New deep data hiding and extraction algorithm using multi-channel with multi-level to improve data security and payload capacity

Multi-channel deep data hiding algorithm (MCDHA)

The MCDHA is the modified version of the previous work (Ahmad, El-Emam & AL-Azawi, 2021) by dividing a secret message SM into n blocks {SM1, SM2, …, SMn}, and distributing these blocks on a multi-channel. Where SMi is the i^th secret message hidden into a cover image of the selected channel index and level number at this channel randomly. Encryption and compression have been applied to the outputs of the hiding process to generate encryption and compression stego-image (ECSI). The process of hiding is continued through a sequence of cover images at the specific channel until the hiding process has reached the main channel. Multi-level images have been applied with progressive enlargement in image size to ensure that security and hidden data are not perceptible (see Fig. 1).

The secret message (SM) is divided into a uniform size of (n)-blocks; each block is hidden in one channel. The hiding process is performed after being encrypted by the XOR key and compressed by the Huffman coding algorithm. Then, the blocks are distributed randomly among channels using the Fibonacci algorithm (Bommala et al., 2020). Finally, the process of hiding is applied using a new approach (NMLSB) based on least significant bit (LSB). The proposed hiding algorithm is applied to grayscale and color images, using the image size’s expansion ratio through the channel’s level, see Figs. 2 and 3.

The deep data hiding approach has been applied using the main channel containing N levels. For each level (i) at the main channel, the cover image at the level (i) is divided into two parts; the first part contains the hiding bits from the stego-image of the main channel at the level (i − 1). In contrast, the second part contains the hiding bits from a stego-image of a sub-channel. This process continues until the main channel’s level (N − 1). The proposed hiding algorithm is based on the new approach using a modified LSB (NMLSB) method with encryption and compression algorithms. Finally, the two parts are joined together to generate a new stego-image created at level (i), see Fig. 4.

The new modification of least significant bits (NMLSB)

The hiding algorithm is carried out within the chain of levels of the main and sub-channels. This process aims to produce a series of stego-images. The last level of the main channel includes a color cover image (RGB image components). This image is used to hide the stego-image from the penultimate level with the last block of the secret message to produce the last stego-image SI_MN, see Fig. 5.

Furthermore, the new modification of least significant bits (NMLSB) is based on the continuous equation to find the number of hidden bits at ith byte (i) (Nbpbi). It is calculated according to Eq. (1). Thus, it appears that the maximum value of (Nbpbi) is reached to three bits per byte if the condition $\left({\sigma }^{((\mathrm{W}\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e}-1)\times \left(\mathrm{W}\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e}-1\right))}>18\mathrm{o}\mathrm{r}\mathrm{E}{\mathrm{D}}^{(\mathrm{W}\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e}\times \mathrm{W}\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e})}>250\right)$ is satisfied.

(1) $$\mathrm{N}\mathrm{b}\mathrm{p}{\mathrm{b}}_{\mathrm{i}}=\{\begin{array}{c}\hfill \\ 1\mathrm{i}\mathrm{f}\left({\sigma }^{(\mathrm{W}\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e}\times \mathrm{W}\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e})}\le 6\right)\hfill \\ 2\mathrm{i}\mathrm{f}\left({\sigma }^{(\mathrm{W}\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e}\times \mathrm{W}\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e})}>6\mathrm{a}\mathrm{n}\mathrm{d}{\sigma }^{(\mathrm{W}\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e}\times \mathrm{W}\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e})}\le 18\right)\hfill \\ 3\mathrm{i}\mathrm{f}\left({\sigma }^{((\mathrm{W}\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e}-1)\times \left(\mathrm{W}\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e}-1\right))}>18\mathrm{o}\mathrm{r}\mathrm{E}{\mathrm{D}}^{(\mathrm{W}\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e}\times \mathrm{W}\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e})}>250\right)\hfill \end{array}$$

where $({\sigma }^{(\mathrm{W}\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e}\times \mathrm{W}\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e})})$ is the standard deviation of window size (Wsize) at the cover image when the (WSize = 3), see Eq. (2).

