In this study, a new algorithm of fractional beta chaotic maps is proposed to generate chaotic sequences for image encryption. The proposed technique generates multi random sequences by shuffling the image pixel position. This technique is used to blur the pixels connecting the input and encrypted images and to increase the attack resistance. The proposed algorithm makes the encryption process sophisticated by using fractional chaotic maps, which hold the properties of pseudo-randomness. The fractional beta sequences are utilized to alter the image pixels to decryption attacks. The experimental results proved that the proposed image encryption algorithm successfully encrypted and decrypted the images with the same keys. The output findings indicate that our proposed algorithm has good entropy and low correlation coefficients. This translates to enhanced security against different attacks. A MATLAB programming tool was used to implement and assess the image quality measures. A comparison with other image encryption techniques regarding the visual inspection and signal-to-noise ratio is provided.

The recent advancements in communications and computer technologies facilitate data transmission over the internet networks [

Fractional calculus plays a major role in many sciences and its applications [

Fractional calculus is utilized to generalize chaotic systems, chaotic maps, optimization, operation, and other theories of chaos and has been employed in an extensive choice of this field including entropy concept [

Therefore, looking for a new chaotic mapping technique, which produces better security performances compared with other methods of image encryption, is very important and superior performance with respect to the trade-offs between the security and efficiency. To further increase, the security performances of the image encryption based on chaos, a new chaotic map based on fractional beta function has been proposed to achieve strong chaotic key generations. The structure of the paper is as follows:

Chaos theory focuses on describing the behavior of a nonlinear dynamic system, which sensitivity to initial conditions is high. The common features between chaotic and encryption models have led to developing different chaos-based image encryption modes. Recently, chaotic map approaches have been used in different ways in image encryption algorithms and related to security applications. Zahmoul et al. [

Hence, the

where κ is a parameter of chaotic map, which is used to adjust the value of β-map and indicate the bifurcation parameter. Beta function is utilized in various types of applications in image processing and engineering, especially in bio-medical signal and image compression, image detection [

The motivation for this study is to employ the concept of the fractional calculus to generalize and improve the β-map. We call the consequence of this generalization the fractional β-map. Then we employ the new look of this function to design a hybrid model for image encryption based on fractional-chaotic maps. Our contribution is to develop a new chaotic map based on fractional-chaotic map is proposed. The proposed fractional-chaos have a large range of bifurcation parameter with the strong chaotic behavior which increases the protection of the image encryption schemes.

In this section, a new mathematical fractional-chaotic maps model has proposed to as a new image encryption algorithm. Consider the polynomial function

The first derivative is known by the formula

Accumulating this yields the general formula

Now, generalize the factorial by the gamma function, we have the generalized calculus

For negative integer power

By using the above conclusion,

The encryption dynamic of the suggested system recognized by the following steps:

Suppose that a picture of size

Produce two dissimilar quasi-random arrangements subsequently constructing numerous recipes of FBM. Via the understanding of the chaotic purpose of the considerable difference in the initial condition, numerous random arrangements might be made;

At this stage, the produced arrangements of the FBM are utilized to waddle the plaintext copy's rows and columns. Organizing elements of Q_pro and Q1_pro whose elements are M * N in matrix system and find Q and Q1 matrices with M * N dimension. The variation procedure affects the original feature pixels by variation them inside columns, utilizing Q1 matrices coefficient's locations. The coefficients of the consequential matrix are formulated, using

Share the consequential matrix into four blocks of equivalent dimension. Transform every block to a quasi-random matrix

The operator

Diffusion imitates the assets that the termination in the data and figures of the plain text is dissolute in the cipher text;

Finally, the encryption procedure upgrades the security of pictures critically. The decryption structure of the process can be investigated as the converse of the encryption scheme utilizing the matching key.

The proposed image encryption algorithm consisted of the following steps:

1-Resize the input image in square equal dimension.

2-Produce different random sequences by using different combinations of beta chaotic maps.

3-The generated sequences are used for shuffling the rows and columns of input image.

4-The substitution process of input image pixels by: f(r) = T(r) mod I. Where

and T is the truncation function.

