1 Introduction

Nowadays, the number of digital images is exponentially growing due to the fast development of information technologies. As a consequence, a large number of images are stored and transmitted every second, where a significant part of these images is considered as private content [56]. To guarantee privacy and security, many researchers and security experts developed several types of algorithms, such as steganography [40, 53, 58], watermarking [3, 5, 10, 29, 34], data hiding [9, 28, 52], and image encryption [19, 21, 30, 38]. Unlike the former types, which tend to hide the secret information into a public image, the image encryption transforms the whole private image into an unrecognized random-like one. Recently, many image encryption schemes have been proposed using various theories and technologies like chaotic maps [19, 25, 31, 41, 55], DNA encoding [7, 12], quantum theory [13, 27], optical systems [24, 46, 54], etc [14, 44]. Among these technologies, chaotic maps are the most popular due to the natural chaos proprieties such as unpredictability, ergodicity, high sensitivity, and a random-like and deterministic behavior. These features perfectly respond to cryptography systems’ needs.

In chaos theory, depending on the number of variables, chaotic maps are divided into two categories: low-dimensional and high-dimensional chaotic maps. Low-dimensional chaotic maps have a simple structure and are more easy to implement. However, most of the low-dimensional maps are proved to be predictable due to their small chaotic range [8, 22, 51]. In contrast, high-dimensional chaotic maps have a more extensive chaotic range but are more complex and consequently more difficult to implement. Discovering some new sources of chaos with better randomness, higher unpredictability, larger chaotic range, and simple structure becomes the main focus of many security researchers [16, 17, 42, 43]. Borgia et al. [2] presented a new one-dimensional chaotic map using a combination of \(\sin \) and \(\arcsin \) functions and then apply the proposed map in the design of a real-time image encryption algorithm. The proposed encryption scheme has a fast speed,yet the used map is topologically conjugated with the logistic map, which has been proved to be easily predictible [23, 39]. In [15], the authors generated a new two-dimensional chaotic map by adjusting the logistic map using the sine map (2D-LASM) and then use the generated map to implement a new encryption system. Unfortunately, the proposed scheme was cryptanlyzed by Feng et al. [11]. In [45], Wang et al. proposed a new color image encryption scheme using a globally customized coupled map lattice. Although the system has a good security level, its encryption speed is too slow.

Motivated by these issues, we propose a new one-dimensional cosine polynomial 1-DCP chaotic map. This new map has a simple mathematical definition and exhibits a high chaotic behavior over an infinite range of its real control parameter values which makes it a perfect candidate for the design of image encryption schemes thanks to its simple structure, high chaotic behavior, and infinite chaotic range. We apply several chaos theory tests to demonstrate the good chaotic performances of the 1-DCP map. Besides, we design a novel efficient image encryption 1-DCPIE scheme based on the proposed map. In 1-DCPIE scheme, we raise the small encryption unit from the pixels level to the rows/columns level which highly increases the encryption speed. Furthermore, we adopt an alternative to the substitution permutation network architecture, where the permutation and substitution phases are merged. This new architecture makes the 1-DCPIE scheme able to withstand separate attacks [47] when only one encryption round is applied and enhances its security and speed. We perform several simulation tests, such as histogram, information entropy, secret key analysis, image sensitivity, and correlation analyses, to prove the high performances of 1-DCPIE.

The rest of the paper is organized as follows. Section 2 presents the 1-DCP chaotic map and evaluates its chaotic behavior. Section 3 describes in detail the proposed 1-DCPIE scheme. In Sect. 4, we simulate the 1-DCPIE scheme and evaluate its performances. Finally, Sect. 5 concludes the paper.

