruction metadata enables deterministic and unambiguous recovery of the original heterogeneous image set during decryption.

It should be noted that the proposed scheme does not require the receiver to infer the number, type, or dimensions of the encrypted images directly from the ciphertext. Instead, the reconstruction metadata maintained during the encryption stage is used to accurately recover the grayscale and color image groups and restore the original image dimensions during decryption.

Decryption

The decryption process of the proposed ECPD-MIE scheme is the exact inverse of the encryption procedure. Using the same secret chaotic parameters, the SHA-256-based plaintext-dependent perturbation mechanism, the entropy-based adaptive diffusion parameter, and the auxiliary reconstruction metadata, the chaotic permutation sequences and diffusion masks generated during encryption are accurately reproduced. Since the permutation and diffusion operations are reversible, the original plaintext images can be recovered without information loss. The detailed decryption steps are described as follows.

2. Cipher image concatenation.

The received cipher images are first categorized into grayscale and color images.

All grayscale cipher images are horizontally concatenated to form the composite grayscale image:

Similarly, the color cipher images are combined to form the composite color image:

To maintain consistency with the encryption stage, both components are merged to reconstruct the unified composite cipher image:

3. Regeneration of chaotic diffusion sequences.

Using the regenerated PWLCM-3, PWLCM-4, and PWLCM-5 parameters, each chaotic system is iterated times to generate the diffusion sequences

These sequences are transformed into 8-bit diffusion matrices as follows:

The generated diffusion sequences remain plaintext-dependent because the PWLCM parameters are dynamically perturbed using the SHA-256 digest of the composite image.

4. Entropy weight factor computation.

The entropy-based weight factor is regenerated as

where EW denotes the Shannon entropy of the permuted image.

The parameter WF serves as a plaintext-dependent adaptive diffusion parameter and is used to reproduce the same diffusion masks generated during the encryption stage. It is not treated as an independent secret-key component.

5. Adaptive chaotic mask generation.

The entropy-based weight factor is incorporated into the chaotic diffusion sequences to regenerate the adaptive diffusion masks:

These masks are identical to those used during encryption and are subsequently employed for the reverse diffusion process.

6. Reverse cross-bit-plane diffusion.

The inverse diffusion process is performed using the regenerated adaptive chaotic masks. Since the XOR operation is self-invertible, associative, and commutative, the original permuted image can be accurately recovered by applying the same diffusion masks generated during the encryption stage.

for all and .

Owing to the associative and commutative properties of the XOR operation, any permutation of the adaptive chaotic masks yields the same recovery result. The ordering is retained solely for implementation consistency with the decryption procedure.

7. Reverse chaotic column permutation.

Using PWLCM-2, the chaotic sequence w is regenerated and discretized as

The inverse column permutation is performed as

The same plaintext-dependent chaotic sequence generated during encryption is reproduced, ensuring accurate recovery of the column permutation stage.

8. Reverse chaotic row permutation.

Using PWLCM-1, the chaotic sequence v is regenerated and discretized as

The inverse row permutation is performed as

Since the same SHA-256-perturbed PWLCM parameters are regenerated during decryption, the original row permutation sequence is reproduced exactly, enabling lossless recovery of the plaintext image.

9. Recovery of original images.

After inverse diffusion and inverse permutation, the recovered composite image is separated into grayscale and color components using the auxiliary reconstruction metadata. Specifically, the parameters maxH, maxW, , , , and are used to determine the boundaries of the grayscale and color image groups and to restore the original dimensions of each image.

The grayscale component is recovered as

while the color component is recovered as

The grayscale component is then partitioned according to the entries in , whereas the color component is first separated into the red, green, and blue channels and subsequently partitioned according to the entries in . The corresponding RGB channels are finally recombined to recover the original color images.

Thus, the recovered image set is given by

which exactly corresponds to the original input images, thereby confirming the correctness, deterministic reconstruction capability, lossless recovery, and reversibility of the proposed ECPD-MIE cryptosystem.

It should be noted that the proposed scheme does not require the receiver to infer the number, type, or dimensions of the encrypted images directly from the ciphertext. Instead, the reconstruction metadata consisting of maxH, maxW, , , , and is maintained together with the decryption parameters. This metadata enables deterministic and unambiguous separation of the grayscale and color image groups and accurate recovery of all individual images.

Simulation results and security analysis

Simulation results for the proposed ECPD-MIE scheme demonstrate its effectiveness in securely encrypting heterogeneous image sets containing both grayscale and color images within a unified framework. Fig 2 presents the overall encryption and decryption results obtained using the proposed scheme, while the detailed performance and security analyses are discussed in the subsequent sections.

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Fig 2. Simulation results:

(a,b) original color images, (e,f) encrypted color images, (i,j) decrypted color images, (c,d,m–p) original grayscale images, (g,h,q–t) encrypted grayscale images, (k,l,u–x) decrypted grayscale images.

https://doi.org/10.1371/journal.pone.0357571.g002

To evaluate the performance of the proposed algorithm, experiments were conducted using eight standard benchmark images obtained from the USC-SIPI Image Database [43]. The experimental image set consists of six grayscale images of size , namely “gray21.512.tiff”, “motion01.512.tiff”, “motion04.512.tiff”, “motion07.512.tiff”, “motion10.512.tiff”, and “ruler.512.tiff”, together with two color images of size , namely “4.1.07.tiff” and “4.1.08.tiff”. The selected images exhibit diverse visual characteristics, including natural scenes, textured regions, motion patterns, and geometric structures, making them suitable for evaluating the effectiveness of heterogeneous multi-image encryption algorithms. Unlike conventional image encryption schemes that generally assume homogeneous image sets, the proposed ECPD-MIE framework simultaneously processes grayscale and color images of different dimensions by first constructing a unified composite image, thereby demonstrating its flexibility for practical multimedia applications.

