^{1}

^{*}

^{2}

In this paper, we propose an effective gray image cryptosystem containing Arnold cat map for pixel permutation and an improved Logistic map for the generation of encryption keys to be used for pixel modification. Firstly, a new chaotic map is designed to show better performance than the standard one in terms of key space range, complexity and uniformity. Generated secret key is not only sensitive to the control parameters and initial condition of the improved map but also strongly depend on the plain image characteristic which provides an effective resistance against statistical and differential attacks. Additionally, to get higher encryption strength of the cryptosystem, both confusion and diffusion processes are performed with different keys in every iterations. Theoretical analysis and simulation results confirm that the proposed algorithm has superior security and effectively encrypts and decrypts the gray images as well.

With the rapid development of information technology and widely use of computer networks, multimedia applications have become much more prevalent than the past. This situation creates security problems of transferring information in communication network and thus, confidentiality of the information is becoming a serious issue nowadays. Among the multimedia information, digital image plays an important role in people’s daily life so the protection of visual information has become a major task. Encryption is an ordinary solution for the protection of information. Most of the available encryption methods such as DES (Data Encryption Standard), AES (Advanced Encryption Standard) and IDEA (International Data Encryption Algorithm) are typically used for text- structure data [

Chaotic systems have important properties of sensitivity to initial conditions and control parameters, pseudo-randomness and ergodicity [

Discrete time chaotic systems have high efficiency comparing with the continuous time chaotic systems [

The proposed cryptosystem utilizes Arnold Cat map (ACM) method that is used to transform all pixel positions of original image to their corresponding positions without changing their values. This part breaks the strong correlation of adjacent pixels and creates a confused image. In diffusion part, all pixels values are modified sequentially through a diffusion function as one pixel change can influences other pixels, which keeps high plain-text and cipher-text sensitivity. The rest of the paper is organized as follows: Section 2 gives brief overview of the standard Logistic map (SLM) with its dynamic defect. Section 3 introduces an improved Logistic map (ILM) with statistical analysis. In Section 4, the proposed cryptosystem is described in detail. Security and performance of the proposed algorithm are analyzed in Section 5. Finally, the conclusions will be discussed in Section 6.

Standard map is one of the simplest discrete systems that exhibit chaos and defined by

where

We modify SLM by adding a new parameter to the map equation as the following

where

by solving the above equations, we get

where the range of

Finally, we obtain the equation of ILM as

where

parameter must be always bigger than

Here,

Lyapunov exponent states a checkable criterion for sensitivity to initial conditions of a nonlinear dynamical system [

A positive Lyapunov exponent indicates that the dynamical system is chaotic [

the Lyapunov spectrums and bifurcation diagrams of the SLM and ILM. Lyapunov coefficients for ILM are always greater than or equal to the values of SLM which shows that ILM has better mixing property.

Randomness means the lack of predictability in a sequence of symbols [

A threshold level of 0.5 is used to generate a bit value “1” or “0” for each

According to the NIST result, it can be concluded that ILM is quite stochastic and generates chaotic sequences which has sufficient randomness.

Test name | p-value | Result |
---|---|---|

Frequency | 0.9362 | Success |

Block frequency | 0.2736 | Success |

Runs | 0.1597 | Success |

Long runs of ones | 0.1484 | Success |

Rank | 0.6484 | Success |

Spectral DFT | 0.3684 | Success |

Non-overlapping templates (m = 9; B = 000000001) | 0.9320 | Success |

Overlapping templates (m = 9) | 0.8690 | Success |

Universal (L = 7; Q = 1280) | 0.3369 | Success |

Linear complexity | 0.0513 | Success |

Serial-1 (m = 5) | 0.9486 | Success |

Serial-2 (m = 5) | 0.9667 | Success |

Approximate entropy (m = 5) | 0.8972 | Success |

Cumulative sums forward | 0.5753 | Success |

Cumulative sums reverse | 0.6476 | Success |

Random excursions (x = +1) | 0.2995 | Success |

Random excursions variant (x = −1) | 0.6508 | Success |

Chaos-based image encryption systems are generally composed of two stages: replacement of pixels called confusion and modification of pixel values called diffusion [

In an ordinary image, adjacent pixels have strong correlation. This strong correlation need to be broken before encryption. Arnold Cat Map is an invertible discrete system that will be used to rearrange the pixel positions of the plain image in a way that the adjacent pixels are far enough from each other. It is represented by

where (x, y) are the pixel position of the plain image with a size of

ACM effectively changes all pixel positions as only linear transformation with simple mod function need to be performed. Furthermore, it has a characteristic of area-preserving which means that if it is iterated enough times, original image reappears. Shortly, ACM can be considered as a permutation method that focuses on the pixel position not the pixel value of the plain image. Hence, the cryptosystem requires a diffusion process to enhance the security.

