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A nonlinear multi-image encryption scheme is proposed by combining the reality-preserving discrete fractional angular transform with the deoxyribonucleic acid sequence operations. Four approximation coefficients of the four images are extracted by performing the two-dimensional lifting wavelet transform. Then, the four approximation coefficients are synthesized to generate a real-valued output with the reality-preserving discrete fractional angular transform. Finally, based on the deoxyribonucleic acid operation and the Logistic-sine system, the real-valued intermedium output will be encrypted to yield the final ciphertext image. To enhance the security of the image encryption algorithm, the initial value of the chaotic system is calculated by the 256-bit binary sequence, which is obtained by taking the statistics information of the plaintext images as the input of SHA-256. Deoxyribonucleic acid sequence operations, as nonlinear processes, could help to improve the robustness of the cryptosystem. Simulation results and security analysis demonstrate the effectiveness of the image encryption algorithm and the capability of withstanding various common attacks.

Images can cover plentiful information and play an essential part in information transmission or processing. To enhance the security of private data in the rapidly developing digital age, more and more people focus on how to design an image encryption algorithm to prevent the leakage of original private information. In the past, many image encryption algorithms have been proposed with different techniques [

In recent years, chaos-based image encryption algorithms have turned into one of the hot topics for their great dynamical performances. Image encryption algorithms combining mathematical transform with chaotic maps have been proposed successively for better security. Derived from one-dimensional (1D) chaotic maps, the Sine and Logistic maps, Hua et al. described the 2D Logistic-sine map and its extension version, the 2D Logistic-adjusted-sine map, to encrypt images for higher robustness and lower time complexity [

Scrambling and diffusion operations in transform domains are effective tools for image encryption, such as the fractional Mellin transform (FrMT) [

However, image encryption schemes in [

The rest of this paper is arranged as follows. In Section

The Logistic-sine system consisting of the Logistic map and the Sine map can be expressed as [

The DFrAT is derived from the discrete fractional Fourier transform and discrete fraction random transform [

The RPDFrAT is defined according to the method of deriving the reality-preserving forms from the fractional transform [

If

where

The RPDFrAT is defined as

There are four nucleobases in a DNA sequence: T (thymine), C (cytosine), A (adenine), and G (guanine). According to the principle of complementary bases pairing, A and

DNA encoding rules.

Rule | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|

00 | A | A | T | T | G | G | C | C |

01 | C | G | C | G | T | A | T | A |

10 | G | C | G | C | A | T | A | T |

11 | T | T | A | A | C | C | G | G |

According to the binary operation rules, the addition, subtraction, and XOR operation rules of the DNA sequences can be realized. For DNA encoding rule 1, the three operation rules of DNA sequences are shown in Tables

DNA addition rule for the DNA encoding rule 1.

+ | T | A | C | G |
---|---|---|---|---|

T | G | T | A | C |

A | T | A | C | G |

C | A | C | G | T |

G | C | G | T | A |

DNA subtraction rule for the DNA encoding rule 1.

– | T | A | C | G |
---|---|---|---|---|

T | A | T | G | C |

A | C | A | T | G |

C | G | C | A | T |

G | T | G | C | A |

DNA XOR rule for DNA encoding rule 1.

XOR | A | G | C | T |
---|---|---|---|---|

A | A | G | C | T |

G | G | A | T | C |

C | C | T | A | G |

T | T | C | G | A |

Table

DNA complementary rules.

Rule 1 | |

Rule 2 | |

Rule 3 | |

Rule 4 | |

Rule 5 | |

Rule 6 |

For higher security, the SHA-256 algorithm is adopted to engender the keys related to the four plaintext images for the chaotic system. The detailed steps are as follows:

where

The proposed image encryption process is shown in Figure

where

The integer sequence produced by the Logistic-sine system with the initial input

The elements in image

The DNA sequence

where

The encrypted DNA sequence

where

Decode the DNA sequence

Generate a sequence

The index sequence

The proposed nonlinear multi-image encryption scheme.

In this paper, the proposed encryption algorithm is symmetric; in a simple way, image decryption is a reverse process of encryption one. Since the keys of encryption and decryption algorithm are consistent and the DNA coding is a reversible process according to relevant rules, the encrypted image can be decrypted with a series of reverse processes. Through inverse scrambling, DNA decoding, inverse RPDFrAT, and inverse 2D LWT, one can recover four plaintext images from the encrypted image.

A series of experiments are implemented on MATLAB (version R2016a) to verify the security and effectiveness of the proposed scheme. Three groups of grayscale images of size

The initial values of the Logistic-sine system are calculated as

Encrypted and decrypted images for test group 1: (a) Peppers, (b) House, (c) Elaine, (d) Bridge, (e) encrypted image, (f) decrypted “Peppers,” (g) decrypted “House,” (h) decrypted “Elaine,” and (i) decrypted “Bridge”.