(2) $${\sigma }^{(\mathrm{W}\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e}\times \mathrm{W}\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e})}=\left(\sqrt{\frac{1}{(\mathrm{W}\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e}\times \mathrm{W}\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e})}{\sum }_{\mathrm{i}=1}^{(\mathrm{W}\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e}\times \mathrm{W}\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e})}{(\mathrm{C}{\mathrm{I}}_{\mathrm{i}}-{{\mu }_{\mathrm{i}}}^{\left(\mathrm{W}\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e}\times \mathrm{W}\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e}\right)})}^2}\right)$$

and $({{\mu }_{\mathrm{i}}}^{(WSize\times WSize)})$ is the mean of window (3 × 3) at the cover image around the ith location, see Eq. (3).

(3) $${\mu }_{\mathrm{i}}^{\left(\mathrm{W}\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e}\times \mathrm{W}\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e}\right)}=\frac{1}{\mathrm{W}\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e}\times \mathrm{W}\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e}}{\sum }_{\mathrm{j}=1}^{\mathrm{W}\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e}\times \mathrm{W}\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e}}\mathrm{C}{\mathrm{I}}_{\mathrm{j}}$$

The metric ( $\mathrm{E}{\mathrm{D}}^{(WSize\times WSize)}$) is the edges detection filter calculated by using Sobel edge detection filter (Tian et al., 2021) with window size $(\mathrm{W}\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e}=3)$, see Eqs. (4)–(6).

(4) $$\mathrm{E}{\mathrm{D}}_{\mathrm{x},\mathrm{C}\mathrm{I}}={\sum }_{i=0}^{WSize-1}{\sum }_{j=0}^{WSize-1}{S}_j{D}_i\ast C{I}_{ji}$$

(5) $$\mathrm{E}{\mathrm{D}}_{\mathrm{y},\mathrm{C}\mathrm{I}}={\sum }_{\mathrm{i}=0}^{\mathrm{W}\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e}-1}{\sum }_{\mathrm{j}=0}^{\mathrm{W}\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e}-1}{\mathrm{S}}_{\mathrm{i}}{\mathrm{D}}_{\mathrm{j}}\ast \mathrm{C}{\mathrm{I}}_{\mathrm{j}\mathrm{i}}$$

(6) $$\mathrm{E}{\mathrm{D}}^{(WSize\times WSize)}=\sqrt{{\left(\mathrm{E}{\mathrm{D}}_{\mathrm{x},\mathrm{C}\mathrm{I}}\right)}^2+{\left(\mathrm{E}{\mathrm{D}}_{\mathrm{y},\mathrm{C}\mathrm{I}}\right)}^2}$$

where the two variables ( $\mathrm{E}{\mathrm{D}}_{\mathrm{x},\mathrm{C}\mathrm{I}}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{E}{\mathrm{D}}_{\mathrm{y},\mathrm{C}\mathrm{I}}$) are calculated based on the Sobel edge detection technique based on horizontal (x-axis) and vertical (y-axis) convolution of a cover image. The Sobel operator for the x-axis is combined with optimal smoothing along the y-axis ( ${\mathrm{S}}_{\mathrm{j}}$) with optimal differencing along the x-axis $({\mathrm{D}}_{\mathrm{i}})$, see Eq. (4), while the Sobel operator for the y-axis is combined with optimal smoothing along the x-axis ( ${\mathrm{S}}_{\mathrm{i}}$) with optimal differencing along the y-axis $({\mathrm{D}}_{\mathrm{j}})$, see Eqs. (7)–(10)

(7) $${\mathrm{S}}_{\mathrm{i}}=\frac{\left(\mathrm{W}\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e}-1\right)!}{\left(\left(\mathrm{W}\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e}-1\right)-\mathrm{i}\right)!},{\mathrm{S}}_{\mathrm{j}}=\frac{\left(\mathrm{W}\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e}-1\right)!}{\left(\left(\mathrm{W}\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e}-1\right)-\mathrm{j}\right)!}$$

(8) $${\mathrm{D}}_{\mathrm{i}}={\mathrm{P}}_{\mathrm{i},\mathrm{W}\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e}-2}-{\mathrm{P}}_{\mathrm{i}-1,\mathrm{W}\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e}-2}$$

(9) $${\mathrm{D}}_{\mathrm{j}}={\mathrm{P}}_{\mathrm{j},\mathrm{W}\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e}-2}-{\mathrm{P}}_{\mathrm{j}-1,\mathrm{W}\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e}-2}$$

(10) $${\text{P}}_{\text{s},\text{r}}=\{\begin{array}{l}\frac{\text{r}!}{\left(\text{l}-\text{s}\right)!\text{*s}!}\text{\hspace{0.17em}\hspace{1em}if\hspace{0.17em}s}\ge \text{0s}\le \text{l}\\ 0\text{\hspace{0.17em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}Otherwise}\end{array}$$

where ( ${\mathrm{P}}_{\mathrm{s},\mathrm{r}}$) is Pascal’s triangle that gives sets of coefficients for a smoothing operator.

