CN112016112A - Method for encrypting image by compounding Fourier transform and differential transform - Google Patents
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Abstract
The invention provides a method for encrypting an image by compounding Fourier transform and differential transform, which is suitable for encryption transmission of important picture information. The method mainly comprises the steps of conducting normalization after reading picture data, conducting two-dimensional fast Fourier transform, obtaining two array matrixes of a real part and an imaginary part respectively, conducting proportional differential transformation on the array matrixes of the real part or the imaginary part, conducting fast normalization, storing the obtained two normalized arrays of the real part and the imaginary part as encrypted image data, sending a fast normalized information factor as one of decryption keys to a remote receiving end, conducting fast normalized inverse transformation and proportional differential inverse transformation decryption on the data, conducting two-dimensional fast Fourier inverse transformation and normalization after compounding the real part and the imaginary part, and obtaining the decrypted and restored image data. The method has the advantages of good encryption safety and high decryption recovery decoding difficulty.
Description
Technical Field
The invention relates to the field of image encryption and restoration, in particular to a method for encrypting an image by compounding Fourier transform and differential transform, which can be applied to the fields of secret communication, encrypted image processing and the like.
Background
As the importance of information security increases, image encryption technology is being paid attention and paid more and more to countries. The traditional image encryption uses the inversion and transformation of pixel data to disturb the spatial position of pixels, but the encryption method often exposes partial spider-web traces of an original image due to the continuity of the image. A chaotic system is also adopted to generate a random sequence for masking, but the method is irrelevant to the information of the image and is difficult to resist plaintext attack.
The fourier transform is a transform from the time domain to the frequency domain, and thus the representation of the image data in the time domain is completely different from the representation in the frequency domain. For example, conventional image data is between 0 and 255, while the data after fourier transform can be up to hundreds of thousands and millions of magnifications. Therefore, the simple corresponding relation is completely used for cracking. Meanwhile, after Fourier transformation, a real part and an imaginary part are generated, and the corresponding relation between the real part and the imaginary part and original image data is more complex.
Based on the reasons, the invention provides a mode of carrying out Fourier transformation on original picture data, decomposing the original picture data into a real part picture and an imaginary part picture, carrying out proportional differential transformation, and then sending the real part picture and the imaginary part picture to a remote terminal, which is completely different from the existing encryption and decryption means, so that the decryption is more complicated, and the security is higher.
It is to be noted that the information invented in the above background section is only for enhancing the understanding of the background of the present invention, and therefore, may include information that does not constitute prior art known to those of ordinary skill in the art.
Disclosure of Invention
The invention aims to provide a method for image encryption by compounding Fourier transform and differential transform, and further solves the problem that the traditional image encryption and decryption algorithm is low in complexity and safety to a certain extent.
According to one aspect of the present invention, there is provided a method for image encryption using fourier transform and differential transform composition, comprising the steps of:
step S10, reading image data, judging image type, converting data type, and normalizing data;
step S20, carrying out fast Fourier transform on the data after the normalization processing, and decomposing the corresponding real part and imaginary part to obtain a real part matrix and an imaginary part matrix;
step S30, performing matrix conversion on the real part matrix to form line data, then performing differential conversion and proportional conversion, and superposing the converted data to obtain a proportional differential superposition matrix;
step S40, matrix transformation is carried out on the proportional differential superposition matrix to obtain a row matrix, and then rapid normalization processing is carried out to obtain a normalization matrix and a normalization information factor;
step S50, the row matrix after the rapid normalization is subjected to data recombination and stored as a real part encrypted image; similarly, fast normalizing the imaginary matrix data, and then performing data recombination to store the imaginary matrix data as an imaginary encrypted image; respectively carrying out remote transmission and sending on the encrypted real part image and the encrypted imaginary part image;
and step S60, receiving the real part encrypted file and the imaginary part encrypted file at the remote terminal, and respectively storing the real part encrypted file and the imaginary part encrypted file as a real part receiving matrix and an imaginary part receiving matrix. Performing row matrix transformation and normalization transformation on the real part receiving matrix and the imaginary part receiving matrix;
step S70, fast normalization inverse transformation is respectively carried out on the real part receiving normalization column matrix and the imaginary part receiving normalization column matrix according to the normalization information factors to obtain a real part inverse transformation matrix and an imaginary part inverse transformation matrix, and then proportional differential inverse transformation is carried out on the real part inverse transformation matrix to obtain a real part proportional differential inverse transformation matrix;
and step S80, compounding the real part proportional differential inverse transformation matrix and the imaginary part inverse transformation matrix to obtain a decrypted complex matrix, then carrying out array recombination, simultaneously carrying out two-dimensional fast Fourier inverse transformation, carrying out fast normalization to obtain a decrypted image matrix, storing the decrypted image matrix as a decrypted image file, and finishing decryption and receiving of the file.
In an exemplary embodiment of the present invention, the reading, image type determination, data type conversion, and data normalization processing of the image data includes:
first, image data is read and determined, and the read data is stored as a matrix a 1.