It is observed that the security of the proposed encryption scheme depends on multiple interacting components, including the initial conditions and control parameters of the five PWLCM systems, the SHA-256 digest of the composite image, and the entropy-based weight factor obtained from the permuted image. In the proposed framework, the SHA-256 hash is employed as a plaintext-dependent perturbation mechanism rather than as an independent secret key. The generated hash dynamically modifies the initial conditions and control parameters of the PWLCM systems, thereby ensuring strong plaintext sensitivity. Consequently, even a slight variation in the input image set produces different chaotic sequences for permutation and diffusion. Together, these components contribute to the generation of highly random chaotic sequences and enhance the resistance of the proposed cryptosystem against brute-force, statistical, differential, and known-plaintext attacks. The parameters used in the experiments are listed in Table 1.

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Table 1. Secret key parameters and plaintext-dependent security parameters.

https://doi.org/10.1371/journal.pone.0357571.t001

Fig 2 illustrates the visual outcomes of the entire encryption and decryption procedure. The original grayscale and color images are transformed into noise-like encrypted images without revealing any perceptual information. During the decryption process, all original images are accurately reconstructed without any observable distortion, thereby demonstrating the correctness and reversibility of the proposed cryptosystem.

An inherent advantage of the proposed ECPD-MIE framework is its capability to simultaneously process grayscale and color images of different dimensions through a unified composite-image architecture. This design improves computational efficiency while preserving the visual characteristics of both image types and eliminates the need for separate encryption procedures for heterogeneous image sets.

Overall, the proposed cryptosystem demonstrates excellent security performance, high reconstruction quality, and efficient heterogeneous multi-image encryption capability. To further validate the security and robustness of the proposed framework, comprehensive security analyses are presented in the following subsections.

Key space analysis

Key space analysis is one of the important criteria used to evaluate the resistance of an encryption algorithm against brute-force attacks. A sufficiently large key space increases the computational complexity required for an attacker to exhaustively search all possible secret keys. In modern cryptographic systems, a key space larger than 2¹²⁸ is generally considered secure against brute-force attacks [44,45].

In the proposed ECPD-MIE scheme, the primary secret key consists of the initial conditions and control parameters of the five PWLCM chaotic systems, namely . Each parameter is represented using double-precision floating-point arithmetic with an effective precision of approximately 10¹⁵ [46]. Therefore, the key space associated with the chaotic parameters can be estimated as

Using the relation , the corresponding key space can be expressed as

In addition to the secret chaotic parameters, the proposed scheme incorporates a SHA-256 hash of the composite image and an entropy-based weight factor. It should be noted that these two components are plaintext-dependent quantities and are not treated as independent secret-key elements. Instead, they dynamically perturb the chaotic systems and diffusion process, thereby enhancing plaintext sensitivity and strengthening resistance against known-plaintext and chosen-plaintext attacks. Consequently, they improve the practical security of the cryptosystem but are not included in the formal key-space computation.

Therefore, the effective secret-key space of the proposed ECPD-MIE scheme is approximately

which is substantially larger than the minimum security requirement of 2¹²⁸ and provides strong resistance against exhaustive brute-force attacks.

Table 2 presents the detailed key-space evaluation of the proposed ECPD-MIE scheme, while Table 3 compares the key space of the proposed method with several recently reported multi-image encryption algorithms [15–21,25–40]. The comparison demonstrates that the proposed scheme provides a sufficiently large key space for resisting brute-force attacks. Furthermore, the incorporation of plaintext-dependent SHA-256-based chaotic perturbation and entropy-controlled diffusion enhances the plaintext sensitivity and practical security of the proposed cryptosystem.

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Table 2. Key space analysis.

https://doi.org/10.1371/journal.pone.0357571.t002

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Table 3. Comparison of total key space.

https://doi.org/10.1371/journal.pone.0357571.t003

Histogram analysis

Histogram analysis is an important statistical tool for evaluating the resistance of image encryption algorithms against statistical attacks. A secure image encryption scheme should produce encrypted images whose histograms are significantly different from those of the corresponding plaintext images and exhibit an approximately uniform distribution [47,48]. In the proposed study, histogram analysis is performed for both grayscale and color images. For color images, the histograms of the red, green, and blue channels are analyzed simultaneously to provide a comprehensive assessment of the statistical characteristics of the encrypted images.

Fig 3 presents the histogram distributions of representative grayscale and color images before and after encryption. The plaintext images exhibit highly non-uniform histogram distributions that reflect the inherent statistical characteristics of the image contents. In contrast, the corresponding encrypted images exhibit nearly uniform histogram distributions without any discernible statistical patterns. For the grayscale images, the encrypted histograms are uniformly distributed across the entire intensity range. Similarly, for the color images, the histograms of the red, green, and blue channels become uniformly distributed after encryption, indicating that the statistical characteristics of the original images have been effectively concealed. These observations demonstrate that the proposed ECPD-MIE scheme successfully removes the statistical information of the plaintext images and provides strong resistance against statistical attacks.

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Fig 3. Histogram analysis results:

(a,b) original color images, (e,f) encrypted color images, (c,d,i–l) original grayscale images, and (g,h,m–p) encrypted grayscale images.

https://doi.org/10.1371/journal.pone.0357571.g003

Histogram variance analysis

Histogram variance is used to evaluate the uniformity of pixel intensity distributions in encrypted images. A lower histogram variance indicates a more uniform distribution, which is desirable for secure image encryption because it reduces statistical leakage and makes statistical attacks more difficult [49,50].

For an image, the histogram variance is defined by Eq. 8 as

(8)

where denotes the number of pixels having intensity level k, and is the mean histogram value given by Eq. 9:

(9)

where denotes the image size.

For grayscale images, the histogram variance is computed directly from the image histogram. For color images, the histogram variance is calculated independently for the red, green, and blue channels, and the average of the three channel variances is reported as the histogram variance of the color image.

