2.1. Josephus Problem

The Josephus problem is a counting-out problem, and it originated from the story of the famous Jewish historian Josephus [26]. When the Jews were invaded by the Romans, Josephus and his friends were hiding in a hole with 39 Jews. These 39 Jews did not want to be caught by the enemies, so they decided to suicide. They stood in a cycle and reported numbers one by one. The man who reported the number 3 must suicide, then restart counting from the next person. However, Josephus and his friends did not want to die. How to be the last two people to be recycled? Josephus carefully calculated and placed him and his friends in the 16th and 31st positions. Finally, they escaped from the death game.

The Josephus problem is described as follows: rearranging M elements into a circle. Then, we repeat looping the circle by deleting the Nth element and restart counting by the N + 1th element [24, 27]. We repeat doing these operations until the last element is selected. The Josephus problem is expressed as a function f (M, N). For example, the solution of the function f(8, 3) is to rearrange the elements {1, 2, 3, 4, 5, 6, 7, 8} in a circle, then loop the circle, and delete the 3rd element. In this Josephus problem, the elements that are sequentially deleted in the circle are {3, 6, 1, 5, 2, 8, 4, 7}. There is another solution to these problems. Still taking f(8, 3) as an example, first we rearrange the elements {1, 2, 3, 4, 5, 6, 7, 8} into a one-dimensional sequence; then, the total of these elements is 8, we calculate mod (3, 8) = 3 and delete the 3rd element which is 3. Then, we rearrange the remaining elements into a new sequence {4, 5, 6, 7, 8, 1, 2}, because the total of the elements in the new sequence is 7, and we calculate mod (3, 7) = 3 and delete the 3rd element in the new sequence which is 6. Repeat doing these operations until the last element is selected. By using this method, the Josephus problem can be solved quickly.

In order to increase its diversity, some scholars had added the starting point S to the original rule and expanded the Josephus function as f(M, N, S). This method can choose the starting point in the Josephus circle, then the Josephus problem is more interesting. On the basis of the expanded Josephus problem, Guo et al. added the cycle direction and the number space to the Josephus problem and extended the Josephus function to f(M, N, S, D, L) [28]. The parameter D is the cycle direction. When D = 1, it represents a clockwise loop will be done, and when D = −1, it represents a counterclockwise loop will be done. The parameter L is the number space, and it represents report one number every L counts.

This paper continues to improve the Josephus problem based on the above expansion methods and proposes a variable-step Josephus problem. This method combines the Josephus problem with the chaotic system and expands the parameter N into a pseudorandom sequence $$. When traversing the Josephus circle, the parameter N in the ith looping is N = $$. Because the elements in sequence N′ have pseudorandomness, they can be expanded infinitely, and the solution of the Josephus problem is greatly increased. For example, when N′ = {1, 2, 3, 4, 5, 6, 7, 8}, the result of the function f(8, {1, 2, 3, 4, 5, 6, 7, 8}, 1, 1, 0) is {1, 3, 6, 4, 5, 2, 7, 8}.

The proposed method can also be used to expand parameters D and L, and it will not be described here. Finally, this expanded Josephus problem will have a solution method in M!

2.2. Hyperchaotic System

3.1. Key Generator

Here, $$ are the given initial values. After the initial values are obtained, the chaotic system is iterated and the values of the first 1000 iterations are discarded to remove the transient effect. Then, four pseudorandom sequences X, Y, Z, and W for scrambling and permutation are obtained.

3.2. Scrambling Method

Scrambling is a commonly used method for changing the position of the pixels. It can be applied to encrypt the sequence with a certain rule. The scrambling method used in this paper is matching the position of the elements in the original sequence with the position of the elements in the pseudorandom sequence s = {s₁, s₂, s₃, …, s_M} of equal length, and then the new sequence $$ is obtained by rearranging the sequence s = {s₁, s₂, s₃, …, s_M} under a certain rule. The original sequence is scrambled to a new sequence according to this rule and the encrypted sequence is obtained. The flowchart of the scrambling process is shown in Figure 2. The decryption process of the scrambling method is the inverse process of encryption, and it will not be described again.

In this paper, the scrambling method is used to scramble the pixel position and the bit position. In this pixel position scrambling method, the parameter N′ in Josephus problem is generated by the hyperchaotic system and the index sequence $$ is obtained. The pixel sequence is scrambled by the rule of s = {1, 2, 3, …, M}⟶$$. In the bit position scrambling method, the bit position in pixel is disordered by the fixed step size Josephus problem. The pixels in the image can be converted to 8-bit binary numbers. So, the total number M is 8, and the index sequence generated by the Josephus problem is s′ = f(8, N). Finally, the bits of the pixel are scrambled by the rules of s = {1, 2, 3, 4, …, 8}⟶$$.

