# Symmetric Image Encryption Algorithm Based on a New Product Trigonometric Chaotic Map

^{1}

^{2}

^{3}

^{*}

*Symmetry*)

## Abstract

**:**

## 1. Introduction

- (1)
- A new product trigonometric chaotic system is constructed. By means of various measures, the chaotic dynamic behavior of the system is analyzed in detail, and the good chaotic characteristics of the system are verified.
- (2)
- A novel symmetric image encryption scheme based on the novel product trigonometric chaotic system is proposed. The system consists of an efficient scrambling process and a secure diffusion operation. The secret keys are generated from the plaintext image by using SHA-256 to resist the chosen-plaintext attacks. Such that any slight change in the plaintext image will affect the whole ciphertext image.
- (3)
- The proposed scheme has been compared with some other recently proposed image encryption schemes. It is verified that the present work outperforms other previously published image encryption schemes and shows better cryptographic performance, while using less computational resources.

## 2. The New Chaotic Product Trigonometric Map

#### 2.1. The Sine Map System

_{n}denotes the state variable of the map at the n-th point of discrete time (n = 1, 2, …) and x

_{1}denotes the initial state value. u denotes the system parameter. The chaotic dynamic behavior of the sine map is similar to that of the logistic map, and its chaotic interval is relatively narrow. Figure 1a,b are the bifurcation diagram and Lyapunov exponent graph of sine map to the system parameter u, respectively, and the range of chaotic parameter is u ∈ [3.4610, 4]. The width of the parameter interval of chaotic behavior is Δu = 0.5390, but there are still some narrow periodic windows in this interval.

#### 2.2. The Proposed Chaotic Product Trigonometric Map

#### 2.2.1. Bifurcation and Lyapunov Exponent Diagram of PTM System

#### 2.2.2. Approximate Entropy and Permutation Entropy of PTM System

#### 2.2.3. The Time-Series and Cobweb Graph of PTM System

^{−12}. The results of Figure 4a show that the evolution of system state values is very sensitive to the initial values, and the slight difference of the initial values causes a great separation between two adjacent orbits.

_{0}= 0.11. From Figure 4b, one can see that the system traverses unlimited non-repetitive chaotic orbits, which proves the existence of a chaotic behavior of system (2) more intuitively.

#### 2.2.4. The NIST Test of PTM System

_{0}= 0.2345 and the system parameter u = 3.9999 to generate a chaotic real number sequence with a length of 10

^{9}, then convert it into a binary pseudo-random sequence with a length of 10

^{9}bits, and divide the sequence into 1000 groups, with each group having a length of 10

^{6}bits for NIST test. Here, the algorithm for converting a chaotic real number x into an 8-bit binary number uIntx is shown in Algorithm 1. Specifically, given a chaotic real number x, we transformed the real value of x to a 64-bit binary string, following the IEEE 754 double precision floating point number standard. Then, the binary digital numbers from 33-th to 40-th in each binary string were sampled as the output of Algorithm 1. Thus, each of the chaotic outputs generates an 8-bit binary numbers.

Algorithm 1: Convert a chaotic real number x into an 8-bit binary number uIntx. |

Input: A real number x |

Output: A byte digital uIntx |

1: Convert the x to a 64-bit binary string b_{1}b_{2}…b_{64} following the IEEE 754 standard; |

2: Intercept 8 digits of the binary x to form unsigned integer: uIntx ← b_{33}b_{34}…b_{40}; |

3: Output the unsigned integer uIntx in binary format. |

## 3. Image Encryption and Decryption Algorithm

#### 3.1. The Encryption Algorithm

**P**to produce a hash value in hexadecimal digit string for the plaintext image. The string is composed of 64 hexadecimal digital symbols, and its shape is as follows: $h={h}_{1}{h}_{2}\dots {h}_{64}$.

_{i}) represents the ASCII value of the character h

_{i}.