2 The proposed chaotic map

Here, we introduce a new one-dimensional cosine polynomial chaotic system (1-DCP) defined by the following equation (Eq. 1):

$$\begin{aligned} \left\{ \begin{array}{ll} f:&{}[-1;1] \longrightarrow [-1;1]\\ &{} x_{n+1}=f(x_{n})=\cos (\mu (x_n^3+x_n))\\ \end{array} \right. \end{aligned}$$
(1)

where \(\mu \) is a real control parameter. Since the new map is bounded by the even function cosine, the sign of \(\mu \) is not significant. Hence we ignore the negative values of \(\mu \). The 1-DCP map exhibits an extremely high chaotic behavior for most of \(\mu \) values. That makes its chaotic region as large as the infinite space of positive real numbers. Therefore, the 1-DCP map fits more the needs of cryptography in terms of large keyspace, complexity, and unpredictability. In the following section, we analyze the performances of the new map through several dynamical systems tests.

2.1 Bifurcation and trajectory analysis

The long-term behavior of a dynamical system (stability, instability, periodicity, or chaos) is visually described through the bifurcation diagram as a function of the parameter values. Figure 1 shows the bifurcation diagram of 1-DCP using different scales. The map first settles into one fixed point when \(0\le \mu <0.56\), and then it enters into a period-doubling phase for \(0.56\le \mu <1.05\), and finally gets into the chaos beyond the point \(\mu ^{*}=1.05\). A few numbers of periodic windows exist in the bifurcation diagram. Figure 2 projects the trajectory of 1-DCP in 2D and 3D phase space. It is seen from the phase diagram that the new map trajectory has a sinusoidal waveform where its periodicity tends to get smaller as the parameter \(\mu \) gets bigger. As a consequence, the trajectory of 1-DCP looks random-like for the high values of \(\mu \).

Fig. 1
figure 1

Bifurcation diagram of 1-DCP using different scales

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2.2 Approximate entropy

The approximate entropy (ApEn) [32, 33] is a statistical test to calculate the complexity and irregularity of dynamical systems. A positive value of the ApEn test reflects the absence of repetitive patterns among the generated orbits. A higher ApEn value means more complexity and unpredictability of the system. As shown in Fig. 3, the ApEn value of 1-DCP map is positive for \(\mu \in [0,1000]\) . Therefore, the generated time series by the new map present no repetitive patterns.Compared to other well-known chaotic maps, the 1-DCP ApEn values are mostly better than the Circle map ApEn values and are comparable to those obtained by the Chebychev map.

Fig. 2
figure 2

Phase diagram of 1-DCP using different values of \(\mu \)

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Fig. 3
figure 3

Approximate entropy analysis and comparison

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2.3 Initial state sensitivity

The sensitivity to the initial values, which is one of the most important characteristics of chaotic systems, is expressed when orbits with infinitesimally close initial values exponentially diverge after a finite number of iterations. The rate of divergence is usually quantified using the Lyapunov exponent(LE) [Eq. (2)] where a positive LE reflects the system’s sensitivity to the initial state and can be comprehended as the existence of a chaotic behavior if the system is bounded. In Fig. 4, we calculate the LE of 1-DCP using Wolf et al. [49] algorithm and plot it against the parameter \(\mu \). As shown in Fig. 4, the 1-DCP map has a large positive LE value for most of \(\mu \) values. In Comparison with some other chaotic maps, the LE of the proposed map is quite better than the LE of the Chebychev and Circle chaotic maps. Besides, to further analyze the sensitivity of the proposed map, we plot two orbits with a \(10^{-16}\) difference between their initial values in Fig. 5a and two orbits with a \(10^{-12}\) difference between their parameter values in Fig. 5b. As shown in these plots, the orbits diverge after only five or six iterations due to the map’s high chaotic behavior.

$$\begin{aligned} \lambda =\lim _{N\rightarrow \infty }\frac{1}{N}\sum _{n=0}^{N}\ln \mid f'(x_{n})\mid \end{aligned}$$
(2)
Fig. 4
figure 4

Lyapunov exponent analysis and comparison

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Fig. 5
figure 5

Sensitivity of 1-DCP to tiny changes At: a initial value \(x_{0}\); b parameter value \(\mu \)