Table 4 presents the histogram variance values of the plaintext and encrypted images. It can be observed that the encrypted images exhibit significantly lower histogram variance values than their corresponding plaintext images, indicating a much more uniform distribution of pixel intensities. This demonstrates that the proposed ECPD-MIE scheme effectively removes the statistical characteristics of the original images and significantly reduces statistical leakage.

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Table 4. Histogram variance analysis.

https://doi.org/10.1371/journal.pone.0357571.t004

Chi-square test analysis

The chi-square test is widely used to evaluate the statistical uniformity of the pixel intensity distribution in encrypted images. It measures the deviation between the observed histogram distribution and the ideal uniform distribution. For a secure image encryption algorithm, the histogram of the encrypted image should closely follow a uniform distribution, resulting in a relatively small chi-square statistic [51,52].

The chi-square statistic is computed using Eq. 10:

(10)

where denotes the observed frequency of intensity level j, and the expected frequency is given by Eq. 11:

(11)

where denotes the image size.

For the proposed ECPD-MIE scheme, the chi-square test is performed for both grayscale and color images. For color images, the chi-square statistic is computed independently for the red, green, and blue channels, and the average value of the three channels is reported.

The chi-square values of the plaintext and encrypted images are presented in Table 5. It can be observed that the encrypted images exhibit significantly improved histogram uniformity compared with their corresponding plaintext images. Furthermore, Table 6 compares the chi-square values of the proposed method with those reported by representative image encryption algorithms. The comparison shows that the proposed ECPD-MIE scheme achieves chi-square values comparable to those of existing state-of-the-art methods while satisfying the critical values corresponding to the 5% and 1% significance levels. These results demonstrate that the proposed encryption scheme effectively conceals the statistical characteristics of the plaintext images and provides strong resistance against statistical attacks.

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Table 5. Chi-square test analysis.

https://doi.org/10.1371/journal.pone.0357571.t005

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Table 6. Comparison of results of encrypted images.

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Pixel correlation analysis

Pixel correlation analysis is employed for evaluating the strength of an encryption technique in reducing the correlations of pixels, which are usually very high in natural images. In plaintext images, it is expected that the correlation of neighboring pixels is very high due to spatial redundancy, but it should be minimized as close to zero as possible using a secure encryption technique [53–55].

The correlation coefficient between two adjacent pixels u and v is defined by Eq 12 as:

(12)

where the covariance and variance are given by Eqs 13–17 as:

(13)

(14)

(15)

(16)

(17)

where P denotes the number of randomly selected pixel pairs. In this study, 10,000-pixel pairs are randomly selected in horizontal, vertical, and diagonal directions to ensure reliable statistical evaluation.

Figs 4 and 5 illustrate the correlation distributions of adjacent pixels for representative grayscale and color images, respectively. The plaintext images exhibit strong linear clustering in the horizontal, vertical, and diagonal directions, indicating the high spatial correlation that naturally exists in digital images. In contrast, the encrypted images display uniformly scattered distributions without any discernible linear relationship, indicating that the proposed ECPD-MIE scheme effectively destroys the statistical dependence among neighboring pixels.

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Fig 4. Pixel correlation results of grayscale images:

(a,c,e,g,i,k) original images, (b,d,f,h,j,l) encrypted images.

https://doi.org/10.1371/journal.pone.0357571.g004

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Fig 5. Pixel correlation results of color images:

(a,c) original images, (b,d) encrypted images.

https://doi.org/10.1371/journal.pone.0357571.g005

The quantitative results presented in Table 7 further confirm this observation. The plaintext images exhibit correlation coefficients close to one in all three directions, whereas the encrypted images exhibit correlation coefficients approaching zero. Furthermore, the comparative results shown in Table 8 demonstrate that the proposed scheme achieves correlation coefficients comparable to or closer to zero than those reported by representative state-of-the-art image encryption algorithms. This excellent decorrelation performance can be attributed to the combined effects of the SHA-256-based plaintext-dependent chaotic perturbation mechanism, the independent PWLCM-based row and column permutation stages, and the entropy-controlled adaptive diffusion process employed in the proposed ECPD-MIE framework.

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Table 7. Pixel correlation analysis.

https://doi.org/10.1371/journal.pone.0357571.t007

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Table 8. Comparison of average adjacent pixel correlation of encrypted images.

https://doi.org/10.1371/journal.pone.0357571.t008

MSE, PSNR, and SSIM analysis

Mean Squared Error (MSE), Peak Signal-to-Noise Ratio (PSNR), and Structural Similarity Index Measure (SSIM) are widely used image quality metrics for evaluating the distortion introduced during encryption and the reconstruction quality achieved after decryption. In the proposed ECPD-MIE scheme, these metrics are computed by comparing the original image with both the encrypted and the corresponding decrypted images to assess the effectiveness and reversibility of the encryption process.

The proposed ECPD-MIE framework evaluates these image quality metrics for both grayscale and color images, thereby demonstrating its capability to securely encrypt and accurately reconstruct heterogeneous image sets within a unified encryption framework.

Mean Squared Error (MSE).

MSE is a measure of the mean of the squared differences between corresponding pixels in two given images. A higher value of MSE between the original and encrypted images implies that a large amount of distortion is introduced during the encryption process, which is a favorable condition for security [56,57]. Similarly, a zero value of MSE between the original and decrypted images implies a perfect reconstruction [56].

The MSE is defined by Eqs. 18 and 19 as:

(18)(19)

where OI, EI, and DI denote the original, encrypted, and decrypted images, respectively, and represents the image size. The measure calculates the distortion between the original and encrypted images, while calculates the distortion between the original and decrypted images.

Peak Signal-to-Noise Ratio (PSNR).

PSNR is derived from MSE and measures the quality difference between two images. A lower PSNR value between the original and encrypted images indicates a greater difference and stronger encryption. In contrast, for successful decryption, the PSNR value between the original and decrypted images tends toward infinity [56].