Multi-channel deep data extraction algorithm (MCDEA)

The main channel contains N levels; these are used to hide and extract (n) blocks of a secret message (SM) beside a sequence of intermediate stego-images. The extraction process has been implemented back-word from level (N) to level (1) for the main channel and from level S to level 1 from each sub-channel., see Fig. 6.

The extraction process at the ith level of the main channel produces two parts; the first part contains the stego-image of the (i − 1)^th level. In contrast, the second part contains the stego-image of the sub-channel at the specific level (i), see Fig. 7.

The secret message is extracted according to the proposed NMLSB after decompression by Huffman coding algorithm and decryption by the XOR key for the main channel and each sub-channel. See Fig. 8.

The (n) blocks of secret messages extracted from the main channel and sub-channels are joined together after reordering blocks’ indexes from random distribution into ascending order of blocks’ indexes using the Fibonacci algorithm21 to obtain a required secret message SM, see Fig. 9.

Implementation of multi-channel deep data hiding algorithm (MCDHA)

Suppose that one byte from stego-image SI is used in the 4^th level of the main channel (SI_M,4). Next, we select one byte from stego-image in the sth level of the ith sub-channel (Sii,s). These bytes are used to perform hiding into the cover image in the 5^th level of the main channel (CI_M,5). In this algorithm, hiding a sequence of bits from each byte into the cover image (CI_M,5) is applied to generate the corresponding stego-image (SI_M,5). The proposed algorithm uses four bits to be hidden into each byte at the cover image. As a result, it generates a stego-image at the next level of the main channel (SI_M,5). Thus, the cover image used for hiding bits is partitioned into two parts; the first part holds the hiding bits from the previous stego-image (SI_M,4). In contrast, the second part holds the hiding bits from the stego-image in the sth level of the sub-channel (SI_i,s), see Fig. 10.

Payload capacity vs. image quality

The mean square error (MSE), peak signal-to-noise ratio (PSNR), and normalized cross-correlation (NCC) are defined in Eqs. (11)–(14), and they are used to evaluate the performance of the suggested hiding algorithm.

(11) $$\mathrm{P}\mathrm{S}\mathrm{N}\mathrm{R}=10\ast {\log }_{10}\frac{255\ast 255}{\mathrm{M}\mathrm{S}\mathrm{E}}$$

(12) $$\mathrm{M}\mathrm{S}\mathrm{E}=\frac{\mathrm{M}\mathrm{S}{\mathrm{E}}_{\mathrm{R}}+\mathrm{M}\mathrm{S}{\mathrm{E}}_{\mathrm{G}}+\mathrm{M}\mathrm{S}{\mathrm{E}}_{\mathrm{B}}}{3}$$where MSEi for i $\in \{\mathrm{R},\mathrm{G},\mathrm{B}\}$ are the mean square error for the Red, Green, and Blue image colors.

(13) $$\mathrm{M}\mathrm{S}{\mathrm{E}}_{\mathrm{k}}=\frac{1}{\mathrm{m}\ast \mathrm{n}}{\sum }_{\mathrm{i}=1}^{\mathrm{m}}{\sum }_{\mathrm{j}=1}^{\mathrm{n}}{\left(\mathrm{C}{\mathrm{I}}_{\mathrm{i}\mathrm{j}}-\mathrm{S}{\mathrm{I}}_{\mathrm{i}\mathrm{j}}\right)}^2;\mathrm{k}\in \{\mathrm{R},\mathrm{G},\mathrm{B}\}$$where (i, j) is the pixel at the cover image and stego-image.

