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A new fractional two-dimensional triangle function combination discrete chaotic map (2D-TFCDM) with the discrete fractional difference is proposed. We observe the bifurcation behaviors and draw the bifurcation diagrams, the largest Lyapunov exponent plot, and the phase portraits of the proposed map, respectively. On the application side, we apply the proposed discrete fractional map into image encryption with the secret keys ciphered by Menezes-Vanstone Elliptic Curve Cryptosystem (MVECC). Finally, the image encryption algorithm is analysed in four main aspects that indicate the proposed algorithm is better than others.

Nowadays, image encryption plays a significant role with the development of security technology in the areas of network, communication, and cloud service. Multifarious chaos-based image encryption algorithms have been developed up to now, such as in [

The theory of the fractional difference has been developed for decades [

ECC is a widely used technology in data security and communication security; it can achieve the same level of security with smaller key sizes and higher computational efficiency [

Many encryption methods based on fractional derivatives have been proposed in recent time, like fractional logistic maps [

In [

Our main purpose is to introduce a new two-dimensional discrete chaotic map based on fractional-order difference and apply it in image encryption. The rest of this paper is organized as follows. In Section

The definitions of the fractional sum and difference are given as follows. Let

Let

For

For the delta fractional difference equation

An elliptic curve (EC)

If

The scalar multiplication over

The order of an EC is defined by the number of points that lie on the EC denoted by

Set

MVECC is one of most significant extensions of ECC; the working principle of MVECC is as follows.

If Andy wants to encrypt and send the message

Any adversary that only has

From [

Let

The bifurcation diagram of the 2D-TFCDM of variable

The bifurcation diagram of the fractional 2D-TFCDM of variable

In Figures

The largest Lyapunov exponent of the 2D-TFCDM of the variable

The largest Lyapunov exponent of the fractional 2D-TFCDM of the variable

By choosing 101 different initial values we can plot

The phase portraits of the 2D-TFCDM for

The phase portraits of the fractional 2D-TFCDM for

The phase portraits of the fractional 2D-TFCDM for

The fractionalized chaotic map can be applied in image encryption. Exploit (

Setting

Calculate

Make

Set

Taking advantage of (

Reversing the above 4 steps, we can remove the effect of permutation to get the original image.

The inverse form of (

The decryption procedure is including 2 parts:

Figure

The proposed encryption method.

The S box.

The original, encrypted, and decrypted images are shown in Figures

Cameraman.

The original figure

The encrypted figure

The decrypted figure

Lena.

The original figure

The encrypted figure

The decrypted figure

Peppers.

The original figure

The encrypted figure

The decrypted figure

Lake.

The original figure

The encrypted figure

The decrypted figure

Dollar.

The original figure

The encrypted figure

The decrypted figure

Columbia.

The original figure

The encrypted figure

The decrypted figure

Lax.

The original figure

The encrypted figure

The decrypted figure

Boat.

The original figure

The encrypted figure

The decrypted figure

Aerial.

The original figure

The encrypted figure

The decrypted figure

The adopted cryptosystem in Section

In the proposed algorithm, the initial values

The quality against any statistical attack is important for a well-designed encryption method; it include 3 aspects as follows.

In an ordinary image, the adjacent pixels are related; therefore the correlation coefficient of adjacent pixels is usually high. A good encryption algorithm should make the correlation coefficients of encrypted image nearly equal to zero. The closer to zero the correlation coefficients is, the better the encryption algorithm is. Formulas (

Correlation coefficients of image.

Image | Original image | Encrypted image | ||||
---|---|---|---|---|---|---|

Horizontal | Diagonal | Vertical | Horizontal | Diagonal | Vertical | |

Cameraman | 0.9276 | 0.9120 | 0.9597 | 0.0119 | −0.0021 | −0.0025 |

Lena | 0.9722 | 0.9527 | 0.9860 | −0.0140 | −0.0086 | −0.0034 |

Peppers | 0.9667 | 0.9382 | 0.9694 | −0.0088 | 0.0080 | −0.0054 |

Lake | 0.9768 | 0.9544 | 0.9748 | −0.0155 | 0.0101 | −0.0088 |

Dollar | 0.8035 | 0.6952 | 0.6938 | 0.0131 | −0.0183 | 0.0263 |

Columbia | 0.9727 | 0.9403 | 0.9705 | 0.0060 | −0.0104 | −0.0093 |

Lax | 0.7889 | 0.7151 | 0.8483 | −0.0107 | 0.0147 | 0.0107 |

Boat | 0.9407 | 0.9158 | 0.9545 | 0.0169 | −0.0074 | −0.0077 |

Aerial | 0.9135 | 0.7952 | 0.8677 | 0.0084 | −0.0123 | −0.0133 |

Cameraman.

The original figure

The encrypted figure

Lena.

The original figure

The encrypted figure

Peppers.

The original figure

The encrypted figure

Lake.

The original figure

The encrypted figure

Dollar.

The original figure

The encrypted figure

Columbia.

The original figure

The encrypted figure

Lax.

