^{1, 2}

^{1}

^{1}

^{2}

In this paper, we have presented a new permutation-substitution image encryption architecture using chaotic maps and Tompkins-Paige algorithm. The proposed encryption system includes two major parts, chaotic pixels permutation and chaotic pixels substitution. A logistic map is used to generate a bit sequence, which is used to generate pseudorandom numbers in Tompkins-Paige algorithm, in 2D permutation phase. Pixel substitution phase includes two process, the tent pseudorandom image (TPRI) generator and modulo addition operation. All parts of the proposed chaotic encryption system are simulated. Uniformity of the histogram of the proposed encrypted image is justified using the chi-square test, which is less than

In any communication system, including satellite and internet, it is almost impossible to prevent unauthorized people from eavesdropping. When information is broadcasted from a satellite or transmitted through the internet, there is a risk of information interception. Security of image and video data has become increasingly important for many applications including video conferencing, secure facsimile, medical and military applications. Two main groups of technologies have been developed for this purpose. The first group is content protection through encryption, for which a key is required for proper decryption of the data. The second group is digital watermarking, which aims to embed a message into the multimedia data. These two technologies could be used complementary to each other [

In secured communications using encryption, which is the focus of the present work, the information under consideration is converted from the intelligible form to an unintelligible structure using certain operations at the transmitter. Data encryption is mainly scrambling the content of data, such as text, image, audio, and video to make the data unreadable, invisible or incomprehensible during transmission. The unintelligible or encrypted form of the information is then transmitted through the insecure channel, that is, internet, to the destination. At the intended recipient side, however, the information is again converted back to an understandable form using decryption operation and thus the information is conveyed securely. It should be noted that the same keys guide both these encryption and decryption operations. Such encryption system is grouped under private key cryptography [

In particular, an image-scrambling scheme transforms an image into another unintelligible image, based on keys only known to the senders and the receivers. The fundamental techniques to encrypt a block of pixels are substitution and permutation. Substitution replaces a pixel with another one; permutation changes the sequence of the pixels in a block to make them unreadable.

In recent years, chaotic maps have been employed for image encryption. Most chaotic image encryptions (or encryption systems) use the permutation-substitution architecture. These two processes are repeated for several rounds, to obtain the final encrypted image. For example, in [

There are however some other chaotic image encryption systems with different structures. For example, Pisarchik and Zanin suggested an algorithm to convert image pixels to chaotic maps coupled to form a chaotic map lattice. The encrypted image is obtained by iterating the chaotic map lattice with secret system parameters and number of cycles [

In this paper, a new permutation-substitution architecture using chaotic maps and Tompkins-Paige algorithm is proposed. Our designed technique for speech scrambling [

The paper is organized as follows. In Section

The word cryptography refers to the science of keeping secrecy of information exchanged between a sender and a receiver over an insecure channel. The objective is achieved by data encryption so that only individuals who have the key can decrypt it. The key

In practice, we need to transmit a reasonable amount of information, which requires a large sample space and that in turn implies a large number of keys. The distribution of a large number of keys is liable to cause horrendous management problems. In a practical system, a cryptanalyst will have to worry about time and facilities. Often, the time taken to solve a permuted sample will be of utmost importance. It is quite likely that the samples need to be secret for a limited period of time, referred to as required cover time. Thus, it is certainly possible for a theoretically insecure system to provide adequate practical security [

With the desirable properties of ergodicity and high sensitivity to initial conditions and control parameters, chaotic maps are suitable for various data encryption schemes. In particular, chaotic maps are easy to be implemented using microprocessors or personal computers. Therefore, chaotic encryption systems generally have high speed with low cost, which makes them better candidates than many traditional ciphers for multimedia data encryption. There are two types of chaotic encryption systems: chaotic stream encryption systems, and chaotic block encryption systems. In chaotic stream encryption systems, a key stream is produced by a chaotic map, which is used to encrypt a plain-text bit by bit. A chaotic block encryption system, on the other hand, transforms a plain-text block by block with some chaotic maps [

In this subsection, we consider nonlinear and chaotic one-dimensional maps

The iterative relation of the tent map is given by [

(a) Tent return map (

Logistic map is a one-dimensional quadratic map defined by

The logistic equation involves two multiplications and one subtraction per iteration, while the tent equation includes one division and on average one subtraction. Meanwhile, the tent map has better chaotic behavior than the logistic map. As mentioned above, the range of control parameter (

Due to the tight relationship between chaos and cryptography, the use of chaotic maps to construct an encryption system has been widely investigated [

Using chaos as a source to generate Pseudorandom bits with desired statistical properties to realize a secret permutation operation [

