CN101739660A - Method for encrypting and decrypting image based on multi-order fractional Fourier transform - Google Patents

Abstract

The invention relates to a method for encrypting and decrypting an image based on multi-order fractional Fourier transform, which belongs to the technical field of information security. The method selects discrete sampling-type discrete fractional Fourier transform as a basic tool; at an encryption end of a system, an original image is equally divided into a plurality of sub-images with the same size; interpolation and discrete fractional Fourier transform of different orders are respectively performed on the row and the column of each sub-image; and the transformed images are superposed to obtain an encrypted image; and at the decryption end of the system, the original image is recovered effectively by establishing a linear equation. The method makes full use of the characteristics of the order change of the fractional Fourier transform, and can be combined with other encrypting methods, thereby expanding key space and improving the security of the system relative to the conventional image encrypting technology on the basis of the fractional Fourier transform. The method of the invention also can be used for encrypting double images or a plurality of images, which provides a new way for the design of future image encryption technology.

Description

A kind of image encryption and decryption method based on multi-order fractional Fourier transform

Technical field

The present invention relates to a kind of image encryption and decryption method, belong to field of information security technology based on multi-order fractional Fourier transform.

Background technology

Image information is vivid, is one of important means of people's expressing information.At present, along with the fast development of network technology, digital image information has been widely used in every field such as military affairs, medical science, finance.But network technology has more inevitably been brought the hidden danger of information security in the information transmission of making.At present, the image information unauthorized theft, alter, propagate and be becoming increasingly rampant, therefore, the technology of protection digital image content safety is arisen at the historic moment.

The notion of Fourier Transform of Fractional Order promptly was suggested as far back as nineteen twenty-nine, was applied to optical field in the eighties in 20th century, became one of the research focus in signal Processing field from the nineties.Fourier Transform of Fractional Order is the generalized form of Fourier transform, it carries out signal Processing on unified time-frequency domain, with respect to traditional Fourier transform, the Fourier Transform of Fractional Order dirigibility is stronger, by being selected by 0 to 2 order, Fourier Transform of Fractional Order can provide signal is transformed to time domain again to frequency domain by spatial transform whole features.Therefore, Fourier Transform of Fractional Order has obtained using widely at information security field, many scholars adopt the kernel function of Fourier Transform of Fractional Order to replace the kernel function of Fourier transform in the information safety system, image encryption technology based on Fourier Transform of Fractional Order has been proposed, the conversion order that utilizes Fourier Transform of Fractional Order is as the key that increases, enlarged the difficulty of key space and illegal stealer's deciphering, the security that has improved system.

But, the just traditional simple extension of existing digital image encryption system based on the Fourier transform secret signalling based on Fourier Transform of Fractional Order, promptly adopt Fourier transform or Fourier in the alternative legacy system of Fourier Transform of Fractional Order or fractional number order Fourier, the order of Fourier Transform of Fractional Order is not fully utilized.

Summary of the invention

A kind of image encryption and decryption method based on multi-order fractional Fourier transform of the present invention is that selection discrete sampling type discrete fraction rank Fourier transform is a basic tool, the conversion order that utilizes a plurality of discrete fractions rank Fourier transform is as the image encryption key, remedied the existing shortcoming that can't make full use of fraction order Fourier transform order based on the image encryption system of Fourier Transform of Fractional Order, enlarge key space, improved the security of system.

The objective of the invention is to be achieved through the following technical solutions.