Next, a picture is judged, that is, if it is a color picture, the picture data is converted into gray data, which is denoted as B1. If it is already a black and white picture, no conversion is needed.
Finally, a normalization process is performed to perform a data type conversion, i.e., to define a matrix C1(m × n) such that each element of C1(m × n) is 1/255 of the corresponding B1(m × n) element.
Where m × n represents the dimension of the matrix, which represents that the matrix has m rows and n columns. C1(m × n) is the normalized matrix.
In an exemplary embodiment of the present invention, performing fast fourier transform on the normalized data matrix, and decomposing the corresponding real part and imaginary part to obtain a real part matrix and an imaginary part matrix includes:
D1(m×n)＝FFT{C1(m×n)}
D1(j,k)＝ejk+ifjk；
E1(j,k)＝ejk；
F1(j,k)＝fjk
wherein C1(m × n) is the normalized matrix, FFT represents two-dimensional fast fourier transform, the obtained two-dimensional complex matrix is denoted as D1(m × n), E1(m × n) is the corresponding real part constituting real part matrix, and F1(m × n) is the corresponding imaginary part constituting imaginary part matrix.
Where i is the imaginary unit, D1(j, k) is the jth row of matrix D1(m × n), the elements of kth column, which is a complex matrix, ejkIs made ofPortion, fjkAs its imaginary part. E1(j, k) is the j-th row and k-th column elements of matrix E1(m × n), F1(j, k) is the j-th row and k-th column elements of matrix F1(m × n), j is greater than or equal to 1 and less than or equal to m, and k is greater than or equal to 1 and less than or equal to n.
In an exemplary embodiment of the present invention, performing matrix transformation on the real part matrix, and then performing differential transformation and proportional transformation to obtain a proportional-differential superposition matrix includes:
H1(i)＝k1G1(i)；
when I is 1, I1(I) is 0;
when I is greater than or equal to 2, I1(I) ═ k2(G1(i)-G1(i-1))/T1；
J1(i)＝I1(i)+H1(i)；
Wherein G1(mn × 1) is a column matrix, which is transformed from the real matrix E1(m × n). H1(mn × 1) is a scaling matrix of G1(mn × 1), H1(i) is the i-th element of matrix H1(mn × 1), G1(i) is the i-th element of matrix G1(mn × 1), and 1 ≦ i ≦ mn. k is a radical of1For the gain factor, see the following example implementation.
I1(mn × 1) is a differential transformation matrix of G1(mn × 1), where k2Is a gain factor, T1The time interval parameter between data is described in detail in the following example. I1(I) is the ith element of matrix I1(mn × 1), I is greater than or equal to 1 and less than or equal to mn. J1(mn × 1) is the final proportional differential superposition matrix, J1(i) is the ith element of the matrix J1(mn × 1), i is 1 ≦ mn.
In an exemplary embodiment of the present invention, performing matrix transformation on the proportional-differential superposition matrix to obtain a row matrix, and then performing fast normalization processing to obtain a normalization matrix and a normalization information factor includes:
M(1)＝max(K1)；
M(2)＝min(K1)；
the K1(1 × mn) is a proportional differential superposition matrix J1(mn × 1), and the row matrix is obtained by row transformation. M is the normalized information factor. M (1) is the maximum value of the matrix K1(1 × mn), and M (2) is the minimum value of the matrix K1(1 × mn). L1(1 × mn) is the matrix after fast normalization, wherein L1(i) is the ith element of the matrix L1(mn × 1), K1(i) is the ith element of the matrix K1(mn × 1), and i is greater than or equal to 1 and less than or equal to mn. max (K1) represents the maximum value for all elements of matrix K1, and min (K1) represents the minimum value for all elements of matrix K1.
In an exemplary embodiment of the present invention, the row matrix after the fast normalization is subjected to data reorganization and stored as a real part encrypted image; similarly, fast normalizing the imaginary matrix data, and then performing data recombination to store the imaginary matrix data as an imaginary encrypted image; respectively carrying out remote transmission and sending on the real part encrypted image and the imaginary part encrypted image comprises the following steps:
firstly, the row matrix L1(1 × mn) obtained after the fast normalization is subjected to data rearrangement, and the row matrix is converted into a data matrix M1(M × n) with M rows and n columns.
Next, the data matrix M1(M × n) is stored as a real-part encrypted image, which is referred to as file2. jpg.
Then, the imaginary matrix F1(m × N) is converted into a column matrix N1(mn × 1), and then fast normalization is performed to obtain O1(mn × 1) and a fast normalization information factor N, where N includes the maximum value and the minimum value of the matrix N1(mn × 1), and are stored in N (1) and N (2), respectively. Namely, it is
N(1)＝max(N1)；
N(2)＝min(N1)；
Where N1(i) is the ith element of matrix N1(mn × 1), M1(i) is the ith element of matrix M1(mn × 1), and 1 ≦ i ≦ mn. max (N1) represents the maximum value for all elements of matrix N1, and min (N1) represents the minimum value for all elements of matrix N1.
