This paper proposes a new family of 4-dimensional chaotic cat maps. This family is then used in the design of a novel block-based image encryption scheme. This scheme is composed of two independent phases, a robust light shuffling phase and a masking phase which operate on image-blocks. It utilizes measures of central tendency to mix blocks of the image at hand to enhance security against a number of cryptanalytic attacks. The mixing is designed so that while encryption is highly sensitive to the secret key and the input image, decryption is robust against noise and cropping of the cipher-image. Empirical results show high performance of the suggested scheme and its robustness against well-known cryptanalytic attacks. Furthermore, comparisons with existing image encryption methods are presented which demonstrate the superiority of the proposed scheme.

The rapid growth in multimedia applications has led to the vast spread of multimedia information across public networks. As a consequence, such information has become vulnerable to eavesdropping. Therefore, the need for safeguarding algorithms has become of major concern. Digital images are amongst the most popular digital media, they are found in a number of applications including military, medical and geographical applications. Due to this wide range of applications, research for developing efficient safeguarding algorithms has grasped the attention of scientists and engineers more than ever.

Cryptography is a field of mathematics and computer science that provide many security services including encryption and data hiding. Data hiding is a process that intends to hide secret information within cover media in such a way that an eavesdropper is incapable to detect the presence of such information within the carrier. On the contrary, encryption is a process that transforms secret information into scrambled data which is totally meaningless to an eavesdropper (Katz

Chaotic systems have a number of important characteristics such as high sensitive dependence on initial conditions and control parameters, large keyspace, unpredictability, ergodicity, and mixing property. Furthermore, with suitable control parameters and initial conditions, they can generate random looking sequences indistinguishable from random sequences. Confusion and diffusion are two important properties of any suitable encryption scheme (Shannon,

Among the large number of image encryption schemes that have appeared in the literature, security flaws in some of these schemes have been revealed by the cryptographic community. Furthermore, the rapid advancement of digital media technology demands the attention of researchers to develop fast and efficient image encryption schemes. Arnold’s cat map (Arnol’d and Avez,

The main contributions of this work are as follows:

The method is simple and efficient.

The encryption scheme is highly sensitive to its key and input image, while the decryption scheme is robust against various alternations such as noise and cropping of cipher-image.

The method is block-based. Based on the block size, there is a tradeoff between the security and the speed of the proposed scheme. However, simulations show that the chosen block size makes the scheme robust to existing attacks, insensitive to cipher-image attacks, and faster than existing schemes.

The paper is organized as follows: Section

Arnold’s cat map (Arnol’d and Avez,

We consider the following path to define a new

It is easy to see that each matrix

For all the experimental results presented in this work, we use the values

In this work, we propose an image encryption scheme that follows Fridrich’s approach. The proposed scheme consists of three phases (i) a preprocessing phase for reshaping the input image, (ii) a shuffling phase for destroying any correlation between adjacent intensity values, and (iii) a masking phase that acts on the shuffle-image to change its intensity values in such a way that a tiny change in one intensity value spreads out to almost all intensity values in the cipher-image. Algorithm

The proposed image encryption scheme Pr-IES

The size of the input image plays an important role in the performance of the proposed scheme. In the preprocessing phase, the input image

This phase aims to destroy correlations between adjacent pixels in the input image. It performs an

The shuffling of the matrix

To illustrate the shuffling phase, we present a one round toy example on the

Traverse

This gives the 1D array

The masking phase acts on the shuffled matrix

Generation of the scrambled matrix

In Bassham

Statistical Test Suite results for a matrix

Statistical test | Set of matrices | |

Result | ||

Frequency | 0.455937 | |

Block-frequency | 0.983453 | |

Cumulative-sums (forward) | 0.350485 | |

Cumulative-sums (reverse) | 0.383827 | |

Runs | 0.779188 | |

Longest-runs | 0.191687 | |

Rank | 0.616305 | |

FFT | 0.494392 | |

Non-overlapping-templates | 0.616305 | |

Overlapping-templates | 0.289667 | |

Universal | 0.494392 | |

Approximate entropy | 0.657933 | |

Random-excursions | 0.324180 | |

Random-excursions variant | 0.706149 | |

Serial 1 | 0.383827 | |

Serial 2 | 0.816537 | |

Linear-complexity | 0.213309 |

In this section, we showcase the efficiency of the proposed scheme. We then evaluate the randomness of cipher-images corresponding to standard test images. Furthermore, we consider cipher-images corresponding to bank of test plain-images.

This section shows the efficiency of the proposed image encryption scheme Pr-IES. Figure

Test plain-images Barbara of size

Figure

The shuffle-images corresponding to the test images Barbara, Lena and Elaine.

Figure

The shuffle-images (top) and cipher-images (bottom) for

Histogram analysis is an important test which shows the distribution of the intensity values of the pixels within an image. A secure image encryption scheme produces cipher-images whose pixel intensity values are uniformly distributed in the interval

The chi-square test results for the cipher-images corresponding to the test images Barbara, Lena and Elaine. This table also reports the chi-square value for a random image.