A kind of image encryption and decryption method based on multi-order fractional Fourier transform of the present invention are divided into image encryption and image deciphering two parts;

The step that realizes image encryption is as follows:

Step 1, will be of a size of the wide and long M and the N equal portions of being divided into respectively of the original image of A * B, obtain the subimage S that size such as M * N is of a size of (A/M) * (B/N) _{(a-1) N+b}, a=1,2 ..., M, b=1,2 ..., N;

Step 2, each subimage S that the information encryption step 1 is obtained _{(a-1) N+b}Each column vector carry out M times of interpolation and p _{2[(a-1) N+b]-1}Contrary discrete fraction rank, rank Fourier transform is carried out N times of interpolation and p to its row vector again _{2[(a-1) N+b]}Contrary discrete fraction rank, rank Fourier transform obtains Y thus _{(a-1) N+b}, its mathematic(al) representation as the formula (1)

$Y_{(a-1)N+b}=\left[\begin{array}{c}{D}_{p_{2[(a-1)N+b]-1},0}\\ D_{p_{2[(a-1)N+b]-1},1}\\ \·\\ \·\\ \·\\ D_{p_{2[(a-1)N+b]-1},M-1}\end{array}\right]\·X_{(a-1)N+b}\·\left[\begin{array}{cccc}{D}_{p_{2[(a-1)N+b]},0}& D_{p_{2[(a-1)N+b]},1}& \·\·\·& D_{p_{2[(a-1)N+b]},N-1}\end{array}\right]---\left(1\right)$

Wherein, X _{(a-1) N+b}For to S _{(a-1) N+b}Column vector be p _{2[(a-1) N+b]-1}Contrary discrete fraction rank, rank Fourier transform, row vector are p _{2[(a-1) N+b]}The contrary discrete fraction rank Fourier transform gained result in rank, diagonal matrix

L wherein ₁=0,1 ..., M-1, l ₂=0,1 ..., N-1 is defined as

${\left[D_{p_{2[(a-1)N+b]},l_1}\right]}_{k,k}=e^{j\&CenterDot;\frac{1}{2}\&CenterDot;\cot \left(\frac{p_{2[(a-1)N+b]}\&CenterDot;\mathrm{\&pi;}}{2}\right)\&CenterDot;[2k({\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="H2009102384791E00011.png"\; state="\{\{state\}\}">l</figure-callout>}}_1\&CenterDot;N)+{({\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="H2009102384791E00011.png"\; state="\{\{state\}\}">l</figure-callout>}}_1\&CenterDot;N)}^2]\&CenterDot;\mathrm{\&Delta;}{t}^2}$

${\left[D_{p_{2[(a-1)N+b]-1},l_2}\right]}_{k,k}=e^{j\&CenterDot;\frac{1}{2}\&CenterDot;\cot \left(\frac{p_{2[(a-1)N+b]-1}\&CenterDot;\mathrm{\&pi;}}{2}\right)\&CenterDot;[2k(l_2\&CenterDot;N)+{(l_2\&CenterDot;N)}^2]\&CenterDot;\mathrm{\&Delta;}{t}^2}$

Step 3, the Y that the image encryption step 2 is obtained _{(a-1) N+b}Addition successively obtains encrypted image Y, and mathematic(al) representation as the formula (2)

$Y=\underset{a=1}{\overset{M}{\mathrm{\&Sigma;}}}\underset{b=1}{\overset{N}{\mathrm{\&Sigma;}}}{Y}_{(a-1)N+b}---\left(2\right)$

The step that realizes the image deciphering is as follows:

Step 1, by l in the encrypted image ₁(A/M)+l during m is capable ₂(B/N)+and a n element, m=1 wherein, 2 ..., A/M, n=1,2 ..., B/N, the unknown number of setting up as the formula (3) is [X _{(a-1) N+b}] _{M, n}, M * N linear equation, a=1 wherein, 2 ..., M, b=1,2 ..., N

$\underset{a=1}{\overset{M}{\mathrm{\&Sigma;}}}\underset{b=1}{\overset{N}{\mathrm{\&Sigma;}}}{\left[D_{p_{2[(a-1)N+b]-1},l_1}\right]}_{n,n}\&CenterDot;{\left[X_{(a-1)N+b}\right]}_{m,n}\&CenterDot;{\left[D_{p_{2[(a-1)N+b]},l_2}\right]}_{m,m}={\left[Y\right]}_{l_1(A/M)+m,l_2(B/N)+n}---\left(3\right)$