(14) $$(\mathrm{N}\mathrm{C}\mathrm{C}{)}_{\mathrm{i},\mathrm{j}}=\frac{{\sum }_{\mathrm{i}}^{\mathrm{m}}{\sum }_{\mathrm{j}}^{\mathrm{n}}\left(\left(\mathrm{C}{\mathrm{I}}_{\mathrm{i}\mathrm{j}}-{\mu }_{\mathrm{C}\mathrm{I}}\right)\times \left(\mathrm{S}{\mathrm{I}}_{\mathrm{i}\mathrm{j}}-{\mu }_{SI}\right)\right)}{\sqrt{{\sum }_{\mathrm{i}}^{\mathrm{m}}{\sum }_{\mathrm{j}}^{\mathrm{n}}{\left(\mathrm{C}{\mathrm{I}}_{\mathrm{i}\mathrm{j}-}{\mu }_{CI}\right)}^2}\sqrt{{\sum }_{\mathrm{i}}^{\mathrm{m}}{\sum }_{\mathrm{j}}^{\mathrm{n}}{\left(\mathrm{S}{\mathrm{I}}_{\mathrm{i}\mathrm{j}}-{\mu }_{SI}\right)}^2}}$$

where ( ${\mu }_{\mathrm{C}\mathrm{I}},\mathrm{a}\mathrm{n}\mathrm{d}{\mu }_{\mathrm{S}\mathrm{I}}$) are the average pixels of cover and stego-images, respectively.

The proposed MCDHEA algorithm aims to make the value of (MSE) low, the value of (PSNR) high, and the value of (NCC) is closed to one. These scale values are necessary to make it difficult for the visual or statistical attacks to notice changes in the cover image. The proposed hiding algorithm is based on bit replacement using NMLSB. This algorithm is compared with previous works such as the MDLSB algorithm by Elshare & EL-Emam (2018), the MLSB algorithm by Ahmad, El-Emam & AL-Azawi (2021), reversible data hiding by Ou et al. (2015), and the noise-like binary image blocks by Li, Li & Yang (2013), see Table 1.

Moreover, the metrics MES, PSNR (dB), and NCC are calculated at the last level of the main-channel SI_M,N of the proposed deep data hiding algorithm. According to these metrics’ results, different payload capacities on three stego-images (from the standard-test-images) are applied to show the system performance. Furthermore, the proposed approach results have been compared with the previous works (Elshare & EL-Emam, 2018; Ahmad, El-Emam & AL-Azawi, 2021; Ou et al., 2015; Li, Li & Yang, 2013). As a result, the average PSNR for the three images using the proposed hiding algorithm is better than the results of the previous algorithms for the same images with the same data hidden capacity. Moreover, the suggested hiding algorithm that generates the “Barbara” stego-image is better than the previous works mentioned above by approximately 8.17%, N/A, 12.51%, and 14.05%, respectively. Furthermore, the Baboon stego-image generated by the proposed algorithm is better than the previous works by approximately 12.37%, 9.40%, 15.57%, and 15.35%, respectively. Finally, the “Peppers” stego-image generated by the suggested algorithm is better than the previous works by approximately 11.28%, 8.08%, 15.92%, and 16.30%, respectively. However, the present work results show the highest average of PSNR is 63.8 (dB), which appears in the “Peppers” stego-image. In contrast, the lowest average of PSNR is 63.18 (dB) in the “Baboon” stego-image. Accordingly, the present work results illustrate that when the payload sharply increases, the PSNR gradually declines. Moreover, the proposed hiding algorithm results confirm that the maximum and the minimum averages of MSE for the three stego-images are 0.037 and 0.0304 for the “Barbara” and “Peppers” stego-images, respectively. In addition, the highest average of Normalized Cross-Correlation NCC results of the three stego-images is 0.99998 for the “Baboon” stego-image.

Accordingly, the suggested hiding algorithm achieves the best performance for all payload capacities.

In addition, the difference (Diff) between the secret message before hiding (SM^H) and the secret message after extraction (SM^E) is checked in Table 1. The results of Diff approved that the suggested algorithm can extract the same secret message before the hiding process by evaluating the difference according to Eq. (15).

