1 Introduction

Smart cities are equipped with smart service delivery systems to make life more comfortable. One of the important characteristics of smart cities is the e-Health ecosystem which can be defined as a collaboration of organizations, governments, individuals, and the use of digital innovation to bring about revolutionary changes in the way healthcare services are delivered [4]. In smart cities, networked technology-based methods can be implemented to prevent and control crises like the Covid-19 pandemic [44]. One of the major areas of innovation includes improvement in the service delivery of smart healthcare systems [20]. Artificial Intelligence (AI) and IoT-based solutions along with proper security frameworks can ensure the efficient implementation of smart health infrastructures [26]. However, together with numerous benefits, the system must assure security, privacy, and confidentiality of medical records [15]. A significant portion of patient data is in the form of images, like, X-ray images, MRI images, etc. Such records have to be transmitted in a confidential and secure environment [23].

Transferring medical images via an insecure network may result in image alteration and can lead to the wrong diagnosis of a patient which could result in a lethal health problem. Despite a tremendous emphasis on developing security solutions for secure information exchange, there had been certain instances where confidential medical records were hacked and modified. For example, on April 17, 2020, Oakland country suffered a leak of a non-public map that includes information about positive Coronavirus cases [10]. Another, data breach has been recorded on April 18, 2020, in which a 16-year old hacker hacks the medical data of Mandrid health service and releases the health information about a politician (who was Coronavirus positive), on his Twitter and Instagram account [11]. As per the widely recognized Protenus breach barometer, more than 41 Million electronic patient record breaches were reported in the first eleven months of the year 2020 alone [1]. The Internet of Things (IoT), which is a new way of controlling the environment [5], is too, not free from security breaches [55]. As IoT provides a common platform for interacting images, video, web browsing, financial transactions, health information, etc. [48], the data collected and managed through IoT is stored in the cloud and each interactive action is completely dependent on the internet as such vulnerable to security threats.

Therefore, the development of a highly secure data transmission mechanism is the need of the hour. Various technologies used to achieve this objective include cryptography and steganography [52, 54]. Cryptography encrypts the data and presents it in an inconceivable manner [29]. Encryption of sensitive information can be done using standard methods, like, Data Encryption Standard (DES), Advanced Encryption Standard (AES), and Rivest Shamir Adleman algorithm (RSA) [31]. These techniques use the secret key or public key to encrypt the data or information. DES and AES have the symmetric key while RSA uses the asymmetric key for the encryption and decryption process [37]. Using standard encryption methods, we have a limitation of the key lengths with a bulk of digital data and thus are more prone to attacks. So, the standard methods are less reliable and insecure methods for encryption purposes. Encryption technique based on chaos theory overcomes the limitation of standard techniques and results in providing good security to the data being transmitted through an insecure network [38]. Chaotic encryption techniques use the initial keys for data encryption. Keys used for encryption are very sensitive to initial conditions, that is, a minute or a bit change in keys results in altogether different values [27].

One of the important parameters of a cryptographic scheme is computational complexity. The computational complexity attains more importance while communicating critical information like patient data etc. Several encryption methods based on chaos theory have been proposed [17, 25, 30] to improve the security of the data. The authors in [30] use the information embedding approach to secure the hidden information, though this approach along with chaotic theory provides good robustness to various attacks, however, the computational complexity of the scheme is high. The authors of [25] review various encryption algorithms and conclude that for real-time applications low computationally complex algorithms are needed. In [17] a Two-dimensional Logistic-Sine-coupling map has been reported. The authors demonstrate that the scheme has better ergodicity and better security. Although the reported techniques provide better security than the standard encryption techniques in terms of security but take more time for algorithm execution. Therefore, there is a need for the development of secure and reliable chaos based algorithms that have less time complexity.

In this paper, we present the IEFHAC method which is a secure and computationally efficient image encryption scheme based on Hessenberg transformation and Chaotic theory. The scheme uses chaotic maps for confusion and Hessenberg transform for diffusion. IEFHAC has 99.66% - 100% NPCR value, 37.39% UACI value, and 0.36 s computational time value. Therefore, the method is highly robust to statistical attacks.

The main contributions of the proposed work are:

  1. i.

    A secure image framework for image encryption IEFHAC has been proposed.