Step 2, obtain formula (4), solve X by the formula in the image decryption step one (3) _{(a-1) N+b}In m capable in the numerical value of n element

$\stackrel{\&RightArrow;}{\hat{X}}=D^{-1}\&CenterDot;\stackrel{\&RightArrow;}{r}---\left(4\right)$

Wherein,

With

Be defined as

${\left[\stackrel{\&RightArrow;}{\hat{X}}\right]}_{(a-1)N+b}={\left[X_{(a-1)N+b}\right]}_{m,n}$

${\left[\stackrel{\&RightArrow;}{r}\right]}_{(a-1)N+b}={\left[Y\right]}_{(a-1)(A/M)+m,(b-1)(B/N)+n}$

${\left[D\right]}_{l_{2}M+{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="H2009102384791E00011.png"\; state="\{\{state\}\}">l</figure-callout>}}_1,[(a-1)N+b]}={\left[D_{p_{2[(a-1)N+b]-1},l_2}\right]}_{n,n}\&CenterDot;{\left[D_{p_{2[(a-1)N+b]},l_1}\right]}_{m,m}$

Step 3, the X that image decryption step two is obtained _{(a-1) N+b}Column vector and the row vector be p respectively _{2[(a-1) N+b]-1}Rank discrete fraction rank Fourier transform and p _{2[(a-1) N+b]}Discrete fraction rank, rank Fourier transform splices original image.

Beneficial effect

1. the image encryption technology that proposes of the present invention has made full use of the order conversion characteristics of Fourier Transform of Fractional Order with respect to existing image encryption technology based on Fourier Transform of Fractional Order, has enlarged key space, has improved the susceptibility of deciphering error to key;

2. the image encryption technology that proposes of the present invention can combine with image encryption method such as double random phase coding, further enlarges the key space of encrypted image;

3. the image encryption and the decryption method of the present invention's proposition can also be applied to digraph and many figure encrypt and decrypt.

Description of drawings

Fig. 1-encrypt the end system structural drawing based on the multi-order fractional Fourier transform image encryption system;

Fig. 2-based on multi-order fractional Fourier transform image encryption system decrypting end system construction drawing;

Fig. 3 (a)-original image, (b)-encrypted image;

Decrypted image under Fig. 4-false key;

Fig. 5 (a)-decrypted image square error is with dp ₁, dp ₂, dp ₃, dp ₄Change curve, (b)-decrypted image square error are with dp ₅, dp ₆, dp ₇, dp ₈Change curve;

Embodiment

A kind of image encryption and decryption method based on multi-order fractional Fourier transform that the present invention proposes comprises image encryption (as shown in Figure 1) and image deciphering (as shown in Figure 2), and the specific implementation step is as follows:

One, encrypting step

(1) row, column of original image is carried out N times and M times of five equilibrium respectively, obtain M * N subimage S that waits size _{(a-1) N+b}, a=1,2 ..., M, b=1,2 ..., N;

(2) the subimage S that will obtain by step () _{(a-1) N+b}Column vector be M times of interpolation and p _{2[(a-1) N+b]-1}Contrary discrete fraction rank, rank Fourier transform, row vector are N times of interpolation and p _{2[(a-1) N+b]}Contrary discrete fraction rank, rank Fourier transform obtains Y _{(a-1) N+b}

(3) according to formula (2), the Y that obtains by step (two) _{(a-1) N+b}Addition obtains encrypted image successively.

Two, decryption step

(1) extract k (B/N)+n element during l in the encrypted image (A/M)+m is capable, l=0 wherein, 1 ..., M-1, k=0,1 ..., N-1, structure linear equation as the formula (3);

(2) solve subimage S according to formula (4) _{(a-1) N+b}Column vector and row vector are p respectively _{2[(a-1) N+b]-1}Rank discrete fraction rank Fourier transform and p _{2[(a-1) N+b]}The corresponding matrix X of rank discrete fraction rank Fourier transform _{(a-1) N+b}Capable n the element numerical value of m, and then obtain X _{(a-1) N+b}In all numerical value;

(3) X that image decryption step two is obtained _{(a-1) N+b}Column vector and the row vector be p respectively _{2[(a-1) N+b]-1}Rank discrete fraction rank Fourier transform and p _{2[(a-1) N+b]}Discrete fraction rank, rank Fourier transform splices original image.

