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Dynamic watermarking scheme for quantum images bad on Hadamard transform
Xianhua Song •Shen Wang •Ahmed A.Abd El-Latif •
Xiamu Niu
Received:23May 2013/Accepted:15January 2014ÓSpringer-Verlag Berlin Heidelberg 2014
Abstract In this paper,a novel watermarking scheme for quantum images bad on Hadamard transform is pro-pod.In the new scheme,a unitary transform controlled by a classical binary key is implemented on quantum image.Then,we utilize a dynamic vector,instead of a fixed parameter as in other previous schemes,to control the embedding process.The dynamic embedding vector is decided by both the carrier quantum image and the watermark image,which is only known by the authorized owner.The propod scheme is analyzed from visual quality,computational complexity,and payload capacity.Analysis and results show that the propod scheme has better visual quality under a higher embedding capacity and lower complexity compared with other schemes pro-pod recently.
Keywords Quantum computation ÁHadamard transform ÁQuantum image ÁWatermarking
1Introduction
For images in conventional computers,there have been many image watermarking algorithms [1,2].Recently,quantum computation has been applied in many fields of computer science [3].The rapid development of quantum computation and quantum computer rais people’s interest to rearch quantum data curity.As with clas-sical data on classical computers,the quantum data such as images will be susceptible to different kinds of abus.In order to guard against their abu,extending similar techniques to quantum images appears necessary.How-ever,quantum images are reprentation patterns on quantum computers [4];the preparation and any pro-cessing about them must satisfy the principles of quantum mechanism.For normal images in classical computers,images are stored as matrices and the processing and watermarking algorithms are bad on matrix operations,but for quantum images,the realization and analys about them rely on the design of quantum circuits Fur-thermore,all operations in quantum computation are unitary transforms described by unitary matrices.The nature of unitary in quantum computation is totally dif-ferent from the classical computations.Therefore,the traditional image watermarking algorithms cannot be directly extended to quantum image watermarking.
Until the arrival of practical quantum computers,the first stage in this direction is building the proposal for capturing and storing the images on quantum computers.Some methods for reprenting quantum images have been propod.The quantum image is reprented by a color that is detected from monochromatic electromagnetic waves through special machines and position.The storing unit was named qubit lattice [5].Latorre [6]brought out a novel method which mapped pixels into the Real Ket of
Communicated by T.Plagemann.
同学聚会祝酒词X.Song ÁS.Wang (&)ÁA.A.Abd El-Latif ÁX.Niu
School of Computer Science and Technology,Harbin Institute of Technology,Harbin 150080,China e-mail:shen.wang@hit.edu
X.Song
Department of Applied Mathematics,Harbin University of Science and Technology,Harbin 150080,China A.A.Abd El-Latif
Department of Mathematics,Faculty of Science,Menoufia University,Shebin El-Koom 32511,Egypt
Multimedia Systems
DOI 10.1007/s00530-014-0355-3
the Hilbert space to complete image compression com-bined with pixel states.Aflexible reprentation of quan-tum image(FRQI)[4]encoded the image color and position into one quantum state which keeps the classical properties of color and position.Bad on FRQI,a multi-channel reprentation for quantum images using RGB a color space is propod in[7].Moreover,Zhou et al.[8] changed the gray-scale of the quantum image in FRQI into the qubit binary string when they designed the quantum image encryption algorithm.An enhanced quantum repre-ntation(NEQR)for digital images is propod in[9], which improves the reprentation of FRQI.In[10],the authors focus on the quantum image reprentation using qutrits(3-level quantum systems).Moreover,a quantum image reprentation for log-polar image(QUALPI)is propod for the storage and processing of images sampled in log-polar coordinates[11].