_{0}, y

_{0}, z

_{0}} produced in Step 3 as the initial value together with the parameter u for the chaotic map to output three pseudo-random integer number sequences X = {x(i)}, Y = {y(j)}, and Z = {z(l)}, respectively, by the pseudo-random integer number generator (PRING). Where, i = 1, 2, …, M; j = 1, 2, …, N; l = 1, 2, …, L = M × N. x(i) ∈ {1, 2, …, M} and x(i) ≠ x(i’) if i ≠ i’. y(j) ∈ {1, 2, …, N} and y(j) ≠ y(j’) if j ≠ j’. z(l) ∈ {1, 2, …, L} and z(l) ≠ z(l’) if l ≠ l’. Algorithm 2 explains the detailed steps of the pseudo-random integer number generator (PRING).

Algorithm 2: Generating a pseudo random integer number sequence. |

Input: x_{0}, u, k, integer M |

Output: A pseudorandom integer number sequence X with length of M |

1: Initialize: flag ← zeros(1, M); X ← zeros(1, M); x ← x_{0}; |

2: Circularly generate M mutually different integers: 1, 2, …, M. |

for i ← 1: M do |

x ← u/4*sin(2π/k*x)*cos(π/k*x); |

j ← mod(floor(x*10^12), M)+1; |

while (flag(j) = 1) do |

x ← u/4*sin(2π/k*x)*cos(π/k*x); |

j ← mod(floor(x*10^12), M)+1; |

end while |

X(i) ← j; flag(j) ← 1; |

end for i |

3: Output X = {X(i)} |

#### 3.2. The Decryption Algorithm

_{0}, y

_{0}, z

_{0}} by using Equation (3a–c).

_{0}, y

_{0}, z

_{0}} as initial values and parameter u for the PTM system (2) to produce pseudo-random number sequences X, Y, and Z.

## 4. Security Analysis and Simulation Results

_{0}, y

_{0}, z

_{0}, u). The simulation was carried out on the Matlab R2021b platform running on a computer with Intel Core i7-9700 @ 3.00GHz processor, 16 GB memory and Windows 10 operating system. In our simulation tests, the secret key parameters {x

_{0}, y

_{0}, z

_{0}} were generated with the plaintext image to be encrypted, and u was set as 5.167.

#### 4.1. Encryption Effect

#### 4.2. Key Space Analysis

_{0}, y

_{0}, z

_{0}} were generated with the 256 bit plaintext image hash value, the parameter u was a double precision real number, which had 15 significant digits after the decimal point, and the total key space was 2

^{256}× 10

^{15}> 2

^{305}. At present, a cryptosystem is secure when the secret key space is larger than or equal to 2

^{100}. Hence, the secret key space of the proposed scheme was large enough to meet the safety requirements.

#### 4.3. Histogram Analysis

_{i}represents the observed occurrence frequency of the i-th level gray; and E

_{i}represents the expected ideal occurrence frequency of the i-th level gray. For a significance level α = 0.05, the critical value for 8-bit gray scale image (I = 256) is equal to χ

^{2}(255, 0.05) = 293.2478. The encrypted images should have a value lower than the critical value 293.2478. We applied the test on some images and their encrypted images, and the experimental results are listed in Table 2.

#### 4.4. Information Entropy

_{i}) is the probability of assurance of instance S

_{i}. For an 8-bit gray image, each pixel value is a random variable, and there are 256 possible values. If the probability of occurrence of each value is equal, then H(S) = 8. Generally speaking, the entropy of the actual image is always less than the ideal value of 8. Therefore, the closer the entropy is to 8, the better the image encryption effect. Table 3 lists the information results of this paper, and lists some comparative results. Therefore, the image encrypted by this method has a very ideal entropy value, and the multi-value is higher than other methods, which indicates that it has better cryptographic performance than the others.

#### 4.5. Correlation Coefficients between Consecutive Pixels

_{i}, y

_{i}) represent a pair of gray values of two adjacent pixels in the image and ${N}_{xy}$ represents the number of total pairs of randomly selected pixels from the image. Some test images were tested, and the experimental results are listed in Table 4, which also lists some comparison results. Compared with the data reported in the literature, this algorithm achieved satisfactory results.

