1.1. Related Work
Image encryption technology won great attention recently because of its complexity.  adopted a novel chaotic block image encryption algorithm based on dynamic random growth technique.  extended a 2D Sine Logistic modulation map for image encryption.  gave a new image encryption algorithm based on non-adjacent coupled map lattices.  considered a novel chaotic image encryption scheme using DNA sequence operations.  concerned an image encryption scheme based on elliptic curve pseudo random and Advanced Encryption System.
Image encryption method often includes space encryption and condenses encryption.  built a color image encryption based on chaotic systems and elliptic curve ElGamal scheme.  studied an optical double-image encryption and authentication by sparse representation.  offered a N-phase logistic chaotic sequence and its application for image encryption.  implied an impulsive synchronization of reaction-diffusion neural networks with mixed delays and its application to image encryption.
However, the appearance of quantum computing brought a great challenge to classic encryption methods. At the same time, quantum encryption also gives us an absolutely secure encryption method. For example,  painted a quantum image encryption based on iterative framework of Frequency-Spatial Domain transforms.  displayed a quantum color image encryption algorithm based on a hyper-chaotic system and quantum Fourier transform.  discussed the research on an E-mail encryption protocol based on quantum teleportation.  featured a quantum image encryption algorithm based on quantum image XOR operations.  stated that encryption faces quantum foe.  put forward a quantum image encryption based on generalized Arnold transform and double random-phase encoding.
In quantum image encryption, the efficiency of transformation plays an important role, especially the Fourier transformation  .  discussed the change from fractional Fourier transformation to quantum mechanical fractional squeezing transformation. This paper tried to introduce an efficient quantum Fourier transformation in image encryption to improve the encryption security and efficiency.
1.2. Organization of the Article
Section 2 defines the general notion of privacy for quantum key distribution. Section 3 contains preliminaries, basic rules and the general model used to analyze the protocol. In Section 4, the protocol is described. Section 5 contains the analysis of privacy protection.
2. Image Security and Image Encryption
2.1. Fourier Transformation in Image Encryption
Through bringing the ideal into this work from a brilliant estimation of this Fourier transform of is required, which denoted by η1/r. It might be illustrated that since one has the estimation
Therefore, for any (as a matter of fact a par holds) as well as one avoids , its value drop off. For the dot not far from distance, for example, 1/r from this fretwork, its significance is still a little positive constant (approximately ). As the length from increases, the significance of the purpose quickly changes into triffling. Because the length between any two matrixes in is at any rate , the normal distribution around each point of are well fell apart.
Let us begin to try to comprehend what the distinguishing difference looks like. Point out that this image matrix D/p compound of translates of the primal image matrix D that’s to say, as for each , consider the set
Then, establishes a division of D/p. what’s more, it could be depicted that since r/p is bigger than the system parameter , the possibility distributed to each under is fundamentally alike, that is, . Intuitively, in addition to system parameter, the normal measure not any more “sees” the discerning construction of D, so distinguished from others, it is not influenced by translations.
It guided us to think about the next dispersion, name it P. A example from P is a pair from which y is sampled , and is such that .
Now the Fourier transform of is presently analyzed. When “a” is zero, the Fourier transform is known as . For universal a, a stock calculation illustrated that the Fourier transform of is granted by
where is outlined as
and gives the only hide vector in to . Put differently, is the vector of constant number of the vector in hide to x when laid out in the foundation of , shrink possibility p. therefore seeing that the Fourier transform is basically , besides that each “hill” has its unique phase as a support for the vector of constant number of the image matrix dot in its center. The visual aspect of those phases is as a termination of a famous dimension of the Fourier transform, given that translation is transmuted with phase to multiplication.
For two real numbers x and , generally, x mod y can be defined as for , is defined as the integer closest to x or, in case of existing two such integers, the small one of the two. For any integer , is written for the cyclic group with addition possibility p.
