The role that chaos plays in cryptography is also influenced by its characteristics. Mutually continuous-time and discrete-time chaotic maps can be distinguished from the mathematical model description of chaotic systems. There are many common models of continuous-time chaotic systems, but two of the most common are the Lorenz system and the Chen system. Arnold maps, logistic maps, sine maps, and Henon maps are examples of models for discrete chaotic systems. Another methodology for categorizing chaotic systems distinguishes between systems with integer forms and systems with fractional forms. The model of a fractional system is more general. Hénon-Lozi type maps [11], for example, are examples of recently proposed typical fractional chaotic maps that display a rich complicated dynamic behaviour. After proposing a hyperchaotic fractional Grassi-Miller map in [12], the authors proceed to implement it in hardware. A chaotic map with fractional order is presented and investigated in [13]. Several books and articles discuss the utilization of chaos for image encryption.

A fast and secure image encryption system was proposed by Khennaoui et al. [14], and it makes use of a 1D chaotic map. The composition of a 1D chaotic system is straightforward, even though the key space is very small. When it comes to protecting images, 1D chaotic maps provide advantages such as increased speed and simpler hardware implementation. Chaotic encryption research excels in the field of image encryption due to the synergy between effective chaotic systems and challenging encryption approaches. In [9], a sine trigonometric function and tent map are utilized to propose a discrete chaotic system. The statistical behaviour is consistent over a large range of parameter values. Piecewise linear chaotic mapping and a trigonometric function were used by Liu et al. [15] to create the chaotic mapping. Li et al. [16] discussed a nonlinear dynamic system that includes a cosine function and finds that it has a lengthy chaotic interval and resilient chaotic qualities. The inverse of a trigonometric function is likewise a trigonometric function because trigonometric functions have unique qualities such as periodicity and boundedness. A 1D piecewise chaotic map and the bisection approach were presented for image encryption in [17]. Elghandour et al. [18] employed a hyperchaotic model to produce a pseudo-random sequence, which they subsequently encrypted using a combination of scrambling and diffusion. Gopalakrishnan et al. [19] generated a new 1D chaotic model using the Beta function and applied it to image encryption; they called their proposal the Beta chaotic map. Zahmoul et al. [20] proposed an innovative image cryptosystem that employs a hybrid chaotic system by combining two 1D chaotic maps. The pseudo-orbits of one-dimensional chaotic systems are employed as the key in an innovative encryption method presented by Alawida et al. [21]. Nepomuceno et al. [22] designed and implemented a one-dimensional sine-powered chaotic map for image encryption. As part of an effective symmetric image encryption scheme, Mansouri et al. [23] proposed a novel 2D chaotic map to expand the available key space. Using a 2D economic chaotic map and a logistic map, Huang et al. [24] devised a method for encrypting images. Askar et al. [25] propose an innovative method of encryption that uses keys generated from either DNA or an image of plaintext. Khan et al. [26] introduced an S-Box and logistic-sine scheme for image encryption.

The research discussed above has led to the proposal of secure cryptosystems. However, it is worth noting that many of these proposals may suffer from either insecurity or impracticality issues, primarily related to time complexity. Therefore, considering the time cost as a crucial factor, we focused on improving the efficiency of the system. To achieve better security, we employed a simple chaotic map available, which exhibits high randomness.

Many encryption algorithms do not offer strong security against classical cryptographic attacks [27–29]. The cryptanalysis of many recently proposed cryptosystems has increased the risk of sensitive information being lost. Therefore, by considering all the weaknesses carefully, we have constructed a secure encryption algorithm to provide image security. We have proposed a multiplication and diffusion-based encryption strategy based on the Hindmarsh-Rose chaotic model [1]. The key generation process is secured by the SHA-256 hashing algorithm. The proposed technique employs an image’s associated encryption key as a substitute for the traditional encryption key, which increases security and reduces the amount of time it takes to decrypt data [30,31]. The encryption method combines a scrambling process with a diffusion process.

The subsequent sections of this manuscript are organised as follows: Section 2 offers a comprehensive analysis of the Hindmarsh-Rose model; Section 3 discusses the construction of the proposed model. In the next two sections, we have shown simulation outcomes and performance analysis, respectively. Finally, the conclusion with some future recommendations is presented in the last section.