In a gray image, each pixel is represented by 8-bit in decimal range [0, 255]. Key must be same format with the pixel in the image to operate diffusion. However, the output of the ILM is a floating-point value. Thus, the following equation is used to obtain encryption key.

The proposed cryptosystem uses a sk parameter which is strongly depends on the pixel values and size of the plain image. It provides different keys even with the same parameters of the cryptosystem and defined by

where normalized image has a size of

Diffusion is a process in which the pixel values of confused image are modified sequentially by mixing the encryption key. Diffusion operation used in the cryptosystem is given by

where

and its value depends on

The confusion and diffusion processes complete the proposed image encryption structure. For instance, Lena image is encrypted by the proposed cryptosystem with a key of

Key space size is the total number of different keys that can be used in a cryptosystem. For an ideal encryption algorithm, it should be larger than

which is sufficient to resist brute-force attack.

Key sensitivity analysis can be observed in two aspects: (i) if slightly different keys are applied to encrypt the same images, then completely different cipher images should be produced; (ii) if a tiny difference exists in decryption key, then the cipher image could not be decrypted correctly. For the first key sensitivity analysis, a test plain of cameraman image is encrypted with a randomly chosen key of

Key | Correlation coefficients between | |||
---|---|---|---|---|

Sk | n | |||

(b)-(c) | 98.765431 | 0.123456 | 2 | 0.00403 |

(b)-(e) | 98.765432 | 0.123457 | 2 | −0.00022 |

In

The proposed cryptosystem should also be sensitive to p, q and n. Cipher images produced by slightly different keys are shown in

These results show that despite being a very small difference at all encryption keys, corresponding cipher images are completely different. For the second case, when a slightly different key is used in decryption, then the cipher image could not be decrypted correctly. Cipher Lena image in

We conclude that the proposed cryptosystem is quite sensitive to all keys and can effectively resist differential attacks.

Key | Correlation coefficients between | ||
---|---|---|---|

sk | |||

(a) | 10.402343 | 98.641976 | |

(b) | 10.398437 | 98.641976 | |

(a)-(b) | 0.999915 | ||

(c)-(d) | 0.004622 |

Figures | Keys | Figures between | Correlation coefficients between | ||
---|---|---|---|---|---|

p | q | n | |||

3(b) | 5 | 4 | 7 | ||

6(a) | 6 | 4 | 7 | 3(b)-6(a) | −0.0019 |

6(b) | 5 | 3 | 7 | 3(b)-6(b) | 0.0028 |

6(c) | 5 | 4 | 6 | 3(b)-6(c) | 0.0044 |

In image processing, histogram is used to represent the distribution of the pixel values in an image. Equal probability of each pixel value creates a uniform histogram which is more robust against statistical attacks in terms of security [

It is clear that the histograms of the cipher images are significantly different than the originals and uniformly distributed even the plain image is purely black.

Information entropy is a measure of uncertainty associated with a random message [

where

Maximum entropy is achieved in the case of a uniform probability distribution. We randomly chose eight test images (Lena, Baboon, Frog, Goldhill, Cat, Landscape, Truck and Clown) which are available on Internet, to be used for entropy analysis. Then these images are encrypted using the proposed cryptosystem. The entropy results for the test images and corresponding cipher images are listed in

Test images | Plain image | Cipher image |
---|---|---|

Lena | 7.229783 | 7.999247 |

Baboon | 7.183162 | 7.999378 |

Frog | 6.994144 | 7.999255 |

Goldhill | 7.705715 | 7.999229 |

Cat | 7.206827 | 7.999833 |

Landscape | 7.353287 | 7.999815 |

Truck | 7.343335 | 7.999842 |

Clown | 7.766840 | 7.999813 |

It is obvious that the entropies of the cipher images are very close to the ideal value, which means that the proposed algorithm is secure against entropy attacks.

