A Novel Strategy for Quantum Image …read.pudn.com/downloads665/ebook/2696136/A Novel Strategy...image information to Moir´e grating, and made the Moir´e image steganography free

Int J Theor PhysDOI 10.1007/s10773-014-2294-3

A Novel Strategy for Quantum Image SteganographyBased on Moire Pattern

Nan Jiang ·Luo Wang

Received: 11 April 2014 / Accepted: 1 August 2014© Springer Science+Business Media New York 2014

Abstract Image steganography technique is widely used to realize the secrecy transmis-sion. Although its strategies on classical computers have been extensively researched, thereare few studies on such strategies on quantum computers. Therefore, in this paper, a novel,secure and keyless steganography approach for images on quantum computers is proposedbased on Moire pattern. Algorithms based on the Moire pattern are proposed for binaryimage embedding and extraction. Based on the novel enhanced quantum representation ofdigital images (NEQR), recursive and progressively layered quantum circuits for embed-ding and extraction operations are designed. In the end, experiments are done to verify thevalidity and robustness of proposed methods, which confirms that the approach in this paperis effective in quantum image steganography strategy.

Keywords Image steganography · Moire pattern · Quantum circuit · Quantumcomputation · Quantum image processing

1 Introduction

A quantum computer based on quantum computation guarantees high-efficiency on pro-cessing tasks which are believed to be difficult to be solved on classical computers [1–4].Therefore, there is vast potential for future development for quantum computation forvarious realms, including quantum image processing.

Image steganography, a subliminal channel technique that hides a private image insidea public image, is applied for secrecy in transmission [5–8]. Different methods to realizeimage steganography have been proposed [5–8]. Among them is image steganography basedon Moire pattern [9–12].

N. Jiang (�) · L. WangCollege of Computer, Beijing University of Technology, Beijing 100124, Chinae-mail: [email protected]

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The Moire effect is an optical phenomenon which occurs when repetitive structures (suchas screens, grids or gratings) are superposed or viewed against each other [13]. That is tosay, under certain condition, a new design occurs and can be recognized by naked eyes whenone pattern presented by a structure is superposed onto another. Researchers utilized theprinciple and properties of Moire patterns to perform image steganography tasks [9–12].

Summarizing the existing researches for image steganography based on Moire pattern[9–12], we find that image steganography based on Moire pattern is a two-step process.First, an embedding operation is employed to embed a secret image (the image that to behidden) into an initial Moire pattern (the original cover image) and results in a Moire pat-tern, which means that the initial Moire pattern is changed according to the secret image,eventually the resulting image is called a Moire pattern. Second, an extraction operation isutilized to restore the secret image from dealing with the initial Moire grating and the Moirepattern.

Munoz-Rodrıguez et al. [9] proposed the principle and model of Moire image steganog-raphy and presented a computational algorithm to get the Moire pattern by adding theimage information to Moire grating, and made the Moire image steganography free ofoptical instruments [9]. Ragulskis et al. [10] raised the concept of stochastic Moire grat-ing and advanced the algorithm in [10] through changing the regular Moire grating tostochastic Moire grating. Ragulskis et al. [11] put forward the Moire image steganographyusing the time-average Moire pattern [11]. Sakyte et al. [12] presented the Moire imagesteganography based on the near-optimal Moire pattern [12].

However, image steganography strategies based on Moire pattern for quantum computershave rarely been researched. Accordingly, this paper compensates this blank by design-ing a steganography algorithm and corresponding quantum circuits, which hides a binaryimage into a gray-scale image. On one hand, the embedding algorithm begins with choos-ing an initial Moire grating, i.e. a stochastic image, as the cover image. Then deform theinitial Moire grating according to the secret image and the deformed Moire grating is theMoire pattern. Finally, alter the Moire pattern to get the stego image. On the other hand,an extraction algorithm is employed to regain the secret image by manipulating the ini-tial Moire grating and the stego image. The quantum image representation we used isNEQR [14].

The rest of the paper is organized as follows. Basic ideas and relative backgrounds arepresented in Section 2. A method for Moire image steganography and corresponding quan-tum circuits are offered in Section 3. In Section 4, some theoretical analyses of the effectof the image steganography strategy by doing essential experiments is provided. Finally, ashort conclusion is given in Section 5.

2 Data Representation

This section offers necessarily quantum data representations in this paper. Section 2.1 pro-vides the principle of the NEQR, the quantum image representation approach utilized here.Section 2.2 introduces the variables used in the proposed strategy.