#### 4.6. Resistance to Differential Attacks

#### 4.7. Time Performance Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Ahmad, J.; Masood, F.; Shah, S.A.; Jamal, S.S.; Hussain, I. A Novel Secure Occupancy Monitoring Scheme Based on Multi-Chaos Mapping. Symmetry
**2020**, 12, 350. [Google Scholar] [CrossRef] [Green Version] - Masood, F.; Boulila, W.; Ahmad, J.; Arshad; Sankar, S.; Rubaiee, S.; Buchanan, W.J. A Novel Privacy Approach of Digital Aerial Images Based on Mersenne Twister Method with DNA Genetic Encoding and Chaos. Remote Sens.
**2020**, 12, 1893. [Google Scholar] [CrossRef] - Buell, D. Modern Symmetric Ciphers—DES and AES. In Fundamentals of Cryptography; Morgan Kaufmann: Burlington, MA, USA, 2021; pp. 123–147. [Google Scholar] [CrossRef]
- Hua, Z.; Zhou, Y.; Huang, H. Cosine-transform-based chaotic system for image encryption. Inf. Sci.
**2019**, 480, 403–419. [Google Scholar] [CrossRef] - Khan, J.S.; Ahmad, J. Chaos based efficient selective image encryption. Multidimens. Syst. Signal Process.
**2019**, 30, 943–961. [Google Scholar] [CrossRef] - Lu, Q.; Yu, L.; Zhu, C. A New Conservative Hyperchaotic System-Based Image Symmetric Encryption Scheme with DNA Coding. Symmetry
**2021**, 13, 2317. [Google Scholar] [CrossRef] - Zhu, S.; Deng, X.; Zhang, W.; Zhu, C. A New One-Dimensional Compound Chaotic System and Its Application in High-Speed Image Encryption. Appl. Sci.
**2021**, 11, 11206. [Google Scholar] [CrossRef] - Masood, F.; Ahmad, J.; Shah, S.A.; Jamal, S.S.; Hussain, I. A Novel Hybrid Secure Image Encryption Based on Julia Set of Fractals and 3D Lorenz Chaotic Map. Entropy
**2020**, 22, 274. [Google Scholar] [CrossRef] [Green Version] - Zhu, S.; Wang, G.; Zhu, C. A Secure and Fast Image Encryption Scheme based on Double Chaotic S-Boxes. Entropy
**2019**, 21, 790. [Google Scholar] [CrossRef] [Green Version] - Shannon, C.E. Communication theory of secrecy systems. Bell Syst. Tech. J.
**1949**, 28, 656–715. [Google Scholar] [CrossRef] - Ouannas, A.; Khennaoui, A.A.; Wang, X.; Pham, V.-T.; Boulaaras, S.; Momani, S. Bifurcation and chaos in the fractional form of Hénon-Lozi type map. Eur. Phys. J. Spec. Top.
**2020**, 229, 2261–2273. [Google Scholar] [CrossRef] - Ouannas, A.; Khennaoui, A.A.; Oussaeif, T.-E.; Pham, V.-T.; Grassi, G.; Dibi, Z. Hyperchaotic fractional Grassi–Miller map and its hardware implementation. Integration
**2021**, 80, 13–19. [Google Scholar] [CrossRef] - Khennaoui, A.A.; Ouannas, A.; Boulaaras, S.; Pham, V.-T.; Azar, A.T. A fractional map with hidden attractors: Chaos and control. Eur. Phys. J. Spec. Top.
**2020**, 229, 1083–1093. [Google Scholar] [CrossRef] - Liu, L.; Miao, S. A new simple one-dimensional chaotic map and its application for image encryption. Multimed. Tools Appl.
**2018**, 77, 21445–21462. [Google Scholar] [CrossRef] - Li, Y.; Li, X.; Liu, X. A fast and efficient hash function based on generalized chaotic mapping with variable parameters. Neural Comput. Appl.