Ultimate principle below in conjunction with the conversion of fractional number order Fourier signal sampling rate is done detailed theoretical validation to embodiment.

For the original image S that is of a size of A * B, to its row row carry out respectively N doubly and M times of five equilibrium be of a size of the subimage S of (A/M) * (B/N) _{(a-1) N+b}, a=1,2 ..., M, b=1,2 ..., N is to subimage S _{(a-1) N+b}The row, column vector carry out N respectively doubly and after M times the interpolation, again interpolation result's row, column is carried out p respectively _{2[(a-1) N+b]}And p _{2[(a-1) N+b]-1}Contrary discrete fraction rank, rank Fourier transform, the gained result can be expressed as so

$Y_{(a-1)N+b}=F_{-p_{2[(a-1)N+b]-1}}\&CenterDot;{\mathrm{\&theta;}}_M\&CenterDot;S_{(a-1)N+b}\&CenterDot;{\mathrm{\&theta;}}_N^T\&CenterDot;F_{-p_{2[(a-1)N+b]}}^T$

$=\left[\begin{array}{c}{D}_{p_{2[(a-1)N+b]-1},0}\\ D_{p_{2[(a-1)N+b]-1},1}\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ D_{p_{2[(a-1)N+b]-1},M-1}\end{array}\right]\&CenterDot;X_{(a-1)N+b}\&CenterDot;\left[\begin{array}{cccc}{D}_{p_{2[(a-1)N+b]},0}& D_{p_{2[(a-1)N+b]},1}& \&CenterDot;\&CenterDot;\&CenterDot;& D_{p_{2[(a-1)N+b]},N-1}\end{array}\right]---\left(5\right)$

In formula (5),

$X_{(a-1)N+b}=F_{-p_{2[(a-1)N+b]-1}}\&CenterDot;S_{(a-1)N+b}\&CenterDot;F_{-p_{2[(a-1)N+b]}}^T$

With

P is represented in expression respectively _{2[(a-1) N+b]-1}Rank and p _{2[(a-1) N+b]}Contrary discrete fraction rank Fourier transform matrix, promptly

${\left[F_{-p_{2[(a-1)N+b]-1}}\right]}_{m,n}=\sqrt{\frac{\sin (p_{2[(a-1)N+b]-1}\mathrm{\&pi;}/2)-j\&CenterDot;\cos (p_{2[(a-1)N+b]-1}\mathrm{\&pi;}/2)}{A}}\&times;$

${\mathrm{<figure-callout\; id="1"\; label="e\; j"\; filenames="H2009102384791E00011.png"\; state="\{\{state\}\}">e</figure-callout>}}^j$

${\left[F_{-p_{2[(a-1)N+b]}}\right]}_{m,n}=\sqrt{\frac{\sin (p_{2[(a-1)N+b]}\mathrm{\&pi;}/2)-j\&CenterDot;\cos (p_{2[(a-1)N+b]}\mathrm{\&pi;}/2)}{B}}\&times;$

${\mathrm{<figure-callout\; id="1"\; label="e\; j"\; filenames="H2009102384791E00011.png"\; state="\{\{state\}\}">e</figure-callout>}}^j$

Wherein, Δ t is the time-domain sampling interval,

$\mathrm{\&Delta;}{u}_{p_{2[(a-1)N+b]-1}}=2\mathrm{\&pi;}\&CenterDot;\sin \left(\frac{p_{2[(a-1)N+b]-1}\mathrm{\&pi;}}{2}\right)/(A\&CenterDot;\mathrm{\&Delta;t})$