  2. ii.

    The presented framework is highly secure with 99.66 to 100% NPCR value and 37.39% UACI value and can resist statistical attacks.

  3. iii.

    The presented algorithm is computationally efficient with an encryption and decryption time of 0.34 and 0.02 s respectively.

The rest of the paper is organized as follows. Section 2 presents a detailed literature review. Section 3 highlights the need for a data security frame. Section 4 and section 5 introduce the preliminaries of chaotic maps and Hessenberg transformation respectively. A detailed description of the proposed technique is provided in Section 6. Section 7 presents the simulation results and at last, the paper ends with concluding comments in Section 8.

2 Related work

Image encryption based on DES, AES, RSA uses the image as a binary data stream that can result in image specification loss [31]. Chaos-based methods use the structure of the algorithm and the performance of chaos theory. Many such approaches could be seen in [30, 40, 51]. In different ways, various chaos-based encryption techniques have been proposed [12, 21, 22, 42]. There are several algorithms where either one or more maps have been used to encrypt the data like in [7], where a chaos-based encryption scheme has been presented. The method has been tested on both standard as well as medical images but has poor security. QR code-based image encryption technique is proposed in [19] where only the grayscale images have been used for the test purpose. A pixel permutation and chaotic map based image encryption technique has been presented in [2]. The scheme has not been evaluated for unified average change intensity (UACI), a well-known objective quality indicator. Pseudo-random number sequence (PRNS), Arnold permutation, and AES technique-based encryption methods have been implemented in [3]. The technique has been divided into two stages where bit-level permutation and block diffusion have been implemented to encrypt the image. The scheme can be further improved to show good security values with less time complexity. A chaotic map based image encryption technique could be seen in [39], in which Piece-Wise Linear Chaotic Map (PWLCM) has been used to permute and diffuse the pixels remaining after blocking the image into 2 × 2 size. The scheme has been tested only on grayscale images and has not been able to provide good security results with less time complexity.

Logistic map based chaotic encryption and embedding technique have been proposed in [36] where the basic logistic map is modified to increase the range of the control parameter. Although the algorithmic structure comes up with a good solution but lacks enhancing the security as evident from evaluated parameters. A multiphase image encryption scheme based on chaos theory is presented in [13]. The technique has been divided into two phases: the shuffling phase and the masking phase. The scheme uses a piecewise linear chaotic map and linear approximation technique in the masking phase. The structure of the algorithm offers good encryption techniques, but the method can be further improved to enhance the security results with less computational complexity. A 3D chaos theory-based image encryption technique has been put forth in [53]. In this, a 3D cat map and pixel frequency have been applied to the image for encryption in a secure way. This technique however has tremendous scope for improvement to achieve better security. Another, 4-D chaotic circuit based image encryption design has been presented [46] to provide security to the transmitting image. The technique has developed a simple algorithm to produce a secure framework, but, can be improved further to increase security values. One more, image encryption based data hiding method has been introduced [28] in which data hiding is done by using comprehensive sensing and discrete fourier transform. In order to encrypt the data Arnold transform has been used to scramble the bits. The proposed algorithm is able to secure data. But, the scheme has used the frequency domain method, thus, has made the algorithm complex.

All the above-mentioned algorithms use only one chaotic map that results in poor security of the secret information being transferred to the receiver. To improve the security of data many encryption techniques utilizing two maps could be seen in the literature. In [14], a privacy-preserving cryptosystem using two chaotic maps has been presented to provide security to the Internet of things (IoT) E-healthcare system. Zaslausky and logistic chaotic maps have been used for encrypting data. However, this method is computationally complex. A similar approach using two chaotic maps could be seen in [18]. Njitackle et al. [34], have presented an image encryption method to protect biomedical images from thefts. The framework has controlled coexisting attractors with two chaotic pair attractors and one periodic pair for improving coupling strength. Though, the technique has improved security to the biomedical images. However, can be improved to get better NPCR value. Another chaotic map based image encryption scheme has been given in [47] to provide security to medical images during transmission. The scheme has used a 2-D trigonometric map using Logistic, Sine, and Cosine maps. Although, the method has been able to enhance security than the standard encryption methods. But, can be enhanced further to upgrade security values. A new chaos theory-based data encryption and embedding technique [24] has come up with a new way of presenting the maps to encrypt the data. In this, the logistic map and tent map have been combined in an alternative manner. The technique gives a glimpse of using different maps in an effective way to encrypt data. The method has been shown to pass various randomness tests.