As two possibility density functions λ1，λ2 on , the statistical length between them is defined as
(as this definition observing, the statistical length ranges in . A allied definition can be showed for sensible random variables. The statistical length satisfies the triangle inequality, it’s to say, for any,
Another significant fact is that the statistical length cannot increase by using a possibility function f, that’s to say,
Retrieve that the normal distribution with variance a2 and mean 0 is the distribution on R illustrated by the density function
where exp(y) denotes ey. Also see from that the summary of two mean = 0 independent variables with variances and is also a normal variable. A vector x and any , let
be a normal function measured by a component of s. if denoted M1 by p. Note that Therefore,
is an n-dimensional chance density function and like what have mentioned above, if apply s to denote s1. Functions are expended to sets in the normal way; that’s to say, for any countable matrix A. For any vector , if defined ， to be a shifted version of . The next example bounds the sum by which can deduce by a little change in .
For all and with and ,
Using the inequality ,
2.2. The Encryption Function
As the option of basis is obvious, if write instead of . For a point , is defined as the only point such that . if denote by the volume of the primal parallelepiped of D or equivalently, the determinant’ absolute value of the matrix with the basis image matrixes of matrix ( is an image matrix invariant, to be exactly, it is free from the option of basis). The double of a image matrix D in , denoted , is the image matrix illustrated by the set of all matrixes such that for all matrixes . In the same way, given a basis of an image matrix, define the double basis as the vector set such that for all where denotes the weight delta, in another word, 1 if and 0 other than. With a little abuse of notation, people often use D for the ϵ x ϵ matrix whose columns are With this notation, find that . Because of that, it shows that . At another case, for a point , is written to illuminate the integer coefficient vector of s.
Because of the iterative step, the algorithm can be expressed as follows. Allow denote The algorithm begin with producing samples from On account of is indeed large, this samples can be computed expeditiously by a unproblematic procedure. The next comes the most essential part of the algorithm: for 1 the algorithm applies samples from to produce samples from by naming the iterative step times. Finally, it ends up with samples from as well as people finish the algorithm by uncomplicated outputting the initial of them. Note the next essential answer: applying samples from , there will have the ability to bring forth the same number of samples from (actually, people could even give forth more than examples). The algorithm would not operate if only generate, in another word, samples.
Now eventually get to depict the iterative step by us. Retrieve that as input the samples from and there will be supposed to give forth a sample from , where . What’s more, r is knowable and assured to be at least , which can be illustrated to illuminate that . From what have illustrated in the former passage, the exact lower related to r does not count much for this summary; it’s adequate to remember that r is adequately larger than (D), and that 1/r is adequately smaller than
The algorithm composed of two primary parts. In that passage, there will be described as a classical algorithm that applying W and the samples from ,
solve . There, what illustrate a encryption algorithm is that, showed an oracle that solves , outputs a example from .
This is the unique encryption element in this essay. People find that the condition is content since
3. Quantum Fourier Transformation for Encryption
3.1. A Fast Quantum Fourier Transformation
In a fast quantum Fourier transformation, the first aim is to produce a quantum announcement in relate to . with formality, it could be described as
Taking account of some possibility distribution P on some image matrix D and its Fourier transform , defined as
where in the second equality. the sum is simply be rewrote as an expectation. By definition, η is -periodic, that’s to say, for any and It can compute an estimation of η to within . If are independent samples from P, and then
where the estimation is to within and poses with possibility exponentially just about 1, presuming that N is a large adequate multinomial.
Let be a negligible function, be an integer, and be a real number. presume that way to an oracle W that solves quantum oracle, given a multinomial number of examples. As for an ϵ-dimensional image matrix D, some , and an integer , there is an algorithm solves if, depicted any point within distance d of D, it outputs mod , the coefficient matrix to x deduced possibility p. Here shows a reduction from to .
There is an effective algorithm for given a image matrix D, a number and an integer , solves given way to an oracle for .