Again, the normalized matrix O1(mn × 1) is stored as an imaginary encrypted image file, denoted file3. jpg.
And finally, remotely transmitting the encrypted file2.jpg and file3.jpg files.
In an example embodiment of the present invention, the real encrypted file and the imaginary encrypted file are received at the remote terminal and stored as a real receive matrix and an imaginary receive matrix, respectively. Then, the row matrix transformation and the normalization transformation of the real part receiving matrix and the imaginary part receiving matrix comprise:
first, the real part encrypted file2.jpg is received at the remote terminal and stored as a real part reception matrix P1(m × n); the imaginary encrypted file3.jpg is received and stored as the imaginary reception matrix Q1(m × n).
Next, P1(m × n) is column-transformed to obtain a real-part reception column matrix R1(mn × 1), and Q1(m × n) is column-transformed to obtain an imaginary-part reception matrix S1(mn × 1).
Finally, the column matrices R1(mn × 1), S1(m × n) are simply normalized as follows:
T1(i)＝R1(i)/255；
U1(i)＝S1(i)/255；
where T1(m × n) is the real-part receive normalized column matrix and U1(m × n) is the imaginary-part receive normalized column matrix, T1(i) is the i-th element of matrix T1(mn × 1), R1(i) is the i-th element of matrix R1(mn × 1), U1(i) is the i-th element of matrix U1(mn × 1), S1(i) is the i-th element of matrix S1(mn × 1), and 1 ≦ i ≦ mn.
In an exemplary embodiment of the present invention, the fast normalized inverse transform is performed on the real part receiving normalized column matrix and the imaginary part receiving normalized column matrix according to the normalized information factor to obtain a real part inverse transform matrix and an imaginary part inverse transform matrix, and then the inverse real part transform matrix is subjected to inverse proportional differential transform to obtain a real part inverse proportional differential transform matrix, where the obtaining of the real part inverse proportional differential transform matrix includes:
V1(i)＝M(2)+(M(1)-M(2))*T1(i)；
W1(i)＝N(2)+(N(1)-N(2))*U1(i)；
X1(1)＝V1(1)；
D(i)＝(V1(i)-k1X(i))/k2
X1(i+1)＝X1(i)+D(i)*T1；
where V1(i) is the ith element of matrix V1(mn × 1) and W1(i) is the ith element of matrix W1(mn × 1)I is more than or equal to 1 and less than or equal to mn. V1(mn × 1) is an inverse real part transform matrix, and W1(mn × 1) is an inverse imaginary part transform matrix. X1(mn X1) is a real part proportional differential inverse transformation matrix, i is more than or equal to 1 and less than or equal to mn-1, k1、k2For the gain factor, T, used in encryption1The time interval parameter between data used for encryption is described in detail in the following examples.
In an exemplary embodiment of the present invention, the combining the real part inverse proportional-differential transform matrix and the imaginary part inverse transform matrix to obtain a decrypted complex matrix, then performing array recombination, simultaneously performing two-dimensional inverse fast fourier transform, and performing fast normalization to obtain a decrypted image matrix, where the storing as the decrypted image file includes:
first, matrix compounding is performed as follows:
xj＝X1(j)；
wj＝W1(j)；
Y1(j)＝xj+wj*i；
wherein X1(mn × 1) is the real part inverse proportional differential transform matrix, W1(mn × 1) is the imaginary part inverse transform matrix, Y1(mn × 1) is the decrypted complex matrix, and i is the imaginary part unit. Y1(j) is the jth variable of matrix Y1(mn × 1), X1(j) is the jth variable of matrix X1(mn × 1), W1(j) is the jth variable of matrix W1(mn × 1), and j is more than or equal to 1 and less than or equal to mn.
Next, the decrypted complex matrix Y1(mn × 1) is recombined to obtain a two-dimensional complex matrix Z1 (mn × n).
Then, two-dimensional inverse fast fourier transform is performed on the two-dimensional complex matrix Z1(m × n) to obtain a matrix a2(m × n), and the matrix a2(m × n) is normalized and converted into data between (0,1), thereby obtaining a normalized matrix B2(m × n).
And finally storing B2 (mxn) as a data file, and recording the data file as file4.jpg, thereby obtaining a final decrypted image.
Advantageous effects
The method for encrypting the image by compounding the Fourier transform and the differential transform, which is provided by the invention, can separate an image into a real part and an imaginary part by adopting a two-dimensional fast Fourier transform mode for separate transmission, and simultaneously performs the compound transform of proportional differential on the basis of the separation of the real part and the imaginary part, thereby further increasing the complexity of image encryption. Meanwhile, decryption is carried out by adopting inverse Fourier transform and fast normalized key and inverse proportional differential transform. Therefore, the whole encryption process has three layers, the first layer is real part and imaginary part separation of Fourier transformation, the second layer is amplification transformation of proportional differentiation, and the third layer is fast normalized key encryption. Therefore, the whole method has high safety and high engineering application value.