The PSNR is computed using Eqs. 20 and 21 as:

(20)(21)

where is the maximum pixel value (255 for 8-bit images). is the PSNR value between original and encrypted images, and is the PSNR value between original and decrypted images.

Structural Similarity Index Measure (SSIM).

SSIM evaluates perceptual similarity by considering luminance, contrast, and structural information. For an effective encryption scheme, the SSIM value between the original and encrypted images should be close to zero, indicating no structural similarity, while the SSIM between the original and decrypted images should be close to one, indicating accurate reconstruction.

The SSIM is defined using Eqs. 22 and 23 as:

(22)(23)

where denotes the mean intensity, the variance, the covariance, and C₁ and C₂ are constants to avoid instability.

The numerical results of the MSE, PSNR, and SSIM analyses for both grayscale and color images are presented in Table 9. The encrypted images exhibit large MSE values, low PSNR values, and SSIM values close to zero, indicating significant visual distortion and effective concealment of the plaintext information. Conversely, the decrypted images achieve zero MSE, PSNR values approaching infinity, and SSIM values approaching one, demonstrating lossless reconstruction and confirming the correctness and reversibility of the proposed ECPD-MIE scheme.

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Table 9. MSE, PSNR, and SSIM analysis.

https://doi.org/10.1371/journal.pone.0357571.t009

Information entropy analysis

Information entropy is a statistical measure used to quantify the randomness and unpredictability of the pixel intensity distribution in an encrypted image. An increase in the value of entropy represents a higher level of unpredictability, thereby improving the robustness against attacks based on statistical methods and entropy attacks. In an 8-bit image, the maximum possible value of entropy is 8 bits; hence, a high value of entropy closer to the optimal value represents a high level of encryption with minimal information leakage [58].

Global information entropy of an image can be computed using Shannon’s entropy, which can be defined by Eq. 24 as

(24)

where represents the probability of occurrence of the pixel intensity level k, and denotes the entropy value in bits.

The computed global entropy values of the plaintext and encrypted grayscale and color images are presented in Table 10. For grayscale images, the entropy is computed directly from the image intensity distribution. For color images, the entropy is computed independently for the red, green, and blue channels, and the average of the three channel entropies is reported. It can be observed that the encrypted images achieve entropy values very close to the theoretical maximum value of 8 bits, indicating excellent randomness and negligible statistical information leakage.

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Table 10. Global Shannon entropy analysis.

https://doi.org/10.1371/journal.pone.0357571.t010

Table 11 compares the global information entropy achieved by the proposed ECPD-MIE scheme with representative state-of-the-art image encryption algorithms. Since information entropy is influenced by the image size, the corresponding image dimensions reported in each study are also included to facilitate a fair comparison. For the proposed scheme, the average entropy values are 7.9973 for images and 7.9993 for images, both of which are very close to the theoretical optimum of 8 bits. These results demonstrate that the proposed ECPD-MIE framework produces highly random ciphertexts and provides strong resistance against entropy-based statistical attacks.

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Table 11. Comparison of global information entropy of encrypted images with different image sizes.

https://doi.org/10.1371/journal.pone.0357571.t011

Although global entropy effectively measures the overall randomness of an encrypted image, it does not fully characterize the randomness within local image regions. Therefore, local Shannon entropy is employed to evaluate the statistical randomness of local image blocks by partitioning the encrypted image into multiple non-overlapping blocks and computing the average entropy of these blocks [59]. The local entropy can be defined by Eq. 25 as follows:

(25)

where n is the number of non-overlapping blocks, denotes the number of pixels in each block, and represents the image block.

In this work, each encrypted image is divided into n = 30 non-overlapping blocks, with each block containing pixels. For significance level of 0.05, the acceptable range of local entropies for the configuration is . The corresponding acceptable ranges for significance levels of 0.01 and 0.001 are and , respectively.

The local Shannon entropy results presented in Table 12 fall within the acceptable ranges corresponding to all three significance levels (0.05, 0.01, and 0.001). These results demonstrate that the encrypted images exhibit excellent local randomness and uniform statistical behavior throughout the image. Consequently, both the global and local entropy analyses confirm that the proposed ECPD-MIE scheme achieves high randomness at both global and local levels, thereby providing strong resistance against entropy-based statistical attacks.

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Table 12. Local shannon entropy analysis.

https://doi.org/10.1371/journal.pone.0357571.t012

Differential attack analysis

Differential attack analysis is employed to evaluate the sensitivity of an image encryption algorithm to slight variations in the plaintext image. A robust image encryption algorithm should exhibit high sensitivity; this means that a slight change to the plaintext image (for example, by changing a single pixel) should result in a significantly different ciphertext image.

To assess this, two metrics that are commonly used are discussed, namely the Number of Pixel Change Rate (NPCR) and the Unified Average Changing Intensity (UACI), which are determined by introducing a pixel change in the original plaintext image and comparing the resultant encrypted image [60, 61].

The NPCR is a measure of the percentage of pixels that are changed due to a small variation in the plaintext, and it is given by the Eq 26

(26)

where the difference matrix is given by the Eq 27

(27)

Here, and are the two different ciphertexts produced by a variation in the plaintext, and is the total number of pixels in the image. The NPCR is thus expected to be close to 99.6094% for an ideal 8-bit image encryption scheme, which implies that all the pixels are changed due to a variation in the plaintext [50,61].

The UACI is a measure of the average intensity difference between two different encrypted images, and it is given by the Eq 28

(28)

This is thus expected to be close to 33.4635% for an ideal 8-bit image encryption scheme, which implies that all the intensity values are changed [50,61].

