An Image Encryption Method Based on Quantum Fourier Transformation

The image security problem is an important area in information security, and image encryption plays a vital role in this day. To protect the image encryption from the attack of quantum algorithm appeared recently, an image encryption method based on quantum Fourier transformation is proposed here. First, the image encryption and Fourier transformation are discussed here, then a encryption function is proposed. Second, a quantum Fourier transformation is introduced to quantum encryption, and the full step of quantum encryption is given as well. Third, the security of the proposed quantum encryption if analyzed, and some propositions are also presented. Lastly, some conclusions are indicated and some possible directions are also listed.


Introduction 1.Related Work
Image encryption technology won great attention recently because of its complexity.[1] adopted a novel chaotic block image encryption algorithm based on dynamic random growth technique.[2] extended a 2D Sine Logistic modulation map for image encryption.[3] gave a new image encryption algorithm based on non-adjacent coupled map lattices.[4] considered a novel chaotic image encryption scheme using DNA sequence operations.[5] concerned an image encryp-

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.

Fourier Transformation in Image Encryption
Through bringing the ideal into this work from PD γ a brilliant estimation of this Fourier transform of PD γ is required, which denoted by η 1/r .It might be illustrated that since ( ) Therefore, ( ) ≈ for any D χ * ∈ (as a matter of fact a par holds) as well as one avoids D * , 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   For two real numbers x and 0 y > , generally, x mod y can be defined as | y y   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 2 p ≥ , p O is written for the cyclic group { } 1 0,1, p − with addition possibility p.As two possibility density functions λ 1 ，λ 2 on R ∈ , the statistical length between them is defined as (as this definition observing, the statistical length ranges in [ ] 0, 2 .A allied definition can be showed for sensible random variables.The statistical length satis-International Journal of Intelligence Science fies the triangle inequality, it's to say, for any, 1 2 3 , , 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 a 2 and mean 0 is the distribution on R illustrated by the density function where exp(y) denotes e y .Also see from that the summary of two mean = 0 independent variables with variances is an n-dimensional chance density function and like what have mentioned above, if apply s to denote s 1 .Functions are expended to sets in the normal way; that's to say, ( ) ( ) for any countable matrix A. For any vector e e e 1 π 2

The Encryption Function
As the option of basis is obvious, if write ( ) χ is defined as the only point ( ) det D the volume of the primal parallelepiped of D or equivalently, the determinant' absolute value of the matrix with the basis image matrixes of matrix (

( )
det D is an image matrix invariant, to be exactly, it is free from the option of basis).The double of a image matrix D in R ∈ , denoted D * , is the image matrix illustrated by the set of all matrixes y R ∈ ∈ such that , 0 x y ∈ for all matrixes x D ∈ .In the same way, given a basis ( ) Pf D , 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 D,r P , solve .This is the unique encryption element in this essay.People find that the condition is content since ( ) ( ) ( )

A Fast Quantum Fourier Transformation
In a fast quantum Fourier transformation, the first aim is to produce a quantum announcement in relate to 1 r η .with formality, it could be described as Taking account of some possibility distribution P on some image matrix D and its Fourier transform where in the second equality.the sum is simply be rewrote as an expectation.By definition, η is D * -periodic, that's to say, ( ) ( )  ( ) where the estimation is to within ( ) and poses with possibility exponentially just about 1, presuming that N is a large adequate multinomial.Let ( ) 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 2 p ≥ , there is an algorithm solves There is an effective algorithm for given a image matrix D, a number ( ) and an integer 2 p ≥ , solves CVP D,d given way to an oracle for The input is a point x in distance d of D. A sequence of points is defined as , , , χ χ χ  as follows.Let ( ) be the coefficient image matrix point to x i .Define.Find that the closest image matrix point to ( ) ( ) more, the length of x i+1 from D is at most d/p i .as well as depicted that this sequence can be computed by applying the oracle.After ∈ steps, there is a point

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 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

( )
M D be brought forth by a new register ( ) . The state acquired by us after the measurement is ( ) In the end, subtract y from the register, and get ( ) Therefore assume any x D ∈ with X r ≤ ∈ .The amplitude squared of- fered to it in Equation ( 13) is and therefore the amplitude is at most In another word, the amplitude squared provided to x by the process is To get this inequality, initially observe that by the simple part, ( ) ( ) what's more, the numerator is in . Therefore, the amplitude squared provided to x is in The l 2 distance between different states r to ( ) ( ) mod here, consider 1 s and 2 s as matrixes in R ∈ -dimensional space.Make Z be the l 2 norm of 1 s .In the next it can be shown that the l 2 length between 1 s and 2 s is at most it is adequate to build that the l 2 distance between different states referring to 1 s and 2 s is exponentially tiny.
Initially, get a good approximation of Z.As far as ( ) s , and so By applying the image matrix s 2 "D/R, get that ( ) ( ) It is verified with an upper relate to the l 2 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 ( ) 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 1 2 r = and the image matrix D R * 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 ( ) Then apply the quantum Fourier transform on p O ∈ .get a state where the am- plitude 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 ( ) M RD for some vector x with X < ∈ .Since x mod ( ) M RD 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 ( ) Let be 2 ∈ matrixes chosen by Obviously, if none of the B ω ′ takes place, then ( ) Therefore, it is necessary to depict that for all i, [ ] ( )

r i M B
−Ω ∈ < .Indeed, fix some i on condition of ( ) . Then the possibility that ( ) )