Drawings
The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate embodiments consistent with the invention and together with the description, serve to explain the principles of the invention. It is obvious that the drawings in the following description are only some embodiments of the invention, and that for a person skilled in the art, other drawings can be derived from them without inventive effort.
FIG. 1 is a flow chart of a method for image encryption using a Fourier transform and differential transform complex according to the present invention;
FIG. 2 is a black-and-white image to be encrypted according to the method provided by the embodiment of the invention;
FIG. 3 is an encrypted transmitted real image according to a method provided by an embodiment of the present invention;
fig. 4 is an imaginary image of encrypted transmission according to a method provided by an embodiment of the invention;
FIG. 5 is a decrypted image with a normal key according to a method provided by an embodiment of the invention;
FIG. 6 shows a key k of a method provided by an embodiment of the present invention20.0011 of the decrypted image;
FIG. 7 shows a key k of a method provided by an embodiment of the present invention20.005 decrypted image;
FIG. 8 shows a key k of a method provided by an embodiment of the invention20.0006 as a decrypted image;
Detailed Description
Example embodiments will now be described more fully with reference to the accompanying drawings. Example embodiments may, however, be embodied in many different forms and should not be construed as limited to the examples set forth herein; rather, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the concept of example embodiments to those skilled in the art. The described features, structures, or characteristics may be combined in any suitable manner in one or more embodiments. In the following description, numerous specific details are provided to provide a thorough understanding of embodiments of the invention. One skilled in the relevant art will recognize, however, that the invention may be practiced without one or more of the specific details, or with other methods, components, devices, steps, and so forth. In other instances, well-known technical solutions have not been shown or described in detail to avoid obscuring aspects of the invention.
The invention provides an image data encryption transmission method combining three-in-one of two-dimensional fast Fourier transform, proportional differential transform and fast normalized transform, and a method for decrypting by sequentially adopting fast normalized inverse transform, proportional differential inverse transform and two-dimensional fast Fourier transform at a decryption end. The method of layer-by-layer encryption ensures that the method provided by the invention has good security. Particularly, the characteristics of dividing the real part and the imaginary part of the Fourier transform into parts and amplifying the parts by discrete accumulation of the parts greatly increase the cracking difficulty of the encryption method.
The method for image encryption by compounding fourier transform and differential transform according to the present invention will be further explained and explained with reference to the drawings. Referring to fig. 1, the method for image encryption by using fourier transform and differential transform composite may include the following steps:
step S10, reading image data, judging image type, converting data type, and normalizing data;
specifically, first, the image data is read and determined. For example, it is assumed that the image file is file1.jpg, but of course, an image file in other format such as png may be used. The matrix A1 is stored after being read out.
Secondly, the picture is judged, that is, if the picture is a color picture, the picture data is converted into gray data, and if the picture is a black-and-white picture, the conversion is not needed. Here, the explanation of the encryption and decryption process is made by taking a black-and-white picture as an example, and a color picture can be regarded as a black-and-white photograph of three primary colors, and the process can be performed three times in the same method. The color image data matrix a1 is converted into black-and-white picture data after being gray-converted, and is referred to as B1. Since a method for converting a color image into black-and-white image data is already known, it will not be described herein.
And finally, carrying out data type conversion on the black-and-white image data. Assuming that B1 contains m rows and n columns, it indicates that the black-and-white data has m × n pixels. Since the types of data in the matrix B1(m × n) are all integer data between 0 and 255, the gradation is expressed. Therefore, it is necessary to first normalize the above data, i.e., define the matrix C1(m × n) such that each element of C1(m × n) is 1/255 of the corresponding B1(m × n) element.
Step S20, carrying out fast Fourier transform on the data after the normalization processing, and decomposing the corresponding real part and imaginary part to obtain a real part matrix and an imaginary part matrix;
specifically, first, a two-dimensional fast fourier transform is performed on the normalized matrix C1(m × n), that is, m × n pieces of data, and the obtained two-dimensional complex matrix is denoted as D1(m × n). The two-dimensional fast fourier transform, i.e. the FFT method, is disclosed in textbooks and will not be described again.
Next, the two-dimensional complex matrix D1(m × n) is decomposed, i.e., each element of the matrix is decomposed into a real part and an imaginary part. The corresponding real part is then assembled into a real part matrix denoted as E1(m × n) and the corresponding imaginary part is assembled into an imaginary part matrix denoted as F1(m × n).
Namely: d1(m × n) ═ FFT { C1(m × n) }
D1(j,k)＝ejk+ifjk；
E1(j,k)＝ejk；
F1(j,k)＝fjk
Where i is the imaginary unit, D1(j, k) is the jth row of matrix D1(m × n), the elements of kth column, which is a complex matrix, ejkIs a real part, fjkAs its imaginary part. E1(j, k) is the j-th row and k-th column elements of matrix E1(m × n), F1(j, k) is the j-th row and k-th column elements of matrix F1(m × n), j is greater than or equal to 1 and less than or equal to m, and k is greater than or equal to 1 and less than or equal to n.
