By considering the principal cascade modulation couple, the authors in [22] proposed a 2D-SCMCI map which can be stated as follows:(1) {x(n+1)=rsin⁡(π[(y(n)+h)ksin⁡(aπx(n))]),  y(n+1)=rsin⁡(π[(kx(n+1)+h)sin⁡(aπx(n))]),  where k signifies the modulation parameter, r, a and h denotes the chaotic parameters, x(n) and y(n) depicts two values at step n. The randomness of the elements of the 2D-SCMCI hyperchaotic map is examined by plotting the values against the total iterations. Fig. 1 shows the iterations plot of the hyperchaotic map for the first 300 values of x and y sequences.

The encryption scheme proposed in [22] was entirely dependent on two operations, the first one the bit-based scrambling, and the second one is the diffusion operation carried by the bitwise XOR. The random number generator utilized to generate the private key was based on the SCMCI hyperchaotic map only. The working strides of the understudy encryption procedure are as follows:

Insert a plain image P of size A×B as the input of the algorithm.

The description of the understudy cryptosystem is shown in Fig. 3.

The originally proposed cryptosystem utilizes the concept of permutation and diffusion with just a 2D-SCMCI hyperchaotic map. The combination of permutation and diffusion is executed by four different keys column-wise permutation, row-wise permutation, forward diffusion, and backward diffusion. The equivalent encryption cipher can be developed for the originally proposed structure to get the loopholes and vulnerabilities. Suppose the keys applied for row-wise and column-wise permutations are renamed as kR and kC respectively and each permutation operation is denoted by P1 and P2 . Consider I denotes the original image to be inserted in the encryption algorithm. Therefore, we can write(6) C1=P1(I, kR),  (7) C2=P2(C1, kC).

The cipher C1 and C2 are the row-wise and column-wise permuted images. The next operation of forward diffusion DF and DB is performed by the keys kF and kB. Hence the operation can be written as(8) C3=DF(C2, kF),  (9) C4=DB(C3, kB).

The cipher C4 is considered the final encrypted version. All these four Eqs. (6)–(9) can be written into one equation by using the property of composition of functions as(10) C=DB(DF(P2(P1(I, kR), kC), kF),  kB).

Eq. (10) is the equivalent description of the complete understudy encryption cipher.

The secure key generation leads to a robust encryption algorithm, but the implementation of the key in the structure also plays an important role. A weak combination of security primitives generates a cryptosystem that can be easily attacked with the help of some classical cryptanalysis assaults. In our case, the combination of permutation and diffusion with input-independent key generation generates the possibility of attack. The cryptanalysis of the cryptosystem is possible only if there exist some vulnerabilities in the implementation structure. The loopholes are determined by examining the encryption operations combinations and key generation of the algorithm. The existing understudy cryptosystem lacks the following points in its structure:

The key generation is only dependent on the 2D-SCMCI hyperchaotic map and does not use any other domain to get the variety of PRNGs for the secret key.

The vulnerabilities listed above make it breakable by some standard cryptanalysis attacks.

The chosen plaintext attack works on the principle that the attacker gets temporary access to the encryption machine and inserts some desired plaintext [20]. The resultant output from these attacks leads to information about the private key. Firstly, the combined operations of the entire structure are detected and then prioritized to make one of them invariant against the encryption procedure. The invariance varies for each cipher operation and gives different outputs according to the implemented operation. The flowchart of the offered chosen-plaintext attack is displayed in Fig. 4.

The full black image with all zero entries makes the diffusion operation invariant. Therefore, we insert a full black image as(11) I=(0⋯0⋮⋱⋮0⋯0)A×B, 

The further attack is executed with the insertion of Eq. (11) into the system (10) and we get(12) C=DB(DF(P2(P1((0⋯0⋮⋱⋮0⋯0), kR), kC), kF),  kB).