5-The decryption process is the reverse of the encrypted using the same key.

The fractional β map

In this section, the performance of the proposed image encryption model is demonstrated using standard test images, commonly known as Lena, Pepper, and Baboon. We employed statistical analyses to assess the model's performance.

The image histogram represents the relationship between the pixel gray level and the frequency of occurrence. In this study, the histograms of different original and encrypted images are illustrated in

The entropy measures the degree of unpredictability of information. The information entropy is calculated for encrypted images to measure the degree of uncertainties; however, any certain degree of predictability will threaten the encryption security [

Images | Plain image | Decrypted image |
---|---|---|

Lena | 7.7534 | 7.9985 |

Peppers | 7.7145 | 7.9980 |

Baboon | 7.7759 | 7.9982 |

Average | 7.7479 | 7.9982 |

The results of

Encryption algorithm | Entropy |
---|---|

Wang et al. [ |
7.9977 |

Zhang et al. [ |
7.9994 |

Liu et al. [ |
7.9995 |

Li et al. [ |
7.9894 |

Proposed FBC maps model | 7.9985 |

The aim of the image encryption algorithm is to decrease the correlation among the image pixels in order to make the prediction of any given pixel from its neighbors more difficult. The next formula indicates the correlation coefficient between each pair:
_{i} and y_{i} from

Image | H | V | D |
---|---|---|---|

Lena | 0.0020 | 0.0019 | −0.0025 |

Pepper | −0.0040 | −0.0045 | −0.0005 |

Baboon | 0.0045 | 0.0061 | 0.0018 |

Algorithm | H | V | D |
---|---|---|---|

Rhouma et al. [ |
0.1257 | 0.0581 | 0.0504 |

Tong et al. [ |
0.0038 | 0.0058 | 0.0133 |

Liu et al. [ |
0.0021 | 0.0046 | 0.0033 |

Li et al. [ |
0.0044 | 0.0015 | 0.00019 |

Zhang et al. [ |
0.0066 | 0.0059 | 0.0008 |

Proposed FBC maps model | 0.0020 | 0.0019 | −0.0025 |

Taking the Lena (256 × 256) image as an experimental object, the comparison of correlation between different encryption algorithms in the horizontal (H), vertical (V) and diagonal (D) directions is illustrated in

The SSIM measure is used to measure the similarity between the input image and the decrypted image. The close pixels have strong SSIM when they are close. The range of SSIM is between −1 and 1.

Images | SSIM |
---|---|

Lena | 0.9305 |

Peppers | 0.9225 |

Baboon | 0.9128 |

The image encryption algorithm primarily depends on the key sensitivity. Even one bit change to the key combinations can produce a different encrypted image. The two measures used for the key sensitivity analysis are the “Number of Changing Pixel Rate (NPCR) and the Unified Averaged Changed Intensity (UACI)”. The NPCR measures the total distinct pixels between two given images, while UACI represents the average of the intensity. These parameters are defined as follows:

Generally, the NPCR value of a ‘good’ chaotic image encryption model needs to be more than 90% and UACI value more than 33%.

Image | NPRC | UACI |
---|---|---|

Lena | 0.9960 | 0.3336 |

Peppers | 0.9964 | 0.3348 |

Baboon | 0.9963 | 0.3342 |

Algorithm | NPRC | UACI |
---|---|---|

Liu et al. [ |
0.9949 | 0.3156 |

Li et al. [ |
0.9949 | 0.3156 |

Zhang et al. [ |
0.9960 | 0.3347 |

Proposed FBC maps model | 0.9960 | 0.3336 |

In this study, a new algorithm of image encryption based on new fractional beta chaotic maps is proposed. All of the experimental results demonstrated that the FBC map algorithm offers a large key space with high sensitivity to all the secret keys. The comparison with other image encryption works indicates that the proposed FBC map model provided the best performance. The proposed FBL map model is preferment and thus valuable for image encryption applications. For future work, further improvements on the encryption system can be assumed including a new fractional chaotic model for image encryption application.

The authors would like to thank the editor office for the deep advice to improve our work.