Table 13 shows the NPCR and UACI results for the proposed ECPD-MIE scheme. The observed values closely resemble theoretical values, which confirm the high sensitivity of the algorithm to the changes in the plaintext. This shows that a change in a single pixel can cause a significant change in the ciphertext. To ensure statistical reliability, NPCR and UACI values are calculated based on 100 experiments with randomly altered pixels. In addition to the average values, the statistical dispersion of the obtained results is evaluated through the standard deviation across heterogeneous image sets. Table 13 presents the average NPCR and UACI values for individual test images, whereas Table 14 summarizes the overall statistical characteristics of the obtained results. As shown in Table 14, the standard deviations of NPCR and UACI are only 0.0040% and 0.0023%, respectively. These small variations indicate that the proposed ECPD-MIE scheme exhibits highly consistent differential attack performance across heterogeneous grayscale and color image sets. Furthermore, the obtained NPCR and UACI values remain very close to their theoretical expectations, thereby confirming the robustness of the proposed cryptosystem against differential attacks.

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Table 13. Differential attack analysis.

https://doi.org/10.1371/journal.pone.0357571.t013

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Table 14. Statistical summary of NPCR and UACI values across heterogeneous image sets.

https://doi.org/10.1371/journal.pone.0357571.t014

Additionally, Table 15 presents a comparative analysis of the proposed method with various state-of-the-art image encryption techniques. As depicted in Table 15, the proposed method achieves competitive NPCR and UACI values compared to various state-of-the-art image encryption techniques. Although some existing methods report slightly higher NPCR or UACI values, the proposed ECPD-MIE scheme provides balanced differential attack performance together with very small statistical variations across heterogeneous image sets. The combination of high average NPCR/UACI values and low standard deviations demonstrates the stability, consistency, and robustness of the proposed cryptosystem against differential attacks.

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Table 15. Comparison of differential attack analysis.

https://doi.org/10.1371/journal.pone.0357571.t015

Key sensitivity analysis

Key sensitivity analysis evaluates the impact of slight variations in secret key parameters on the resulting encrypted images. A secure encryption algorithm must exhibit high key sensitivity, such that even an infinitesimal change in any key parameter produces a completely different ciphertext.

In the proposed ECPD-MIE scheme, the secret keys are dynamically generated using a combination of PWLCM-based chaotic parameters and the SHA-256 hash of the plaintext image. Due to the inherent sensitivity of chaotic systems to initial conditions and the strong avalanche effect of the hash function, even a minimal variation in any key parameter leads to a significantly different chaotic sequence. Consequently, the generated ciphertext becomes entirely uncorrelated with that produced using the original key.

To evaluate this property, each key parameter is perturbed independently while keeping all other parameters constant. In this study, a very small perturbation of 10^-15 is introduced to all chaotic initial conditions and control parameters. The quantitative evaluation is performed for all key parameters, including , to comprehensively assess the sensitivity of the proposed encryption scheme.

For instance, only representative key parameters z₁ and e₁ are considered for the purpose of illustration, as all the key parameters show similar characteristics of sensitivity. Figs 6 and 7 show the visual results of the sensitivity analysis of the key parameters for grayscale and color images, respectively. The results show that even a slight change in the key parameters leads to a highly dissimilar encrypted image, which does not show any visual similarity to the encrypted images obtained by using the original keys. The difference images confirm this as the changes are uniformly distributed over the entire image.

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Fig 6. Key sensitivity results in changing key from z₁ to .

https://doi.org/10.1371/journal.pone.0357571.g006

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Fig 7. Key sensitivity results in changing key from e₁ to .

https://doi.org/10.1371/journal.pone.0357571.g007

For quantitative analysis, the NPCR, UACI, and PSNR measures are obtained between the ciphertexts obtained using the original key and perturbed keys. The results obtained are shown in Table 16. The NPCR and UACI measures are close to their respective ideal values (approximately 99% and 33%, respectively), implying a high degree of pixel-level changes due to key perturbation. Moreover, the PSNR measures are found to be very low, further confirming the high degree of dissimilarity between the obtained ciphertexts using similar keys.

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Table 16. Key sensitivity analysis.

https://doi.org/10.1371/journal.pone.0357571.t016

Overall, both the visual and quantitative analyses suggest the presence of key sensitivity for the proposed ECPD-MIE approach. The considerable changes observed in the resulting ciphertexts upon slight modifications to the key may be attributed to the robustness offered by the proposed approach against brute-force attacks and other key-related and differential cryptanalyses.

Noise attack analysis

In actual communication and storage situations, encrypted images often receive different kinds of noise due to channel interference, transmission errors, and other disturbances. Therefore, besides ensuring security, it is also important for an image encryption algorithm to ensure resilience to noise attacks [62]. Noise attack analysis of an image encryption technique checks for the resilience of the encryption technique in reconstructing acceptable plaintext images from noisy ciphertexts.

To assess the robustness of the proposed ECPD-MIE method, noise is added to the fused encrypted image, and the resulting noisy ciphertext is decrypted using appropriate secret keys. The resulting composite image is then decomposed into individual grayscale and color images for evaluation. In this research, two different noise models are taken into consideration, which are widely used in image processing. They are Gaussian noise and salt & pepper noise.

The quality of the decrypted images is quantitatively evaluated using metrics such as MSE, PSNR, and SSIM in comparison to their corresponding original plaintext images. In this regard, a smaller value of MSE indicates a smaller reconstruction error, and larger values of PSNR and SSIM imply better image quality.

The results of the Gaussian noise perturbation are shown in Fig 8, where Gaussian noise of variances 0.0001, 0.0003, and 0.0005 is added to the encrypted image. The subplots of Fig 8(a-h), Fig 8(i-p), and Fig 8(q-x) represent the images obtained by adding Gaussian noise of variances 0.0001, 0.0003, and 0.0005, respectively. Even when the intensity of the noise is increased, the quality of the images obtained upon decryption is a testimony to the robustness of the proposed scheme.

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Fig 8. Gaussian noise attack results:

(a-h) variance  =  0.0001, (i-p) variance  =  0.0003, (q-x) variance  =  0.0005.

https://doi.org/10.1371/journal.pone.0357571.g008

Quantitative results are presented in Table 17. The proposed method has achieved low MSE values along with high PSNR and SSIM values, thereby indicating that the images obtained after the decryption process are similar to the original images even in the presence of noise.

