The generation method of the key stream and the structure of the algorithm determine the security of the cryptosystem. The classical chaotic map has simple dynamic behavior and few control parameters, so it is not suitable for modern cryptography. In this paper, we design a new 2D hyperchaotic system called 2D simple structure and complex dynamic behavior map (2D-SSCDB). The 2D-SSCDB has a simple structure but has complex dynamic behavior. The Lyapunov exponent verifies that the 2D-SSCDB has hyperchaotic behavior, and the parameter space in the hyperchaotic state is extensive and continuous. Trajectory analysis and some randomness tests verify that the 2D-SSCDB can generate random sequences with good performance. Next, to verify the excellent performance of the 2D-SSCDB, we use the 2D-SSCDB to generate a keystream for color image encryption. In the encryption algorithm, the encryption algorithm scrambles and diffuses simultaneously, increasing the cryptographic system’s security. The horizontal correlation, vertical correlation, and diagonal correlation of ciphertext are −0.0004, −0.0004 and 0.0007, respectively. The average information entropy of the ciphertext is 7.9993. In addition, the designed encryption algorithm reduces the correlation between the three channels of the color image. Security analysis shows that the color image encryption algorithm designed using 2D-SSCDB has good security, can resist standard attack methods, and has high efficiency.
Image is an essential carrier of information. Due to the openness of the network environment, digital images are inevitably subject to various illegal attacks during the transmission process, and essential information in images will be stolen [
Because of the large amount of image data and the strong correlation between adjacent pixels, traditional advanced encryption standard (AES), triple data encryption algorithm (3DES), and other methods are not suitable for image encryption, these methods cannot reduce the correlation of adjacent pixels in the image, and the efficiency is very slow [
The chaos system is divided into a one-dimensional chaotic system, a two-dimensional chaotic system, and a high-dimensional chaotic system. A one-dimensional chaotic system has the characteristics of simple structure and fast generation of keystream [
In order to balance the performance of the high-dimensional and one-dimensional chaotic systems, it is the best choice to use a 2D chaotic system to generate a key stream. Many 2D chaotic systems have been proposed. Although they can exhibit excellent performance, they are pretty complex, and the parameter space in the hyperchaotic state is discontinuous [
Furthermore, to verify the practicality of the 2D-SSCDB, we propose a color image encryption algorithm based on the 2D-SSCDB. In the encryption algorithm, the secret key of the cryptosystem is generated from the plaintext. In the encryption stage, scrambling and diffusion are carried out simultaneously, which increases the algorithm’s security, and the attacker needs to break the scrambling and diffusion operations simultaneously. The experimental results verify that the 2D-SSCDB can be well applied in chaotic image encryption.
In this paper, a 2D hyperchaotic system with simple structure, complex dynamic behavior and continuous chaotic parameter space is proposed, called 2D-SSCDB. The 2D-SSCDB is derived from Sine map, Logistic map, the defined of the 2D-SSCDB is shown in
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
The Lyapunov exponent analysis of the 2D-SSCDB is shown in
At
In order to generate chaotic sequences with excellent performance, the range of parameters we choose in the cryptosystem is