A thorough survey of literature led us to conclude that using two maps is a good approach to improve security because it reorders data more randomly and cracking of two maps with more than two initial control parameters or initial conditions is difficult. A critical analysis of state-of-the-art techniques [2, 3, 13, 14, 18, 28, 34, 36, 39, 46, 47, 53], reveals that there is a need of developing a more secure solution that is computationally efficient and which applies to both grey as well as color images. Towards this end, we provide an efficient solution, IEFHAC which is an image encryption framework based on Hessenberg transform and Chaotic theory. We show that the proposed scheme is more secure than state-of-the-art schemes and is computationally efficient.

3 Need for a security framework for healthcare data

The Covid-19 pandemic has put the world in difficult times. As of 16th January 2021, COVID-19 confirmed deaths 2,03,21,06 deaths worldwide [50]. A proper diagnosis while keeping a physical distance is the need of the hour. In such a scenario the use of smart healthcare is assisting the radiologists and other front-line workers to help prevent possible breakdown of healthcare facilities. This mechanism uses the internet as a medium for interaction and transmission of significant patient data. Medical imagery like Computed Tomography (CT) scans and X-ray imagery are of significant use for diagnosing COVID-19 patients. Such image data is frequently exchanged through the internet on a smart health platform. However, sharing critical personal patient information is quite risky as healthcare data breaches are exponentially increasing throughout the world. Thus, there is a need of securing this patient information (Medical imagery) to ensure secure and attack-free communication of critical data. Towards this end, we have developed an efficient image encryption solution IEFHAC for encrypting the medical imagery and hence allowing secure transfer of data which could be quite useful under testing times of Covid-19.

4 Cryptographic encryption based on chaotic maps

Encrypted data using chaotic maps appear to be raw, noise, or distortion and its behavior is unpredictable. Related behavior of cryptography and chaos theory has made its place for secure data transmission via an insecure network. Standard encryption techniques and chaotic encryption techniques show almost similar behavior like its sensitivity to initial conditions, random behavior, and wobbly phases but the only difference is the encryption transformation definition in phase space [35]. Cryptography is defined on a finite set of integers, while as, the chaotic system is defined on a set of real numbers. The general model for encryption is confusion and diffusion [43] shown in Fig. 1.

Fig. 1
figure 1

General Model for Image Encryption [43]

Full size image

Confusion permutes the pixels of an image with chaotic maps whereas diffusion substitutes pixel values of an image with suitable values.

4.1 Encryption based on logistic map

The Logistic map is the simplest and the easiest encryption technique [25, 27, 33, 41]. It is a function of a nonlinear system generated by iterating the equation shown in Eq. 1 up to (x-1). The map shows the chaotic behavior in the range of x0 ϵ [0,1] for an initial condition [x0] and control parameter r ϵ [0,1].

$$ {x}_{n+1}=r\ast x\ast \left(1-{x}_n\right) $$
(1)

4.2 Encryption based on sine map

A Sine map is another chaotic encryption map [35]. The iteration from [0, x-1] generates the random values based on Eq. 2. In this map, the control parameter μ can be any real positive number and initial condition x0 ϵ [0,1].

$$ {x}_{n+1}=\upmu \ast \sin \left(\uppi \ast {x}_n\right) $$
(2)

4.3 Bifurcation diagram

Bifurcation shows the behavioral change of the system as the change of the parameter. The bifurcation plot draws phase lines for various parameter values. It is a sudden change in the qualitative global features of a system as the parameter changes. A small change in a system parameter leads to a large and quantitative change in the systems’ behavior. Bifurcation in a dynamical system is a sudden quantitative change as a parameter is varied continuously. Here, quantitative is not the location of a fixed point but the change in the number of fixed points or stability that is a sudden qualitative change in the overall global behavior of the system. The bifurcation plot for the Logistic map and Sine map is shown in Figs. 2 and 3. Bifurcation of the Logistic map shows that the system behaves randomly in the range of [3.57, 4]. While the Sine map’s bifurcation diagram shows random behavior from less than 0.5.