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Table 17. Gaussian noise attack analysis.

https://doi.org/10.1371/journal.pone.0357571.t017

Likewise, the robustness of the proposed scheme in the presence of salt & pepper noise is verified by incorporating noise density levels of 5%, 10%, and 25% in the encrypted image. The corresponding results are depicted in Fig 9, where Fig 9a–9h, 9i–9p, and 9q–9x represent the results for different noise density levels. As depicted, even in the presence of impulse noise, the quality of the decrypted images is satisfactory in terms of structural features.

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Fig 9. Salt & Pepper noise attack results:

(a-h) noise level  =  5%, (i-p) noise level  =  10%, (q-x) noise level  =  25%.

https://doi.org/10.1371/journal.pone.0357571.g009

The quantitative result for the proposed method for the salt & pepper noise attack is shown in Table 18. The result shows that the proposed method has a low MSE value and high PSNR and SSIM values compared to existing methods. This proves that the proposed encryption algorithm can efficiently handle the impact of impulse noise during the transmission process.

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Table 18. Salt and Pepper noise attack analysis.

https://doi.org/10.1371/journal.pone.0357571.t018

In general, it can be inferred from both visual and quantitative assessments that the proposed ECPD-MIE scheme exhibits excellent resilience under both Gaussian and salt & pepper noise perturbations. The proposed scheme’s ability to produce high-quality decrypted images under noisy circumstances is a testament to the reliability of the proposed framework for image encryption.

Cropping attack analysis

Cropping attacks, also known as data loss attacks, are commonly used to test the strength of an image encryption system against potential losses of pieces of information during transmission and/or storage. In these types of attacks, a portion of the encrypted image is removed and then decrypted using the relevant decryption keys. A good image encryption system should be able to maintain structural details in the decrypted image even if large pieces of ciphertext are missing.

In this research work, the cropping attacks are carried out on the aggregate encrypted image, which includes both grayscale and color information. In this regard, the excised segments of the ciphertext are cropped from the top-left portion of the aggregate encrypted image. The cropped ciphertext is retained for further investigation. In this regard, the cropping factors 1/16, 1/8, 1/4, and 1/2 are applied to the encrypted image for the simulation of various levels of loss. The cropped ciphertext images are decrypted using the same secret keys as those generated during the encryption process. The cropped encrypted images and their corresponding decrypted outcomes, including the aggregate encrypted image and the grayscale and color components, are retained for further investigation.

The visual results of the cropping attack are shown in Figs 10 and 11. As shown, the decrypted images are the combination of the cropped ciphertext and the decrypted image. Even though the visual quality of the decrypted images is compromised due to the cropping of the ciphertext, the structural and visual information is maintained, even for a cropping factor of 1/2. This is an indicator that the scheme has some redundancy and is robust against the cropping attack.

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Fig 10. Cropping attack results:

(a) 1/16th cropped combined output, (b) 1/8th cropped combined output, (c-j) 1/16th cropped decrypted outputs, (k-r) 1/8th cropped decrypted outputs.

https://doi.org/10.1371/journal.pone.0357571.g010

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Fig 11. Cropping attack results:

(a) 1/4th cropped combined output, (b) 1/2nd cropped combined output, (c-j) 1/4th cropped decrypted outputs, (k-r) 1/2nd cropped decrypted outputs.

https://doi.org/10.1371/journal.pone.0357571.g011

To perform a quantitative analysis, the PSNR between original images and decrypted images obtained from cropped ciphertexts is calculated. The results for both grayscale and color images are presented in Table 19. As expected, PSNR decreases with an increase in cropping ratios due to increased information loss. However, PSNR remains within an acceptable range for both cases, demonstrating that the quality of decrypted images remains significantly similar to original images.

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Table 19. Cropping attack analysis.

https://doi.org/10.1371/journal.pone.0357571.t019

Overall, the experimental outcomes verify the robustness of the proposed ECPD-MIE approach against the cropping attacks. The capability to restore the meaningful image content from the partially disclosed ciphertext validates the reliability and practicability of the proposed multi-image encryption approach.

Execution time analysis

Execution time analysis represents a significant performance criterion for evaluating the practicality of image encryption algorithms. In addition to ensuring a high level of security, a good encryption technique must also prove its effectiveness in terms of minimal computational overhead and high-speed performance for encryption and decryption operations. In this paper, the performance of the proposed ECPD-MIE technique is evaluated in terms of overall execution time and processing speeds.

In the proposed framework, multiple grayscale and color images are first combined into a single composite image and then processed through a unified encryption structure. Therefore, the reported execution time corresponds to the total time required to encrypt and decrypt all images collectively, rather than processing each image individually. As shown in Table 20, the total encryption time and decryption time for the combined dataset are 15.413728 s and 13.719226 s, respectively.

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Table 20. Comparison of execution time and processing speed.

https://doi.org/10.1371/journal.pone.0357571.t020

To further evaluate efficiency, encryption and decryption speeds are computed as the ratio of the total input data size to the corresponding processing time, expressed in megabytes per second (MB/s), by Eqs 29 and 30 as follows:

(29)

(30)

Based on the total input size of 1.8750 MB, the proposed scheme achieves an encryption speed of 0.1216 MB/s and a decryption speed of 0.1367 MB/s. These results indicate that the proposed method maintains consistent processing performance for both encryption and decryption operations.

Table 20 shows a comparison of our proposed algorithm with other existing multi-image encryption algorithms. It can be seen that most of the algorithms give their results for execution time based on per image whereas our algorithm gives execution time based on total time to process multiple images. Though there is a difference in the way these algorithms have been evaluated, our proposed method performs well since it can handle heterogeneous images at once.