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
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
Number | Statistical test | ||||||||
---|---|---|---|---|---|---|---|---|---|
Result | Result | Result | Result | ||||||
1 | Longest run of ones | 0.455937 | Success | 0.699313 | Success | 0.574903 | Success | 0.419021 | Success |
2 | Overlapping template matching | 0.171867 | Success | 0.911413 | Success | 0.213309 | Success | 0.983453 | Success |
3 | Random excursions variant | 0.253551 | Success | 0.148094 | Success | 0.122325 | Success | 0.949602 | Success |
4 | Rank | 0.991468 | Success | 0.616305 | Success | 0.616305 | Success | 0.383827 | Success |
5 | Frequency | 0.419021 | Success | 0.023545 | Success | 0.015598 | Success | 0.085587 | Success |
6 | Universal | 0.911413 | Success | 0.983453 | Success | 0.045675 | Success | 0.657933 | Success |
7 | Random excursions | 0.739918 | Success | 0.804337 | Success | 0.911413 | Success | 0.804337 | Success |
8 | Block frequency | 0.779188 | Success | 0.616305 | Success | 0.494392 | Success | 0.066882 | Success |
9 | Cumulative sums | 0.657933 | Success | 0.534146 | Success | 0.122325 | Success | 0.191687 | Success |
10 | Runs | 0.851383 | Success | 0.350485 | Success | 0.383827 | Success | 0.051942 | Success |
11 | Serial | 0.816537 | Success | 0.779188 | Success | 0.534146 | Success | 0.383827 | Success |
12 | Spectral | 0.191687 | Success | 0.911413 | Success | 0.816537 | Success | 0.383827 | Success |
13 | Approximate entropy | 0.699313 | Success | 0.494392 | Success | 0.534146 | Success | 0.739918 | Success |
14 | Non-overlapping template matching | 0.534146 | Success | 0.883171 | Success | 0.779188 | Success | 0.108791 | Success |
15 | Linear complexity | 0.108791 | Success | 0.153763 | Success | 0.350485 | Success | 0.616305 | Success |
In order to explore the application of the 2D-SSCDB to image encryption, we propose a color image encryption algorithm based on 2D-SSCDB, which we call simple structure-color image encryption (SS-CIE). The SS-CIE is a symmetric encryption algorithm, and the decryption process is the reverse process of encryption. The SS-CIE is an encryption algorithm that performs scrambling and diffusion at the same time. This design structure increases the security of the algorithm.
The plaintext image is
Input:
Output:
Step 1: Connect the three channels of the plaintext in turn to obtain a new plaintext
Step 2: Get an initial fine-tuning key
Step 3: Bring
Step 4: Randomly give the initial key of the cryptosystem,
Step 5: Let
Step 6: Generate the encryption matrix of the cryptosystem
Step 7: Set a sorting function
According to the sorting function, generate two sorting matrices
Step 8: The encryption process is described as,
The decryption algorithm is the inverse process of the encryption algorithm. The decryption algorithm is described as follows.
Input:
Output:
Step 1: Convert ciphertext
Step 2: According to the secret keys
Step 3:The decryption process is described as follows,
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
Key analysis is divided into key sensitivity analysis and key space analysis. The secret key of SS-CIE are
When the key space of the algorithm exceeds
In order to test the key sensitivity of the image encryption algorithm in this paper, the key is changed
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
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
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
Image | Plaintext | Ciphertext | |||||
---|---|---|---|---|---|---|---|
H | V | D | H | V | D | ||
Lena | R | 0.9798 | 0.9893 | 0.9697 | −0.0012 | −0.0004 | −0.0002 |
G | 0.9689 | 0.9824 | 0.9554 | 0.0012 | −0.0014 | 0.0005 | |
B | 0.9325 | 0.9574 | 0.9181 | −0.0023 | −0.0023 | −0.0002 | |