With the study of the reprentations for quantum images,people begin to rearch quantum image water-marking strategies.Bad on FRQI,Iliyasu et al.[12] propod a quantum images watermarking and authenti-cation(WaQI)algorithm by means of restricted geometric transformations[
13].The strategy is cure,keyless,and blind.In[14],Zhang et al.propod a watermarking scheme for quantum images bad on quantum Fourier transform(QFT)and FRQI.The watermark strategy is ud tofind the owner of the carrier image efficiently.However, the whole computation process of scheme[14]cannot keep the images in a normalized quantum state,while normali-zation is a basic property of quantum state.The reason is that their embedding parameter is afixed parameter. Moreover,the complexities of[12,14]are both O(n2)for an2n92n sized image,which are not efficient enough.
In this paper,a novel dynamic watermarking scheme for quantum images bad on an elementary quantum gate, Hadamard gate[15],is propod.The propod scheme utilizes a dynamic matrix to control embedding process during the embedding process,instead of afixed parameter as in other schemes.The matrix is an optimal solution of an optimization equation which is built to make the water-marked image have better visual quality.Simulations conducted show that the propod scheme has better visual quality under a bigger capacity compared to other schemes. Moreover,the propod scheme has lower computational complexity compared to the existing schemes[12,14].
The rest of this paper is organized as follows:Sect.2 gives a brief background of quantum computation,the flexible reprentation of quantum images(FRQI).The propod dynamic watermark
ing scheme is given in Sect.
3.Section4is devoted to simulation results and analysis. Finally,Sect.5concludes this paper.2Background on quantum computation, reprentation of quantum images
In this ction,we give a brief overview of quantum computation and theflexible reprentation of quantum images[4],which are the basis for the propod scheme.
2.1Background on quantum computation
A quantum computer is a physical machine that can accept input states which reprent a coherent superposition of many different possible inputs and subquently evolve them into a corresponding superposition of outputs. Quantum ,a quence of unitary trans-formations,affects simultaneously each element of the superposition,generating a massive parallel data process-ing albeit within one piece of quantum hardware[15].A quantum bit,or qubit,is a unit vector in a two-dimensional complex vector space for which a particular basis,denoted by0j i;1j i;has beenfixed.A quantum circuit provides a visual reprentation of how a complicated multi-qubit quantum computation can be decompod into a quence of simpler,usually1-qubit and2-qubit,quantum gates.In general,a given unitary matrix,which specifies some desired quantum computation,will admit
many different, but equivalent,decompositions depending on the t of primitive quantum gates ud,and the skill of the quantum circuit designer in composing tho gates in an intelligent way.Some notations for the elementary quantum gates and their corresponding matrices are shown in Fig.1.
In our watermarking scheme given below,quantum networks for arithmetic operation should be designed.As we all know,for classical computation,it is easy to implement arithmetic operations.However,any unitary operation in quantum computer is invertible.That is why quantum networks effecting elementary arithmetic opera-tions such as addition,multiplication,and exponentiation cannot be directly deduced from their classical Boolean counterparts.
An explicit construction of veral elementary quantum ,plain adder,adder modulo N,controlled-multiplier modulo N,and exponentiation modulo N is designed in[16].A quantum network is a quantum com-puting device consisting of quantum logic gates who computational steps are synchronized in time.The outputs of some of the gates are connected by wires to the inputs of others.Inputs are encoded in binary form in the compu-tational basis of lected qubits often called a quantum register,or simply a register.The addition of two registers a j i and b j i can be written as Eq.(1)
a;b
j i!a;aþb
j i:ð1Þ
X.Song et al.
To prevent overflows,the cond register (initially loa-ded in state b j i )should be sufficiently ,if both a and b are encoded on n qubits,the cond register should be of size n ?1.In addition,the network described here also requires a temporary register of size n -1,initially in state 0j i ;to which the carries of the addition are
provisionally written.The basic sub networks of carry and
sum operations for plain addition network are demon-strated in Fig.2.And the operation of the full addition network is illustrated in Fig.3.If we rever the action of the above network (i.e.,if we apply each gate of the net-work in the reverd order)with the input (a ,b ),the output will produce (a ,a -b )when a C b .When a \b ,
the
Fig.1Notations and
corresponding matrices for basic quantum gates.a NOT gate,b pha gate,c controlled NOT gate,d Toffoli gate,e Hadamard
gate
Fig.2Basic sum and carry operations for plain addition network.a The sum operation,b the carry operation [16
]
Fig.3Plain adder network.The position of a thick black bar on the right-or left-hand side of basic carry and sum networks.A network with a bar on the left side reprents the reverd quence of elementary gates embedded in the same network with the bar on the right side [16]
Dynamic watermarking scheme for quantum images
法文output is (a ,2n ?1-(b -a )),where n ?1is the size of the cond register.