2D LWT needs less memory space, has time-frequency localization capability, and can be calculated more efficiently. In the proposed scheme, four detail components including approximation coefficient

PSNR values for different subimages in test group 1.

Image | PSNR (dB) | Resolution ratio | |||
---|---|---|---|---|---|

Peppers | House | Elaine | Bridge | ||

23.8991 | 28.0304 | 24.7152 | 24.8944 | ||

24.5692 | 29.7437 | 27.3227 | 28.2862 | ||

28.2129 | 29.0744 | 28.1203 | 26.6629 | ||

23.5328 | 28.2238 | 25.2936 | 25.4404 | ||

24.801 | 31.7226 | 28.7284 | 32.4214 | ||

20.4607 | 30.3214 | 26.8095 | 28.8676 | ||

23.4738 | 29.5268 | 28.4828 | 26.9537 | ||

44.819 | 32.0278 | 35.0628 | 46.598 |

PSNR values for different subimages in test group 2.

Image | PSNR (dB) | Resolution ratio | |||
---|---|---|---|---|---|

Baboon | Lax | Woman | Barbara | ||

18.0353 | 22.4549 | 17.3189 | 23.0936 | ||

20.0236 | 24.0307 | 18.4896 | 24.6340 | ||

18.4570 | 25.3754 | 20.1867 | 25.5969 | ||

18.5045 | 23.2808 | 18.3846 | 24.2337 | ||

23.3347 | 27.9507 | 20.4353 | 23.7978 | ||

21.9994 | 24.9871 | 17.5307 | 25.2391 | ||

20.0223 | 26.7670 | 20.4971 | 25.8953 | ||

21.7028 | 44.8666 | 23.4088 | 42.3084 |

PSNR values for different subimages in test group 3.

Image | PSNR (dB) | Resolution ratio | |||
---|---|---|---|---|---|

Couple | Airfield | Flowers | Lake | ||

23.7612 | 21.2786 | 21.2690 | 22.9572 | ||

26.7980 | 23.8957 | 23.1842 | 25.1712 | ||

27.1773 | 23.9516 | 23.9700 | 24.5594 | ||

24.7250 | 22.5693 | 21.9644 | 23.4695 | ||

27.7772 | 21.7138 | 19.9242 | 31.5062 | ||

26.2791 | 21.7087 | 19.0926 | 27.0977 | ||

27.0403 | 21.1431 | 19.4538 | 26.6705 | ||

38.9034 | 43.1816 | 36.8264 | 29.8761 |

The histogram is one of the important statistical assessment tools for cryptosystem. By comparing the characteristics of the plaintext image histograms and the ciphertext histograms, one can analyze the ability of the proposed scheme to homogenize encrypted image histograms. Figures

Histograms for three test groups’ images: (a) “Peppers,” (b) “House,” (c) “Elaine,” (d) “Bridge,” (e) “Baboon,” (f) “Lax,” (g) “Woman,” (h) “Barbara,” (i) “Couple,” (j) “Airfield,” (k) “Flower,” and (l) “Lake”.

Histogram for encrypted images: (a) encrypted image of test group 1, (b) encrypted image of test group 2, and (c) encrypted image of test group 3.

Besides, the chi-square

Chi-square test result.

Test group | Decision | |||
---|---|---|---|---|

1 | 218.7031 | Pass | ||

2 | 271.3438 | Pass | ||

3 | 250.125 | Pass |

In this part, we will discuss the ability of the proposed scheme to eliminate the correlation among adjacent points in images. To calculate the correlation coefficient, 1000 pairs of adjacent pixels in horizontal, vertical, and diagonal directions are extracted casually from the three test groups’ images and their corresponding encrypted ones, respectively. The correlation distributions of test group 1 in the horizontal direction are shown in Figure

Correlation distribution results in the horizontal direction for test group 1: (a) “Peppers,” (b) “House,” (c) “Elaine,” (d) “Bridge,” and (e) encrypted image of test group 1.

Correlation coefficients of adjacent pixels.