The original figure

The encrypted figure

Boat.

The original figure

The encrypted figure

Aerial.

The original figure

The encrypted figure

With the sharp contrast of data between original image and encrypted image, Table

Compared with other algorithm, we can observe that most correlation coefficients of encrypted image are nearer to 0 in Table

Comparison of correlation coefficients of image.

Algorithm | Image | Original image | Encrypted image | ||||
---|---|---|---|---|---|---|---|

Horizontal | Vertical | Diagonal | Horizontal | Vertical | Diagonal | ||

Proposed | Lena | 0.9722 | 0.9527 | 0.9860 | −0.0140 | −0.0086 | −0.0034 |

[ | Lena | 0.9503 | 0.9755 | 0.9275 | −0.0226 | 0.0041 | 0.0368 |

[ | Lena | 0.927970 | 0.926331 | 0.839072 | −0.010889 | −0.018110 | −0.006104 |

[ | Lena | 0.946 | 0.973 | 0.921 | −0.0055 | −0.0075 | 0.0026 |

[ | Lena | 0.9569 | 0.9236 | 0.9019 | 0.0042 | −0.0043 | 0.0163 |

Histogram reflects the distribution of colors inside the image. The adversary can get some effective information from the regularity of histogram. Therefore, a well-designed image encryption method should make the pixel value of encrypted image distribute uniformly. Figure

Cameraman.

The original image

The encrypted image

The decrypted image

Lena.

The original figure

The encrypted figure

The decrypted figure

Peppers.

The original figure

The encrypted figure

The decrypted figure

Lake.

The original figure

The encrypted figure

The decrypted figure

Dollar.

The original figure

The encrypted figure

The decrypted figure

Columbia.

The original figure

The encrypted figure

The decrypted figure

Lax.

The original figure

The encrypted figure

The decrypted figure

Boat.

The original figure

The encrypted figure

The decrypted figure

Aerial.

The original figure

The encrypted figure

The decrypted figure

Information entropy defines the randomness and the unpredictability of information in an image. It is defined by

Information entropy.

Image | Original image | Encrypted image |
---|---|---|

Cameraman | 7.0097 | 7.9974 |

Peppers | 7.5739 | 7.9976 |

Dollar | 6.9785 | 7.9992 |

Lax | 6.8272 | 7.9993 |

Aerial | 6.9940 | 7.9992 |

Lena | 7.2185 | 7.9993 |

Lake | 7.4845 | 7.9993 |

Columbia | 7.2736 | 7.9992 |

Boat | 6.9391 | 7.9972 |

From Table

Comparison of information entropy.

Algorithm | Image | Original image | Encrypted image |
---|---|---|---|

Proposed | Lena | 7.2185 | 7.9993 |

[ | Lena | 7.2072 | 7.9973 |

[ | Lena | Undefined | 7.9972 |

[ | Lena | Undefined | 7.987918 |

[ | Lena | 7.447144 | 7.988847 |

The different range between two images is measured by two criteria: number of pixels change rate (NPCR) and unified average changing intensity (UACI). They are defined as follows:

We encrypt the image by the keys

Comparison of key spaces.

Algorithm | Proposed | [ | [ | [ |
---|---|---|---|---|

Key spaces | | | ≈2^{273} | |

NPCR and UACI between Figures

Image | NPCR and UACI | |
---|---|---|

NPCR (%) | UACI (%) | |

Figure | 99.61 | 31.26 |

Figure | 97.02 | 30.23 |

Figure | 99.60 | 31.03 |

Figure | 99.61 | 31.01 |

Figure | 99.62 | 31.27 |

The test of key sensitivity.

The correct keys

In contrast with other algorithm, the key space of proposed algorithm is larger than others.

Most NPCR are near to 99.61% and most of UACI are higher than 30% in Table

By encrypting two same images with only one pixel difference, the attackers can obtain effective information by comparing the two encrypted images. Therefore an encryption method designed against differential attack should ensure that the two encrypted images are completely different even if there is only a pixel difference in the original image.

In Table

NPCR and UACI between cipher-images with slightly different plain-images.

Image | NPCR and UACI of Cameraman | |||
---|---|---|---|---|

NPCR (1-round %) | UACI (1-round %) | NPCR (2-round %) | UACI (2-round %) | |

Figure | 4.84 | 1.64 | 99.57 | 33.61 |

Figure | 81.43 | 27.39 | 99.62 | 33.56 |

Figure | 80.87 | 27.19 | 99.59 | 33.51 |

Figure | 6.82 | 2.28 | 99.59 | 33.46 |

NPCR and UACI between cipher-images with slightly different plain-images.

Image | NPCR and UACI of Lena | |||
---|---|---|---|---|

NPCR (1-round %) | UACI (1-round %) | NPCR (2-round %) | UACI (2-round %) | |

Figure | 1.21 | 0.41 | 99.59 | 33.39 |

Figure | 95.06 | 31.93 | 99.59 | 33.53 |

Figure | 94.90 | 31.93 | 99.60 | 33.48 |

Figure | 1.71 | 0.58 | 99.63 | 33.40 |

NPCR and UACI between cipher-images with slightly different plain-images.