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3 Image encryption scheme

Here, we propose a new fast image encryption scheme 1-DCPIE based on the 1-DCP chaotic map. Unlike most of the existing encryption schemes, we combine the permutation and the substitution stages to significantly increase the encryption speed by reducing the number of loops over the pixels. Besides, the combination of these two phases makes the new scheme able to withstand the separate attacks. Assuming a plain image P, the encryption process is divided into two modules: a row phase followed by a column phase. To encrypt the image rows, we iterate over the image rows \(P_i\) starting from the first to the last row wherein each iteration we encrypt the row \(P_i\) and another row \(P_{\mathrm{EP}(i)}\) indicated by a key stream EP. Indeed, we mask these two rows values using the predecessor row, a pseudo-random numbers sequences generated from the 1-DCP chaotic map, and the modulo operation. Then, we scramble the row \(P_i\) pixels using the circular shifting operation. Finally, we encrypt the columns by transposing the obtained image and reapplying the rows encryption phase. Figure 6 shows the encryption flowchart.

Fig. 6
figure 6

Flowchart of the proposed image encryption scheme 1-DCPIE

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3.1 The encryption steps

The detailed encryption steps of the proposed scheme are described as follows

(1):

Read the plain image P, round keys \(x_{ir},\mu _{ir}\) where \(i\in \lbrace 1,2,3,4\rbrace \), \(r\in \lbrace 1,2,\dots , T\rbrace \), and T is the total number of encryption rounds.

(2):

Set the round counter \(r=1\).

(3):

Set M as the total number of rows and N as the total number of columns.

(4):

Use the 1-DCP map to generate encryption positions sequence EP of length M using \(x_{1r}\) and \(\mu _{1r}\) as initial conditions and then apply Eq. 3 to normalize EP values.

$$\begin{aligned} \mathrm{EP}= \lbrace \mathrm{EP}_{i}|\mathrm{EP}_{i}=(\mathrm{EP}_{i}\times 10^{7})~\mathrm{mod}~M\rbrace \end{aligned}$$
(3)
(5):

Calculate \(x_{5r}\) using the following equation:

$$\begin{aligned} x_{5r}=x_{1r}+\mathrm{mean}(P/\lbrace P_{1},P_{\mathrm{EP}(1)}\rbrace ) \end{aligned}$$

where \(\mathrm{mean}(P/\lbrace P_{1},P_{\mathrm{EP}(1)}\rbrace )\) is the average pixels value of the image P excluding the first and the EP(1) row.

(6):

Generate sequences Y, Z of length N with the 1-DCP map using \(x_{3r},x_{5r}\) and \(\mu _{3r}\), respectively, as initial conditions. Then, apply Eqs. 4 and 5 to normalize the generated sequences.

$$\begin{aligned} Y= & {} \lbrace Y_{j}|Y_{j}=(Y_{j}\times 10^{7})~\mathrm{mod}~256\rbrace \end{aligned}$$
(4)
$$\begin{aligned} Z= & {} \lbrace Z_{j}|Z_{j}=(Z_{j}\times 10^{7})~\mathrm{mod}~256\rbrace \end{aligned}$$
(5)
(7):

Starting from the first to the last row, encrypt in each iteration two rows \(P_{i}\) and \(P_{\mathrm{EP}(i)}\) as follows.

$$\begin{aligned} \left\{ \begin{array}{l} P_{i}=\mathrm{circshift}((P_{i}+f(i)+\mathrm{pred}(i))~\mathrm{mod}~256,\mathrm{EP}(i))\\ P_{\mathrm{EP}(i)}=(P_{\mathrm{EP}(i)}+f(i)+\mathrm{pred}(i))~\mathrm{mod}~256\\ \end{array} \right. \end{aligned}$$

The \(\mathrm{circshift}(P_{i},n)\) function is a circular shifting operation of the row \(P_{i}\) to the right n times, and the \(f(i), \mathrm{pred}(i)\) functions are defined as follows

$$\begin{aligned} f(i)= \left\{ \begin{array}{ll} Y &{} ~\mathrm{if}~i\ne 1\\ Z &{} ~\mathrm{if}~i=1 \\ \end{array} \right. \mathrm{pred}(i)= \left\{ \begin{array}{ll} P_{M} &{} ~\mathrm{if}~i=1\\ P_{i-1} &{} ~\mathrm{else} \\ \end{array} \right. \end{aligned}$$
(8):