To find the equilibrium points, we set the derivatives in the above equations to zero and solve for x, y, and z:

where x is the membrane potential, y and z are the gating variables for the two potassium currents, a, b, c, d, r, s, and x0 are model parameters, and I is an external current input.

We will use the following parameter values for the stability analysis: a=1.0, b=3.0, c=1.0, d=5.0, r=0.001, s=4.0,  x0=−1.6, I=4.0.

To determine the fixed points of the model, we solve the equations dxdt=dydt=dzdt=0 simultaneously. This gives us the following three fixed points:

To examine the stability of the fixed points, compute the Jacobian matrix J evaluated at each fixed point. The Jacobian matrix is given by:

where x, y, and z are the values of the fixed point. We evaluate J at every fixed point and calculate the eigenvalues. The eigenvalues of J for each fixed point are:

Fixed point 1: (−10.8647,−0.0378+0.2021i)

The real part of all eigenvalues is negative, so the fixed point is stable.

Fixed point 2: (1.6439,−1.1188+0.2491i)

One eigenvalue has a positive real part, so the fixed point is unstable.

Fixed point 3: (−6.2145,−1.9401+0.0000i)

The real part of both eigenvalues is negative, so the fixed point is stable.

Therefore, we have one unstable fixed point and two stable fixed points. This means that the system can exhibit different types of behaviour depending on the initial conditions. If the initial conditions are near the unstable fixed point, the system will diverge and exhibit chaotic behaviour. If the initial conditions are instead near a stable fixed point, the system will converge towards that point and exhibit stable behaviour. The behaviour of a dynamical system is largely determined by its fixed points, which are values of the system’s variables that do not change over time. Stable fixed points act as attractors, pulling the system towards them, while unstable fixed points act as repellers, pushing the system away from them.

To examine the execution of the Hindmarsh-Rose model we have performed simulations based on different times. The spiking-bursting behaviour findings are explained in Fig. 1.

The National Institute of Standards and Technology (NIST) developed a widely used suite of tests for measuring the random behaviour of time series. Each sequence being evaluated is 1,000,000 bits in length, hence testing many sequences is necessary. Random performance of time series may be measured with the help of the p-value. The standard deviation is set to = 0.01. We produced 500 chaotic real number categorizations, individually with a length of 125,000 real numbers, to assess the stochastic recital of sequences produced using the chaotic map. For the NIST evaluation, one hundred sequences of length one million bits are obtained. Table 1 summarizes our experimental findings and comparative results with the existing chaotic map. Each p-value is bigger than 0.01, and the run test has a minimum pass rate of 96%, as can be seen from the test outcome. All chaotic sequences created by system (1) have been shown to pass the NIST test in experiments.

The initial conditions and the key parameters of the proposed encryption algorithm are generated by inserting the input image in the SHA-256 algorithm. The results generated from SHA-256 are utilized as key parameters of the Hindmarsh-rose chaotic model. The first step was using the SHA-2 256 hash method to get the encryption key from the hash of the plaintext picture. The initial state value of the chaotic Hindmarsh-Rose system was determined by dividing the hash string into four parts, each of which was then mapped to a decimal larger than 0 and less than 1.

In this study, the colour digital image is encrypted by combining the operation of diffusion and invertible matrix multiplication generated from the Hindmarsh chaotic map. The notion of secure key generation from SHA-256 makes the encryption secure against statistical attacks. The major operations involved in the cryptosystem are matrix multiplication and diffusion. The array for diffusion is generated from the Hindmarsh-Rose model. The arrays constructed for the matrices are filtered through the inverse operation to make the decryption possible. The steps of the proposed encryption are as follows:

Step 1: The size of the input image is m×n×3 in the encryption.

Step 2: The layers of the plain image are separated into red, green, and blue channels.

Step 3: Each layer is divided into blocks of 2×2 matrices.

Step 4: The invertible matrices generated from the Hindmarsh-rose model are then multiplied with the plain image matrices, respectively.

Step 5: The multiplicated results are then diffused with the key arrays generated from the Hindmarsh-Rose model.

Step 6: The resultants are then concatenated as cipher images.