(15) $\begin{array}{c}\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}=\sum _{\mathrm{i}=1}^{\left|\mathrm{S}\mathrm{M}\right|}\left|{\mathrm{S}\mathrm{M}}_{\mathrm{i}}^{\mathrm{H}}-{\mathrm{S}\mathrm{M}}_{\mathrm{i}}^{\mathrm{E}}\right|\end{array}$

where ( ${\mathrm{S}\mathrm{M}}_{\mathrm{i}}^{\mathrm{H}}$) is the ith character of the secret message before the hiding process. Furthermore, Fig. 11 shows the PSNR values obtained by the suggested algorithm and RDH algorithm by Li, Li & Yang (2013) for the first 50 images in the UCID database with the payload capacity of 250 × 10³ bit. Results settle that the suggested algorithm has a more considerable PSNR value for all images. The maximum difference between the two algorithms equals 10.266 (dB). Furthermore, it appears in the image index (32). In contrast, the minimum difference between the two algorithms equals 8.547 (dB), appearing in the image index (37).

Structural similarity index measure (SSIM)

The structural similarity index measure (SSIM) is a model of comparison metric to check the structural similarity between two images. It is calculated using Eq. (16).

(16) $$\mathrm{S}\mathrm{S}\mathrm{I}\mathrm{M}(\mathrm{C}\mathrm{I},\mathrm{S}\mathrm{I})=\frac{\left(2{\mu }_{\mathrm{C}\mathrm{I}}{\mu }_{\mathrm{S}\mathrm{I}}+{\left(\left(2^{24}-1\right)\ast 0.01\right)}^2\right)\left(2{\sigma }_{\mathrm{C}\mathrm{I},\mathrm{S}\mathrm{I}}+{\left(\left(2^{24}-1\right)\ast 0.03\right)}^2\right)}{\left({\mu }_{\mathrm{C}\mathrm{I}}^2+{\mu }_{\mathrm{S}\mathrm{I}}^2+{\left(\left(2^{24}-1\right)\ast 0.01\right)}^2\right)\left({\sigma }_{\mathrm{C}\mathrm{I}}^2+{\sigma }_{\mathrm{S}\mathrm{I}}^2+{\left(\left(2^{24}-1\right)\ast 0.03\right)}^2\right)}$$

where μ_Ic and μ_Is metrics represent the average of the cover and stego images, respectively, σIcIs metric represents the covariance between the cover and the stego images, and ( ${\sigma }_{\mathrm{I}\mathrm{c},}^2{\sigma }_{\mathrm{I}\mathrm{s}}^2$) represents the variance between the cover and the stego images (Elshare & EL-Emam, 2018).

Structural similarity index measure (SSIM) is introduced in this study to check the image quality and attack resistance by calculating the similarity between the cove-images and corresponding steg-images. Comparisons with the previous hiding algorithms (Elshare & EL-Emam, 2018) and (Li, Li & Yang, 2013) have been made by using fifty color images selected randomly within the sizes (512 × 384 and 384 × 512) from the UCID v2 database for three payload capacities (20%, 30%, and 40%). It found that the suggested hiding algorithm achieved excellent performance see Table 2.

Figure 12 shows the SSIM values obtained by the suggested algorithm for the first 50 images in the UCID database with a payload capacity of 250 × 10³ bit. Results show that the maximum SSIM equals 0.9998, appearing in the image indexes (46–48, 49–50).

Euclidean norm test

The Euclidean norm is defined in Eq. (17); this test demonstrates that the suggested algorithm works well against visual attacks. The distance (D) between the corresponding color {R, G, B} of both cover and stego-images is calculated.

(17) $$\mathrm{D}=\sqrt{{\left({\mathrm{R}}_{\mathrm{C}\mathrm{I}}-{\mathrm{R}}_{\mathrm{S}\mathrm{I}}\right)}^2+{\left({\mathrm{G}}_{\mathrm{C}\mathrm{I}}-{\mathrm{G}}_{\mathrm{S}\mathrm{I}}\right)}^2+{\left({\mathrm{B}}_{\mathrm{C}\mathrm{I}}-{\mathrm{B}}_{\mathrm{S}\mathrm{I}}\right)}^2}$$

Results of Euclidean norm (D) appeared in Figs. 13 and 14 for two-color images with image size (512 × 512) with the payload percentages are 10%, 30%, and 40%. The purpose of the suggested algorithm is to compare the results with the previous works: the MDLSB algorithm (Elshare & EL-Emam, 2018), and (El-Emam & Al-Zubidy, 2013).