Cipher-image | |

Cipher-Barbara | 262.5859 |

Cipher-Lena | 248.8477 |

Cipher-Elaine | 222.1147 |

Random image | 235.4453 |

Histograms of the test images Barbara, Lena and Elaine (top) and their corresponding cipher-images (bottom).

A secure image encryption scheme generates cipher-images almost free of any correlation. The correlation coefficients

Table

Correlation coefficients of the test plain-images, shuffle-images and cipher-images for

Image | Adjacency | Plain-image | Shuffle-image | Cipher-image |

Barbara | Horizontal | 0.956279 | ||

Vertical | 0.971464 | 0.003786 | 0.007816 | |

Diagonal | 0.935520 | |||

Lena | Horizontal | 0.972826 | 0.006197 | 0.001692 |

Vertical | 0.986398 | 0.020036 | ||

Diagonal | 0.962357 | |||

Elaine | Horizontal | 0.994613 | 0.015765 | |

Vertical | 0.993920 | 0.008746 | ||

Diagonal | 0.989842 | 0.003508 |

Figure

Point plots of the intensity values of randomly chosen pairs of horizontally, vertically and diagonally adjacent pixels in the plain-image Lena (top), its corresponding shuffle-image (middle) and cipher-image (bottom).

Information entropy (Shannon,

Entropy measures for the test plain-images Barbara, Lena, Elaine and their corresponding cipher-images.

Image | Entropy | |

Plain-image | Cipher-image | |

Barbara | 7.6019 | 7.9971 |

Lena | 7.4455 | 7.9993 |

Elaine | 7.5029 | 7.9998 |

To further showcase the randomness of the proposed image encryption scheme we measure the entropy over local cipher-images blocks (Wu

Average entropy of image blocks.

Image | Plain-image | Cipher-image | ||||

Barbara | 5.7160 | 6.5322 | 7.0868 | 7.1766 | 7.8076 | 7.9549 |

Lena | 4.9910 | 5.6328 | 6.2260 | 7.1763 | 7.8098 | 7.9550 |

Elaine | 4.7618 | 5.3754 | 5.9626 | 7.1759 | 7.8095 | 7.9546 |

Random | 7.1750 | 7.8097 | 7.9542 | 7.1738 | 7.8090 | 7.9542 |

In this section, we evaluate the randomness of cipher-images generated by the proposed scheme Pr-IES using the STS proposed by the National Institute for Standards and Technology (NIST) (Bassham

Statistical Test Suite results for 100 cipher-images, each of length 2097152 bits.

Statistical test | Cipher-images | |

Result | ||

Frequency | 0.911413 | |

Block-frequency | 0.366918 | |

Cumulative-sums (forward) | 0.924076 | |

Cumulative-sums (reverse) | 0.851383 | |

Runs | 0.334538 | |

Longest-runs | 0.419021 | |

Rank | 0.816537 | |

FFT | 0.108791 | |

Non-overlapping-templates | 0.897763 | |

Overlapping-templates | 0.739918 | |

Universal | 0.994250 | |

Approximate entropy | 0.657933 | |

Random-excursions | 0.534146 | |

Random-excursions variant | 0.846579 | |

Serial 1 | 0.719747 | |

Serial 2 | 0.191687 | |

Linear-complexity | 0.289667 |

In this section, we report the running speed of the proposed image encryption scheme Pr-IES in MATLAB on a desktop machine with an Intel® Core™ i7-4770 processor and 8GB of memory, running Windows 10. Table

Running time of the proposed encryption scheme.

Size | Encryption time in seconds |

0.0644554 | |

0.2422222 | |

1.0021399 |

Encryption time versus image size.

In this section, we evaluate the security level of the proposed scheme. We show that the proposed scheme Pr-IES is highly sensitive to a slight modification in the plain-image. We further show that the scheme has a large keyspace, and it is highly sensitive to its secret key and control parameters. Moreover, we analyse the security of the proposed scheme under cipher-image scenario and chosen plain-image scenario. In addition to that, we demonstrate the robustness of its decryption to various alterations in the cipher-image.

Differential analysis of an image encryption scheme investigates the affect of a slight modification in the plain-image on the corresponding cipher-image. In this section, we measure the sensitivity of the proposed image encryption against slight modification in the plain-image. The Number of Pixels Change Rate (NPCR) and Unified Average Changing Intensity (UACI) are two measures used to evaluate the strength of image encryption schemes against differential attacks (Wu

According to Wu

Acceptance intervals for the null hypothesis with different levels of significance (Wu

Parameter | Size | 0.05-level | 0.01-level | 0.001-level |

NPCR | ||||

UACI | ||||

We evaluate the robustness of the proposed image encryption scheme by considering two plain-images

NPCR and UACI measures between cipher-images

Measures | Cipher-images of Barbara | Cipher-images of Lena | Cipher-images of Elaine | ||||||

Min | Mean | Max | Min | Mean | Max | Min | Mean | Max | |

NPCR | 99.5483 | 99.6093 | 99.6796 | 99.5819 | 99.6110 | 99.6399 | 99.5972 | 99.6089 | 99.6252 |

UACI | 33.2820 | 33.4913 | 33.6932 | 33.3231 | 33.4388 | 33.5638 | 33.4136 | 33.4649 | 33.5160 |

We further evaluate the robustness of the proposed scheme by subjecting each of the first 100 test images from (BOWS2) to the plain-image sensitivity test. For each plain-image we repeat the test 100 times, where each time we make a change to the least significant bit of a randomly chosen intensity value of the original plain-image. It turns out that the pass rate for the NPCR is

NPCR (left) and UACI (right) measures for plain-image sensitivity of the proposed scheme. Each point represent an NPCR/UACI measure resulting from repeating the test 100 for each test image in (BOWS2).