2.2Flexible reprentation of quantum image
Inspired by the pixel reprentation for images in classical computers,a flexible reprentation for quantum images on quantum computers is propod in [4],in which the quantum image corresponding to a classical image of size 2n 92n is defined by a quantum encoding state about image color and ,
I ðh Þj i ¼12X
22n
À1i ¼0ðcos h i 0j i þsin h i 1j iÞ i j i ;h i 20;p 2
h i ;i ¼0;1;...;22n À1;
ð2Þ
wherein,cos h i 0j i ?sin h i 1j i encodes the color information and i j i encodes about the corresponding positions of the quantum images.The position information includes two parts:the vertical and horizontal coordinates.Considering a quantum image in 2n -qubit system,
i j i ¼y j i x j i ¼y n À1y n À2;...;y 0j i x n À1x n À2;...;x 0j i ;x ;y 20;1;...;2n À1f g ;ð3Þ
y j ;x j
2f 0j i ;1j ig ;j ¼0;1;...;n À1;
here y n À1y n À2...y 0j i encodes the first n -qubits along the
vertical location and x n À1x n À2...x 0j i encodes the cond n -qubits along the horizontal axis.The FRQI state is a normalized ,
I ðh Þj i k k ¼12ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX
22n À1i ¼0ðcos 2h i þsin 2h i Þv u u t ¼1:ð4Þ
Furthermore,the authors in [4]propod a polynomial
preparation theorem which proves the existence of a uni-tary preparation process using a polynomial number of simple operators transforming quantum computers from the initial state to FRQI state.
3Propod watermarking scheme
Here,we propo a new quantum watermarking scheme for
quantum images bad on Hadamard transform.In the embedding algorithm,we propo a dynamic vector a ,which is solved by an optimization equation to ensure the best image quality.The quantum circuit of the scheme is compod of the simplest quantum ,Hadamard gate,NOT gate,Pha gate,controlled NOT gate,and Toffoli gate.The concrete procedures of the embedding and the extraction are given hereafter.3.1Watermark image’s embedding procedure
The propod watermarking embedding procedure is given by the following steps and can be en in Fig.4.3.1.1Quantum images preparation
Given a classical carrier quantum image of size 2n 92n ,who FRQI is
C j i ¼12n X 22n
À1i ¼0
ðcos h i 0j i þsin h i 1j iÞ i j i
¼
X
22n À1i ¼0
c i j i i j i ð5Þ
and a watermarking image of size 2n 92n ,who FRQI is
W j i ¼12n
X
22n
À1j ¼0
ðcos u j 0j i þsin u j 1j iÞ j j i ¼
X
22n À1j ¼0
w j
j j i ;
ð6Þ
where c i j i ¼12n cos h i 0j i þsin h i
1j i ðÞand w j ¼1
n cos u j 0j i þsin u j 1j i ÀÁare just simplified forms.
英文发音规则
Note The size of the images is not limited for square images 2n 92n ;here,we lect the exponential of 2is just for easily coding it to FRQI.Moreover,the watermark
is
Fig.4General framework for the embedding procedure of the propod quantum image watermarking scheme
X.Song et al.
not necessarily the same size of the carrier image,but the maximum size of it cannot exceed that of the carrier image.If the size of the watermark is smaller than that of the carrier image,the remaining qubits can be replaced by basis state 0j i :
3.1.2Hadamard transforms
First,construct unitary transform HT controlled by a binary key K ,where,K =k 1k 2,…,k 2n ?1,k 1=1,k i [{0,1},i =2,3,…,2n ?1,k 1=1is to ensure the transformation of color qubit.