Fig. 2
figure 2

Bifurcation Plot of the Logistic Map

Full size image
Fig. 3
figure 3

Bifurcation Plot of the Sine Map

Full size image

4.4 Lyapunov exponent

Quality of chaos is described by the sensitive dependence on the initial conditions. A small change in the initial condition leads us to different dynamic values. Chaotic exponent divergence is shown by the Lyapunov exponent. A Lyapunov exponent is a number that tells us how sensitive a system is and estimates the behavior of the chaotic system. So, it gives us more information about whether the system has a butterfly effect or not. The Lyapunov exponent for both the Logistic map and Sine map is shown in Figs. 4 and 5. The diagram indicates that the maps are good for the encryption purpose of transmitted data via an insecure network after processing.

Fig. 4
figure 4

Lyapunov Exponent of the Logistic Map

Full size image
Fig. 5
figure 5

Lyapunov Exponent of the Sine Map

Full size image

5 Hessenberg transformation using household matrix

The concept of Hessenberg was given by Karl Hessenberg and the method was entitled to his name. Hessenberg Transformation (HT) is a change in a square matrix from one form to another form with different values [6, 45, 49]. The resultant matrix could be either a square matrix having elements equal to zero below the first sub-diagonal, called the upper Hessenberg matrix, or the square matrix having elements equal to zero above the first super-diagonal, called lower Hessenberg matrix. Any square matrix ‘A’, can be converted to a Hessenberg matrix ‘H’ by multiplying household matrix ‘U’ and its transpose, in the manner displayed in Eq. 3.

$$ H={U}^T AU $$
(3)
$$ Where,\kern0.5em {U}^TU=I $$
(4)

The household matrix is an orthogonal matrix [32, 33] that satisfies the condition shown in Eq. 4. The household matrix can be calculated by the approach shown in Eq. 5.

$$ U=I-2W{W}^T $$
(5)
$$ \mathrm{Where}^{\prime}\mathrm{I}^{\prime}\mathrm{is}\ \mathrm{an}\ \mathrm{identity}\ \mathrm{matrix},\kern0.5em W=\left[\begin{array}{c}{x}_1\\ {}{x}_2\\ {}{x}_3\\ {}.\\ {}.\\ {}{x}_n\end{array}\right]\ and\ {W}^T=\left[{x}_1\kern0.5em {x}_2\kern0.5em {x}_3\kern0.75em .\kern0.5em .\kern0.5em {x}_n\right] $$
(6)

To form vector ‘W’, there is a need for determining x0 that can be calculated by the method shown in Eq. 7, aji are the elements of a symmetric matrix. The computation of H is described in Example 1.

$$ {x}_1=0,{x}_2=\frac{a_{21}-\alpha }{2r},\kern0.5em and\kern0.5em {x}_n=\frac{a_{n1}}{2r}\ \mathrm{f} or\ each\ n=3,4,\dots, n $$
(7)
$$ Where,\kern0.5em \alpha =-\mathit{\operatorname{sign}}\left({a}_{21}\right)\sqrt{\sum \limits_{j=2}^n{a}_{j1}^2} $$
(8)
$$ and,\kern0.5em r=\sqrt{\frac{1}{2}\left({\alpha}^2-{a}_{21}\alpha \right)} $$
(9)

Example 1

Consider a matrix, \( A=\left[\begin{array}{ccc}1& 5& 7\\ {}4& 0& 6\\ {}3& 3& 1\end{array}\right] \) and symmetric matrix, \( S=\left[\begin{array}{ccc}1& 4& 3\\ {}4& 1& 2\\ {}3& 2& 1\end{array}\right] \) to find the Hessenberg matrix by using household transformation.

To find the household matrix (orthogonal matrix), we have to find out the values of x1, x2, and x3.

Here, x1 = 0, \( {x}_2=-\frac{1}{\sqrt{10}} \), and \( {x}_3=\frac{3}{\sqrt{10}} \) . Thus, \( W=\left[\begin{array}{c}0\\ {}-\frac{1}{\sqrt{10}}\\ {}\frac{3}{\sqrt{10}}\end{array}\right] \) and \( {W}^T=\left[0\kern0.5em -\frac{1}{\sqrt{10}}\kern0.5em \frac{3}{\sqrt{10}}\right] \).