Using chaos as a source to generate Pseudorandom pixels with desired statistical properties to realize a secret substitution operation [

Using two chaotic maps in both permutation and substitution [

The fundamental techniques to encrypt a block of symbols are confusion and diffusion. Confusion can make ambiguous the relationship between the plain-text and the cipher-text. Diffusion can spread the change throughout the whole cipher-text. Substitution, which replaces a symbol with another one, is the simplest type of confusion, and permutation that changes the sequence of the symbols in the block is the simplest method of diffusion. These techniques together are still the foundations of encryption [

In designing private key cryptographic techniques, permutation methods are considered as important building blocks in conjunction with Pseudorandom sequence generators for selecting a specific permutation key. First, a

A permutation matrix is an identity matrix with the rows and columns interchanged. It has a single

Therefore, the

In cryptography, a substitution cipher is a method of encryption by which blocks of plain text are replaced with cipher-text according to a regular system; the blocks may be single or several letters. The receiver deciphers the text by performing an inverse substitution. Substitution ciphers can be compared with permutation ciphers. In a permutation cipher, the blocks of the plain-text are rearranged in a different and usually quite complex order, but the blocks themselves are left unchanged. By contrast, in a substitution cipher, the blocks of the plain-text are retained in the same sequence as in the cipher-text, but the blocks themselves are altered.

A permutation-only encrypted system is insecure against attacks [

The block diagram of the proposed chaotic image encryption system is illustrated in Figure

Block diagram of the chaotic image encryption system.

It is assumed that

There are three steps in the design of the permutation subsystem, which are explained as follows. First, a

The block diagram of the system is illustrated in Figure

Block diagram of the chaotic pixel permutation unit.

Chaotic pixel permutation is used as the target permutation matrix to implement 1D and 2D image permutation [

Generation of the chaotic random bit sequence is done as follows [

A sample histogram of a logistic map with

Let

Tompkins-Paige algorithm gives a one-to-one correspondence between the integers and the permutation. As an example, the simple permutation of nine elements of

In this example, the target permutation of these

In this paper, the Tompkins-Paige algorithm is applied to

As mentioned earlier,

The main idea behind the present work is that an image can be viewed as an arrangement of 2D pixels [

The image can be seen as a

In row permutation, according to Figure

In column permutation, the pixels of all columns are rearranged with respect to the permutation matrix/vector according to the

We extend the basic concept of

In identical permutation approach, the pixels of all rows are first rearranged, with respect to the permutation matrix/vector and the pixels of all columns are then rearranged with respect to the same permutation matrix/vector. The encrypted images that appear as a random noisy image are shown in Section

In a different permutation approach, the pixels of all rows are first rearranged, with respect to the first permutation matrix/vector and the pixels of all columns are then rearranged, with respect to the second permutation matrix/vector. The encrypted images that appear as a random noisy image are also shown in Section

In the permutation part of the system, pixel positions are displaced without changing their gray level values. Hence, the histogram of the permuted image is similar to the histogram of the plain-image. The permuted image however cannot resist against "statistical" and “known plain text” attacks [

Block diagram of the chaotic pixel substitution unit.

There are two main subunits in the substitution unit, tent Pseudorandom image generator and modulo addition. A

There are two options to generate a chaotic Pseudorandom image. A chaotic random generator along with a simple threshold detector similar to Section

Tent map is chosen as a chaotic system instead of a logistic map, since its probability density function (PDF) is uniform and implementation is almost simple.

Control parameter and initial condition of the map is determined by

Real values of chaotic sequences are generated by iterations of the map:

255 threshold levels in the range [

TPRI output seems to be a noisy image and its histogram is uniform.

It is desirable to decrease intelligibility of the encryption image. That is achievable with a substitute operation such that the final histogram becomes uniform and correlation between pixels is reduced. The permuted image could therefore be mixed with a noise image, TPRI. Modulo addition/subtraction is more suitable than XOR/XNOR operation. In this research, modulo

In the decryption side, to recover the

The proposed chaotic image encryption along with individual permutation and substitution has been simulated using MATLAB tools. In order to verify the exact operation of the proposed encryption system, and according to the process map of the system, that is, Figures

The proposed encryption system includes two major units, chaotic pixels permutation unit and chaotic pixels substitution unit. Three processes called logistic random bit sequence (LRBS) generator, Pseudorandom number calculator, and Tompkins-Paige algorithm are used to perform the pixel permutations.

First, the logistic map to generate a string of bits uses a

A sample of identity and permutation matrix.

Subsequently, as shown in Figure

Afterward, an

Row chaotic pixel permutation of Lena image.

Column chaotic pixel permutation of Lena image.