A meaningful image has a property of strong correlation between adjacent pixels since its pixel values are close to each other. A cipher image with sufficiently low pixel correlation should be produced after the encryption. To evaluate the correlation coefficients for all the pairs of the adjacent pixels in diagonal direction, the following formula is used

where

diagonally adjacent pixels. The results of the correlation coefficients using four test images and their corresponding cipher images are given in

Generally, if only one pixel change in the plain image causes a significant change in the cipher image, then the image cryptosystem will resist the differential attack efficiently. Two common approaches, namely, NPCR (Number of Pixels Change Rate) and UACI (Unified Average Changing Intensity) are used to test the influence of only one pixel change in the plain image over the whole cipher image. They are defined [

Test images | Plain image | Cipher image |
---|---|---|

Flowers | 0.89561 | −0.00348 |

Cameraman | 0.89237 | 0.01957 |

Liberty statue | 0.95275 | −0.02114 |

Landscape | 0.97320 | 0.01522 |

where

With W and H are the width and height of the cipher image.

It is obvious that our proposed scheme is stable under NPCR and UACI analysis and highly sensitive at plain image even in the first iteration.

n | Proposed scheme | Reference [ | ||
---|---|---|---|---|

NPCR | UACI | NPCR | UACI | |

1 | 99.62 | 33.45 | 0.422 | 0.136 |

2 | 99.55 | 33.51 | 81.19 | 27.38 |

3 | 99.58 | 33.57 | 99.60 | 33.39 |

4 | 99.63 | 33.44 | 99.59 | 33.47 |

5 | 99.62 | 33.45 | 99.63 | 33.48 |

A powerful image cryptosystem should resist data loss during transmission.

For noise attack analysis, Lena image is encrypted and then “Salt & Pepper” with 1% noise is added to create noisy encrypted image. The result of the noise attack is shown in

In order to evaluate the running speed of the proposed cryptosystem, enough number of test images are encrypted. Then, we have analyzed the average encryption/decryption rate of the proposed algorithm on Intel Core i7 3.4 GHz CPU with 4 GB RAM running on Windows 7 by using MATLAB 7.9 software. The average execution time for the results can be found in

In this section, we will compare the performance of the SLM and ILM in the proposed cryptosystem with the same key parameters. The effects of these two maps on the cipher images will be evaluated under the same conditions. Histogram, entropy and correlation coefficient analysis are performed for the corresponding cipher images. Lena image is used for both encryption processes. Cryptosystem-1 uses the SLM as a key generator with the parameters of

From the histogram results, it is obvious that the output of the Cryptosystem- 1 is not as uniform as the Cryptosystem-2 and vulnerable to statistical attacks.

Entropy values and correlation coefficients between plain and cipher images are listed in

Visual and numerical results show that the positive contribution and validity of the ILM in the proposed cryptosystem.

Test images | Average confusion & diffusion time (sec) | Decryption time (sec) | Encryption rate (Mbps) | Decryption rate (Mbps) |
---|---|---|---|---|

128 × 128 | 0.0211 | 0.0238 | 6.21 | 5.50 |

256 × 256 | 0.0842 | 0.0947 | 6.22 | 5.53 |

512 × 512 | 0.3368 | 0.3804 | 6.22 | 5.51 |

1024 × 1024 | 1.3190 | 1.4869 | 6.35 | 5.64 |

Cryptosystem-1 | Cryptosystem-2 | |
---|---|---|

Entropy | 7.987539 | 7.999285 |

Correlation Coefficient | 0.01039 | −0.00141 |

An efficient gray image cryptosystem based on chaos is proposed in this paper. The entire range of the control parameter of the improved map can be used to build the key space due to the having unlimited value of control parameter. A small change in the plain image or any parameters of the cryptosystem will provide totally different keys even with the same encryption key is used. Both confusion and diffusion processes are iterated with different keys in order to get higher encryption strength of the cryptosystem. Security and performance analysis is performed numerically and visually. Both theoretical and simulation results are satisfactory and show that the proposed cryptosystem is highly secure thanks to its large key space, high sensitivity to the encryption keys and plain images. The implementation of the proposed cryptosystem using a digital hardware is possible direction for our future work.

Oğraş, H. and Türk, M. (2017) A Robust Chaos-Based Image Cryptosystem with an Improved Key Generator and Plain Image Sensitivity Me- chanism. Journal of Information Security, 8, 23-41. http://dx.doi.org/10.4236/jis.2017.81003