Image | HD | VD | DD | |
---|---|---|---|---|

Peppers | 0.9470 | 0.9489 | 0.9045 | |

House | 0.9647 | 0.9768 | 0.9457 | |

Elaine | 0.9564 | 0.9541 | 0.9389 | |

Bridge | 0.9036 | 0.9286 | 0.8776 | |

Proposed scheme | Encrypted image | −0.0043 | −0.0011 | 0.0052 |

Ref. [ | Encrypted image | 0.0187 | 0.0495 | −0.0246 |

Ref. [ | Encrypted image | −0.0142 | −0.0092 | 0.0157 |

Baboon | 0.7079 | 0.8365 | 0.6947 | |

Lax | 0.7250 | 0.5911 | 0.5719 | |

Woman | 0.9217 | 0.8801 | 0.8507 | |

Barbara | 0.9070 | 0.8567 | 0.8100 | |

Proposed scheme | Encrypted image | −0.0024 | −0.0034 | −0.0070 |

Ref. [ | Encrypted image | −0.2044 | 0.0392 | −0.0143 |

Ref. [ | Encrypted image | −0.0011 | −0.0039 | −0.0136 |

Couple | 0.9421 | 0.9106 | 0.8758 | |

Airfield | 0.9360 | 0.9249 | 0.9008 | |

Flowers | 0.9826 | 0.9697 | 0.9580 | |

Lake | 0.9330 | 0.9370 | 0.9046 | |

Proposed scheme | Encrypted image | −0.0023 | 0.0015 | −0.0063 |

Ref. [ | Encrypted image | −0.0132 | 0.0142 | 0.0110 |

Ref. [ | Encrypted image | 0.0114 | 0.0126 | −0.0021 |

In the proposed scheme, the DNA operation and the chaotic system are utilized to scramble the pixels of the test images and change the value of pixels, which contribute to providing a lower correlation between any two adjacent pixels. Besides, compared with the schemes in [

If the input variable confirms the uniform distribution, the global Shannon entropy will reach the maximal value, which represents a great uncertainty. The entropy of an image with

The maximal global Shannon entropy for a grayscale image of 256-level is 8 bits. However, considering the weakness of the global Shannon entropy including inaccuracy, inconsistency, and low efficiency, Wu et al. introduced another indicator, namely, local Shannon entropy [

Global and local Shannon entropy analysis for the encrypted images.

Test group | Global Shannon entropy (bits) | Local Shannon entropy (bits) | Local Shannon entropy critical values (bits) | ||
---|---|---|---|---|---|

1 | 7.9978 | 7.9030 | Pass | Pass | Pass |

2 | 7.9972 | 7.9026 | Pass | Pass | Pass |

3 | 7.9973 | 7.9029 | Pass | Pass | Pass |

The DNA sequence operation based on Logistic-sine chaotic system in our proposed scheme can randomly change the pixel intensity value as well as contribute to uniform distributions of the encrypted images. As the results shown in Table

In this segment, we will evaluate the ability of this presented scheme to resist brute-force attacks [

In our proposed scheme, the keys are composed of

MSE curve of the key: (a)

Decrypted images with deviated keys: (a)

To measure whether this image encryption scheme can be against differential attacks or not, the number of pixels change rate (NPCR) and the uniform average change intensity (UACI) are employed as usual [

Tables

NPCR test results.

Theoretical NPCR critical values | NPCR(%) | ||
---|---|---|---|

Test group 1 | Test group 2 | Test group 3 | |

99.65 | 99.68 | 99.66 | |

UACI test result.

Theoretical UACI critical values | UACI(%) | ||
---|---|---|---|

Test group 1 | Test group 2 | Test group 3 | |

33.55 | 33.58 | 33.54 | |

The white Gaussian noise (WGN) with zero-mean and unit standard deviation is added to the encrypted image as

Noise attacks analysis in the encrypted image of test group 1: (a–d) decrypted images of “Peppers,” “House,” “Elaine,” and “Bridge,” respectively, with the noise intensity

The MSE and PSNR values of test group 1 with the white Gaussian noise.

The intensity of the white Gaussian noise | Image | MSE | PSNR (dB) |
---|---|---|---|

Decrypted “Peppers” | 1152.9237 | 17.5128 | |

Decrypted “House” | 848.3636 | 18.8450 | |

Decrypted “Elaine” | 830.1441 | 18.9393 | |

Decrypted “Bridge” | 1128.4652 | 17.6059 | |

Decrypted “Peppers” | 1672.5266 | 15.8971 | |

Decrypted “House” | 1287.4921 | 17.0334 | |

Decrypted “Elaine” | 1203.8451 | 17.3251 | |

Decrypted “Bridge” | 1687.9173 | 15.8573 | |

Decrypted “Peppers” | 2329.1915 | 14.4588 | |

Decrypted “House” | 2106.6591 | 14.8949 | |

Decrypted “Elaine” | 2074.7117 | 14.9612 | |

Decrypted “Bridge” | 2486.3427 | 14.1752 | |

Decrypted “Peppers” | 2848.1021 | 13.5852 | |

Decrypted “House” | 2562.9442 | 14.0434 | |

Decrypted “Elaine” | 2379.4283 | 14.3661 | |

Decrypted “Bridge” | 3111.0023 | 13.2018 |

Moreover, considering a practical situation that the encrypted image may be deliberately occluded by unaccredited attackers in transmission, the encrypted image is partially cropped with different sizes to analyze the ability to resist the occlusion attacks. The corresponding decryption images are shown in Figure

Occlusion attack results.