$\mathrm{\&Delta;}{u}_{p_{2[(a-1)N+b]}}=2\mathrm{\&pi;}\&CenterDot;\sin \left(\frac{p_{2[(a-1)N+b]}\mathrm{\&pi;}}{2}\right)/(B\&CenterDot;\mathrm{\&Delta;t})$

θ _MAnd θ _NRepresent M times and N times of interpolation operation respectively, it can be expressed as

Diagonal matrix

L wherein ₁=0,1 ..., M-1, l ₂=0,1 ..., N-1 is defined as

${\left[D_{p_{2[(a-1)N+b]},l_1}\right]}_{k,k}=e^{j\&CenterDot;\frac{1}{2}\&CenterDot;\cot \left(\frac{p_{2[(a-1)N+b]}\&CenterDot;\mathrm{\&pi;}}{2}\right)\&CenterDot;[2k({\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="H2009102384791E00011.png"\; state="\{\{state\}\}">l</figure-callout>}}_1\&CenterDot;N)+{({\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="H2009102384791E00011.png"\; state="\{\{state\}\}">l</figure-callout>}}_1\&CenterDot;N)}^2]\&CenterDot;\mathrm{\&Delta;}{t}^2}$

${\left[D_{p_{2[(a-1)N+b]-1},l_2}\right]}_{k,k}=e^{j\&CenterDot;\frac{1}{2}\&CenterDot;\cot \left(\frac{p_{2[(a-1)N+b]-1}\&CenterDot;\mathrm{\&pi;}}{2}\right)\&CenterDot;[2k(l_2\&CenterDot;N)+{(l_2\&CenterDot;N)}^2]\&CenterDot;\mathrm{\&Delta;}{t}^2}$

If with the Y shown in the formula (5) _{(a-1) N+b}Addition can obtain successively

$Y=\underset{a=1}{\overset{M}{\mathrm{\&Sigma;}}}\underset{b=1}{\overset{N}{\mathrm{\&Sigma;}}}{Y}_{(a-1)N+b}$

$=\underset{a=1}{\overset{M}{\mathrm{\&Sigma;}}}\underset{b=1}{\overset{N}{\mathrm{\&Sigma;}}}\left[\begin{array}{c}{D}_{p_{2[(a-1)N+b]-1},0}\\ D_{p_{2[(a-1)N+b]-1},1}\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ D_{p_{2[(a-1)N+b]-1},M-1}\end{array}\right]\&CenterDot;X_{(a-1)N+b}\&CenterDot;\left[\begin{array}{cccc}{D}_{p_{2[(a-1)N+b]},0}& D_{p_{2[(a-1)N+b]},1}& \&CenterDot;\&CenterDot;\&CenterDot;& D_{p_{2[(a-1)N+b]},N-1}\end{array}\right]---\left(6\right)$

Therefore, by X _{(a-1) N+b}In capable, the n column element of m can obtain M * N linear equation as the formula (7)

$\underset{a=1}{\overset{M}{\mathrm{\&Sigma;}}}\underset{b=1}{\overset{N}{\mathrm{\&Sigma;}}}{\left[D_{p_{2[(a-1)N+b]-1},l_1}\right]}_{n,n}\&CenterDot;{\left[X_{(a-1)N+b}\right]}_{m,n}\&CenterDot;{\left[D_{p_{2[(a-1)N+b]},l_2}\right]}_{m,m}={\left[Y\right]}_{l_1(A/M)+m,l_2(B/N)+n}---\left(7\right)$

Therefore, in the system of linear equations that linear equation constitutes as the formula (7) by M * N, suppose

For by unknown parameter [X _{(a-1) N+b}] _{M, n}The vector that constitutes, so

$\stackrel{\&RightArrow;}{\hat{X}}=D^{-1}\&CenterDot;\stackrel{\&RightArrow;}{r}$

Wherein,

With

Be defined as

${\left[\stackrel{\&RightArrow;}{r}\right]}_{(a-1)N+b}={\left[Y\right]}_{(a-1)(A/M)+m,(b-1)(B/N)+n}$