Image | NPCR and UACI of Peppers | |||
---|---|---|---|---|

NPCR (1-round %) | UACI (1-round %) | NPCR (2-round %) | UACI (2-round %) | |

Figure | 4.83 | 1.64 | 99.56 | 33.44 |

Figure | 81.44 | 27.35 | 99.62 | 33.50 |

Figure | 6.12 | 2.03 | 99.57 | 33.42 |

Figure | 6.83 | 2.32 | 99.60 | 33.50 |

NPCR and UACI between cipher-images with slightly different plain-images.

Image | NPCR and UACI of Lake | |||
---|---|---|---|---|

NPCR (1-round %) | UACI (1-round %) | NPCR (2-round %) | UACI (2-round %) | |

Figure | 1.21 | 0.41 | 99.61 | 33.46 |

Figure | 95.09 | 31.88 | 99.61 | 33.42 |

Figure | 94.92 | 31.89 | 99.60 | 33.46 |

Figure | 1.71 | 0.58 | 99.59 | 33.47 |

NPCR and UACI between cipher-images with slightly different plain-images.

Image | NPCR and UACI of Dollar | |||
---|---|---|---|---|

NPCR (1-round %) | UACI (1-round %) | NPCR (2-round %) | UACI (2-round %) | |

Figure | 1.21 | 0.41 | 99.64 | 33.42 |

Figure | 95.08 | 32.00 | 99.60 | 33.49 |

Figure | 94.90 | 31.93 | 99.61 | 33.48 |

Figure | 1.71 | 0.57 | 99.61 | 33.41 |

NPCR and UACI between cipher-images with slightly different plain-images.

Image | NPCR and UACI of Columbia | |||
---|---|---|---|---|

NPCR (1-round %) | UACI (1-round %) | NPCR (2-round %) | UACI (2-round %) | |

Figure | 94.83 | 31.96 | 99.61 | 33.47 |

Figure | 93.48 | 31.40 | 99.48 | 33.36 |

Figure | 0.96 | 0.32 | 99.60 | 33.45 |

Figure | 1.11 | 0.38 | 99.61 | 33.51 |

NPCR and UACI between cipher-images with slightly different plain-images.

Image | NPCR and UACI of Lax | |||
---|---|---|---|---|

NPCR (1-round %) | UACI (1-round %) | NPCR (2-round %) | UACI (2-round %) | |

Figure | 1.21 | 0.40 | 99.62 | 33.39 |

Figure | 95.06 | 31.99 | 99.61 | 33.49 |

Figure | 94.92 | 31.87 | 99.58 | 33.41 |

Figure | 1.70 | 0.58 | 99.62 | 33.48 |

NPCR and UACI between cipher-images with slightly different plain-images.

Image | NPCR and UACI of Boat | |||
---|---|---|---|---|

NPCR (1-round %) | UACI (1-round %) | NPCR (2-round %) | UACI (2-round %) | |

Figure | 4.83 | 1.60 | 99.59 | 33.47 |

Figure | 81.46 | 27.45 | 99.62 | 33.59 |

Figure | 80.82 | 27.24 | 99.58 | 33.48 |

Figure | 6.82 | 2.32 | 99.62 | 33.61 |

NPCR and UACI between cipher-images with slightly different plain-images.

Image | NPCR and UACI of Aerial | |||
---|---|---|---|---|

NPCR (1-round %) | UACI (1-round %) | NPCR (2-round %) | UACI (2-round %) | |

Figure | 1.21 | 0.41 | 99.61 | 33.49 |

Figure | 95.06 | 31.93 | 99.62 | 33.43 |

Figure | 94.91 | 31.88 | 99.61 | 33.53 |

Figure | 1.71 | 0.57 | 99.61 | 33.52 |

From Table

Comparison of NPCR and UACI of image.

Algorithm | Image | NPCR (%) | UACI (%) |
---|---|---|---|

Proposed | Lena | 99.60 | 33.48 |

[ | Lena | 99.61 | 33.53 |

[ | Lena | 99.6429 | 33.3935 |

[ | Lena | 99.6304 | 33.5989 |

[ | Lena | 99.932 | 39.520 |

[ | Lena | 75.62561 | 34.84288 |

[ | Lena | 99.6091 | 33.5038 |

[ | Lena | 99.6330 | 34.1319 |

In Section

Fractional 2D-TFCDM is obtained from the 2D-TFCDM. After that, we found new chaotic dynamics behaviors from the fractionalized map. Moreover, the map can be converted into image encryption algorithm as an application. Finally, the encryption effect is analysed in 4 main aspects; we find the proposed scheme is superior to others almost anywhere in comparison. As far as we know, the proposed image encryption algorithm has never been reported before.

The authors declare that they have no conflicts of interest.

The project was supported by the National Natural Science Foundation of China (Grant nos. 61072147, 11271008).