2.1 The Representation of Quantum Images

In order to represent images on quantum computers, the novel enhanced quantum represen-tation of digital images (NEQR) proposed in [14] is introduced. The NEQR model containsthe color information and corresponding position information of every pixel in an image. It

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uses two entangled qubit sequences to store the gray-scale and site information, and storesthe whole image in the superposition of the two qubit sequences.

In [14], (Y,X) is employed to represent the position information of a pixel. Zhang andet al. reported that when the gray range of an image has a value of 2q , a binary sequenceC0YXC

1YX . . . C

q−2YX C

q−1YX encodes the gray-scale value f (Y,X) of the corresponding pixel

(Y,X) as in (1):

f (Y,X) = C0YXC

1YX . . . C

q−2YX C

q−1YX , Ck

YX ∈ {0, 1} (1)

The representative expression of a quantum image for a 2n × 2n, where n is an integer andn > 0, image can be written as in (2) [14]:

|I 〉 = 1

2n

2n−1∑

Y=0

2n−1∑

X=0

|f (Y,X)〉|YX〉 = 1

2n

2n−1∑

Y=0

2n−1∑

X=0

q−1⊗i=0

|CiYX〉|YX〉 (2)

Figure 1 shows a 4 × 4 NEQR image model.The NEQR employs the basis state of qubit sequence to represent the gray scale of pixels

[14]. Different gray scales can be distinguished in NEQR because different basis states ofqubit sequence are orthogonal [14].

2.2 Data Notations

1. Secret image f is the image that to be hidden. It is a 2n×2n binary image with f (Y,X)

denoting the gray-scale of corresponding pixel (Y, X).2. Initial Moire grating I1 is the original cover image. It is a 2q -gray-scale

image, that can be selected freely by users. I1(Y, X) is represented asC0I1(Y,X)

C1I1(Y,X)

. . . Cq−2I1(Y,X)

Cq−1I1(Y,X)

.3. Stego image I2 is the cover image that has already accommodated a secret image. It is

represented as C0I2(Y,X)

C1I2(Y,X)

. . . Cq−2I2(Y,X)

Cq−1I2(Y,X)

.4. Decrypted image fd is the restored secret image extracting from I2.

Fig. 1 A 4 × 4 NEQR quantumimage. f (Y,X) denotes thegray-scale value of pixel (Y,X),which is stored as the basis state|f (Y,X)〉 of a qubit sequence

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3 Image Steganography Based on Moire Effect

3.1 Embedding Operation

The embedding operation of the image steganography strategy based on the Moire patternis designed as the following.

(1) Choose an initial Moire grating as a cover image. It can be selected freely by users.(2) Use deformation operation to create the Moire pattern by dealing with the initial Moire

grating and a secret image.(3) Use denoising operation to transform the Moire pattern into the stego image.

3.1.1 Deformation Operation

The deformation operation is shown in (3) which is the computational algorithm of gettingthe Moire pattern from embedding a secret image f into an initial Moire grating I1.

I1(Y,X) =⎧⎨

⎩

I1(Y, X), f (Y,X) = 0I1((Y − 1),X), f (Y,X) = 1 and Y > 0I1((2q − 1),X), f (Y,X) = 1 and Y = 0

(3)

The effect of (3) is that when f (Y,X) = 0, I1(Y, X) remains unchanged; otherwise, useI1((Y − 1),X) to substitute for I1(Y, X), i.e., use the previous pixel to substitute for thecurrent pixel. In fact, the bottom two lines of (3) can be merged into the form of I1((Y −1)mod2q,X) because if Y = 0,

I1((Y − 1)mod2q,X) = I1((−1)mod2q,X) = I1((2q − 1),X).

When (3) is calculated, the secret image has been embedded into an initial Moire grating,and the initial Moire grating becomes a Moire pattern.

Fig. 2 The circuit for Deformation operation

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The quantum circuits of the Deformation operation is summarized as a mod-ule Def ormation(q) which uses C-SWAP gates to exchange I1(Y,X) and I1((Y −1)mod2q,X) when f (Y,X) = 1 as shown in Fig. 2.

3.1.2 Denoising Operation

When extracting, the value of a watermark bit is depended on whether I2(Y,X) = I1(Y, X)

or not (see Section 3.2). If I2(Y,X) = I1(Y,X), f d(Y,X) = 0, otherwise fd(Y,X) = 1.However, there is a big problem which makes lots of noise when (5) (will be described inthe following) is used to get the restored secret image. That is because if two conditions aresatisfied:

(1) f (Y,X) = 1: according to (3), I2(Y, X) = I1((Y − 1)mod2q, X);(2) I1(Y, X) = I1((Y − 1)mod2q,X): then I2(Y, X) = I1(Y, X),

according to (5), fd(Y,X) = 0, which contradicts f (Y,X) = 1.Hence, a Denoising operation was added in the Embedding operation to eliminate noise.