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 p O .A parameters setting undertakes both safety and right is the next.pick 2 P > to be some initial number between 2 ∈ and 2 2 ∈ make ( )( ) for some arbitrary constant 2 q > The chance distribution x is selected to be ( ) α ψ ∈ for ( ) ( ) if the bit is 0 and In case of decryption, the decryption of a pair (a, b) is 0 if 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 .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 p O and an integer k ≥ 0, define k x * as the distribution got- ten by adding up k , whose addition is operated in p O (for k = 0 we define 0 x * as the distribution that is incessantly 0).For a chance distribution λ on T define f likely.For an component p a O ∈ , a is defined as the integer a if and as the integer pa otherwise.Differently, a reshowed the distance of a from 0. likely, for x T ∈ , x is defined as x for [ ] 0,1 2 x ∈ and as 1 -x other than.
Let a > 0. Presume that for any Next, the possibility of decryption error will be decreased.In another word, for any bit { } 0,1 c ∈ , 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 For the option of parameters it contains that for any for some trifling function δ(∈ ).By way of showing that this distribution is near uniform, compute its l 2 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 l 2 norm of M g is given by ( ) In the end, the expected length from the uniform distribution is ( ) ( ) ( ) A ω for any ω q Y.
In the next, describe the distinguisher Z. distribution gives a R that is either U or x A ω for some s q Y. m samples is taken from ( ) 1 , m i i i a b = from R. Let As the length from D * increases, the significance of the purpose quickly changes into triffling.Because the length between any two matrixes in D * is at any rate distribution around each point of D * are well fell apart.Let us begin to try to comprehend what the distinguishing difference , D p r p P looks like.Point out that this image matrix D/p compound of p ∈ translates of the primal image matrix D that's to say, as for each of D/p.
gives the only hide vector in D * to χ .Put differently, ( ) T x is the vector of constant number of the vector in D * hide to x when laid out in the foundation of D * , shrink possibility p. therefore seeing that the Fourier transform , p r f , 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.
coefficient vector of s.Because of the iterative step, the algorithm can be expressed as follows.Allow On account of 3 r ∈ is indeed large, this samples can be computed expeditiously by a unproblematic procedure.The next comes the most essential part of the algorithm: for 3 ,3 1, i = ∈ ∈ −  1 the algorithm applies people finish the algorithm by uncomplicated outputting the initial of them.Note the next essential answer: applying c ∈ samples from i D,r P , there will have the ability to bring forth the same number of samples c ∈ from people could even give forth more than c ∈ examples).

.
There, what illustrate a encryption algorithm is that, showed an oracle that solves

∈∈
and y D * ∈ It can compute an estimation of η to within ( ) independent samples from P, and then ( ) coefficient ma- trix to x deduced possibility p.Here shows a reduction from CVP to

.
the image matrix is at most d p∈ .An algorithm is applied for solving the closest matrix.This outputs a image matrix point Da within distance is the image matrix point closest to 1 χ ∈+ and one tried to retrieve 1 a a ∈+ = realizing and a ∈ mod p (by applying the oracle), one can recover .This finishes the algorithm for Da 1 is the closest point to 1x x =As the option of r, ( By combining the discussion above this could be done.Initially, it depicts an algorithm W' that, showed samples from , is described how to use W' and the shown samples from D P γ in order to solve and then apply quantum Fourier transformation.
By scaling, presume without decline in amount of generality that d = ∈ .Let adequate integer, presume that log R is multinomial in the image matrix D. The initial task is to build a state exponentially near to users of the encryption system partake some fixed (and trustworthy) random options of 1 , , m a a  Next, the public key require made of 1 , , m b b  with possibility at least 1 -δ.From the aspect, it is closer to 0 than to 2p      and hence the decryption is International Journal of Intelligence Science right.
are selected from x A ω , and ( ) , a b is an encryption of 0 with the public key ( ) 1 , m i i i a b = .likewise, define ( ) u P ω to be the acceptance possibili- ty of W', where ( ) are in Y. Therefore, it is adequate to show a distinguisher Z that separates between U and x

=
p a b = be the possibility with input ( where the possibility is picked on ( ) , a b with the public key ( ) 1 , m i i i a b = as an encryption bit 0.
what's more, it could be depicted that since r/p is bigger than the system parameter It guided us to think about the next dispersion, name it P.A example from P is a pair ( ) , a y from which y is sampledP D + is presently analyzed.When "a"is zero, the Fourier transform is known as p r f .For universal a, a stock calculation illustrated that the Fourier transform of Now eventually get to depict the iterative step by us.Retrieve that as input the ∈ samples from D,r P and there will be supposed to give forth a sample from The algorithm would not operate if only generate, in another word, 2 c ∈ samples.D,r P , where ( ) 1 r r ap = ∈ .What's more, r is knowable and assured to be at least ( ) that the state is within l 2 dis- DOI: 10.4236/ijis.2018.8300481 International Journal of Intelligence Science ( ) that's to say, ( )In case of encryption, first select a random set S uniformly between all 2 m