While some other existing techniques may result in relatively faster performance, their reliance is mainly on the use of a relatively simple architecture, processing of individual images, or security functions with fewer parameters. Conversely, the novel ECPD-MIE technique involves the use of more than one PWLCM chaotic function, generation of keys using SHA-256, and application of entropy-controlled diffusion all in one-pass permutation-diffusion approach.

In addition, unlike other schemes based on DNA coding, compressive sensing, or multiple transformation rounds, the proposed technique does not use iterative or blocked approaches, thus keeping the computational complexity linear. Processing of both gray and colored images under one roof reduces redundancy and increases the scalability of the proposed method.

As a result, the evaluation of the execution time and speed clearly shows that the newly introduced ECPD-MIE model succeeds in maintaining a proper equilibrium between the computation efficiency and the security robustness. The capability to handle diverse images simultaneously makes it applicable in multimedia secure communications.

NIST statistical attack analysis

Statistical randomness is a fundamental requirement for a secure cryptographic system, since any statistical regularity present in the ciphertext may be exploited to launch statistical attacks. To evaluate the randomness characteristics of the encrypted images generated by the proposed ECPD-MIE scheme, representative statistical tests from the NIST SP800−22 suite are employed. The NIST SP800−22 statistical test suite, developed by the National Institute of Standards and Technology, provides a collection of statistical tests for evaluating the randomness of binary sequences.

Since the NIST tests operate on binary sequences, each encrypted image is converted into a one-dimensional binary sequence by representing every pixel using its 8-bit binary representation. In the proposed ECPD-MIE framework, six grayscale images of size and two color images of size are combined to construct a unified composite image of dimensions . Consequently, the binary sequence length used for the NIST statistical analysis is

The NIST SP800−22 test suite consists of a comprehensive collection of statistical tests designed to evaluate the randomness characteristics of binary sequences. In the revised analysis, the encrypted bit sequences are evaluated using an expanded set of NIST tests, including Frequency (Monobit), Block Frequency, Runs, Longest Run of Ones, Binary Matrix Rank, Discrete Fourier Transform (FFT), Non-Overlapping Template Matching, Overlapping Template Matching, Universal Statistical, Linear Complexity, Serial, Approximate Entropy, Cumulative Sums (Forward), Cumulative Sums (Reverse), and Random Excursions Variant tests.

Each test produces a p-value, which represents the probability that the tested sequence could be generated by a truly random process. According to the NIST SP800−22 standard, a sequence passes a statistical test if its p-value is greater than or equal to 0.01.

The results obtained are presented in Table 21. It can be observed that all computed p-values are greater than the NIST acceptance threshold of 0.01, indicating that the encrypted bit sequences successfully pass all applied statistical tests. The Frequency and Block Frequency tests confirm a balanced distribution of zeros and ones, while the Runs and Longest Run tests verify the randomness of consecutive bit patterns. The Binary Matrix Rank and FFT tests demonstrate the absence of structural dependencies and periodic patterns in the encrypted sequences.

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Table 21. NIST SP800−22 statistical test results of encrypted images.

https://doi.org/10.1371/journal.pone.0357571.t021

Furthermore, the Non-Overlapping Template, Overlapping Template, Universal Statistical, Linear Complexity, Serial, and Approximate Entropy tests confirm the high complexity and unpredictability of the generated ciphertext. The Cumulative Sums and Random Excursions Variant tests further demonstrate the absence of statistically significant deviations from the behavior expected of truly random sequences.

Overall, the successful passing of all applied NIST SP800−22 statistical tests demonstrates that the encrypted outputs generated by the proposed ECPD-MIE scheme exhibit strong statistical randomness, high unpredictability, and no observable statistical bias. These results confirm the suitability of the proposed cryptosystem for resisting statistical attacks and further strengthen the security of the proposed heterogeneous multi-image encryption framework.

Cryptanalysis

Cryptanalysis is performed for the assessment of security robustness of the proposed ECPD-MIE scheme. A robust image encryption scheme needs to effectively counter statistical, differential, brute-force, and data loss attacks, besides being highly sensitive to the key and maintaining randomness. In this work, a series of cryptanalysis tests are performed on the proposed scheme, which include statistical analysis, assessment of the differential attack, key sensitivity, evaluation of the scheme under noisy and cropped conditions, randomness testing through statistical tests based on the guidelines of the National Institute of Standards and Technology, and assessment of the scheme for classical models of a cryptographic attack. Mathematically, the encryption process is defined as follows:

where P is the plaintext image, C is the ciphertext image, and K is the secret key used for encryption. A robust encryption scheme needs to ensure that a small variation in the plaintext or the key results in a considerably different ciphertext. This is defined as follows:

These properties ensure strong sensitivity to both plaintext and key variations, which are validated through the analyses presented in this section.

Differential attack resistance.

The strength of the proposed scheme against differential attacks is also evaluated based on NPCR and UACI criteria. NPCR values converge towards the optimal value of nearly 99%, whereas the evaluated UACI values approximate the optimal value close to 33%. This implies that any variation in a pixel of the plaintext image results in a substantial change in the corresponding ciphertext, thereby meeting the criterion:

This demonstrates the strong diffusion characteristics of the scheme.

Key space and key sensitivity analysis.

The key space for the proposed scheme is sufficiently large due to the incorporation of multiple chaotic parameters, entropy-based parameters, and hash-based keys, thereby making it more resilient to brute-force attacks. Moreover, key sensitivity analysis is performed by introducing a small perturbation (10^-15) in each key parameter. The resulting ciphertexts show significant deviations compared to the ciphertexts obtained using the original keys. Quantitative analysis using NPCR, UACI, and PSNR measures also prove that a small change in any key parameter results in a ciphertext that is completely uncorrelated with the original one, satisfying the condition:

This shows that the proposed scheme is highly sensitive to the key parameters.