HT ¼H 2n þ1i ¼2H k i ;H k i ¼H ;I ;&
预告犯k i ¼1encore是什么意思
k i ¼0;i ¼2;3;...;2n þ1;
where H is a Hadamard matrix and I is a 2-D identity
matrix.
Then,execute HT on the carrier quantum image C j i ;getting vector HT(C j i )as follows:
HT ðC j iÞ¼12X
22n
À1i ¼0
HT ðcos h i 0j i þsin h i 1j iÞ i j i
¼
X
22n
À1i ¼0
x c i j i i j i ;ð7Þ
where x c i j i ¼
12n
HT cos h i 0j i þsin h i 1j i ðÞis still a simple
reprentation.
3.1.3Watermark processing
The quantum watermark image W j i is procesd by the following unitary transforms:
P i ¼I X
22n
À1j ¼0;j ¼i
j j i j h j !þPh a i ðÞ i j i i h j ;
ð8ÞI ¼
1001
;Ph a i ðÞ¼e i a
i
0e i a i
ted熊;where I is a 2-D identity matrix and Ph(a i )is a pha gate.Therefore,applying P k and P l P k to quantum watermark image W j i ;gives us
P k W j i ðÞ¼P k 1
2n X 22n À1j ¼0cos u j 0j i þsin u j 1j i j j i
!
¼12n
amount什么意思X
22n
À1j ¼0;j ¼k
cos u j 0j i þsin u j 1j i ÀÁ j j i "þe i a k cos u k 0j i þsin u k 1j i ðÞ k j i Ã
;
ð9ÞP l P k W j i ðÞ¼P l P k W j i ðÞðÞ
¼12X
22n
À1j ¼0;j ¼k ;l
cos u j 0j i þsin u j 1j i ÀÁ j j i "þe i a k cos u k 0j i þsin u k 1j i ðÞ k j i
þe i a k cos u l 0j i þsin u l 1j i ðÞ l j i Ã
:
ð10Þ
From Eq.(10),it is clear that
P W j i ðÞ¼Y 22n À1
j ¼0
P j !W j i ðÞ
¼12n
X
22n
À1j ¼0
e i a j cos u j 0j i þsin u j 1j i ÀÁ j j i ¼PW j i ;ð11Þ
where a ¼a 0;a 1;...;a 22n À1ðÞis a dynamic vector instead of a fixed parameter as in [8].3.1.4Embedding watermarking
The procesd quantum watermark image PW j i is embedded into the coefficients HT(C j i )according to the following model:
HT ðCW j iÞ¼HT ðC j iÞþPW j i
¼snowman
X
22n À1i ¼0
x mc i j i i j i ;
ð12Þ
where x mc i j i is a mark of state HT(CW j i ).
水落管HT(C j i )reprents the qubits of the carrier image and HT(CW j i )reprents the qubits after embedding watermark.Quantum image is the reprentation of pattern of image in quantum computer,so the whole implementation process about it must satisfy the principal of quantum mechanics.A quantum network of the embedding process on a quantum image of size 292is demonstrated in Fig.5and can be understood as follows:at first,two registers are needed to encode the carrier image and the watermark image.Then,HT operation is executed on the quantum carrier image C j i ;and a quence of pha gates Ph(a )=P decided by a is implemented on the watermark image.Finally,the adder quantum network shown in Fig.3is implemented on the carrier image and the
watermark image.
Note When we extract the watermark image,we should execute inver HT to HT(CW j i ),that is,inHT HT CW j i ðÞf g ¼inHT HT C j i ðÞþPW j i f g
¼C j i þinHT PW j i f g ;
ð13Þ
where inHT is the inver HT operator.If we want to make the watermarked quantum image more similar to the
Dynamic watermarking scheme for quantum images