The MSE and PSNR values of test group 1 with data loss.

Data loss in cipher | Image | MSE | PSNR (dB) |
---|---|---|---|

Figure | Decrypted “Peppers” | 1631.2076 | 16.0057 |

Decrypted “House” | 1370.5680 | 16.7618 | |

Decrypted “Elaine” | 1635.5325 | 15.9942 | |

Decrypted “Bridge” | 1864.8458 | 15.4244 | |

Figure | Decrypted “Peppers” | 2244.3347 | 14.6199 |

Decrypted “House” | 2024.8464 | 15.0669 | |

Decrypted “Elaine” | 1945.2161 | 15.2411 | |

Decrypted “Bridge” | 2267.7465 | 14.5739 | |

Figure | Decrypted “Peppers” | 2288.5194 | 14.5353 |

Decrypted “House” | 1838.9888 | 15.4850 | |

Decrypted “Elaine” | 1810.6396 | 15.5525 | |

Decrypted “Bridge” | 2441.6336 | 14.2540 | |

Figure | Decrypted “Peppers” | 2775.3518 | 13.6976 |

Decrypted “House” | 2400.4469 | 14.3279 | |

Decrypted “Elaine” | 2252.1406 | 14.6048 | |

Decrypted “Bridge” | 3011.6712 | 13.3427 |

The four classical attacks are ciphertext-only attack, chosen-ciphertext attack, known-plaintext attack, and chosen-plaintext attack (CPA). Among them, CPA is the most forceful attack; thus, if the proposed image encryption scheme can resist CPA, it can also perform well in withstanding the other three typical attacks [

The execution efficiency of one image cryptosystem is an important practical issue that needs to be considered. In this part, we will give the computation complexity analysis. In the proposed nonlinear multi-image cryptosystem, the computation complexity is mainly related to the scrambling and diffusion operations. One real-valued intermediate encrypted image from RPDFrAT will be scrambled and diffused by the DNA sequence operation and chaotic system, so the first time-consuming part in computation is the operation of multiplying floating point numbers for the generation of chaotic sequences. Hence, the time complexity is

The proposed nonlinear multi-image encryption scheme has been compared with the preexisting chaos-based schemes [

Comparison of different image encryption algorithms.

Algorithms | Types of encryption | Chaotic maps | DNA sequence operation |
---|---|---|---|

Proposed scheme | Multi-image | Logistic-sine chaotic map | Yes |

Reference [ | Multi-image | Henon map, logistic map | No |

Reference [ | Multi-image | PWLCM system | No |

Reference [ | Multi-image | PWLCM system | No |

Reference [ | Multi-image | PWLCM system | Yes |

Reference [ | Single image | 2D logistic chaotic map | Yes |

Reference [ | Single image | 3D Lorenz chaotic system, Chen’s 4D hyperchaotic system | Yes |

Comparison results for encrypted single “Elaine” of size

Algorithms | Information entropy (bits) | NPCR(%) | UACI(%) |
---|---|---|---|

Proposed scheme | 7.9978 | 99.63 | 33.52 |

Reference [ | 7.9974 | 99.61 | 33.46 |

Reference [ | 7.9976 | 99.62 | 33.41 |

Reference [ | 7.9992 | 99.63 | 33.54 |

Reference [ | 7.7196 | 99.59 | 33.43 |

Comparison of key space results.

Algorithms | Proposed scheme | Ref. [ | Ref. [ | Ref. [ | Ref. [ | Ref. [ |
---|---|---|---|---|---|---|

Key Space |

Reference [

A nonlinear multi-image encryption scheme is presented. The main features of multiple original images are extracted with the 2D lifting wavelet transform, and the information of original images is compressed into a small amount of data as well. Next, the reality-preserving discrete fractional angular transform is employed to produce a real-valued intermediate output, which is convenient to transfer, display, and store. Ultimately, the scrambling-diffusion operations are conducted with the combination of the deoxyribonucleic acid sequence operations and Logistic-sine chaotic system, which promises a bright prospect with the development of DNA computer. The proposed lossy multi-image encryption scheme could greatly improve the encryption efficiency at the cost of the quality of decryption images. Moreover, the lossy multi-image encryption scheme is robust and secure against various attacks where the deoxyribonucleic acid sequence operations are nonlinear and the main keys are associated with the original images.

The raw/processed data required to reproduce these findings can be available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

This work was supported by the National Natural Science Foundation of China (Grant nos. 62041106 and 61861029), the Major Academic Discipline and Technical Leader of Jiangxi Province (Grant no. 20162BCB22011), the Cultivation Plan of Applied Research of Jiangxi Province (Grant no. 20181BBE58022), the Foundation of Jiujiang University (Grant no. 2015LGYB03), and the foundation of the Education Department of Jiangxi Province (Grant no. GJJ190203).