The Euclidean norm distances of two images have been calculated individually; the minimum and the maximum average of the norm for three payloads are 392 and 413 at the “Peppers” and “Baboon” images, respectively. In addition, the smallest and the most significant difference between the average norm of the proposed algorithm and the others are 137 and 203 at the “Baboon” image for the works (Elshare & EL-Emam, 2018) and (El-Emam & Al-Zubidy, 2013) respectively.

Furthermore, the most significant difference in the Euclidean norm reached D = 360.35 at the image “Pepper,” with a load ratio of 40%. It appeared between the proposed algorithm and the previous work (El-Emam & Al-Zubidy, 2013). At the same time, the slightest difference in the Euclidean norm reached D = 9.51 at the image “Baboon” with a load ratio of 10%. It appeared between the proposed algorithm and previous work (Elshare & EL-Emam, 2018).

The difference between adjacent pixels

The difference between adjacent pixels of a cover image and its corresponding stego image was calculated by using Eqs. (18) and (19), where the scales ( ${\mathrm{D}}_{\mathrm{i},\mathrm{j}}^{\mathrm{c}\mathrm{I}}$ and ${\mathrm{D}}_{\mathrm{i},\mathrm{j}}^{\mathrm{s}\mathrm{I}}$) represent the absolute difference in the adjacent horizontal pair to CI and SI (El-Emam, 2015).

(18) $${\mathrm{D}}_{\mathrm{i},\mathrm{j}}^{\mathrm{C}\mathrm{I}}=|\mathrm{C}{\mathrm{I}}_{\mathrm{i},\mathrm{j}}-\mathrm{C}{\mathrm{I}}_{\mathrm{i},\mathrm{j}+1}|$$

(19) $${\mathrm{D}}_{\mathrm{i},\mathrm{j}}^{\mathrm{S}\mathrm{I}}=|\mathrm{S}{\mathrm{I}}_{\mathrm{i},\mathrm{j}}-\mathrm{S}{\mathrm{I}}_{\mathrm{i},\mathrm{j}+1}|$$

where ( $\mathrm{C}{\mathrm{I}}_{\mathrm{i},\mathrm{j}}$ and $\mathrm{S}{\mathrm{I}}_{\mathrm{i},\mathrm{j}}$) are two pixels at the location (i, j) in the cove-image and the corresponding stego-images, respectively. In this test, three-color stego-images with size (512 × 512) from the last level of the main channel with a payload capacity equal to 40% of the image size are used. The difference values ( ${\mathrm{D}}_{\mathrm{i},\mathrm{j}}^{\mathrm{C}\mathrm{I}}\mathrm{a}\mathrm{n}\mathrm{d}{\mathrm{D}}_{\mathrm{i},\mathrm{j}}^{\mathrm{S}\mathrm{I}}$) belong to the interval [−255, +255], and the frequency of each of these different values was counted. Figures 15 and 16 illustrate a graph of the pixel difference value on the X-axis and the frequency on the Y-axis. Results show that the difference value between the stego-image and the cover image is closed.

Using the visual information fidelity (VIF) model

The VIF test is used to assess the loyalty of the stego-image to the corresponding cover image. This test is based on the reference field (RD), natural scene statistics NSS, and the mixture of Gaussian scale (GSM) with the image distortion (ID) and the human visual system (HVS) (Han et al., 2013) and (El-Emam, 2015). VIF test is calculated using Eq. (20), where this equation includes two mutual information measures. The first measure of the VIF test is based on the information exchanged between inputs and outputs of HVS channels without channel distortion. In contrast, the second measure of the VIF test is based on the information exchanged between the inputs and the outputs of the distorted HVS channels. The output of the HVS channels is a stego-image.