**2016**, 28, 1405–1415. [Google Scholar] [CrossRef] - Yu, W.; Yu, T. Analysis of chaotic characteristics of trigonometric function system. Mod. Phys. Lett. B
**2020**, 34, 2050210. [Google Scholar] [CrossRef] - Elghandour, A.N.; Salah, A.M.; Elmasry, Y.A.; Karawia, A.A. An Image Encryption Algorithm Based on Bisection Method and One-Dimensional Piecewise Chaotic Map. IEEE Access
**2021**, 9, 43411–43421. [Google Scholar] [CrossRef] - Gopalakrishnan, T.; Ramakrishnan, S. Image Encryption Using Hyper-chaotic Map for Permutation and Diffusion by Multiple Hyper-chaotic Maps. Wirel. Pers. Commun.
**2019**, 109, 437–454. [Google Scholar] [CrossRef] - Zahmoul, R.; Ejbali, R.; Zaied, M. Image encryption based on new Beta chaotic maps. Opt. Lasers Eng.
**2017**, 96, 39–49. [Google Scholar] [CrossRef] - Alawida, M.; Samsudin, A.; Teh, J.S.; Alkhawaldeh, R.S. A new hybrid digital chaotic system with applications in image encryption. Signal Process.
**2019**, 160, 45–58. [Google Scholar] [CrossRef] - Nepomuceno, E.G.; Nardo, L.G.; Arias-Garcia, J.; Butusov, D.N.; Tutueva, A. Image encryption based on the pseudo-orbits from 1D chaotic map. Chaos
**2019**, 29, 061101. [Google Scholar] [CrossRef] - Mansouri, A.; Wang, X. A novel one-dimensional sine powered chaotic map and its application in a new image encryption scheme. Inf. Sci.
**2020**, 520, 46–62. [Google Scholar] [CrossRef] - Huang, H.; Yang, S.; Ye, R. Efficient symmetric image encryption by using a novel 2D chaotic system. IET Image Process.
**2020**, 14, 1157–1163. [Google Scholar] [CrossRef] - Askar, S.; Karawia, A.; Al-Khedhairi, A.; Al-Ammar, F. An Algorithm of Image Encryption Using Logistic and Two-Dimensional Chaotic Economic Maps. Entropy
**2019**, 21, 44. [Google Scholar] [CrossRef] [Green Version] - Khan, J.S.; Boulila, W.; Ahmad, J.; Rubaiee, S.; Rehman, A.U.; Alroobaea, R.; Buchanan, W.J. DNA and Plaintext Dependent Chaotic Visual Selective Image Encryption. IEEE Access
**2020**, 8, 159732–159744. [Google Scholar] [CrossRef] - Lu, Q.; Zhu, C.; Deng, X. An Efficient Image Encryption Scheme Based on the LSS Chaotic Map and Single S-Box. IEEE Access
**2020**, 8, 25664–25678. [Google Scholar] [CrossRef] - Zhu, S.; Zhu, C. Security Analysis and Improvement of an Image Encryption Cryptosystem Based on Bit Plane Extraction and Multi Chaos. Entropy
**2021**, 23, 505. [Google Scholar] [CrossRef] - Zhu, S.; Zhu, C. An efficient chosen-plaintext attack on an image fusion encryption algorithm based on DNA operation and hyperchaos. Entropy
**2021**, 23, 804. [Google Scholar] [CrossRef] - Stoyanov, B.; Kordov, K. Image Encryption Using Chebyshev Map and Rotation Equation. Entropy
**2015**, 17, 2117–2139. [Google Scholar] [CrossRef] - Yan, X.; Wang, X.; Xian, Y. Chaotic image encryption algorithm based on arithmetic sequence scrambling model and DNA encoding operation. Multimed. Tools Appl.
**2021**, 80, 10949–10983. [Google Scholar] [CrossRef]

