The Lyapunov exponent is one of the most effective methods to test whether the nonlinear dynamical system is in chaotic or hyperchaotic state. If a two-dimensional chaotic system has two positive Lyapunov exponents, the system is hyperchaotic in this parameter space. Hyperchaotic behavior has more complex dynamical behavior compared to chaotic behavior. The calculation formula of Lyapunov exponent is shown in Eq. (2), (2)λ=limT→+∞1T∑t=0T|f′(xt)|.

The Lyapunov exponent analysis of the 2D-SSCDB is shown in Fig. 1, and compare the Lyapunov exponent with Logistic Map and cascade modulation couple with two 1D-chaotic map (2D-SCMCI) [39].

At η=2, the 2D-SSCDB exhibits weak dynamic behavior when μ≤6.921, the 2D-SSCDB exhibits strong dynamic behavior when μ>6.921. At η=5, the 2D-SSCDB presents a chaotic state, and the behavior of hyperchaotic states alternately appears. With η increasing gradually, the period window gradually disappears. At η=7.2, the period window disappears completely, when μ<2.151, the 2D-SSCDB presents a chaotic state, and when μ≥2.151, the 2D-SSCDB presents a hyperchaotic state. Thereafter, the 2D-SSCDB exhibits stable kinetic behavior with η increasing gradually. Compared with Logistic Map, the 2D-SSCDB has more complex dynamic behavior. Compared with the 2D-SCMCI, the parameter space of the 2D-SSCDB in hyperchaotic state is continuous. Therefore, the 2D-SSCDB has better expressiveness.

In order to generate chaotic sequences with excellent performance, the range of parameters we choose in the cryptosystem is η≥7.2, and μ≥3. At this time, the state of 2D-SSCDB is hyperchaotic, which means that the sequence generated by it has excellent randomness.

Phase diagram analysis describes the trajectory of a nonlinear dynamical system. The larger the area occupied by the phase diagram, the better the chaotic performance of the nonlinear dynamical system. The phase diagram analysis of the 2D-SSCDB is shown in Fig. 2. The initial value are x0=0.465651321, and y0=0.32131654. Phase diagram analysis shows that the 2D-SSCDB occupies a larger space compared to 2D-SCMCI [39] and 2D-SLMM [40]. Therefore, the 2D-SSCDB has better ergodicity and can take all the values of the phase space.

The NIST statistical test suite was used to test the randomness of the sequences generated by the 2D-SSCDB.

NIST contains 15 tests, and when the P-value is greater than 0.01, the random sequence passes the test. The NIST test of the 2D-SSCDB is shown in Table 1, where x0=0.465651321 and y0=0.32131654. The parameters for the 2D-SSCDB are η=7.2,μ=7.2 and η=15,μ=8.9. NIST tests show that the sequences generated by the 2D-SSCDB have good randomness.

The plaintext image is P∈Mm×n×3, the red channel of the plaintext is PR∈Mm×n, the green channel of the plaintext is PG∈Mm×n, and the blue channel of the plaintext is PE∈Mm×n. The algorithm of the SS-CIE is described as follows,

Input: P (PR, PG, PG)

Output: C (C∈Mm×n×3)

Step 1: Connect the three channels of the plaintext in turn to obtain a new plaintext P∈Mm×3n, P=[PR,PG,PB].

Step 2: Get an initial fine-tuning key k by P, where k=∑i=1m∑j=13nP(i,j)/1010×π. If k=0, then we set k=100.

Step 3: Bring k into the Logistic iteration for 50 times to get the final fine-tuning key k1(k1=z50) by Eq. (3), (3)zn+1=3.99999×zn×(1−zn),n=1,2,3,…,50,z0=k.

Step 4: Randomly give the initial key of the cryptosystem, x0 (x0∈(0,1)), y0 (y0∈(0,1)), μ (μ∈[3,+∞)) and η (η∈[7.2,+∞)). Set the new keys which are xn0=x0+k1, yn0=y0+k1, μn=μ+k1, and ηn=η+k1.

Step 5: Let xn0, yn0, μn and ηn be the initial values and parameters of the 2D-SSCDB, iteratively generate chaotic sequences are X and Y, where X∈M1×3mn and Y∈M1×3mn. It should be noted that the generated sequence needs to discard the initial sequence, so that the generated sequence is sufficiently chaotic, and the number of discarded sequences set in this paper is 300.

Step 6: Generate the encryption matrix of the cryptosystem D (D∈Mm×3n), D=floor(X×1010)mod256.

Step 7: Set a sorting function A=f(a,L), which can truncate the first L sequences from a one-dimensional vector a. Sort the intercepted sequence, and find the position of the sorted sequence in the original sequence, and return as A.

According to the sorting function, generate two sorting matrices S1 and S2, where S1=f (Y,m) and S2=f (Y(m+3n:end),3n).

Step 8: The encryption process is described as, (1)

C(S1(1),S2(1))=(P(1,1)+D(S1(1),S2(1)))mod256.

The decryption algorithm is the inverse process of the encryption algorithm. The decryption algorithm is described as follows.

Input: C (C∈Mm×n×3), x0, y0, μ, η, and k1

Output: P (P∈Mm×n×3)

Step 1: Convert ciphertext C to new ciphertext C∈Mm×n×3→C∈Mm×3n.

Step 2: According to the secret keys x0, y0, μ, η, and k1, the encryption matrix D of the cryptosystem can be generated, and two sorting matrices S1 and S2 are generated.

Step 3:The decryption process is described as follows, (1)

P(1,1)=(C(S1(1),S2(1))−D(S1(1),S2(1)))mod256

The encryption simulation experiment is carried out using the image encryption algorithm based on chaos theory proposed in this paper.