There, the household matrix can be calculated using Eq. 5 i.e.,

$$ {\displaystyle \begin{array}{c}U=\left[\begin{array}{ccc}1& 0& 0\\ {}0& 1& 0\\ {}0& 0& 1\end{array}\right]-2\left[\begin{array}{c}0\\ {}-\frac{1}{\sqrt{10}}\\ {}\frac{3}{\sqrt{10}}\end{array}\right]\ \left[0\kern0.5em -\frac{1}{\sqrt{10}}\kern0.5em \frac{3}{\sqrt{10}}\right]\\ {}U=\left[\begin{array}{ccc}1& 0& 0\\ {}0& 0.8& 0.6\\ {}0& 0.6& -0.8\end{array}\right]\ \mathrm{and}\ {U}^T=\left[\begin{array}{ccc}1& 0& 0\\ {}0& 0.8& 0.6\\ {}0& 0.6& -0.8\end{array}\right]\end{array}} $$

Hence, the upper Hessenberg matrix using household can be calculated by Eq. 3 i.e.,

$$ {\displaystyle \begin{array}{c}H=\left[\begin{array}{ccc}1& 0& 0\\ {}0& 0.8& 0.6\\ {}0& 0.6& -0.8\end{array}\right]\left[\begin{array}{ccc}1& 5& 7\\ {}4& 0& 6\\ {}3& 3& 1\end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ {}0& 0.8& 0.6\\ {}0& 0.6& -0.8\end{array}\right]\\ {}H=\left[\begin{array}{ccc}1& 5& 7\\ {}5& 1.8& 5.4\\ {}0& -2.4& 2.8\end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ {}0& 0.8& 0.6\\ {}0& 0.6& -0.8\end{array}\right]\\ {}H=\left[\begin{array}{ccc}1& 8.2& -2.6\\ {}5& 4.68& -3.24\\ {}0& -0.24& -3.68\end{array}\right]\end{array}} $$

6 Proposed algorithm (IEFHAC)

The use of the smart healthcare system in smart cities is increasing exponentially as it provides timely access to the patient records that not only provide on-time treatment to the patient but also keeps the doctor out of patients’ contact in critical present pandemic-like situations. However, the use of the internet for such important records can cause risk to the data that needs to be secure during transmission. Keeping this in view, the proposed IEFHAC framework has been presented. The proposed IEFHAC technique can be represented by a block diagram which is shown in Fig. 6. For confusion, pixel permutation is done with the help of Random Sequence Generation (RSG) using two chaotic maps i.e., Logistic chaotic map and Sine chaotic map already discussed in section 4. The output for one iteration of the one map is the initial value for the second map, and the first iteration value of the second map is further used as the initial value of the first map, so on and so forth [refer to Fig. 7]. In this way, the outputs of the overall iterations for both maps are different and hence balance key length requirement as compared to the single map or the double maps, as the initial value is set once, while in the proposed method the initial value itself goes on varying. The generated chaotic sequence is then XORed with the pixels of an image and hence creates more confusion among the pixels. The process of confusion is shown in Algorithm I.

Fig. 6
figure 6

Block Diagram of the Proposed IEFHAC Algorithm

Full size image
Fig. 7
figure 7

Block Diagram for Confusion Process

Full size image

The bifurcation plot for the proposed random sequence with the chaotic maps is shown in Fig. 8. From the diagram, one can see that the presented method shows the more behavioral change in the system with a change in the parametric value. Further, the Lyapunov exponent of the proposed way is presented in Fig. 9 that shows the system is more sensitive to initial conditions.

Fig. 8
figure 8

Bifurcation Plot for Generated Random Sequence

Full size image
Fig. 9
figure 9

Lyapunov Exponent for Generated Random Sequence

Full size image

In the process of diffusion, pixel values of the confused image are modified with the Hessenberg transformation described in section 5. The confused image (CI) is transformed twice with the Hessenberg principle, once directly and secondly by doing rotation or transpose before and after the processing of the confused image shown in Fig. 10. Then, the XOR operation is performed between these two transformed images. Algorithm II describes the diffusion process with the Hessenberg transformation.