The input is a point x in distance d of D. A sequence of points is defined as as follows. Let be the coefficient image matrix point to xi. Define. Find that the closest image matrix point to therefore what’s more, the length of xi+1 from D is at most d/pi. as well as depicted that this sequence can be computed by applying the oracle.
After steps, there is a point whose length to the image matrix is at most . An algorithm is applied for solving the closest matrix. This outputs a image matrix point Da within distance of . Therefore, Da is the image matrix point closest to and one tried to retrieve realizing and mod p (by applying the oracle), one can recover . proceeding this process, one could recover . This finishes the algorithm for Da1 is the closest point to
As the option of r, ( it’s adequate to depict an efficient algorithm for . By combining the discussion above this could be done. Initially, it depicts an algorithm W’ that, showed samples from . The next, it is described how to use W’ and the shown samples from in order to solve
3.2. Quantum Encryption Step
Repeating the process illustrated above times, the system state is described as an -fold tensor product of the state in Equation (12), which might be understood as
For it indicates that the state is within l2 distance of
Therefore, for the goal it can be presumed that it is generated the state in Equation (14).
The next step, applying the LLL foundation reduction algorithm, a base can be acquired for D of length at most and let be brought forth by a new register Let denote the state that . The state acquired by us after the measurement is
In the end, subtract y from the register, and get
Therefore assume any with . The amplitude squared offered to it in Equation (13) is By The denominator is and therefore the amplitude is at most
In another word, the amplitude squared provided to x by the process is . Then the denominator is
To get this inequality, initially observe that by the simple part, , and then apply quantum Fourier transformation. what’s more, the numerator is in . Therefore, the amplitude squared provided to x is in
The l2 distance between different states r to
here, consider and as matrixes in -dimensional space. Make Z be the l2 norm of . In the next it can be shown that the l2 length between and is at most Z. it is adequate to build that the l2 distance between different states referring to and is exponentially tiny.
Initially, get a good approximation of Z. As far as
, each key in the definition of
, and so
By applying the image matrix s2"D/R, get that
It is verified with an upper relate to the l2 distance between the two matrixes. Applying the normal monotonicity of s,
There will be an effective quantum algorithm that, offered any n-dimensional image matrix D, a number , and an oracle that handles , outputs a sample from .
By scaling, presume without decline in amount of generality that . Let be a big adequate integer, presume that log R is multinomial in the image matrix D. The initial task is to build a state exponentially near to
As a state on log R qubits, that is a multinomial number in the input scale. In order to do in this way, initially, it is used with and the image matrix to make the state
Then, this is exponentially relate to
An then, calculate x mod M(D*) in a new register and get
applying the CVP oracle, recover x from x mod M(D*). This admits us to uncompute the primal register and get
Then, this state is exponentially close to the recommended state (25).
In the next step, apply the quantum Fourier transform. To begin with, applying the mapping between and , rewrite (25) as
Then apply the quantum Fourier transform on . get a state where the amplitude of t for te, ZR is proportional to
where the last equality follows from Equation (26). Therefore, the crucial state can be fairly written as
Look at that Therefore, according to the image matrix RD, and get that this state is exponentially close to
Quantify this state and get x mod for some vector x with . Since x mod is within of the image matrix RD, and , recuperate x by using. The answer of the algorithm is x.
Presume without deprivation of generalization that the vector is or thogonal to H. There is,
Let be matrixes chosen by For , let be the event that
Obviously, if none of the takes place, then . Therefore, it is necessary to depict that for all i, . Indeed, fix some i on condition of such that . Then the possibility that
is at most . This indicated that , as commanded.
4. Security Analysis and Proposition of Quantum Encryption
Let ϵ be the security parameter of encryption system. The encryption system is parameterized by two integers m，p and a possibility distribution x on . A parameters setting undertakes both safety and right is the next. pick to be some initial number between and make for some arbitrary constant The chance distribution x is selected to be for , that’s to say, is such that . For instance, it can be chosen as . In the next illustration, all additions are operated in , i.e., possibility p.―Private key: select uniformly randomly. The private key is .