The secret key

Figure

The experimental results presented in Fig.

Bitwise xor between cipher-images

Histograms of the two images of Fig.

In this section, we show that the proposed scheme is robust against cipher-image and plain-image analysis. In a cipher-image attack, the intruder has only access to the cipher-image. Since the above tests show that no useful information about the plain-image can be gained from the corresponding cipher-image, we conclude that the proposed scheme is robust against this type of attack. In a plain-image attack, the intruder can choose any part of the plain-image and request its corresponding cipher-image part. The aim of this attack is to reconstruct some other plain-image parts. The fact that the chaotic map possesses the one-way property due to floating point errors makes the inverse computation very difficult. Furthermore, since the proposed scheme is highly dependent on its secret key, one cannot predict further outputs of the 4D cat map. Thus, the scheme is robust against this type of attacks.

NPCR and UACI measures between cipher-images

Measures | Cipher-images of Barbara | Cipher-images of Lena | Cipher-images of Elaine | |||

NPCR | 99.6338 | 99.6124 | 99.6002 | 99.6101 | 99.6215 | 99.6066 |

UACI | 33.6906 | 33.4831 | 33.3946 | 33.4988 | 33.4807 | 33.4435 |

Earlier, we have shown that the proposed scheme is highly sensitive to its secret key, and it is also highly sensitive to a tiny change in its input plain-image. That is, a change in a plain-image intensity value spreads over all intensity values in the corresponding cipher-image. In this section, we show that a change in intensity values in the cipher-image affects only few intensity values in the corresponding plain-image. The importance of this feature is that with distortion of cipher-images due to salt and pepper noise or data loss one can still successfully recover the corresponding plain-image. Figure

The reconstructed plain-image Lena resulting from subjecting its corresponding cipher-image to a

The reconstructed plain-image Lena (bottom) resulting from subjecting its corresponding cipher-image to a

In this section, we compare the performance of the proposed scheme Pr-IES with existing ones. Figure

The NPCR and UACI pass rates of the proposed scheme and some existing schemes. The pass rates for the schemes under comparison are quoted from Hua

Scheme | Pass rate | |

NPCR | UACI | |

WWZ (Wang |
||

ZBC1 (Zhou |
||

XLLH (Xu |
||

LSZ (Liu |
||

HZ (Hua and Zhou, |
||

ZBC2 (Zhou |
||

WZNA (Wu |
||

CSL (Cao |
||

HZH (Hua |
||

Pr-IES |

NPCR (left) and UACI (right) measures for cipher-images generated by existing image encryption schemes and the proposed scheme. The measures for existing schemes are obtained from Hua

We further compare the correlation coefficients between adjacent pixels of the proposed scheme and existing ones. Figure

Adjacent intensity values correlation coefficients for the test image Lena and corresponding cipher-images generated by existing schemes and the proposed scheme. The values for existing schemes are obtained from Hua

Table

Running time in seconds for encrypting a single image by existing schemes and the proposed encryption scheme Pr-IES. The running times for the schemes under comparison are quoted from Hua

Image size | ||||

(Diaconu, |
0.0579 | 0.2224 | 0.9731 | 3.8377 |

(Ping |
0.0902 | 0.3440 | 1.3357 | 5.3223 |

(Chai |
0.2757 | 0.9810 | 3.8539 | 15.4565 |

(Hua and Zhou, |
0.1531 | 0.6347 | 2.4913 | 9.9185 |

(Xu |
0.0247 | 0.1164 | 0.4924 | 20.144 |

(Zhou |
0.0933 | 0.3843 | 1.4824 | 5.8175 |

(Liao |
0.0323 | 0.1440 | 0.5510 | 2.0864 |

(Hua |
0.0244 | 0.0949 | 0.4010 | 1.9857 |

Pr-IES | 0.0217 | 0.0645 | 0.2422 | 1.0021 |

It is evident from the obtained results that the proposed scheme has superiority over existing schemes and competitive with others.

We propose a new family of 4D chaotic cat maps. As an application of these maps, we present a novel block-based image encryption scheme utilizing them. This scheme consists of a light shuffling phase and a masking phase which uses measures of central tendency for mixing the image blocks. While encryption is highly sensitive to the secret key and the input image, decryption is robust against noise and cropping of the cipher-image. Simulations show that the proposed scheme generates cipher-images possessing high randomness properties. Furthermore, the scheme is shown to be robust against differential cryptanalysis. With respect to existing works, the proposed scheme is shown to have superior performance over existing image encryption algorithms and to be competitive with others.

The authors are grateful to the anonymous referees whose remarks helped improve the presentation of this work.