Step S30, performing matrix conversion on the real part matrix to form line data, then performing differential conversion and proportional conversion, and superposing the converted data to obtain a proportional differential superposition matrix;
specifically, the real part matrix E1(m × n) is first converted into a column matrix G1(mn × 1), i.e., a two-dimensional matrix is converted into a one-dimensional matrix vector.
Next, the column matrix G1(mn × 1) is scaled to obtain the matrix H1(mn × 1) by scaling the corresponding matrix elements, i.e., scaling
H1(i)＝k1G1(i)；
Where H1(i) is the ith element of matrix H1(mn × 1), G1(i) is the ith element of matrix G1(mn × 1), and 1 ≦ i ≦ mn. k is a radical of1For the gain factor, see the following example implementation.
The column matrix G1(mn × 1) is subjected to differential transformation to obtain a matrix I1(mn × 1), which is calculated as follows:
when I is 1, I1(I) is 0;
when I is greater than or equal to 2, I1(I) ═ k2(G1(i)-G1(i-1))/T1；
Wherein k is2Is a gain factor, T1The time interval parameter between data is described in detail in the following example. I1(I) is the ith element of matrix I1(mn × 1), I is greater than or equal to 1 and less than or equal to mn.
Finally, the proportional differential data are superimposed to obtain a final proportional differential superimposed matrix, which is denoted as J1(mn × 1), and the calculation process is as follows:
J1(i)＝I1(i)+H1(i)；
wherein J1(i) is the ith element of the matrix J1(mn × 1), i is greater than or equal to 1 and less than or equal to mn.
Step S40, matrix transformation is carried out on the proportional differential superposition matrix to obtain a row matrix, and then rapid normalization processing is carried out to obtain a normalization matrix and a normalization information factor;
specifically, the proportional-differential superposition matrix is first line-transformed to obtain a line matrix denoted as K1(1 × mn).
Next, the row matrix K1(1 × mn) is simply normalized to obtain a normalized matrix L1(1 × mn) and a normalized information factor M. The normalization information factor M includes the maximum value of the matrix K1(1 × mn), which is denoted as M (1), and the minimum value of the matrix K1(1 × mn) is stored in M (2). Where the transformation relationship between K1(1 xmn) and L1(1 xmn) can be described as follows:
M(1)＝max(K1)；
M(2)＝min(K1)；
where L1(i) is the ith element of matrix L1(mn × 1), K1(i) is the ith element of matrix K1(mn × 1), and i is greater than or equal to 1 and less than or equal to mn. max (K1) represents the maximum value for all elements of matrix K1, and min (K1) represents the minimum value for all elements of matrix K1.
Step S50, the row matrix after the rapid normalization is subjected to data recombination and stored as a real part encrypted image; similarly, fast normalizing the imaginary matrix data, and then performing data recombination to store the imaginary matrix data as an imaginary encrypted image; and then the real part encrypted image and the imaginary part encrypted image are remotely transmitted and sent respectively.
Specifically, the row matrix L1(1 × mn) obtained by the fast normalization is rearranged, and the row matrix is converted into a data matrix M1(M × n) with M rows and n columns.
Next, the data matrix M1(M × n) is stored as a real-part encrypted image, which is referred to as file2. jpg.
Then, for the imaginary matrix F1(m × N), the imaginary matrix F1(m × N) is first converted into a column matrix N1(mn × 1), and then fast normalization is performed to obtain a normalized matrix O1(mn × 1) and a fast normalization information factor N, where N includes the maximum value and the minimum value of the matrix N1(mn × 1), and the maximum value and the minimum value are respectively stored in N (1) and N (2).
Namely, it is
N(1)＝max(N1)；
N(2)＝min(N1)；
Where N1(i) is the ith element of matrix N1(mn × 1), M1(i) is the ith element of matrix M1(mn × 1), and 1 ≦ i ≦ mn. max (N1) represents the maximum value for all elements of matrix N1, and min (N1) represents the minimum value for all elements of matrix N1.
Again, the normalized matrix O1(mn × 1) is stored as an imaginary encrypted image file, denoted file3. jpg. It is worth to be noted that, the imaginary part data matrix is not encrypted in a proportional differential mode, and if the security level needs to be improved and the decoding difficulty needs to be increased, the imaginary part data matrix can also be encrypted in a real part mode. It is not described here again, and at the same time, since the real part data is differentially encrypted, the requirements of general secret communication can be satisfied.
And finally, remotely transmitting the encrypted file2.jpg and file3.jpg files.
And step S60, receiving the real part encrypted file and the imaginary part encrypted file at the remote terminal, and respectively storing the real part encrypted file and the imaginary part encrypted file as a real part receiving matrix and an imaginary part receiving matrix. And then performing column matrix transformation and normalization transformation on the real part receiving matrix and the imaginary part receiving matrix.