${\left[D\right]}_{l_{2}M+{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="H2009102384791E00011.png"\; state="\{\{state\}\}">l</figure-callout>}}_1,[(a-1)N+b]}={\left[D_{p_{2[(a-1)N+b]-1},l_2}\right]}_{n,n}\&CenterDot;{\left[D_{p_{2[(a-1)N+b]},l_1}\right]}_{m,m}$

Below in conjunction with drawings and Examples the present invention is elaborated:

In this emulation experiment, adopt shown in Fig. 3 (a), to be of a size of 256 * 256 Lena image, the row and column of image carries out the twice five equilibrium respectively and obtains 4 width of cloth subimages, after the column vector of every width of cloth subimage is carried out 2 times of interpolations, carries out p respectively ₁=0.41, p ₃=0.43, p ₅=0.45 and p ₇Contrary discrete fraction rank ,=0.47 rank Fourier transform after the capable vector of every width of cloth subimage carries out 2 times of interpolations, is carried out p respectively ₂=0.42, p ₄=0.44, p ₆=0.46 and p ₈Contrary discrete fraction rank ,=0.48 rank Fourier transform, system encryption end output encrypted image is shown in Fig. 3 (b).

At the system decrypts end, if the fraction order Fourier transform order that deciphering the is adopted fraction order Fourier transform order multiplexing with encrypting end do not wait, deciphering will make a mistake so.Fig. 4 has provided and has worked as p ₁(p when making a mistake ₁=0.4) decrypted image.Fig. 5 (a) has provided the decrypted image square error respectively with p ₁, p ₂, p ₃, p ₄Error (dp ₁, dp ₂, dp ₃, dp ₄) curve map that changes.Fig. 5 (b) has provided the decrypted image square error respectively with p ₅, p ₆, p ₇, p ₈Error (dp ₅, dp ₆, dp ₇, dp ₈) curve map that changes.

Can find by above simulation result, based on multi-order fractional Fourier transform image encryption system system, each fraction order Fourier transform order can be as the key of image encryption, and when each key produces 0.01 error, the capital causes decrypted image to occur and the bigger error of original image, therefore, the image encryption technology that the present invention proposes has enlarged key space with existing image encryption technology based on Fourier Transform of Fractional Order relatively, has very high security.

Claims (1)

1. image encryption and decryption method based on a multi-order fractional Fourier transform is characterized in that: be divided into image encryption and image deciphering two parts;

The step that realizes image encryption is as follows:

Step 1, will be of a size of the wide and long M and the N equal portions of being divided into respectively of the original image of A * B, obtain the subimage S that size such as M * N is of a size of (A/M) * (B/N) _{(a-1) N+b}, a=1,2 ..., M, b=1,2 ..., N;

Step 2, each subimage S that the information encryption step 1 is obtained _{(a-1) N+b}Each column vector carry out M times of interpolation and p _{2[(a-1) N+b]-1}Contrary discrete fraction rank, rank Fourier transform is carried out N times of interpolation and p to its row vector again _{2[(a-1) N+b]}Contrary discrete fraction rank, rank Fourier transform obtains Y thus _{(a-1) N+b}, its mathematic(al) representation as the formula (1)

$Y_{(a-1)N+b}=\left[\begin{array}{c}{D}_{p_{2[(a-1)N+b]-1},0}\\ D_{p_{2[(a-1)N+b]-1},1}\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ D_{p_{2[(a-1)N+b]-1},M-1}\end{array}\right]\&CenterDot;X_{(a-1)N+b}\&CenterDot;\left[\begin{array}{cccc}{D}_{p_{2[(a-1)N+b]},0}& D_{p_{2[(a-1)N+b]},1}& \&CenterDot;\&CenterDot;\&CenterDot;& D_{p_{2[(a-1)N+b]},N-1}\end{array}\right]---\left(1\right)$