Transpose the obtained image matrix \(P=P^t\) to encrypt the columns and then repeat steps 3–7 using \(x_{2r}+\mathrm{mean}(\lbrace X_{ir}\rbrace )\), \(x_{4r},\mu _{2r},\mu _{4r}\) instead of \(x_{1r},x_{3r},\) \(\mu _{1r},\mu _{3r}\) where \(i \in \lbrace 1,2,3,4\rbrace \) and \(\mathrm{mean}(\lbrace X_{ir}\rbrace )\) is the average of round key parts.

(9):

Transpose back the cipher image \(P=P^t\).

(10):

Increment the rounds counter \(r=r+1\) and repeat steps 3–9 until \(r>T\).

3.2 The decryption steps

The decryption of cipher images is easily achieved by inverting the encryption steps as follows.

(1):

Read the cipher image P, round keys \(x_{ir},\mu _{ir}\) where \(i\in \lbrace 1,2,3,4\rbrace \), \(r\in \lbrace 1,2,\dots , T\rbrace \), and T is the total number of encryption rounds.

(2):

Set the round counter \(r=T\).

(3):

Transpose the cipher image matrix \(P=P^t\) to decrypt the columns.

(4):

Set M as the total number of rows and N as the total number of columns.

(5):

Use the 1-DCP map to generate encryption positions sequence EP of length M by setting \(x_{2r}+\mathrm{mean}(X_r)\) and \(\mu _{2r}\) as initial conditions and then apply Eq. 3 to normalize EP values.

(6):

Generate sequence Y of length N with the 1-DCP map using \(x_{4r}\) and \(\mu _{4r}\), respectively, as initial conditions. Then, apply Eq. 4 to normalize the obtained sequence.

(7):

Starting from the last to the second row, decrypt in each iteration two rows \(P_i\) and \(P_{\mathrm{EP}(i)}\) as follows.

$$\begin{aligned} \left\{ \begin{array}{l} P_{i}=(\mathrm{circshift}(P_{i},\mathrm{EP}(i))-Y-\mathrm{pred}(i))~\mathrm{mod}~256\\ P_{\mathrm{EP}(i)}=(P_{\mathrm{EP}(i)}-Y-\mathrm{pred}(i))~\mathrm{mod}~256\\ \end{array} \right. \end{aligned}$$
(8):

Calculate \(x_{5r}\) using the following equation:

$$\begin{aligned} x_{5r}=x_{1r}+\mathrm{mean}(P/\lbrace P_{1},P_{\mathrm{EP}(1)}\rbrace ) \end{aligned}$$
(9):

Generate sequence Z of length N with the 1-DCP map using \(x_{5r}\) and \(\mu _{4r}\), respectively, as initial conditions. Then, apply Eq. 5 to normalize the obtained sequence.

(10):

decrypt the first row \(P_1\) and the row \(P_{\mathrm{EP}(1)}\) using the following equation:

$$\begin{aligned} \left\{ \begin{array}{l} P_{1}=(\mathrm{circshift}(C_{1},\mathrm{EP}(1))-Z-P_M)~\mathrm{mod}~256\\ P_{\mathrm{EP}(1)}=(C_{\mathrm{EP}(1)}-Z-P_M)~\mathrm{mod}~256\\ \end{array} \right. \end{aligned}$$
(11):

Transpose the obtained image matrix \(P=P^t\) to decrypt the rows and then repeat steps 4–10 using \(x_{1r},x_{3r},\mu _{1r},\mu _{3r}\) instead of \(x_{2r}~+~\mathrm{mean}(\lbrace X_{ir}\rbrace )\), \(x_{4r},\mu _{2r},\) \(\mu _{4r}\).

(12):

Decrement the encryption rounds counter \(r=r-1\) and repeat steps 3–11 until \(r<1\).