If f (Y,X) = 1 and I1(Y,X) = I1((Y − 1)mod2q,X), in order to avoid the possibility ofI2(Y,X) = I1(Y, X), the last qubit of I2 is inverted, as shown in (4).

Cq−1I1(Y,X)

= 1 − Cq−1I1(Y,X)

, if f (Y,X) = 1 and I1(Y, X) = I1((Y − 1)mod2q,X)) (4)

The quantum circuits of the denoising operation is summarized as a moduleDenoising(q) which consists 4 steps as shown in Fig. 3.

(1) Judge whether the condition in (4) is satisfied by adding q Toffoli gates. If I1(Y,X) =I1((Y − 1)mod2q,X), all the qubits Ck

I1((Y−1)mod2q ,X), k = 0, 1, . . . , q− 1 will be set

to 0.(2) Add q NOT gates to inverse Ck

I1((Y−1)mod2q ,X), k = 0, 1, . . . , q − 1. Accordingly,

if the condition in (4) is satisfied, all the qubits f (Y,X) and CkI1((Y−1)mod2q ,X)

, k =0, 1, . . . , q − 1 will be 1.

Fig. 3 The circuit for denoising operation

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(3) Add a Toffoli gate to inverse the supplementary |0〉 input into |1〉 if the condition in(4) is satisfied.

(4) Add a C-NOT gate to inverse Cq−1I1(Y,X)

if the supplementary qubit is 1.

3.1.3 Integrated Embedding Circuits

The integrated quantum circuit, i.e. the Embedding(q) module, is the series connection ofthe Def ormation(q) and the Denoising(q) as shown in Fig. 4.

Figure 5 gives an example for Embedding operation which uses 8, i.e., q = 3, gray-scaleinitial Moire grating.

3.2 Extraction Operation

This operation aims at getting the decrypted image fd from I1 and I2. According to theembedding operation, all the pixels should be checked whether I2(Y,X) = I1(Y,X). If so,fd(Y,X) = 0; otherwise, fd(Y,X) = 1 as shown in (5).

fd (Y,X) ={

0, I2(Y,X) = I1(Y,X)

1, I2(Y, X) �= I1(Y,X)(5)

The Extraction quantum circuits are defined as the Extraction(q) module. The inputsinclude f (Y,X), I1(Y,X), I2(Y, X) and an extra 1 input, and (q) represents the gray-scaleof I1(Y,X) and I2(Y, X). The output is fd(Y,X) coming from the extra 1 input. The processof circuits generation is given below.

(1) Add q C-NOT gates to compare I2(Y,X) and I1(Y, X) qubit by qubit. If I2(Y,X) =I1(Y, X), all the target qubits C0

I2(Y,X), C1I2(Y,X), . . . , C

q−2I2(Y,X), C

q−1I2(Y,X) will be set to

0.(2) Add q NOT gates to invert C0

I2(Y,X), C1

I2(Y,X), . . . , C

q−2I2(Y,X)

, Cq−1I2(Y,X)

.(3) Add a Toffoli gate to gain fd(Y,X). If I2(Y,X) = I1(Y,X), the control qubits

C0I2(Y,X), C

1I2(Y,X), . . . , C

q−2I2(Y,X), C

q−1I2(Y,X) are all 1 and the target qubit is inverted to 0;

otherwise, the target qubit remains unchanged 1 (Fig. 6).

An example is given for using 8 gray-scale initial Moire grating as shown in Fig. 7.

Fig. 4 The circuit for embedding operation

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Fig. 5 An example using 8 gray-scale initial Moire grating for the circuit for embedding operation

4 Simulation-Based Experiments and Analysis

This section gives some simulation-based experiments and analysis of the results and per-formance of the proposed steganography strategy. All experiments are simulated on theMATLAB 7.12.

4.1 Visual Effects

Figure 8 gives some examples to show the visual effects of the proposed scheme. (a)-(d)and (e)-(h) utilize “Lena” and “Cameraman” as the cover image, i.e. initial Moire grating,respectively.

PSNR (Peak Signal-to-Noise Ratio) is applied to display the accuracy of proposedsteganography methods. Assuming that two m × n images I and J , I (Y,X) and J (Y,X)

representing the pixel value of pixel (Y,X) where the max pixel value is 2q−1, MSE (MeanSquared Error) is defined as (6), based on which PSNR is defined as (7) [15].