Specifically, the real part encrypted file2.jpg is received at the remote terminal, and is stored as a real part receiving matrix, which is denoted as P1(m × n); the imaginary encrypted file3.jpg is received and stored as an imaginary reception matrix, denoted as Q1(m × n).
Secondly, the real part receiving matrix and the imaginary part receiving matrix are subjected to column-column transformation, namely the dimension of the matrix is changed, and the two-dimensional matrix is converted into a one-dimensional column matrix. Wherein, the P1(m × n) is subjected to column transformation to obtain a real part receiving column matrix R1(mn × 1), and the Q1(m × n) is subjected to column transformation to obtain an imaginary part receiving matrix S1(mn × 1).
Finally, the column matrices R1(mn × 1) and S1(m × n) are simply transformed into normalized real-part receiving normalized column matrix T1(m × n) and imaginary-part receiving normalized column matrix U1(m × n) in the following manner:
T1(i)＝R1(i)/255；
U1(i)＝S1(i)/255；
where T1(i) is the i-th element of matrix T1(mn × 1), R1(i) is the i-th element of matrix R1(mn × 1), U1(i) is the i-th element of matrix U1(mn × 1), S1(i) is the i-th element of matrix S1(mn × 1), and 1 ≦ i ≦ mn.
And step S70, respectively carrying out rapid normalization inverse transformation on the real part receiving normalization column matrix and the imaginary part receiving normalization column matrix according to the normalization information factors to obtain a real part inverse transformation matrix and an imaginary part inverse transformation matrix, and then carrying out proportional differential inverse transformation on the real part inverse transformation matrix to obtain a real part proportional differential inverse transformation matrix.
Specifically, first, according to the normalization factors M and N, the real-part receiving normalized column matrix T1(mn × 1) and the imaginary-part receiving normalized column matrix U1(mn × 1) are subjected to the following fast normalized inverse transformation to obtain a real-part inverse transformation matrix V1(mn × 1) and an imaginary-part inverse transformation matrix W1(mn × 1), which are described as follows:
V1(i)＝M(2)+(M(1)-M(2))*T1(i)；
W1(i)＝N(2)+(N(1)-N(2))*U1(i)；
where V1(i) is the ith element of matrix V1(mn × 1), W1(i) is the ith element of matrix W1(mn × 1), and 1 ≦ i ≦ mn.
Next, inverse proportional-differential transformation is performed on the real inverse transformation matrix V1(mn × 1) to obtain a real inverse proportional-differential transformation matrix X1(mn × 1), and the specific transformation process is as follows:
X1(1)＝V1(1)；
D(i)＝(V1(i)-k1X(i))/k2
X1(i+1)＝X1(i)+D(i)*T1；
wherein i is more than or equal to 1 and less than or equal to mn-1, k1、k2For the gain factor, T, used in encryption1Time interval parameter between data used for encryptionFor details, see the examples below.
And step S80, compounding the real part proportional differential inverse transformation matrix and the imaginary part inverse transformation matrix to obtain a decrypted complex matrix, then carrying out array recombination, simultaneously carrying out two-dimensional fast Fourier inverse transformation, carrying out fast normalization to obtain a decrypted image matrix, storing the decrypted image matrix as a decrypted image file, and finishing decryption and receiving of the file.
Specifically, the real part inverse proportional differential transform matrix X1(mn × 1) and the imaginary part inverse transform matrix W1(mn × 1) are first combined to obtain a decrypted complex matrix denoted as Y1(mn × 1), and the combining process is as follows:
xj＝X1(j)；
wj＝W1(j)；
Y1(j)＝xj+wj*i；
where i is the imaginary unit. Wherein Y1(j) is the jth variable of matrix Y1(mn × 1), X1(j) is the jth variable of matrix X1(mn × 1), W1(j) is the jth variable of matrix W1(mn × 1), and j is more than or equal to 1 and less than or equal to mn.
Secondly, the decrypted complex matrix Y1(mn × 1) is recombined to obtain a two-dimensional complex matrix Z1 (mn × n), that is, the one-dimensional column matrix is recombined into a two-dimensional matrix.
Then, the two-dimensional complex matrix Z1(m × n) is subjected to two-dimensional inverse fast fourier transform to obtain a matrix a2(m × n), and a method of the inverse transform is known from published textbooks and will not be described again here. The matrix a2(m × n) is normalized and converted into data between (0,1), and a normalized matrix B2(m × n) is obtained.
Finally, B2(m × n) is stored as a data file, which is recorded as file4.jpg, thereby obtaining the final decrypted image. By contrast, it can be seen that the decrypted image almost coincides with the original image, and even if the two undecrypted images are intercepted, the original image information cannot be obtained at all because the distortion degree is too large.
Case implementation and computer processing result analysis
In step S10, the image file is selected as a screen shot of the desktop of the home notebook computer. Since it is a color image, it is converted into a black-and-white image, and then it is shown in fig. 2. It contains 204 rows of m and 803 columns of n.