4 Experimentation and security analysis

In this section, we perform a set of image encryption security tests to analyze the security and performances of the 1-DCPIE scheme. The experimentation environment is powered by an Intel I7-7700HQ processor, 16 GB RAM and MATLAB 2018. Besides, we choose the simulation plain images from the well-known ‘Miscellaneous’ USC-SIPI dataset.

4.1 Histogram analysis

In the statistical attacks, an illegitimate person can hack the scheme by discovering any eventual relation between plain and cipher images pixels’ intensity level. Therefore, the encryption scheme must produce cipher images with a uniform equal distribution. Figure 7 shows the pixels intensity levels histogram of plain/cipher images of grayscale and color ‘Lena’ image. We can notice that the histogram of the cipher images is flat and equally distributed. Hence, it does not leak any useful information to the attacker. Besides, we use the Chi-square test to prove the cipher image’ histogram uniformity further. The Chi-square is calculated as follows.

$$\begin{aligned} \chi ^2=\sum \limits _{i=0}^{255}\frac{(O_i-E_i)^2}{E_i} \end{aligned}$$

where i is the gray-level intensity, \(O_i\) is the observed occurrence frequency of the gray level i, and \(E_i\) is the expected occurrence frequency of the gray level i. Assuming a significance level \(\alpha =0.05\), the critical value for 8-bit grayscale image is equal to \(\chi ^2(255,0.05)=293.2478\). The Chi-square test value of the generated cipher image by 1-DCPIE is equal to 218.5039 which is lower than the critical value and hence implies that the generated cipher image has a uniform distribution.

Fig. 7
figure 7

Grayscale intensity histograms of plain and cipher ‘Lena’ images

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4.2 Information entropy analysis

The Shannon entropy [35] can quantify the information uncertainty in a given message m. In image encryption, we apply Shannon entropy to measure information randomness in the produced cipher image C. The theoretical ideal value of information entropy is equal to 8, and the closest is the resulted entropy to the ideal value, the better is the scheme security. The information entropy is calculated as follows

$$\begin{aligned} H(C)=-\sum _{i=0}^{255}p(C_{i})\log _2 p(C_{i}) \end{aligned}$$

where \(p(C_{i})\) is the occurrence frequency of the pixel value intensity i in the image C. We apply the entropy test 1000 times using random secret keys to encrypt ‘Lena’ image by the 1-DCPIE and some recent state of the art schemes, and then we report the results in Table 1. The results show that the new scheme produce cipher images with high entropy using only one encryption round.

Table 1 Entropy test comparison
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Table 2 Local Shannon entropy test results
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In a recent study, Wu et al. [50] proposed the local Shannon entropy (LSE) , which is a new information randomness test more strict and more suitable for image encryption. Unlike traditional Shannon entropy, the LSE test measures the average entropy value of k non-overlapping random local blocks. The LSE is calculated as follows

$$\begin{aligned} \overline{H_{k,T_{B}}}(C)=\sum _{i=1}^{k}\frac{H(C_{i})}{k} \end{aligned}$$

where \(C_{i},i\in \lbrace 1,2,\dots ,k\rbrace \) are k randomly selected blocks, \(T_{B}\) is the block size, and the function \(H(C_{i})\) calculates the entropy of block \(C_i\). According to the authors, the accepted interval of the LSE test for 8-bit grayscale images is [7.901901; 7.903037]. Table 2 shows the results of the LSE test of the 1-DCPIE and some other schemes applied to the USC-SIPI image database. As one can see, the new scheme achieves an acceptable LSE score in most of the dataset images.