Fig. 10
figure 10

Diffusion Process Block Diagram

Full size image
figure a
figure b

6.1 Encryption process

This section includes the prologue for the computation of the encryption process in which the initial values are set once for the single map and for the rest one it is generated by itself. The encryption process of the proposed method is described in Algorithm III.

figure c

The encryption process of the proposed method is also applicable to color images, the only difference is that the image is separated into three planes and the method of encryption is processed on each plane individually. And the encrypted planes are concatenated to form the final encrypted image.

The following algorithm along with Fig. 11 explains the confusion and diffusion process. The detailed process is explained as follows:

  1. 1.

    To confuse image pixels both Logistic and Sine maps need two initial values (Vi = 1, 2, 3, 4) to generate a random sequence (Sn = 1, 2, …, n).

  2. 2.

    At first, two initial values (V1, V2) for the first map (Logistic map, Eq. 1) are given directly to generate the first sequence value (S1).

  3. 3.

    For the next two initial values (V3, V4) of a second map (Sine map, Eq. 2), one value is the previously generated sequence value (V3 = S1) and the other initial value (V4) is given directly to generate the second sequence value (S2).

  4. 4.

    For the generation of third sequence value (S3), the first map (Logistic map, Eq. 1) uses one value (V1 = S2) as previously generated value (S2) from a second map (Sine map), and another initial value (V2) directly and so on.

  5. 5.

    Generated chaotic sequence is converted into matrix form to XOR it with Image matrix for confused image formation.

  6. 6.

    Resultant confused image is transformed with the Hessenberg principle (Explained in Example 1 of section 5) as shown in Fig. 10 to form an encrypted image.

Fig. 11
figure 11

Example of Encryption Process

Full size image

6.2 Decryption process

This section masquerades the decryption process. The encrypted image formed after confusion and diffusion process can be decrypted back to acquire the original image. The decryption process for recovering back the original image is the reverse of the encryption process that is shown in Algorithm IV.

figure d

The decryption process for color images is the reverse step of the encryption method.

7 Simulation results and discussions

IEFHAC has been tested on various images (Medical as well as general images) taken from the University of Southern California – Signal and Image Processing Institute (UPC-SIPI), Uncompressed Color Image Database (UCID), and Open Access Biomedical Image (OPENi) databases. However, the Covid-19 patient’s medical image has been taken from the GITHUB image dataset [9]. Because of the space limitations, analysis of randomly chosen nine images each size 256 × 256 is displayed, shown in Fig. 12. The technique is applicable to both gray images as well as color images of different image sizes with different formats. Since the encryption and decryption of color images are the same as gray-scale images, therefore, the scheme is evaluated only on test gray-scale images (refer to Fig. 12). However, to show the applicability of the proposed method to the color images as well, the NPCR and UACI values are given for both grayscale images and color images given in section 7.2. Encryption of the image is performed with initial conditions shown in Table 1.

Fig. 12
figure 12

General Images used for Encryption

Full size image
Table 1 Initial Values used for the Proposed Algorithm
Full size table

In this section, we present the experimental results of the proposed technique to evaluate the system for security and computational complexity. The scheme has been evaluated in terms of the histogram, the Number of Pixels Change Rate (NPCR), Unified Average Change Intensity (UACI), computational time, etc.

7.1 Histogram analysis

Histogram analysis displays the pixel’s intensity distribution of the digital image. An encrypted image should have a uniform distribution. When an image shows random behavior, i.e., it has a uniform distribution, then it is difficult for an anonymous adversary to attack or modify the secret information. The histograms of the plain images and resultant encrypted images, using the proposed scheme are shown in Fig. 13. It can be seen that encrypted image histogram show uniformity among different pixels, indicating that the proposed algorithm is robust and can prevent leakage of secret information from any statistical attacks. The histogram analysis has been also done on a number of images randomly chosen from UPC-SIPI and the result shows a similar/ nearly uniform behavior as that of the images displayed.