Wherein, X _{(a-1) N+b}For to S _{(a-1) N+b}Column vector be p _{2[(a-1) N+b]-1}Contrary discrete fraction rank, rank Fourier transform, row vector are p _{2[(a-1) N+b]}The contrary discrete fraction rank Fourier transform gained result in rank, diagonal matrix

L wherein ₁=0,1 ..., M-1, l ₂=0,1 ..., N-1 is defined as

${\left[D_{p_{2[(a-1)N+b]},l_1}\right]}_{k,k}=e^{j\&CenterDot;\frac{1}{2}\&CenterDot;\cot \left(\frac{p_{2[(a-1)N+b]}\&CenterDot;\mathrm{\&pi;}}{2}\right)\&CenterDot;[2k(l_1\&CenterDot;N)+{(l_1\&CenterDot;N)}^2]\&CenterDot;{\mathrm{\&Delta;t}}^2}$

${\left[D_{p_{2[(a-1)N+b]-1},l_2}\right]}_{k,k}=e^{j\&CenterDot;\frac{1}{2}\&CenterDot;\cot \left(\frac{p_{2[(a-1)N+b]-1}\&CenterDot;\mathrm{\&pi;}}{2}\right)\&CenterDot;[2k(l_2\&CenterDot;N)+{(l_2\&CenterDot;N)}^2]\&CenterDot;{\mathrm{\&Delta;t}}^2}$

Step 3, the Y that the image encryption step 2 is obtained _{(a-1) N+b}Addition successively obtains encrypted image Y, and mathematic(al) representation as the formula (2)

$Y=\underset{a=1}{\overset{M}{\mathrm{\&Sigma;}}}\underset{b=1}{\overset{N}{\mathrm{\&Sigma;}}}{Y}_{(a-1)N+b}---\left(2\right)$

The step that realizes the image deciphering is as follows:

Step 1, by l in the encrypted image ₁(A/M)+l during m is capable ₂(B/N)+and a n element, m=1 wherein, 2 ..., A/M, n=1,2 ..., B/N, the unknown number of setting up as the formula (3) is [X _{(a-1) N+b}] _{M, n}, M * N linear equation, a=1 wherein, 2 ..., M, b=1,2 ..., N

$\underset{a=1}{\overset{M}{\mathrm{\&Sigma;}}}\underset{b=1}{\overset{N}{\mathrm{\&Sigma;}}}{\left[D_{p_{2[(a-1)N+b]-1},l_1}\right]}_{n,n}\&CenterDot;{\left[X_{(a-1)N+b}\right]}_{m,n}\&CenterDot;{\left[D_{p_{2[(a-1)N+b]},l_2}\right]}_{m,m}={\left[Y\right]}_{l_1(A/M)+m,l_2(B/N)+n}---\left(3\right)$

Step 2, obtain formula (4), solve X by the formula in the image decryption step one (3) _{(a-1) N+b}In m capable in the numerical value of n element

$\stackrel{\&RightArrow;}{\hat{X}}=D^{-1}\&CenterDot;\stackrel{\&RightArrow;}{r}---\left(4\right)$

Wherein,

With

Be defined as

${\left[\stackrel{\&RightArrow;}{\hat{X}}\right]}_{(a-1)N+b}={\left[X_{(a-1)N+b}\right]}_{m,n}$

${\left[\stackrel{\&RightArrow;}{r}\right]}_{(a-1)N+b}={\left[Y\right]}_{(a-1)(A/M)+m,(b-1)(B/N)+n}$

${\left[D\right]}_{l_{2}M+l_1,[(a-1)N+b]}={\left[D_{p_{2[(a-1)N+b]-1},l_2}\right]}_{n,n}\&CenterDot;{\left[D_{p_{2[(a-1)N+b]},l_1}\right]}_{m,m}$

Step 3, the X that image decryption step two is obtained _{(a-1) N+b}Column vector and the row vector be p respectively _{2[(a-1) N+b]-1}Rank discrete fraction rank Fourier transform and p _{2[(a-1) N+b]}Discrete fraction rank, rank Fourier transform splices original image.

Images