4.3 Plain image sensitivity

The chosen plain text attacks (CPA) are very common and effective attacks against encryption schemes having low sensitivity to the plain image [6, 48, 57]. Hence, a secure image encryption scheme should always have an avalanche effect for minor changes in the plain images. To test the plain image sensitivity, we use the Number of Pixels Change Rate (NPCR) and the Unified Average Changing Intensity (UACI) measures. These tests quantitatively evaluate the difference between the cipher images C and \(C^{\prime }\) generated from two identical plain images having only a slight difference. The NPCR and UACI are defined as follows

$$\begin{aligned} \mathrm{NPCR}(C,C^{\prime })= & {} \frac{\sum _{i=1}^{M}\sum _{j=1}^{N}D_{ij}}{M\times N}\\ \mathrm{UACI}(C,C^{\prime })= & {} \frac{\sum _{i=1}^{M}\sum _{j=1}^{N}|C_{ij}-C_{ij}^{\prime }|}{N\times M\times F} \end{aligned}$$

where \(D_{ij}=0\) if \(C_{ij}=C^{\prime }_{ij}\) and \(D_{ij}=1\) if \(C_{ij}\ne C^{\prime }_{ij}\) and F is the biggest possible pixel value. According to Wu et al. [4], the NPCR and UACI values with regard to a significance level \(\alpha \) must be greater than \(N_{\alpha }^{*}\) for the NPCR and should belong to \([U_{\alpha }^{*-},U_{\alpha }^{*+}]\) for the UACI. The NPCR/UACI thresholds values are calculated as follows

$$\begin{aligned}&N_{\alpha }^{*}=\frac{F-\phi ^{-1}(\alpha )}{F+1}\times \sqrt{\frac{F}{M\times N}}\\&\left\{ \begin{array}{l} U_{\alpha }^{*-}=\mu _{u}+\phi ^{-1}(\frac{\alpha }{2})\times \sigma _{u}\\ U_{\alpha }^{*+}=\mu _{u}-\phi ^{-1}(\frac{\alpha }{2})\times \sigma _{u}\\ \end{array} \right. \end{aligned}$$

where

$$\begin{aligned} \mu _{u}=\frac{F+2}{3F+3}, ~\sigma _{u}^{2}=\frac{(F+2)(F^{2}+2F+3)}{18(F+1)^{2}\times F \times M \times N} \end{aligned}$$

and \(\phi ^{-1}(.)\) is the inverse cumulative density function. Table 3 contains the NPCR/UACI thresholds for different image sizes where the significance level is set to \(\alpha =0.05\). We apply the NPCR and UACI tests on the 1-DCPIE scheme, where we slightly modify one random pixel. Figures 8 and 9 show the NPCR/UACI scores obtained by the proposed scheme and some other encryption methods. Figures 10 and 11 show the NPCR and UACI values evolution as a function of the encryption round number r. Besides, since the proposed scheme uses the average function, we reapply the NPCR and UACI tests on ‘Lena’ image by adding one to a random pixel and subtracting one from another to get the same average value. The results are reported in Table 4. In comparison, we can notice that the proposed scheme has quite similar results as the other methods. However, the minimum number of encryption rounds is remarkably fewer.

Table 3 Expected NPCR and UACI values
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Fig. 8
figure 8

NPCR test results and comparison

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Fig. 9
figure 9

UACI test results and comparison

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Fig. 10
figure 10

NPCR evolution over encryption rounds number

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Fig. 11
figure 11

UACI evolution over encryption rounds number

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Table 4 NPCR/UACI values when changing two pixels
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Fig. 12
figure 12

NPCR and UACI analysis results of 1-DCPIE encryption secret key sensitivity

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Fig. 13
figure 13

a Plain ‘Lena’ image; decryption using slightly modified b \(x_{1}\); c \(x_{2}\); d \(x_{3}\); e \(x_{4}\); f \(\mu _{1}\); g \(\mu _{2}\); h \(\mu _{3}\); i \(\mu _{4}\); j original key