Goldhil | R | 0.9783 | 0.9722 | 0.9583 | 0.0005 | −0.0006 | 0.0016 |
G | 0.9815 | 0.9831 | 0.9699 | −0.0006 | 0.0012 | 0.0013 | |
B | 0.9837 | 0.9840 | 0.9723 | −0.0014 | −0.0010 | 0.0005 | |
White | R | 1 | 1 | 1 | −0.0022 | −0.0013 | 0.0020 |
G | 1 | 1 | 1 | 0.0012 | −0.0001 | 0.0015 | |
B | 1 | 1 | 1 | −0.0029 | 0.0011 | −0.0014 | |
Black | R | 1 | 1 | 1 | 0.0023 | −0.0012 | 0.0002 |
G | 1 | 1 | 1 | 0.0006 | 0.00006 | −0.0016 | |
B | 1 | 1 | 1 | −0.0003 | 0.0003 | 0.0046 | |
Average | 0.9854 | 0.9890 | 0.9786 | −0.0004 | −0.0004 | 0.0007 |
Algorithms | SS-CIE | Algorithm in [ |
Algorithm in [ |
Algorithm in [ |
---|---|---|---|---|
Horizontal | −0.0082 | −0.0013 | 0.0057 | |
Vertical | −0.0128 | 0.0004 | 0.0061 | |
Diagonal | −0.0012 | 0.0078 | −0.0031 |
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
Ours | Image | ( |
( |
( |
---|---|---|---|---|
Plaintext | Lena | 0.8785 | 0.6763 | 0.9105 |
Goldhil | 0.9390 | 0.9000 | 0.9736 | |
White | 1 | 1 | 1 | |
Black | 1 | 1 | 1 | |
Ciphertext | Lena | −0.0020 | 0.0021 | 0.0017 |
Goldhil | −0.0018 | −0.0001 | 0.0026 | |
White | −0.0012 | −0.0023 | −0.00041 | |
Black | −0.0032 | −0.0037 | 0.0004 |
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,
The information entropy analysis of SS-CIE is shown in
Image | Plaintext | Ciphertext | |
---|---|---|---|
Lena | R | 7.2531 | 7.9993 |
G | 7.5952 | 7.9993 | |
B | 6.9686 | 7.9993 | |
Goldhil | R | 7.6101 | 7.9996 |
G | 7.5544 | 7.9996 | |
B | 7.5540 | 7.9996 | |
White | R | 0 | 7.9992 |
G | 0 | 7.9992 | |
B | 0 | 7.9993 | |
Black | R | 0 | 7.9992 |
G | 0 | 7.9992 | |
B | 0 | 7.9992 | |
Average | 3.71128 | 7.9993 |
Algorithms | SS-CIE | Algorithm in [ |
Algorithm in [ |
Algorithm in [ |
---|---|---|---|---|
Information entropy | 7.9895 | 7.9973 | 7.9979 |
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. [
Image | NPCR (%) | PASS/NO PAASS | UACI (%) | PASS/NO PAASS | |
---|---|---|---|---|---|
Lena | R | 99.6040 | PASS | 33.4707 | PASS |
G | 99.6101 | PASS | 33.4394 | PASS | |
B | 99.6063 | PASS | 33.4214 | PASS | |
Goldhil | R | 99.6281 | PASS | 33.3890 | PASS |
G | 99.6250 | PASS | 33.4765 | PASS | |
B | 99.6021 | PASS | 33.4699 | PASS | |
White | R | 99.6219 | PASS | 33.4023 | PASS |
G | 99.6109 | PASS | 33.4161 | PASS | |
B | 99.6253 | PASS | 33.3908 | PASS | |
Black | R | 99.6231 | PASS | 33.4272 | PASS |
G | 99.6067 | PASS | 33.3784 | PASS | |
B | 99.6116 | PASS | 33.4496 | PASS |
There will be noise attacks and clipping attacks in the transmission of ciphertext. The effect of SS-CIE against clipping attack is shown in
The running environment of the SS-CIE is Windows 10, matlab 2020, i3-10105F. The efficiency analysis is shown in
Algorithms | SS-CIE | Algorithm in [ |
Algorithm in [ |
Algorithm in [ |
---|---|---|---|---|
Time/s | 1.0930 | 19.1400 | 8.0130 | 4.2120 |
In this paper, a new 2D hyperchaotic system is proposed, called 2D-SSCDB. The 2D-SSCDB has a simple structure and can generate complex dynamic behavior. Through Lyapunov exponent, phase diagram analysis, NIST test, sensitivity analysis, and comparative analysis, it is verified that the 2D-SSCDB has good performance and can generate random numbers with better performance. In order to verify the practicability of the 2D-SSCDB, combined with the 2D-SSCDB system, we propose a color image encryption algorithm. Scrambling and diffusion are carried out at the same time in this algorithm. Through key analysis, information entropy analysis, statistical analysis, and other methods, it is verified that the cryptographic system has high security and can resist common attack methods. The 2D-SSCDB is a better candidate for keystream generation based on chaotic image encryption.