The decryption of the ciphertext is performed in the same step in a reverse manner. The detailed working strides of the decryption process are as follows:

Step 1: The cipher image of size m×n×3 is inserted as the input of the decryption algorithm.

Step 2: Inverse diffusion is applied to the layer of the cipher image.

Step 3: The resultant from Step 2 is then multiplied with the inverse of the private key constructed from the Hindmarsh-Rose model.

Step 4: The outcome layers from Step 3 are then combined into one plain image.

The working mechanism of the proposed encryption work is shown in Fig. 2.

The histogram of an image provides a visual representation of how the image’s pixel values are distributed. Plaintext images typically exhibit non-normal distribution shapes in their histograms. The histograms of an encrypted image should be uniformly distributed for higher security. The statistical histograms of Baboon, Parrots, Peppers, and Tulip test images and their enciphered counterparts are displayed in Fig. 4.

The horizontal coordinates in Fig. 4 represent pixel value and the vertical coordinate denotes the frequency occurrence of each pixel. The histograms of the encrypted and plaintext versions of the image are very different from one another. The histogram of the enciphered ciphertext image is normally dispersed, even though the histogram of the plain image was not. Consequently, the encrypted image is secure against attacks based on statistical analysis.

Furthermore, we can use the Chi-square test to quantify the histogram’s uniformity distribution. The Chi-square χ2 of an image can be determined as follows:

where n denotes the number of grayscale levels in the image, Oi is the frequency with i-th gray level that has been detected, and Ei is the expected standard frequency with i-th gray level that has been observed for an image with dimensions M×N, Ei=M×N×n. The critical value for an 8-bit grayscale image (n=256) at a significance level, α=0.05 is χ2(255, 0.05)=293.2478. For an enciphered 8-bit grayscale image, the value must be less than 293.2478. Numerous test images and their enciphered counterparts are used in the test, and the findings are summarized in Table 2.

From Table 2, the Chi-square results of all the enciphered images are smaller than the crucial value. However, the Chi-square estimate of enciphered images produced by our proposed work is lower than that of similarly prepared images from other encryption schemes.

There is a strong correlation among neighbouring pixels in meaningful plaintext images. As such, a strong encryption method needs to be able to break the link among adjacent pixels. A correlation coefficient provides a quantitative calculation of the degree to which neighbouring pixels are correlated with one another. In this work, we used the correlation coefficient, a measure for examining the degree to which neighbouring image pixels share common characteristics.

where (xi,yi) is a pair of integers representing the average grey amount of a sample of neighbouring pixels in the image, and Nxy is the total quantity of sampled pixel bands. Since the rxy correlation coefficient has a smaller absolute value, there is less of a connection between the two sets of pixels. Correlation coefficients for diagonally adjacent pixels in the enciphered image are explained in Table 3.

The comparison outcomes are shown in Table 3. The proposed approach yields good results, especially when compared to state-of-art techniques. In this study, we present a method that utilizes the Hindmarsh Rose model to create confusion and diffusion among pixels.

Fig. 5 plots the distribution of neighbouring pixel values for the Parrots, making it easy to see how their values are related to one another. Fig. 5 shows that, in the original Parrots image, neighbouring points are typically distributed along or around the 45-degree line, signifying that the estimates of nearby pixels are the same or very similar. Yet, there is a significant variance in value between neighbouring pixels, as seen by the fact that the ciphertext image’s adjacent points are not centred on the 45-degree line. As a result, the image’s pixel correlation is essentially broken by the encryption technique.

Information entropy can be utilized to calculate how unpredictable or random data is. The greater the information entropy, the more difficult it is to forecast or understand the underlying information. The following formula may be used to determine an information source’s entropy:

where S=s1,s2,…,sn is the information source and pi is the probability that si will occur. The maximum information entropy principle states that an information source has the highest possible entropy, or log2⁡(n), when all possible states si have the same probability pi=(1/ n). An 8-bit grayscale image’s data source has 256 grey levels, making n=256. As log2⁡(256)=8 is the highest possible entropy for a grayscale image, this is the case. Hence, the larger the uncertainty and the significant the robustness of an enciphered image, the nearer its information entropy is to 8. Table 4 shows the entropy of encrypted examples of typical test images using this approach and other previously published schemes.