3.3. Pixel Permutation

This paper uses a 2-bit binary number addition and subtraction method to perform a permutation operation on the pixels. In the 2-bit binary numbers, there is a special addition and subtraction rule regardless of the carry and borrow in the addition and subtraction process. Only the last two binary digits in the operation result are retained. For example, ‘10’ + ‘11’ = ‘01’ and ‘01’ − ‘10’ = ‘11’. There are 32 possible cases when using this addition and subtraction rules. These 32 cases are shown in Table 1.

When permuting two pixels with the 2-bit binary number addition and subtraction rules, the pixels should first be converted into two 8-bit binary numbers, and then the 8-bit binary numbers are split into 2-bit binary numbers. For example, using 225 and 108 for two-bit binary number addition and subtraction, first 225 and 101 should be converted into 8-bit binary digits “11100001” and “01101100”, respectively, and then the two numbers are split into 2-bit binary numbers for addition. The result of the calculation is “00001101”. We convert this number into a decimal number, and the result is 13. The two-bit binary addition and subtraction operations are reciprocal process.

3.4. Encryption Process

The flowchart of the encryption process is shown in Figure 3. The decryption process of the encryption algorithm is the inverse process of encryption, which will not be described again.

4.1. Key Analysis

4.1.3. Decryption Key Sensitivity Analysis

The sensitivity of the key is more obvious in the decryption process. When the key is changed slightly, the decrypted image is significantly different from the original image. The plain cameraman image and decrypted images when the keys change slightly are shown in Figure 5. Generally, NPCR, UACI, MSE, and PSNR are used to measure the differences between these images. MSE (mean square error) and PSNR (peak signal-to-noise ratio) are two indicators to measure the similarity of two images. In general, when MSE ≤ 30 dB, there is no significant difference between the two images. When MSE > 30, the closer the MSE value is to 30, the smaller the difference between the two images. PSNR reflects the magnitude of image distortion. The larger the PSNR, the smaller the image distortion. MSE (mean square error) and PSNR (peak signal-to-noise ratio) are shown in the following equations:

$$ (9)

$$ (10)

When the key changes slightly, the indicators of the difference between the decrypted image and the original image are shown in Table 3. By comparison, the algorithm has strong sensitivity to key decryption. When the key changes slightly, the image cannot be decrypted.

Table 3. Key sensitivity of decryption process.

Initial values	NPCR	UACI	MSE	PSNR
x1 + 10−10	99.5895	33.9695	11341	7.5841
y1 + 10−10	99.5819	33.2737	11509	7.5201
z1 + 10−10	99.5560	33.4204	11582	7.4928
w1 + 10−10	99.6200	33.4349	115891	7.4903

4.2. Differential Attack Analysis

The differential attack analysis refers to making minor changes to the original image and encrypting it, then analyzing the cipher image and analyzing its sensitivity to plaintext. NPCR (number of pixels change rate) and UACI (unified average changing intensity) are used to measure the ability of resisting differential attacks. Table 4 lists the values of NPCRs and UACIs between the cipher images and the cipher images when the original image changes by 1 bit. The data in Table 4 are close to the theoretical values. This reflects that there is a strong correlation between the cipher image encrypted by this encryption algorithm and the original image. Even if the original image changes slightly by 1 bit, the cipher image encrypted by this encryption algorithm will undergo a thorough change. This algorithm is effective against differential attacks.

4.3. Histogram Analysis

The histogram is an indicator of statistics in the image, which reflects the total pixel numbers of each value in the image [30]. The histograms of the original cameraman 256 × 256 image and brain 256 × 256 image and their cipher images are listed in Figure 6. By comparing and analyzing the histograms of the original images and the cipher images, it can be seen that the distribution of the pixel values of the original images is relatively centralized, which has certain statistical characteristics and has no resistance to brute force attacks. However, the pixels of cipher image are distributed more uniform, which breaks the rule of distribution of pixels and does not have statistical characteristics. The attacker cannot use statistical characteristics to restore the original information of the image, so it can resist statistical attacks well.

The histogram obeys the chi-square distribution with 255 degrees of freedom. Given a significant level of α, make $$, namely, $$, accept the hypothesis. When a significant levels α = 0.01, 0.05, and 0.1, there are $$. The chi-square distribution of some images is shown in Table 5. The commonly used significant level is α = 0.05, and all the cipher images in Table 5 passed the test. By comparison, it can be seen that the algorithm greatly changes the histogram distribution of the image and has a good ability to break the statistical characteristics of the plain images.

4.4. Information Entropy Analysis

Table 6 lists some information entropies of the images encrypted by the proposed encryption algorithm. By comparison, the cipher images have good information entropies, which is close to the ideal value of 8, and the cipher images have good randomness.

In formula (13), k represents the total blocks. S₁, S₂,…,S_k are nonoverlapping blocks with T_B pixels randomly selected in the image, and H(S_i) means the entropy of the chosen pixels in S_i. When k = 30 and T_B = 1936, the results of the local Shannon entropy are shown in Table 7. It can be seen from the results that the cipher image possesses high randomness.