In step S20, performing fast fourier transform on the normalized data, and decomposing the corresponding real part and imaginary part to obtain a real part matrix and an imaginary part matrix;
in step S30, k is selected1＝1，k2＝0.001，T1Converting the real part matrix to obtain a proportional differential superposition matrix;
in step S40, performing matrix transformation on the proportional-derivative superposition matrix to obtain a row matrix, and then performing fast normalization processing to obtain a normalization matrix and a normalization information factor;
in step S50, performing data reconstruction on the row matrix after the fast normalization, and storing the row matrix as a real part encrypted image as shown in fig. 3; similarly, the imaginary matrix data is quickly normalized and then data is reconstructed, and the data is stored as an imaginary encrypted image as shown in fig. 4.
In step S60, the real encrypted file and the imaginary encrypted file are received at the remote terminal and stored as a real receiving matrix and an imaginary receiving matrix, respectively. And then performing column matrix transformation and normalization transformation on the real part receiving matrix and the imaginary part receiving matrix.
In step S70, a pick k is set as in step S301＝1，k2＝0.001，T1The real part inverse proportional differential transform matrix is obtained at 0.001.
In step S80, after performing two-dimensional inverse fast fourier transform, performing fast normalization to obtain a decrypted image matrix, and storing the decrypted image matrix as a decrypted image file is shown in fig. 5.
When the decryption parameter k is selected in step S701＝1，k2＝0.0011，T1When the value is 0.001, the image effect obtained by decryption is as shown in fig. 6.
When the decryption parameter k is selected in step S701＝1，k2＝0.005，T1When the value is 0.001, the image effect obtained by decryption is as shown in fig. 7.
When in stepThe decryption parameter k selected in step S701＝1，k2＝0.0006，T1When the value is 0.001, the image effect obtained by decryption is as shown in fig. 8.
It can be seen that if the parameters are slightly different, the decryption effect is very different, especially when k is larger2Below the key, the image quickly darkens and disappears as shown in fig. 8. The above result is also the case that the decryption method is completely open, and completely different encryption effects can be achieved only by slightly different key parameters. If only the encrypted data is intercepted, as shown in fig. 3 and 4, a completely black image appears, and useful information cannot be obtained at all. Therefore, the method provided by the invention has high engineering application value.
Other embodiments of the invention will be apparent to those skilled in the art from consideration of the specification and practice of the invention disclosed herein. This application is intended to cover any variations, uses, or adaptations of the invention following, in general, the principles of the invention and including such departures from the present disclosure as come within known or customary practice within the art to which the invention pertains. It is intended that the specification and examples be considered as exemplary only, with a true scope and spirit of the invention being indicated by the following claims.
Claims (9)
1. A method for encrypting an image by compounding Fourier transform and differential transform is characterized by comprising the following steps:
step S10, reading image data, judging image type, converting data type, and normalizing data;
step S20, carrying out fast Fourier transform on the data after the normalization processing, and decomposing the corresponding real part and imaginary part to obtain a real part matrix and an imaginary part matrix;
step S30, performing matrix conversion on the real part matrix to form line data, then performing differential conversion and proportional conversion, and superposing the converted data to obtain a proportional differential superposition matrix;
step S40, matrix transformation is carried out on the proportional differential superposition matrix to obtain a row matrix, and then rapid normalization processing is carried out to obtain a normalization matrix and a normalization information factor;
step S50, the row matrix after the rapid normalization is subjected to data recombination and stored as a real part encrypted image; similarly, fast normalizing the imaginary matrix data, and then performing data recombination to store the imaginary matrix data as an imaginary encrypted image; respectively carrying out remote transmission and sending on the encrypted real part image and the encrypted imaginary part image;
and step S60, receiving the real part encrypted file and the imaginary part encrypted file at the remote terminal, and respectively storing the real part encrypted file and the imaginary part encrypted file as a real part receiving matrix and an imaginary part receiving matrix. Performing row matrix transformation and normalization transformation on the real part receiving matrix and the imaginary part receiving matrix;
step S70, fast normalization inverse transformation is respectively carried out on the real part receiving normalization column matrix and the imaginary part receiving normalization column matrix according to the normalization information factors to obtain a real part inverse transformation matrix and an imaginary part inverse transformation matrix, and then proportional differential inverse transformation is carried out on the real part inverse transformation matrix to obtain a real part proportional differential inverse transformation matrix;
and step S80, compounding the real part proportional differential inverse transformation matrix and the imaginary part inverse transformation matrix to obtain a decrypted complex matrix, then carrying out array recombination, simultaneously carrying out two-dimensional fast Fourier inverse transformation, carrying out fast normalization to obtain a decrypted image matrix, storing the decrypted image matrix as a decrypted image file, and finishing decryption and receiving of the file.
2. The method of claim 1, wherein the image data reading, image type judging, data type converting and data normalizing processing comprises:
first, image data is read and determined, and the read data is stored as a matrix a 1. Next, a picture is judged, that is, if it is a color picture, the picture data is converted into gray data, which is denoted as B1. If it is already a black and white picture, no conversion is needed. Finally, a normalization process is performed to perform a data type conversion, i.e., to define a matrix C1(m × n) such that each element of C1(m × n) is 1/255 of the corresponding B1(m × n) element. Where m × n represents the dimension of the matrix, which represents that the matrix has m rows and n columns. C1(m × n) is the normalized matrix.