The results demonstrate that ciphertext image information entropy is quite near to the maximum value and hence the proposed scheme is secure against entropy attack.

The strong sensitivity of the ciphertext to the plaintext and the secret keys is a feature of an effective encryption algorithm. A comparison of the enciphered image’s sensitivity to the original image or secret keys can be carried out with either NPCR or UACI. Mathematically, NPCR and UACI are written as:

where M and N refer to the image’s row and column coordinates. The sensitivity of the encryption technique is relative to the square root of the product of NPCR and UACI. An NPCR of 99.6094% and a UACI of 33.4635% are considered good for image encryption.

After comparing the UACI and NPCR values of two different enciphered images for the key sensitivity study, we found that the respective encryption keys differed by just one parameter on the order of 1015. Tabulated below are the outcomes of the experiments. Experimental findings demonstrate that UACI and NPCR values are close to the ideal values, showing that the cryptosystem is highly sensitive to the specifics of every key parameter. The proposed scheme has higher key sensitivity than the results from [29,30]. Table 5 summarizes the findings from the experiments.

Through, key space analysis, one can analyse the total number of possible keys. In this study, the parameters and starting results of the chaotic model are the original keys to the algorithm. Eleven parameters of double precision {a, b, c, d, r, s, x0,x(0),y(0),z(0), Iext} make up the key set if the system parameter is ignored. There are fifteen distinct binary options for each parameter. This results in a total key space of 1015∗11=10165 > 2249. According to [29], an encryption algorithm is considered secure if its key space is larger than 2100. Hence, the proposed technique has a necessarily large key space to resist brute-force attacks.

In evaluating our algorithm’s efficacy, we utilize MSE and PSNR. The MSE shows how far off the target image is from the original. MSE is measured as:

where M1 denotes the row number and M2 denoted the column number, I0(i,j) is the value of the plain-image pixel at the position (i,j), and Ie(i,j) is the value of the enciphered image pixel at the position (i,j). PSNR is written as:

where Imax is the highest possible pixel quantity in the image. When assessing the encrypted version of an image to the original, the PSNR should be low. Images of Baboon, Parrots, Peppers, and Tulip are encrypted, and the PSNR (dB) of these encrypted images is calculated to compute the quality of the encryption. MSE and PSNR values are illustrated in Table 6. From the Table, it is evident that the proposed scheme is more secure than another encryption scheme.

While transmitting encrypted images, a good encryption technique should be able to deal with a certain amount of data loss. A visually recognizable decrypted image may be recovered even when noise or data loss corrupts the encrypted image, demonstrating the algorithm’s resilience.

To evaluate the algorithm’s robustness against data loss, we first encrypt a 256×256 image in the top left corner, then divide it into 32×32, 64×64, and 128×128 sub-blocks and decode each one separately. Fig. 6 depicts the decryption effect. The experimental findings demonstrate that the method is even capable of successfully retrieving the plain image even when the sliced region is as large as 128×128. The technique described in [30] can only handle a maximum data loss size of 8×16 in enciphered images, in contrast. Using the techniques described in [30], we find that when our enciphered image has a data loss with dimensions 128×128, the deciphered image is identical to the enciphered image with a data loss of 8×16. Our technique outperforms the scheme proposed in [30] in terms of resilience to data loss.

The proposed approach encrypts data in three distinct steps: generating Hindmarsh-Rose chaotic secret key streams, encryption of pixels using invertible matrices, and pixel diffusion carried by bitwise XOR operation. As part of the algorithm’s time-cost analysis, we encrypt and decrypt a 256-by-256 grayscale Baboon image. Table 7 displays the average encryption and decryption times (in seconds) from some different studies. Table 7 also includes statistics on the time cost of several chaos-based algorithms taken from recently published work. The outcomes show that the proposed method is quicker at both encrypting and decrypting than the methods discussed in [30,31].

The strength of the proposed encryption scheme can be measured by evaluating it against classical cryptanalysis attacks. When the system is subjected to the chosen plaintext or chosen ciphertext attack then the attacker might try to insert some images trying to recover the private keys from the system. As the proposed encryption scheme utilized the algorithm of SHA-256 based on the input of the algorithm, therefore, the output against each image would be different. Therefore, the proposed structure can resist all types of classical attacks due to its nature of the complex design.