―Public Key: for select m matrixes from the uniform distribution. Also select elements referring to x . The public key is offered by where
In case of encryption, first select a random set S uniformly between all subsets of The encryption is if the bit is 0 and
if the bit is 1.
In case of decryption, the decryption of a pair (a, b) is 0 if is closer to 0 than to possibility p. whereas, the decryption is 1.
Apparently, the public key size is and the encryption procedure multiplies the scale of a message by a element of . As a matter of fact, it is probable to decrease the size of the public key to . Presume all users of the encryption system partake some fixed (and trustworthy) random options of Next, the public key require just made of . This tranformation does not influence the safety of the encryption system.
Next, illustrate that under a sure condition on x , m, and p , the possibility of decryption problem is tiny. Latterly, depict that the option of parameters meets this condition. There exists a desire to insert some additional notation. As for a distribution x on and an integer k ≥ 0, define as the distribution gotten by adding up k , whose addition is operated in (for k = 0 we define as the distribution that is incessantly 0). For a chance distribution λ on T define f likely. For an component , is defined as the integer a if
and as the integer p - a otherwise. Differently, reshowed
the distance of a from 0. likely, for , is defined as x for and as 1 - x other than.
Let a > 0. Presume that for any , x’ meets that
Next, the possibility of decryption error will be decreased. In another word, for any bit , if apply the protocol above to pick private and public keys, encrypt c, and then decrypt the answer, then the final result is c with possibility at least 1 - δ.
Initially, think about an encryption of 0. It is offered by (a, b) for and
Therefore, is exactly , with distribution function . referring to the supposal, is less than with possibility at least 1 - δ. From the aspect, it is closer to 0 than to and hence the decryption is right.
For the option of parameters it contains that for any ,
for some trifling function δ( ).
For a selection of l components from G, let be the distribution sum of , i.e., 1
By way of showing that this distribution is near uniform, compute its l2 norm, and observe that it is very approach to 1/|G|. From this it will keep up that the distribution must be approach to the distribution function. The l2 norm of Mg is given by
In the end, the expected length from the uniform distribution is
For , let be the possibility with input where are selected from , and is an encryption of 0 with the public key . likewise, define to be the acceptance possibility of W’, where are selected from , and is now selected randomly from . The assumption on W’ says that
By an averaging line of reasoning in of the s are in Y. Therefore, it is
adequate to show a distinguisher Z that separates between U and for any ω q Y.
In the next, describe the distinguisher Z. distribution gives a R that is either U or for some s q Y. m samples is taken from from R. Let be the possibility with input where the possibility is picked on with the public key as an encryption bit 0. Likewise, let be the possibility with input where the possibility is picked over the option of as a uniform component of . While W’ is applied as a multinomial number of times, the distinguisher Z reckon both and up to an habit-forming error
of . If the two estimation is different from each other more than , Z will be accepted, or Z will be rejected.
Besides, to some very fundamental definitions referring to image matrixs, it must be made from the normal distribution on D of width r, denoted The possibility of distribution image matrix of each is partial to exp . It is mentioned here the system parameter (D). This is a positive real number related to any image matrix is an error parameter can be safely omitted here. Inaccurate to say, it lets the smallest r beginning with which like a continuous normal distribution. For example, for , matrixes picked from have norm about with high possibility. By comparing, for enough small r, offers almost all its mass to the primal 0, whereas not commanded for this part, a clear list of definitions can be seen in part 2.
The key of the encryption algorithm is called as the iterative step. Its input form a number r which is promised to be larger than , and nc examples from in which c is stable. Its output is an example from the distribution , for . Find that since , to make the shifting matrixes of norm into smaller matrixes of norm , the process prefers to using the quantum oracle.
This research was supported by the National Natural Science Foundation of China (No. 71471102), and Yichang University Applied Basic Research Project in China (Grant No. A17-302-a13).
Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this paper.