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4.4 Secret key analysis

In cryptography, the keyspace should be as large as possible to resist brute-force attacks where an illegitimate person can decrypt a cipher image by trying all the possible keys. In the 1-DCPIE scheme, we set the combination \(\lbrace x_{ir},\mu _{ir}\rbrace \) (\(i \in \lbrace 1,2,3,4\rbrace \), \(r\in \lbrace 1,2,\dots ,T\rbrace \) and T is the total number of encryption rounds). In the IEEE-754 standard [20], the precision of the double float type is equal to \(10^{-16}\). However, the precision of the (1-DCP) control parameter \(\mu \) is equal to \(10^{-12}\). Therefore, the 1-DCPIE secret keyspace is approximated as \((10^{16\times 4}\times 10^{12\times 4})^{T}\approx 2^{392\times T}\), which is largely sufficient to handle brute-force attacks [1, 36]. Moreover, the secret key must be sensitive to any minor modification to produce a totally different cipher image whenever it is altered. We apply the NPCR/UACI tests to quantify the difference between the output cipher images encrypted using almost identical secret keys. The obtained results of modifying any part in the key are all in the acceptable range of NPCR/UACI (see Fig. 12). Besides, in Fig. 13 we illustrate the results of key sensitivity in the decryption process where we slightly change a part of the key and try to decrypt the cipher image. We can notice that only the original key can decrypt the cipher image.

4.5 Speed analysis

Nowadays, many image encryption schemes achieve a sufficient security level. However, their encryption algorithms are too complicated, which mainly affect their speed and hence make them inapplicable in real communications. In the new scheme, we aim to increase the processing speed by significantly reducing the chaotic map use, raising the small encryption unit from the pixels level to the rows/columns level, and combining the substitution and permutation stages. In Table 5, we compare the theoretical and experimental speeds of the 1-DCPIE scheme and some recently published encryption schemes which have been simulated in the same running environment.

Table 5 Speed analysis test results (ms)
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4.6 Correlation analysis

The adjacent pixels values in the image data type are often close and highly correlated. This image characteristic is a big obstacle for cryptography where an attacker can try to hack the scheme by finding eventual relations between correlation in the plain and cipher images. Therefore, the encryption scheme must break any existing high correlation among adjacent pixels. To test the resistance of the new scheme against this type of attack, we choose 2000 random pixels in both plain and cipher images. Then, we calculate the correlation coefficient of the chosen pixels by the following equation:

$$\begin{aligned} \mathrm{corr}(x,y) = \frac{\mathrm{cov}(x,y)}{\sqrt{D(x)}\times \sqrt{D(y)}} \end{aligned}$$

where

$$\begin{aligned}&\mathrm{cov}(x,y)=\frac{1}{N}\sum _{i=1}^{N}(x_{i}-E(x))(y_{i}-E(y))\\&D(x)=\sum _{i=1}^{N}(x_{i}-E(x))^{2},~E(x)=\frac{1}{N} \sum _{i=1}^{N}x_{i} \end{aligned}$$

The test results are reported in Table 6. The correlation test results prove that the proposed scheme produces cipher images with no correlation.

Table 6 Correlation test analysis
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4.7 Interference resistance analysis

In reality, the transmitted data over communication canals are sometimes altered by the effect of noise interference signals or deliberately by attackers. Thus, some parts of the received cipher images can be lost. An effective encryption scheme must be able to decrypt the rest of the affected cipher image and retrieve the corresponding plain image. Figure 14 illustrates the capacity of 1-DCPIE to decrypt noisy cipher images.

Fig. 14
figure 14

Interference resistance analysis: a cipher image with \(5\%\) data loss; b cipher image with \(10\%\) data loss; c cipher image affected by \(2.5\%\) of ‘Salt & Pepper’ noise; d cipher image affected by \(5\%\) of ‘Salt & Pepper’ noise; e decryption of image a; f decryption of image b;g decryption of image c; h decryption of image d

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5 Conclusion

In this paper, we have developed a new one-dimensional chaotic (1-DCP) map defined by a simple iterative mathematical equation. The new chaotic map exhibits a very high chaotical behavior over a large interval of its positive real control parameter. Therefore, this map fits more the needs of cryptography, such as large keyspace, unpredictability, and speed than some predictable low-dimensional maps or slower high-dimensional maps. Through several analytical tests, we demonstrate the high chaotic behavior of the proposed map. Besides, we design a new efficient image encryption scheme based on the 1-DCP map. In this scheme, we merged the permutation and substitution stages to improve encryption speed and security. The experimental analysis results demonstrate that the 1-DCPIE scheme is more secure and faster than other recently proposed encryption schemes.