(20) $\begin{array}{c}\mathrm{V}\mathrm{I}\mathrm{F}=\frac{{\sum }_{\mathrm{j}\in \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{\_}\mathrm{b}\mathrm{a}\mathrm{n}\mathrm{d}}{\sum }_{\mathrm{i}\in \mathrm{b}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{k}}{\log }_2\left(\frac{{\left({\sigma }_{\mathrm{j}\mathrm{i}}^{\mathrm{C}\mathrm{I},\mathrm{S}\mathrm{I}}\right)}^2}{\left({\left({\sigma }_{\mathrm{j}\mathrm{i}}^{\mathrm{S}\mathrm{I}}\right)}^2\times {\left({\sigma }_{\mathrm{j}\mathrm{i}}^{\mathrm{C}\mathrm{I}}\right)}^2-{\left({\sigma }_{\mathrm{j}\mathrm{i}}^{\mathrm{C}\mathrm{I},\mathrm{S}\mathrm{I}}\right)}^2+{\sigma }_{\mu }^2\times {\left({\sigma }_{\mathrm{j}\mathrm{i}}^{\mathrm{C}\mathrm{I}}\right)}^2\right)}+1\right)}{{\sum }_{\mathrm{j}\in \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{\_}\mathrm{b}\mathrm{a}\mathrm{n}\mathrm{d}}{\sum }_{\mathrm{i}\in \mathrm{b}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{k}}{\log }_2\left(\frac{{\left({\sigma }_{\mathrm{j}\mathrm{i}}^{\mathrm{C}\mathrm{I}}\right)}^2}{{\sigma }_{\mu }^2}+1\right)}\end{array}$

where ( ${\sigma }_{\mathrm{j}\mathrm{i}}^{\mathrm{C}\mathrm{I}}\mathrm{a}\mathrm{n}\mathrm{d}{\sigma }_{\mathrm{j}\mathrm{i}}^{\mathrm{S}\mathrm{I}}$) are the standard deviation of the cover image (CI) and stego-image (SI) in the ith block at the jth sub-band, respectively. The covariance of CI and SI in the ith block at the jth sub-band is defined in ( ${\sigma }_{\mathrm{j}\mathrm{i}}^{\mathrm{C}\mathrm{I},\mathrm{S}\mathrm{I}}$) is, see Han et al. (2013).

In Table 3, the VIF measure is reported to demonstrate visual information fidelity. In this work, testing on three images from the database of the standard color images (Imtiaz, 2019) with the sizes (256 × 256) pixels are applied. The proposed hiding process is studied using the payload capacities and the VIF metric. The measurement of results confirmed that the suggested algorithm is better than the typical reference (El-Emam, 2015). The suggested algorithm is working adeptly.

Testing chi-square (χ2) attack

The primary purpose of the suggested hiding algorithm is to hide a secret message SM in the color image without realizing the presence of hidden data. Chi-square test is used in this situation to check color uniformity in stego-images by verifying how the expected ( ${\mathrm{E}}_{\mathrm{i}}$) and observed ( ${\mathrm{O}}_{\mathrm{i}}$) frequencies of stego pixels are structured (Ahmad, El-Emam & AL-Azawi, 2021).

(21) $${\chi }_{s-1}^2={\sum }_{i=0}^{s-1}\frac{{\left(O_i-E_i\right)}^2}{E_i}$$

where (s) is the number of classes in the stego-image, (s − 1) is a degree of freedom, and ( ${\mathrm{E}}_{\mathrm{i}}$) is the expected frequency of ( ${\mathrm{i}}^{\mathrm{t}\mathrm{h}}$) pair see Eq. (22).

(22) $${\mathrm{E}}_{\mathrm{i}}=\frac{1}{2}\underset{?\mathrm{c}\mathrm{o}\mathrm{l}\mathrm{o}\mathrm{r}}{\mathrm{f}\mathrm{r}}\left\{{\mathrm{P}}_{2\mathrm{i}},{\mathrm{P}}_{2\mathrm{i}+1}\right\},\mathrm{\forall }\mathrm{i}=0,\dots ,\mathrm{s}-1$$

where, $\left\{{\mathrm{P}}_{2\mathrm{i}},{\mathrm{P}}_{2\mathrm{i}+1}\right\}$ is the ${\mathrm{i}}^{\mathrm{t}\mathrm{h}}$ pair of pixels $\{{\mathrm{P}}_0,{\mathrm{P}}_1,....,{\mathrm{P}}_{255\}}$. The frequency observed at the ( ${\mathrm{i}}^{\mathrm{t}\mathrm{h}}$) color is shown in Eq. (23).