Fig. 13
figure 13

Histogram of Original and Encrypted Image for General and Medical Images

Full size image

To test the uniformity of histogram, a chi-square test has been used which is computed by the following equation [46].

$$ {\boldsymbol{\upchi}}^{\mathbf{2}}=\sum \limits_{\boldsymbol{d}=\mathbf{0}}^{\mathbf{2}\mathbf{55}}\frac{{\left({\boldsymbol{f}}_{\boldsymbol{d}}-\boldsymbol{s}\right)}^{\mathbf{2}}}{\boldsymbol{s}} $$
(10)

Where fd is the pixel value frequency (0–255), s = (r × c)/256 is the size of an image in which r and c are the image dimensions. The lower chi-square value represents better uniformity. The Chi-square value of the above-shown histogram for different gray images is reported in Table 2. The analyzed test confirms the histogram uniformity of the stated encrypted images.

Table 2 χ2 values of Fig. 13 for different gray-scale images
Full size table

7.2 Differential analysis

Differential analysis checks the sensitivity of the encrypted image to its original image and secret key. The strength of the encrypted image has been checked with the two well-known standard criteria: NPCR and UACI tests. NPCR checks the pixel modification in an original image with the encrypted image. It estimates the sensitivity of the encrypted image about its original image and initial values/parameters. The NPCR is described in Eq. 11 where m × n is the image size. Here, D is the number of uneven values that can be found with Eq. 12 in which I0 is the original image and Ie is the encrypted image.

$$ NPCR=\frac{1}{mn}\sum \limits_{i,j=1}^{mn}D\left(i,j\right)\times 100\% $$
(11)
$$ D=\left\{\begin{array}{c}0, if\ {I}_0={I}_e\\ {}1, if\ {I}_0\ne {I}_e\end{array}\right. $$
(12)

UACI is the average intensity change between the original image and the encrypted image. UACI test can be evaluated with Eq. 13. Here, I0 and Ie are the original and encrypted images with m × n dimensions.

$$ UACI=\frac{1}{mn}\left(\sum \limits_{i,j=1}^{mn}\frac{\left|{I}_0\left(i,j\right)-e\left(i,j\right)\right|}{255}\right)\times 100\% $$
(13)

As per [16] for a secure cryptosystem, NPCR and UACI should be near 99.61% and 33.44% respectively. The evaluation values for gray-scale and color test images are shown in Tables 3 and 4 and its comparison with the existing techniques for NPCR and UACI are given in Table 5. To make the analysis precise, the study has considered six images from Fig. 12 for color image encryption analysis. The calculated examinations show that the proposed algorithm has better sensitivity values than the existing state-of-the-art algorithms. Additionally, the proposed technique has been also tested on various medical images randomly taken from the OPENi medical image repository, in which the average NPCR is near about 99.76⁒ and UACI value is 36.52⁒ that shows the excellence of the scheme.

Table 3 NPCR and UACI Values for different gray-scale images
Full size table
Table 4 NPCR and UACI Values for different color images
Full size table
Table 5 Comparison of NPCR and UACI Values
Full size table

7.3 Keyspace analysis

The number of keys that can be used in the encrypted technique is the keyspace of the algorithm. IEFHAC uses two chaotic maps having two keys each. Total combinations of different keys that can be used are 1042 if the average precision set is 10−14, which means the proposed scheme guarantees to withstand any brute-force attack.

7.4 Key sensitivity analysis

Key sensitivity refers to a one-bit change in the key that leads to altogether different results. Wrong key substitution in the algorithm results in an incorrect decrypted image. The general and medical test images have been used to test the sensitivity of the key for which the image is decrypted with a wrong key (bit change in a logistic key: from 0.90 to 0.91) and the effects of only nine images are displayed, shown in Fig. 14. From the diagram, it can be observed that the decrypted image is different and wrong in comparison to the original one that shows the high sensitivity of the proposed algorithm.

Fig. 14
figure 14

Key Sensitivity test: Decryption with Original and Wrong Key

Full size image

7.5 Computational time analysis

Computational time determines the time a system takes to complete the process of encryption/decryption that is helpful to check the system’s efficiency. This analysis is carried out on an Intel®, Core™ i3-3120M, CPU @ 2.50GHz, 4 GB RAM running Windows 10. Simulation results for time analysis for encryption and decryption are shown in Table 6. The time required to encrypt an image is the encryption time, whereas, decryption time is the image decryption processing time. For encryption, too many processing steps are required to complete the task while as in decryption, the method uses generated random sequence directly which has already been generated in the encryption process instead of generating it again, as the sequence will be the same either processing it once or twice. Hence, the encryption processing time is more than the decryption computation time. Time taken to complete the encryption/decryption process can be calculated by dividing the size of data by the speed shown in Eq. 14. It is clear from the table that the proposed technique takes less time for encryption and decryption and hence is less complex in terms of speed.