Taking Lena as an example, the image encryption and decryption results are shown in Fig. 3. Taking Black as an example, the image encryption and decryption results are shown in Fig. 4. The initial keys for SS-CIE are set to x0=0.465651321, y0=0.32131654, μ=7.2, and η=7.2. Visual analysis shows that the SS-CIE is visually secure, and the ciphertext is transformed into an unrecognizable noise image.

Key analysis is divided into key sensitivity analysis and key space analysis. The secret key of SS-CIE are x0 (x0∈(0,1)), y0 (y0∈(0,1)), μ (μ∈[3,+∞)), η (η∈[7.2,+∞)), and k1, if the calculation accuracy of the computer is 10−15, the secret key space of x0, y0, μ, η, and k1 are 1015, 1015, 1015×1015, 1015×1015, and 1015. The secret key space of SS-CIE is keyspace=1015×1015×(1015×1015)×(1015×1015)×1015=10105≈2348.

When the key space of the algorithm exceeds 2100, the algorithm is considered to be resistant to brute force attacks. So the SS-CIE can resist brute force attacks.

In order to test the key sensitivity of the image encryption algorithm in this paper, the key is changed 10−15, and the simulation results are shown in Fig. 5. The initial keys for SS-CIE are set to x0=0.465651321, y0=0.32131654, μ=7.2, and η=7.2. Key sensitivity analysis shows that the SS-CIE is sensitive to keys, and the decryption system also has good sensitivity.

The histogram can be used to visually see the distribution characteristics of the pixel value of an image. Taking Goldhill as an example, the histogram analysis of SS-CIE is shown in Fig. 6.

Histogram analysis shows that the ciphertext histogram distribution of SS-CIE is uniform, which means that the ciphertext pixel value distribution is random, and the attacker cannot obtain any useful information from the ciphertext, so SS-CIE can Resist statistical attacks.

Generally, the correlation between adjacent pixels of the raw unprocessed image is high. We hope that the correlation performance between adjacent pixels of the image will be reduced after encryption, so that the ciphertext image cannot be cracked through the correlation analysis of adjacent pixels. Taking Goldhill as an example, the correlation analysis of the SS-CIE is shown in Figs. 7 and 8.

When the adjacent correlation is small, the correlation image shows a divergent state. When the adjacent correlation is large, the image presents an aggregated state. Therefore, visually, the ciphertext of SS-CIE has less correlation. To further verify the accuracy of the correlation, the calculation formula of the correlation is (4)rρ=cov(x,y)D(x)⋅D(y).

Table 2 is the correlation analysis results of SS-CIE. And compared with some classic algorithms (Refs. [43–45]), the comparison results are shown in Table 3.

The correlation coefficient of the plaintext image is large, and the correlation is very strong. However, the correlation coefficient of adjacent pixels of the ciphertext image is very small, and the correlation is low. It shows that after the encryption of the algorithm in this paper, the ciphertext image can well resist the attack of statistical analysis. In addition, the correlation comparison with other algorithms shows that the SS-CIE is more resistant to statistical analysis.

Using the method of correlation analysis to analyze the correlation between the three channels, a safe algorithm can not only reduce the correlation between adjacent pixels, but also reduce the correlation between the three channels of the plaintext image. The correlation analysis of the three channels is shown in Table 4.

The experimental results show that the SS-CIE effectively reduces the correlation between the three channels, and the attacker cannot infer the information of the remaining channels through the ciphertext value of one channel, indicating that the algorithm has good security.

The amount of information contained in an image can be reflected by information entropy. The greater the information entropy, the better the encryption effect of the algorithm and the more hidden information. The calculation formula of information entropy is, (5)H=∑i=0255p(gi)log2⁡1p(gi).

The information entropy analysis of SS-CIE is shown in Table 5, and the information entropy comparison results with other algorithms are shown in Table 6 (Refs. [43–45]).

The information entropy analysis shows that the plaintext image carries a lot of information, while the ciphertext image obtained by the SS-CIE has a small amount of information (the information entropy is close to 8). The ciphertext image presents a random noise image. The information entropy comparison results show that the ciphertexts of SS-CIE have better randomness and less information, so the SS-CIE is more resistant to statistical attacks, and the algorithm has better security.

A secure algorithm is sensitive to the plaintext. Even if the plaintext changes slightly, two distinct ciphertexts will still be obtained. NPCR and UACI are two indicators to evaluate the ability of the algorithm to resist differential attack. Wu et al. proposed in the Ref. [46] that when the value of NPCR exceeds 99.5893%, the value of UACI is between 33.3730% and 33.5541%, indicating that the algorithm can resist differential attacks. The ability of SS-CIE to resist differential attacks is shown in Table 7. Differential attack analysis shows that SS-CIE has excellent resistance to differential attacks, and the attacker cannot crack the algorithm by analyzing the characteristics of the two ciphertexts.

There will be noise attacks and clipping attacks in the transmission of ciphertext. The effect of SS-CIE against clipping attack is shown in Fig. 9. The effect of SS-CIE against noise attack is shown in Fig. 10. Noise attack and clipping attack show that SS-CIE has good robustness.

The running environment of the SS-CIE is Windows 10, matlab 2020, i3-10105F. The efficiency analysis is shown in Table 8. Efficiency analysis shows that SS-CIE requires about 1.0930 s to encrypt color images with 512 * 512. Compared with other algorithms, SS-CIE has higher efficiency. SS-CIE not only has high security, but also runs efficiently.