Computational complexity analysis

The complexity of the proposed ECPD-MIE algorithm depends on three main processes: the generation of the chaotic sequence, the permutation-diffusion process, and the permutation step. If we assume that the size of the concatenated image, consisting of grayscale and color images, is , then we have representing the total pixel count. In the case of color images, the RGB components are included in the proposed structure; however, the complexity is determined by the total pixel count L.

Chaotic sequence generation.

The proposed scheme employs five PWLCM systems for generating the required pseudo-random sequences for the permutation and diffusion stages. PWLCM-1 and PWLCM-2 are employed for the row and column permutations, where the number of iterations required is M and N, respectively. Similarly, PWLCM-3, PWLCM-4, and PWLCM-5 are employed for the diffusion stage, where the required iterations for each system are L. Thus, the total iterations required for the chaotic sequence generation can be given by the equation:

As , the equation can be approximated as:

As the number of iterations for each PWLCM system is constant, the computational complexity for the chaotic sequence generation can be given by:

Permutation complexity.

The permutation stage includes chaotic circular permutations for the concatenated image along both the row and column axes. In the row permutation stage, each of the M rows are subject to a circular shift over the N columns, while in the column permutation stage, each of the N columns are subject to a circular shift over the M rows. Since each pixel is accessed and shifted only once for each permutation stage, and the computational efficiency of circular shift functions, the computational complexity for the permutation stage can be given by:

Diffusion complexity.

The diffusion stage implements entropy-controlled masking and cross-bit-plane diffusion using various chaotic sequences. The XOR operation is performed on the pixel by a predetermined number of chaotic matrices and the entropy weight factors. The number of operations on the pixel is constant, and the complexity of the diffusion is given by the formula:

Overall encryption and decryption complexity.

Thus, by combining all the stages, the overall computational complexity for the entire encryption procedure can be stated as:

It should be noted that the proposed ECPD-MIE scheme processes a unified composite image rather than encrypting individual images separately. Therefore, the computational complexity depends on the total number of pixels in the composite image rather than on the number of constituent images. If multiple heterogeneous images are combined to form a composite image of dimensions , the permutation and diffusion stages are executed only once on the resulting composite image. Consequently, the computational complexity remains irrespective of the number of grayscale and color images included in the composite image.

The decryption procedure involves the use of the inverse diffusion and the inverse permutation. This has the same computational complexity as the encryption procedure. Thus, the computational complexity for the decryption procedure can be stated as:

Similarly, during decryption, the chaotic diffusion sequences generated by PWLCM-3, PWLCM-4, and PWLCM-5 are regenerated only once for the entire composite image. Therefore, even when encrypting or decrypting a large number of heterogeneous images, the computational cost scales linearly with the total pixel count rather than with the number of images. For example, a set of 100 heterogeneous images having a combined size of pixels requires processing of only 4,194,304 pixels, thereby preserving the linear complexity of the proposed scheme.

The proposed ECPD-MIE scheme follows a single-pass permutation-diffusion architecture, which eliminates the need for iterative rounds, block-wise transformations, or computationally expensive operations such as sorting, matrix decomposition, or transform-domain processing.

From the comparative analysis presented in Table 22, it can be observed that:

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Table 22. Comparative computational complexity analysis of multi-image encryption schemes.

https://doi.org/10.1371/journal.pone.0357571.t022

• Most modern chaos-based multi-image encryption schemes achieve linear complexity O(L), primarily due to pixel-wise permutation-diffusion structures.
• However, schemes incorporating transform-domain operations (e.g., DWT, quaternion transforms), matrix decompositions (e.g., SVD), compressive sensing, or neural network-based processing exhibit higher computational complexity of due to additional mathematical operations.
• DNA-based schemes, although theoretically linear, introduce higher computational overhead due to encoding and decoding operations at the bit level.

In contrast, the proposed ECPD-MIE scheme avoids such computationally intensive operations and processes both grayscale and color images within a unified framework using a single-pass structure.

Furthermore, the use of a unified composite-image framework enables efficient scalability for large heterogeneous image sets, since the encryption and decryption procedures depend on the total number of pixels rather than the number of individual images.

Comparison analysis

To further verify the effectiveness of the proposed ECPD-MIE scheme, a comprehensive comparison with several state-of-the-art image encryption schemes is conducted. The results of this comparison are presented in Table 20. It summarizes the total time required for encryption and decryption, and the speed of both processes under similar system conditions.

Based on the outcomes, it is obvious that the proposed scheme achieves competitive execution times along with high processing speeds. The increased efficiency of the proposed method can be attributed to the single-pass permutation-diffusion configuration and the use of a unified image processing approach, which avoids the need for repeated passes, block-based processing, and the differentiation between grayscale and color images. This method greatly minimizes the computational cost compared to a wide variety of existing methods.

On the contrary, a variety of conventional encryption techniques are based on multi-round permutation-diffusion structures, DNA encoding operations, or compressive sensing methods, which increase the complexity of computation and delay the computation process. While some techniques may provide faster computation in specific scenarios, they often compromise scalability or ease of implementation. The proposed ECPD-MIE scheme provides a balanced trade-off between efficiency and security.

Besides, as opposed to many existing techniques which deal with grayscale and color images separately, the proposed method efficiently handles several heterogeneous images within a unified framework, thus improving the processing efficiency and reducing redundancy. This unified processing scheme improves the temporal efficiency and simplicity of implementation, making it suitable for real-time processing.

Apart from that, the scheme also demonstrates a strong security feature, as shown by statistical analysis, resistance to differential attacks (NPCR and UACI), key sensitivity, robustness against noise and cropping attacks, and randomness testing based on the NIST model. It is possible that some schemes may perform better than the proposed scheme in a particular criterion, but they may not perform well in all criteria.

From the above comparative analysis, it can be concluded that the proposed ECPD-MIE scheme offers a good compromise between computational efficiency and cryptographic strength. The linear time complexity, fast computing capability, and robust security features of the proposed scheme make it a viable option for secure multi-image encryption in communication systems.