4.5. Correlation Analysis

For the original images, the values of adjacent pixels in most regions are very close. The correlation between the values of adjacent pixels in the image is very strong. Figure 7 shows the correlation analysis of the cameraman plain image and cipher image in three directions. Breaking the strong correlation between adjacent pixels is of great significance to resist statistical attacks. The calculation method of correlation coefficient between adjacent pixels is shown in formula (14).

We randomly selected 5000 pairs of pixels to calculate the correlation coefficients of the original image and cipher image in horizontal, vertical, and diagonal directions. The statistical results are shown in Table 8. Statistical results show that in the original image, the correlations between randomly selected pixels are very strong, while in the cipher image, the correlation coefficients between pixels are close to 0. The proposed algorithm can better disturb the correlations between pixels, so it can better resist statistical attacks.

4.6. Noise Attack Analysis

In the process of information transmission, it is often interfered by various kinds of noise, resulting in signal distortion. Common noise interferences include Gaussian noise, Poisson noise, and salt and pepper noise. The recovery ability of cipher disturbed by noise is one of the standards to evaluate the performance of encryption algorithm. Here, the salt and pepper noise with intensity of 0.01, 0.05, and 0.1 are interfered to the cipher, and the decrypted images are shown in Figure 8.

Correlations, NPCR, UACI, MSE, and PSNR are used to measure the recovery degree of the algorithm suffered from noise attack, and the details are shown in Table 9. According to the data in Table 9, the encryption algorithm has a strong recovery ability against the noise interference and can resist the noise attack very well.

4.7. Data Loss Attack Analysis

Data loss attacks are attacks that intercept cipher images and delete some data. A certain amount of data are lost after the cipher image is attacked. If the recovery ability of the decryption algorithm is bad, the decrypted image of the cipher image after losing information cannot provide enough effective information. The test and analysis of data loss attacks refers to deleting some pixels of cipher image, then comparing, and analyzing the decrypted image with the original image through corresponding decryption algorithm, and its statistic recovery degree. Figure 9 shows the cipher images and corresponding decrypted images after data loss attacks.

Correlations, MSE, PSNR, NPCR, and UACI are as indicators to measure the similarity of the two images. Table 10 lists the indicators of decrypted cameraman images after data loss attacks. Through comparison, it can be seen that the algorithm has recovery ability when it is attacked by data loss and has resistance ability to data loss attacks.

4.8. Classical Attack Analysis

There are four classical types of attacks: cipher-only attack (COA), known-plaintext attack (KPA), chosen-cipher attack (CCA), and chosen-plaintext attack (CPA). Among these four classical types of attacks, the COA refers to the attack performed under the cipher is known; the KPA refers to the attack under attacker masters a certain plaintext and the corresponding cipher; the CCA refers to when the attacker uses this encryption algorithm to construct the cipher corresponding to plaintext; the CPA refers to when the attacker knows the plaintext corresponding to a certain number of ciphers. In these four attack methods, if the encryption algorithm can effectively resist CPA, it must be able to resist the other three classical types of attack methods.

Attacker mostly use special image, such as white and all black to attack the image encryption methods and find the key. In this proposed algorithm, the scramble process and the pixel permutation process all related to the hyperchaotic system, and the initial values of the hyperchaotic system are very sensitive to the plain images; this has been proven in the key analysis and differential attack analysis, and so this algorithm can resist the CPA very well. The proposed algorithm can effectively resist the four classic attacks. Figure 4 shows the encryption results of all-black and all-white images of size 256 × 256. Tables 6 and 8 show the information entropy and three kinds of correlation of images.

4.9. Performance Comparison Analysis

This section lists and analyzes the security analysis of some image encryption algorithms in the references. The comparison analysis of the literature results can illustrate the security of the encryption algorithm very well. Table 11 lists some security analysis indicators of the encryption algorithm. It can be shown in Table 11 that the proposed encryption algorithm has good encryption performance and can be applied in the field of image encryption.

4.10. Randomness Test of Cipher Image

SP800 is a series of guidelines on information security issued by the National Institute of Standards and Technology (NIST). In the NIST standard series of documents, although NIST SP is not a formal statutory standard, in actual work, it has become a de facto standard and authoritative guide widely recognized by the United States and the international information security community. This paper selects 14 test methods given in SP800-22 Revision la to test the random characteristics of cipher images encrypted by the proposed algorithm. The test results of the 14 test indicators and cipher random performance are shown in Table 12. When the test result is greater than 0.01, it means that the test object is random. Therefore, the image encrypted with this encryption algorithm has good random performance.

4.11. Complexity Analysis

4.12. Test Images of USC-SIPI Miscellaneous Database

In order to further verify the performance of this scheme. The proposed scheme has been tested on the USC-SIPI miscellaneous database. Performance evaluations include differential attack, correlation coefficients, and information entropy. The results are shown in Table 14.