In Figure

Row-Column chaotic 2D pixels permutation of Lena image (identical permutation matrixes for row and column).

Row-Column chaotic 2D pixels permutation of Lena image (different permutation matrixes for row and column).

A

Finally, the

Proposed Chaotic Encrypted Image of Lena with 256 gray scales and its image histogram.

In this section, the performance of the proposed chaotic image encryption system is analyzed. The security analysis presented in this section is based on the performance of only one round of operation of the proposed encryption system including a

The histogram of the plain-image is illustrated in Figure

The proposed chaotic image encryption system should be resistant to statistical attacks. Correlation coefficients of pixels in the encrypted image should be as low as possible [

Comparison of correlation coefficients of the proposed methods.

Correlation coefficient | Plain-image (Figure | Proposed permuted image (Figure | Proposed tent Pseudorandom image (Figure | Proposed encrypted image (Figure |
---|---|---|---|---|

Horizontal (H) | ||||

Vertical (V) | ||||

Diagonal (D) | 0.769 | 0.054 | ||

Average (H, V, D) |

Meanwhile, the correlation coefficients of the proposed methods (Table

Comparison of correlation coefficient of the proposed method and the other methods.

Correlation coefficient | Mao et al. [ | Zhang et al. [ | Gao et al. [ | Zhou et al. [ | Proposed encrypted image (Figure |
---|---|---|---|---|---|

Horizontal (H) | |||||

Vertical (V) | |||||

Diagonal (D) | |||||

Average (H, V, D) |

The encrypted image should be significantly different to the original one. To quantify this requirement, three measures are used: mean absolute error (MAE), the number of pixel change rate (NPCR), and unified average changing intensity (UACI) [

The performance of each stage of the difference between permuted/encrypted and plain-images is measured by the mean absolute error (MAE) criterion in

A comparison of MAE of different methods.

Proposed Methods | MAE |
---|---|

Row Permutation | |

Column Permutation | |

Substitution and Permutation |

The NPCR is the percentage of corresponding pixels with different gray levels in two images. Let

Considering two Pseudorandom images, the expected value of NPCR is found to be

As shown in Table

Comparison of NPCR and UACI criteria of proposed method and the others.

Criteria (expected value) | Mao et al. [ | Zhang et al. [ | Gao et al. [ | Zhou et al. [ | Proposed method first round |
---|---|---|---|---|---|

NPCR ( | 37% | 21.5% | NA | 25.0% | 99.7% |

UACI ( | 9% | 2.5% | NA | 8.5% | 29.3% |

Key space should be sufficiently large to make brute-force attack infeasible. Key space is the total number of different keys that can be used in the encryption system. The keys of the proposed system in this paper consist of permutation key,

Comparison of Key length of proposed method and the others.

Key | Mao et al. [ | Zhang et al. [ | Gao et al. [ | Zhou et al. [ | Proposed Method |
---|---|---|---|---|---|

Length (Bin.) | |||||

Length (Dec.) | NA |

It is possible to increase the number of bits for total key in hardware implementation. However, by increasing the key length, volume of hardware is increased and consequently speed of the system is decreased. With respect to the speed of the today’s computers, the key space size should be more than

In this paper, we presented a new permutation-substitution image encryption architecture using chaotic maps and Tompkins-Paige algorithm. The proposed encryption system included two major parts, chaotic pixels permutation and chaotic pixels substitution. A logistic map was used to generate a bit sequence, which was in turn used to generate Pseudorandom numbers in Tompkins-Paige algorithm, in pixel permutation phase. Pixel substitution phase, included two processes, the tent Pseudorandom image (TPRI) generator and modulo addition operation. A tent map was used to produce a Pseudorandom image that was mixed with the permuted image.

The permutation and substitution operations needed two different keys,

The image was a

All parts of the proposed chaotic encryption system were simulated using a computer code. The histogram of the encrypted image was approximated a uniform distribution. The uniformity was justified by the chi-square test. Chi-square value shows that the distribution of the histogram of the encrypted image is uniform. The vertical, horizontal, and diagonal correlation coefficients, as well as their average and RMS values for the proposed encrypted image were calculated. The individual values and their average and RMS values of correlation coefficients were lower than the corresponding values from previous research by a factor between

To quantify the difference between encrypted image and corresponding plain-image, three measures were used: mean absolute error (MAE), number of pixel change rate (NPCR), and unified average changing intensity (UACI). It was concluded that the NPCR and UACI criteria of the proposed system were satisfactory when compared to other research results as was the security performance of the proposed system. All these results were obtained in only one round of encryption process.

The authors would like to thank Dr. Mehrnaz Shoushtarian, for her useful comments and suggestions.