(23) $${\mathrm{O}}_{\mathrm{i}}=\mathrm{f}\mathrm{r}\left({\mathrm{C}}_{\mathrm{i}}\right)\mathrm{\forall }\mathrm{i}=0,\dots ,\mathrm{s}-1$$

The probability $({Pr}_{{\chi }^2,\mathrm{s}-1})$ based on Chi-square value ( ${\chi }^2$) with $(\mathrm{s}-1)\mathrm{d}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{e}\mathrm{e}$ of freedom is calculated using Eq. (24).

(24) $${Pr}_{{\chi }^2,\mathrm{s}-1}={\left(2^{\frac{\mathrm{s}-1}{2}}\mathrm{\Gamma }\left(\frac{\mathrm{s}-1}{2}\right)\right)}^{-1}{\int }_{{\chi }^2}^{\mathrm{\infty }}{\left(\mathrm{X}\right)}^{\frac{\mathrm{s}-1}{2}-1}{\mathrm{e}}^{-\frac{\mathrm{X}}{2}}\mathrm{d}\mathrm{X}$$

where the Gamma ( $\mathrm{\Gamma }$(.)) that appears in Eq. (24) is a factorial function, see Eq. (25).

(25) $$\mathrm{\Gamma }(\mathrm{Z})={\int }_0^{\mathrm{\infty }}{\mathrm{X}}^{\mathrm{Z}-1}{\mathrm{e}}^{-\mathrm{X}}\mathrm{d}\mathrm{X}$$

The Chi-square attack on a stego-image with randomly distributed secret messages was studied. The proposed hiding algorithm verified comparisons between the two cover images (Baboon and Peppers) and their corresponding stego-images. The probability (Pr) values at the beginning of the block at the color image are irregular. With the block size increasing, the Pr value eventually drops to zero.

Figures 17A–17D illustrates the probabilities’ results concerning the block sizes. The maximum block size reached 20 and 25 for the Baboon image and 25 and 50 for the Peppers image when the secret message length equals 25% and 50%, respectively.

Therefore, steganalysis cannot detect SM due to the identical Pr for the stego-images and their corresponding cover images.

Estimate the probability of detection error ( ${\mathbf{P}}_{\mathbf{E}}$)

The proposed multi-channel based on a deep data hiding algorithm can imperceptibly hide secret data and resist attacks even in largely hidden payloads. The algorithm’s performance in data hidden can be checked by calculating the probability detection error (PE) expressed in equation Eq. (26).

(26) $${\mathrm{P}}_{\mathrm{E}}=min\left(\frac{1}{2}\left({\mathrm{P}}_{\mathrm{F}\mathrm{A}}+{\mathrm{P}}_{\mathrm{M}\mathrm{D}}\right)\right)$$

Let the metrics (#CISI and #SICI) be the number of cover images recognized as stego-images and the number of stego-images recognized as cover images, respectively. Therefore the probability of false alarms (PFA) and miss detection (PMD) are calculated in Eqs. (27) and (28).

(27) $${\mathrm{P}}_{\mathrm{F}\mathrm{A}}=\frac{\mathrm{\#}\mathrm{C}{\mathrm{I}}_{\mathrm{S}\mathrm{I}}}{\mathrm{\#}\mathrm{C}\mathrm{I}}$$

(28) $${\mathrm{P}}_{\mathrm{M}\mathrm{D}}=\frac{\mathrm{\#}\mathrm{S}{\mathrm{I}}_{\mathrm{C}\mathrm{I}}}{\mathrm{\#}\mathrm{S}\mathrm{I}}$$

where $P_{FA},P_{MD}\in \left[0,1\right]$

The detection error (PE) value range from 0 to 0.5. When PE = 0, discovering secret data is optimal, while PE = 0.5 means perfect security and difficulty accessing secret data.

Figure 18 shows the probability detection error (PE) as a function of the payload capacity represented by the number of bits per pixel (bpp). This function measures the area under the curve (AUC), where excellent security is reached when the area of AUC is large. The proposed deep data hiding algorithm result shows that the mean value of PE equals 0.467. This result is better than the previous works (Elshare & EL-Emam, 2018; Ahmad, El-Emam & AL-Azawi, 2021) by about 3.8% and 2.2%, respectively. Moreover, the excellent security percentage reaches 98% when bpp = 0.05, whereas the worst security percentage reaches 86% when bpp = 0.4.