MSE = 1

mn

m−1∑

i=0

n−1∑

j=0

[I (i, j)− J (i, j)]2 (6)

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Fig. 6 The circuit for extractionoperation

PSNR = 20 × log10

(2q − 1√MSE

)(7)

In Fig. 8 the PSNR values for pairs of initial Moire grating and corresponding stegoimage is calculated. It can be seen that although the PSNR is about 30db, the cover imageand the stego image are identical to human eyes. Furthermore, in general scenario, ordinaryusers can only get the Moire pattern which looks more normal without the comparison ofthe initial Moire grating. Users will not suspect that some information are hidden in theimage they get.

4.2 Robustness Analysis

If there are no noises and attacks, the effect of the steganography based on Moire pattern isexcellent and error-free. This section discusses what happens if there are some noises andattacks. After extracting the decrypted image from the attacked stego image, BER (Bit ErrorRate) is calculated to analyze the robustness of the steganography.

4.2.1 Robustness Performance under Noises

Adding noise is a common operation. Many image processing softwares, printing and scan-ning operations, and so on can add noise into an image intentionally or unintentionally. In

Fig. 7 An example using 8 gray-scale initial Moire grating for thecircuit for extraction operation

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Fig. 8 Examples using 256 gray-scale initial Moire grating for 256 × 256 binary image steganography. a tod are the case where “Lena” as the initial Moire grating and e to h are the example where “Cameraman” asthe initial Moire grating

this section, salt & pepper noises are applied with different density into 256 × 256 “Lena”stego image, as shown in Fig. 9a–d. The corresponding extracted results from the noisedstego images are shown in Fig. 9e–h.

The robustness of the proposed quantum steganography method under the salt & peppernoise is high, because the extracted results can be easily recognized, although the stegoimages have been in great chaos, especially the comparison between (c) and (g), and (d) and

Fig. 9 Stego images and the corresponding extracted results with salt & pepper noises for the density of0.05, 0.10, 0.15 and 0.20

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Fig. 10 The relationship between the value of BER and the density of the salt & pepper noise

(h) in Fig. 9. The relationship between the value of BER and the noise density is shown inFig. 10.

It can be seen that the relationship is nearly linear. It is worthwhile to note that thegradient is less than 1, about 0.3

0.5 = 0.6. That is to say, although the salt & pepper noiseaffects approximately density ∗ 100 % pixels of the stego image, only 60 % of them aremaintained in the decrypted image.

The reason to this phenomenon is explained as the following. According to (5), theextraction operation determines the value of fd(Y,X) as 0 if I2(Y,X) is equal to I1(Y, X),otherwise determines fd(Y,X) as 1. If I2(Y, X) and I1(Y, X) are unequal both before andafter adding noise, the extracted value fd(Y,X) is not erroneous.

Fig. 11 Stego images and the corresponding extracted images being cropped 25, 50, 75 and 100 lines

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Fig. 12 The relationship between the value of BER and the number of lines cropped

4.2.2 Robustness Performance under Attack

Cropping is a common attack. In this section, cropping is being carried on nonadjacentparallel lines, where lines being cut off are replaced by those nonadjacent parallel linesin (b) of Fig. 8. In the experiment, 0 to 100 lines are considered. Figure 11a–d show thecropped stego images. The corresponding extraction results are given in Fig. 11e–h.

It can be seen that with some lines being cropped, some nonadjacent parallel black linesattach to all restored images in Fig. 11. Although black parallel lines appear, the contentand meaning of the secret image still can be observed easily after being cropped, even if100256 ≈ 39 % of the stego image is cropped (see Fig. 11h).

The relationship between the value of BER and the number of lines cropped is shownin Fig. 12 which indicates that the quantum image steganography strategy has a won-derful robustness when suffers cropping by lines. The BER value is just over 0.1 evenwhen the number of lines being cropped reaches 100. The color value of pixel (Y, X)

of the restored image relies on if the corresponding grey-scale value of the cover imageand the stego image are the same. Therefore, if a line of the cover image replacesthe corresponding line of the stego image, all pixels of the line of the restored imagebecome black. However, nonadjacent black lines do not affect the visibility of the restoredimage.

5 Conclusion

In this paper, a quantum image steganography approach based on Moire pattern in theform of quantum circuits is proposed. The Embedding circuit embeds the secret binaryimage into an initial Moire grating to generate the Moire pattern, while the Extraction cir-cuit gets restored binary image according to initial Moire grating and the Moire pattern.Simulation-based experiments show that the visual effects and robustness of our approachesare dependable.

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Acknowledgments This work is supported by the Beijing Municipal Education Commission Science andTechnology Development Plan under Grants No. KM201310005021, and the Graduate Technology Fund ofBJUT under Grants No. YKJ-2013-10282.

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