**Figure 1.**Bifurcation diagram and Lyapunov exponent of sine map system. (

**a**) Bifurcation of system state versus parameter u; (

**b**) the graph of Lyapunov exponent versus parameter u.

**Figure 2.**Bifurcation diagram and Lyapunov exponent of the PTM chaotic system. (

**a**) Bifurcation of system state versus parameter u; (

**b**) Lyapunov exponent curve with parameter u. (

**c**) Bifurcation scene with parameter k; (

**d**) the graph of Lyapunov exponent versus parameter k.

**Figure 3.**Comparison of ApEn and PeEn of sequences generated by two systems. (

**a**) The approximate entropy (ApEn) and (

**b**) permutation entropy (PeEn).

**Figure 4.**Time-sequences and cobweb chart of PTM system. (

**a**) Two time-series with two initial values have slight differences (x

_{0}= 0.23 and y

_{0}= 0.23 + 10

^{−12}); (

**b**) the cobweb graph.

**Figure 6.**The standard test images and their encrypted ones. (

**a**) The plaintext image cameraman. (

**b**) The plaintext image peppers. (

**c**) The plaintext all-white image. (

**d**) The plaintext all-black image. (

**e**) The encrypted image cameraman. (

**f**) The encrypted image peppers. (

**g**) The encrypted all-white image. (

**h**) The encrypted all-black image.

**Figure 7.**Plaintext/encrypted images and their histograms. (

**a**) Plaintext image lena. (

**b**) Histogram of (

**a**). (

**c**) Encrypted image lena. (

**d**) Histogram of (

**c**). (

**e**) Plaintext image mandrill. (

**f**) Histogram of (

**e**). (

**g**) Encrypted image mandrill. (

**h**) Histogram of (

**g**). (

**i**) Plaintext image boat. (

**j**) Histogram of (

**i**). (

**k**) Encrypted image boat. (

**l**) Histogram of (

**k**).

**Figure 8.**Horizontal, vertical, and diagonal correlation point diagrams of plaintext image Peppers and encrypted image Peppers. (

**a**) Original Peppers horizontal; (

**b**) Original Peppers vertical; (

**c**) Original Peppers diagonal; (

**d**) Encrypted Peppers horizontal; (

**e**) Encrypted Peppers vertical; adn (

**f**) Encrypted Peppers diagonal.

NIST Statistical Test Item | p-Value | Pass Rate | Results |
---|---|---|---|

Frequency (monobit) | 0.763677 | 986/1000 | passed |

Block Frequency (m = 128) | 0.745908 | 990/1000 | passed |

Cumulative Sums (Forward) | 0.984881 | 988/1000 | passed |

Cumulative Sums (Reverse) | 0.599693 | 987/1000 | passed |

Runs | 0.195864 | 993/1000 | passed |

Longest Run of Ones | 0.820143 | 985/1000 | passed |

Rank | 0.016149 | 994/1000 | passed |

FFT | 0.014754 | 987/1000 | passed |

Non-Overlapping Templates (m = 9, B = 000000001) | 0.711601 | 990/1000 | passed |

Overlapping Templates (m = 9) | 0.953089 | 986/1000 | passed |

Universal | 0.410055 | 991/1000 | passed |

Approximate Entropy (m = 10) | 0.725829 | 987/1000 | passed |

Random-Excursions (X = −4) | 0.663542 | 628/631 | passed |

Random-Excursions Variant (X = −9) | 0.422753 | 622/631 | passed |

Serial Test 1 (m = 16) | 0.877083 | 996/1000 | passed |

Serial Test 2 (m = 16) | 0.848027 | 989/1000 | passed |

Linear complexity (M = 500) | 0.329850 | 992/1000 | passed |

Images | χ^{2} of Plaintext Images | χ^{2} of Encrypted Images (This Work) | χ^{2} of Encrypted Image (Ref. [17]) |
---|---|---|---|

Lena (256 × 256) | 3.0666 × 10^{4} | 217.8984 | 230.1484 |

Cameraman (256 × 256) | 1.1097 × 10^{5} | 219.4609 | 234.3047 |

Lena (512 × 512) | 1.5802 × 10^{5} | 249.7266 | 239.7539 |

Cameraman (512 × 512) | 4.1853 × 10^{5} | 261.8965 | 278.0410 |

Barbara (512 × 512) | 9.5552 × 10^{4} | 227.0996 | 253.9297 |

Boat (512 × 512) | 3.8397 × 10^{5} | 207.9766 | 246.9434 |

Mandrill (512 × 512) | 2.1137 × 10^{5} | 241.0781 | 245.0137 |

Image Name | Image Size | Ours | Ref. [20] | Ref. [29] | Ref. [30] |
---|---|---|---|---|---|