The row-wise and column-wise permutation according to the bit arrangement, becomes invariant because all the zero bit remains zero after the permutation operation. Therefore, Eq. (12) can be modified as(13) C=DB(DF((0⋯0⋮⋱⋮0⋯0), kF),  kB),  (14) C=DB((0⋯0⋮⋱⋮0⋯0)⊕kF,  kB ),  (15) C=(0⋯0⋮⋱⋮0⋯0)⊕kF⊕ kB, 

The forward and backward diffusion keys kF⊕ kB can be combined as k. Hence, Eq. (15) can be written as(16) C=(0⋯0⋮⋱⋮0⋯0)⊕k=k.

The diffusion key makes the utilization of the previous bits. Therefore, the process of decryption of forward and back diffusion is applied to the original diffusion key K. The retrieved diffusion key eliminates the effect of forward and backward diffusion. Hence, after getting the diffusion key, the system (10) is reduced to only a permutation-based system, which can be written as:(17) C=P2(P1(I, kR), kC), 

The following procedure involves the insertion of other plain images into the exact permutation keys. The permutation keys are extracted accurately with the insertion of 257 plain images. The first plain image is inserted as(18) I1=(01⋯01⋯⋮0⋮1⋱⋯ 255255⋮255)A×B.

All the other plain images are inserted by shuffling I1 in row and column direction 256 times. After the insertion of all the images, the resultant gives all the possible permutation positions of the understudy cryptosystem.

The experimental outcomes are performed on grayscale images used in the originally proposed algorithm in [22], with dimensions 256 × 256. The visual results are shown in Fig. 5. From the depicted results we can see that the retrieved data is the same as the original data. The exactness of the retrieved data depicts the vulnerability of the cryptosystem against cryptanalysis attacks. The images in Figs. 5a–5d are termed “Image 1”, “Image 2”, “Image 3”, and “Image 4” respectively.

In digital images, the histogram can be used to directly represent the distribution of all pixel values by counting the number of pixels with the same pixel value. Histograms of meaningful images typically show fluctuations because of the unequal distribution of pixel values. We can say that if two images visually look the same and their histograms are also equivalent then we can say that the original and retrieved possess zero error. By comparison, the encrypted image has a relatively flat histogram with a consistent distribution of pixel values. The results of the histogram for original, encrypted, and recovered “Images 1” to “Image 4” are shown in Fig. 6. The same sharp peaks in the original and retrieved data depict the exactitude of the proposed attack and the vulnerabilities of the cryptosystem.

There is a high connection between neighboring pixels in meaningful images, and the images include much redundant information. Instead, cipher images are entirely devoid of redundancy and have no association with neighboring pixels. To prevent cracking the encryption technique by analysis of the correlation between neighboring pixels, an image encryption algorithm should be designed to break this strong correlation. The correlation coefficients of the original, encrypted, and retrieved images are listed in Table 1. From the enumerated values, we can observe that the correlation between original and recovered images is very similar. Therefore, we can say that the pixel of the recovered image is the same as the pixel of the original image.

The correlation diagram for the original and recovered “Image 1” are displayed in Fig. 7. The patterns in the correlation diagram of the original and retrieved images indicate the error-free recovery of the data applying the proposed attack.

The entropy of an image is an effective metric for identifying the unpredictability of the distribution of pixel values since it signifies the uncertainty of the information contained within the image. The formula for determining the entropy of image data is(19) H(m)=∑i=02n−1⁡p(mi)log1p(mi),  where p(mi) shows the probability of the symbol mi . The ideal value for entropy is 8. The results of the entropy analysis are shown in Table 2. In the case of our understudy scheme, the entropy of encrypted result approaches 8. After the successful implementation of the suggested attack, the entropy of the original and recovered becomes the same due to the exact recovery of the image pixels.

In this segment of the manuscript, we have performed the time execution analysis of the proposed chosen plaintext on different sizes of images. The proposed attack can be retrieved the key from different sizes of images in very less time. The results are shown in Table 3. The depicted results indicate that our proposed attack can retrieve private keys in very less time, which indicates the reliability of our proposed algorithm.