$$ Time=\frac{Data\ size}{speed} $$
(14)

Where, \( Data\ size=\frac{cycles}{second}\ and\ Speed=\frac{cycles}{byte} \)

Table 6 Computation Time Comparison with the Existing Techniques
Full size table

7.6 Randomness analysis

The randomness experiment for the proposed algorithm has been tried with the NIST statistical test suite (SP 800–22) [8]. This test suite includes almost 15 computable tests. The results for these tests for IEFHAC are shown in Table 7.

Table 7 NIST Statistical tests
Full size table

The proposed algorithm has passed all the tests that show the randomness of the scheme.

7.7 Decrypted image quality analysis

To evaluate the quality of a decrypted image, various image quality analysis has been used. The investigation includes the Structural Similarity Index (SSIM), Normalized Cross-Correlation (NCC), and Bit Error Rate (BER). SSIM checks the alikeness between the original image and the decrypted image. It can be calculated with Eq. 15 [14], where μ represents the intensity of an image, σ represents the standard deviation, and σ of (x, y) represents the cross-correlation of an image. NCC checks the relation between the original and decrypted image. It can be calculated through Eq. 16 [14] in which wo and wx are the original and encrypted image bits. The decrypted image should have SSIM and NCC values close or equal to 1 which means the image is decrypted perfectly. BER represents the error between the original image and the encrypted image. Here, the BER should be near or equal to 0 which indicates the image is decrypted without any error. The mathematical representation of BER [42] is shown in Eq. 17 wherein Nerr and Nbits symbolize the number of error bits and the number of pixels respectively.

$$ SSIM\left(x,y\right)=\frac{\left(2{\mu}_x{\mu}_y+{C}_1\right)\left(2{\sigma}_{xy}+{C}_2\right)}{\left({\mu}_x^2+{\mu}_y^2+{C}_1\right)\left({\sigma}_x^2+{\sigma}_y^2+{C}_2\right)} $$
(15)
$$ NCC=\frac{\sum \limits_{i=1}^m\sum \limits_j^n{w}_o\left(i,j\right){w}_x(i.j)}{\sum \limits_{i=1}^m\sum \limits_{j=1}^n{\left[{w}_o\left(i,j\right)\right]}^2} $$
(16)
$$ BER=\frac{N_{err}}{N_{bits}} $$
(17)

The average estimated and comparison values of these quality analysis tests for a number of images randomly selected from the USC-SIPI image database are shown in Table 8.

Table 8 Image Quality Value Results Between Original and Decrypted Image
Full size table

8 Conclusion

Smart health systems are distinguished by the use of Information and Communication Technology (ICT) in healthcare. The implementation of technology-based solutions in healthcare can lead to better diagnosis and treatment of diseases. This will bring about an overall improvement in the quality of care provided to the patients. In smart e-health systems, the security of medical images plays an instrumental role in proper service delivery. Image data security has become a major concern to mitigate the confidentiality issues related to sensitive medical data during transmission in critical situations. Ensuring the security of medical images is imperative for the efficient implementation of Smart health infrastructures. In this paper, we present IEFHAC, a secure and less computationally complex image encryption technique based on two chaotic maps and Hessenberg household transformation. The algorithm comprises two stages: the confusion stage and the diffusion stage. The confusion stage randomizes the image using a Logistic map and Sine map, whereas, the diffusion stage transforms the confused-image with the help of the Hessenberg principle. The chaotic sequence generation method makes the system strong against attacks and it is difficult for the anonymous person to find the exact key for decryption because one of the keys itself is random. Furthermore, the Hessenberg principle transforms the confused-image with altogether different pixel values that add more security to the system. The scheme shows better performance in terms of security as well as time and can withstand many attacks, like, differential attacks, statistical attacks, etc. IEFHAC shows better results than the already existing algorithms in terms of NPCR, UACI, and computational time with 99.66% - 100%, 37.39%, and 0.36 s respectively. Thus, IEFHAC could be used for real-life medical applications in smart healthy cities.