5.1.10 | 256 × 256 | 7.99665 | 7.99720 | 7.99717 | 7.99680 |

5.1.11 | 256 × 256 | 7.99717 | 7.99730 | 7.96999 | 7.99710 |

5.1.12 | 256 × 256 | 7.99727 | 7.99540 | 7.99757 | 7.99730 |

5.1.13 | 256 × 256 | 7.99707 | 7.99630 | 7.99735 | 7.99680 |

5.1.14 | 256 × 256 | 7.99711 | 7.99730 | 7.99674 | 7.99690 |

5.2.08 | 512 × 512 | 7.99931 | 7.99920 | 7.99934 | 7.99920 |

5.2.09 | 512 × 512 | 7.99930 | 7.99900 | 7.99930 | 7.99940 |

5.2.10 | 512 × 512 | 7.99928 | 7.99870 | 7.99926 | 7.99930 |

7.1.01 | 512 × 512 | 7.99932 | 7.99800 | 7.99929 | 7.99930 |

7.1.02 | 512 × 512 | 7.99944 | 7.99490 | 7.99931 | 7.99930 |

7.1.03 | 512 × 512 | 7.99935 | 7.99830 | 7.99925 | 7.99940 |

7.1.04 | 512 × 512 | 7.99926 | 7.99850 | 7.99923 | 7.99940 |

7.1.05 | 512 × 512 | 7.99924 | 7.99880 | 7.99929 | 7.99930 |

7.1.06 | 512 × 512 | 7.99932 | 7.99900 | 7.99933 | 7.99930 |

7.1.07 | 512 × 512 | 7.99924 | 7.99870 | 7.99931 | 7.99910 |

7.1.08 | 512 × 512 | 7.99930 | 7.99880 | 7.99923 | 7.99920 |

7.1.09 | 512 × 512 | 7.99926 | 7.99850 | 7.99219 | 7.99920 |

Elaine | 512 × 512 | 7.99926 | 7.99930 | 7.99922 | 7.99930 |

5.3.01 | 1024 × 1024 | 7.99985 | 7.99930 | 7.99983 | 7.99980 |

5.3.02 | 1024 × 1024 | 7.99977 | 7.99920 | 7.99981 | 7.99990 |

Testpat | 1024 × 1024 | 7.99983 | 7.98470 | 7.99982 | 7.99980 |

Algorithm | Image Name | Horizontal | Vertical | Diagonal |
---|---|---|---|---|

This work | 5.1.10 | 0.001403 | 0.000645 | −0.002410 |

Ref. [29] | 5.1.10 | −0.002971 | −0.000897 | 0.003682 |

Ref. [30] | 5.1.10 | −0.007100 | 0.008500 | 0.000200 |

This work | 5.1.11 | −0.010029 | 0.002503 | −0.000842 |

Ref. [29] | 5.1.11 | 0.001757 | −0.010444 | 0.001124 |

Ref. [30] | 5.1.11 | −0.004800 | −0.001700 | 0.006800 |

This work | 5.1.12 | −0.000273 | −0.000764 | −0.001682 |

Ref. [29] | 5.1.12 | 0.009575 | −0.002502 | −0.000582 |

Ref. [30] | 5.1.12 | 0.005500 | −0.004900 | 0.000100 |

This work | 5.1.13 | 0.002899 | −0.000105 | −0.001305 |

Ref. [29] | 5.1.13 | 0.000347 | 0.004691 | −0.009999 |

Ref. [30] | 5.1.13 | 0.003800 | 0.002500 | 0.003200 |

This work | 5.1.14 | 0.004723 | 0.000035 | −0.000283 |

Ref. [29] | 5.1.14 | 0.008773 | −0.011971 | 0.000220 |

Ref. [30] | 5.1.14 | 0.000400 | 0.000400 | 0.001200 |

This work | 5.2.08 | −0.001405 | −0.002724 | 0.0009704 |

Ref. [29] | 5.2.08 | −0.002389 | −0.003528 | −0.003059 |

Ref. [30] | 5.2.08 | 0.004100 | 0.001400 | 0.000054 |

This work | 5.2.09 | −0.003732 | 0.002767 | 0.000471 |

Ref. [29] | 5.2.09 | 0.000783 | −0.003316 | −0.000207 |

Ref. [30] | 5.2.09 | −0.001700 | −0.001800 | −0.001900 |

This work | 5.2.10 | 0.003098 | −0.001703 | −0.001175 |

Ref. [29] | 5.2.10 | −0.006168 | −0.007614 | 0.000369 |

Ref. [30] | 5.2.10 | 0.000007 | 0.002100 | 0.001200 |

This work | 7.1.01 | 0.001635 | −0.001531 | 0.000747 |

Ref. [29] | 7.1.01 | −0.002843 | 0.000667 | 0.004116 |

Ref. [30] | 7.1.01 | −0.000100 | 0.001300 | −0.001300 |

This work | 7.1.02 | 0.002013 | 0.000773 | −0.000288 |

Ref. [29] | 7.1.02 | −0.003666 | −0.001386 | −0.001295 |

Ref. [30] | 7.1.02 | 0.000900 | 0.001600 | 0.005700 |

This work | 7.1.03 | 0.000500 | 0.000885 | −0.003690 |

Ref. [29] | 7.1.03 | −0.002931 | −0.004124 | 0.003147 |

Ref. [30] | 7.1.03 | 0.000100 | 0.000200 | 0.003100 |

This work | 7.1.04 | 0.000826 | −0.000919 | −0.001786 |

Ref. [29] | 7.1.04 | −0.004028 | −0.001065 | −0.000901 |

Ref. [30] | 7.1.04 | −0.001400 | 0.000811 | −0.003100 |

This work | 7.1.05 | −0.002312 | 0.001432 | 0.001277 |

Ref. [29] | 7.1.05 | 0.001735 | −0.003046 | −0.002081 |

Ref. [30] | 7.1.05 | −0.002400 | −0.000700 | 0.003400 |

This work | 7.1.06 | 0.001373 | 0.001590 | −0.005810 |

Ref. [29] | 7.1.06 | −0.001395 | −0.003363 | −0.001516 |

Ref. [30] | 7.1.06 | 0.000832 | 0.001700 | 0.001800 |

This work | 7.1.07 | −0.002871 | −0.000073 | 0.000955 |

Ref. [29] | 7.1.07 | −0.000608 | 0.000682 | −0.000090 |

Ref. [30] | 7.1.07 | 0.003900 | 0.002100 | 0.002500 |

This work | 5.3.01 | −0.000665 | 0.000425 | 0.000494 |

Ref. [29] | 5.3.01 | 0.000606 | 0.000090 | 0.002417 |

Ref. [30] | 5.3.01 | 0.000400 | 0.002600 | 0.001200 |

This work | 5.3.02 | −0.000417 | −0.000375 | −0.000678 |

Ref. [29] | 5.3.02 | 0.000502 | 0.001669 | −0.000435 |

Ref. [30] | 5.3.02 | −0.000377 | −0.000474 | −0.000301 |

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**MDPI and ACS Style**

Lu, Q.; Yu, L.; Zhu, C.
Symmetric Image Encryption Algorithm Based on a New Product Trigonometric Chaotic Map. *Symmetry* **2022**, *14*, 373.
https://doi.org/10.3390/sym14020373

**AMA Style**

Lu Q, Yu L, Zhu C.
Symmetric Image Encryption Algorithm Based on a New Product Trigonometric Chaotic Map. *Symmetry*. 2022; 14(2):373.
https://doi.org/10.3390/sym14020373

**Chicago/Turabian Style**

Lu, Qing, Linlan Yu, and Congxu Zhu.
2022. "Symmetric Image Encryption Algorithm Based on a New Product Trigonometric Chaotic Map" *Symmetry* 14, no. 2: 373.
https://doi.org/10.3390/sym14020373