3. The method of claim 1, wherein the fast fourier transform is performed on the normalized data matrix, and the decomposition of the corresponding real part and imaginary part to obtain the real part matrix and imaginary part matrix comprises:
D1(m×n)＝FFT{C1(m×n)}
D1(j,k)＝ejk+ifjk；
E1(j,k)＝ejk；
F1(j,k)＝fjk
wherein C1(m × n) is the normalized matrix, FFT represents two-dimensional fast fourier transform, the obtained two-dimensional complex matrix is denoted as D1(m × n), E1(m × n) is the corresponding real part constituting real part matrix, and F1(m × n) is the corresponding imaginary part constituting imaginary part matrix. Where i is the imaginary unit, D1(j, k) is the jth row of matrix D1(m × n), the elements of kth column, which is a complex matrix, ejkIs a real part, fjkAs its imaginary part. E1(j, k) is the j-th row and k-th column elements of matrix E1(m × n), F1(j, k) is the j-th row and k-th column elements of matrix F1(m × n), j is greater than or equal to 1 and less than or equal to m, and k is greater than or equal to 1 and less than or equal to n.
4. The method of claim 1, wherein performing matrix transformation on the real part matrix, and performing differential transformation and proportional transformation to obtain a proportional-differential superposition matrix comprises:
H1(i)＝k1G1(i)；
when I is 1, I1(I) is 0;
when I is greater than or equal to 2, I1(I) ═ k2(G1(i)-G1(i-1))/T1；
J1(i)＝I1(i)+H1(i)；
Wherein G1(mn × 1) is a column matrix, which is transformed from the real matrix E1(m × n). H1 (m)n × 1) is the scaling matrix of G1(mn × 1), H1(i) is the ith element of matrix H1(mn × 1), G1(i) is the ith element of matrix G1(mn × 1), 1 ≦ i ≦ mn. k is a radical of1Is a gain factor. I1(mn × 1) is a differential transformation matrix of G1(mn × 1), where k2Is a gain factor, T1Is a time interval parameter between data. I1(I) is the ith element of matrix I1(mn × 1), I is greater than or equal to 1 and less than or equal to mn. J1(mn × 1) is the final proportional differential superposition matrix, J1(i) is the ith element of the matrix J1(mn × 1), i is 1 ≦ mn.
5. The method of claim 1, wherein the matrix transformation of the proportional-differential superposition matrix to obtain a row matrix is performed, and then the fast normalization processing is performed to obtain a normalization matrix and a normalization information factor, the method comprising:
M(1)＝max(K1)；
M(2)＝min(K1)；
the K1(1 × mn) is a proportional differential superposition matrix J1(mn × 1), and the row matrix is obtained by row transformation. M is the normalized information factor. M (1) is the maximum value of the matrix K1(1 × mn), and M (2) is the minimum value of the matrix K1(1 × mn). L1(1 × mn) is the matrix after fast normalization, wherein L1(i) is the ith element of the matrix L1(mn × 1), K1(i) is the ith element of the matrix K1(mn × 1), and i is greater than or equal to 1 and less than or equal to mn. max (K1) represents the maximum value for all elements of matrix K1, and min (K1) represents the minimum value for all elements of matrix K1.
6. The method for image encryption by compounding fourier transform and differential transform as claimed in claim 1, wherein the fast normalized row matrix is subjected to data reorganization and stored as a real encrypted image; similarly, fast normalizing the imaginary matrix data, and then performing data recombination to store the imaginary matrix data as an imaginary encrypted image; respectively carrying out remote transmission and sending on the real part encrypted image and the imaginary part encrypted image comprises the following steps:
firstly, the row matrix L1(1 × mn) obtained after the fast normalization is subjected to data rearrangement, and the row matrix is converted into a data matrix M1(M × n) with M rows and n columns. Next, the data matrix M1(M × n) is stored as a real-part encrypted image, which is referred to as file2. jpg. Then, the imaginary matrix F1(m × N) is converted into a column matrix N1(mn × 1), and then fast normalization is performed to obtain O1(mn × 1) and a fast normalization information factor N, where N includes the maximum value and the minimum value of the matrix N1(mn × 1), and are stored in N (1) and N (2), respectively. Namely, it is
N(1)＝max(N1)；
N(2)＝min(N1)；
Where N1(i) is the ith element of matrix N1(mn × 1), M1(i) is the ith element of matrix M1(mn × 1), and 1 ≦ i ≦ mn. max (N1) represents the maximum value for all elements of matrix N1, and min (N1) represents the minimum value for all elements of matrix N1. Again, the normalized matrix O1(mn × 1) is stored as an imaginary encrypted image file, denoted file3. jpg. And finally, remotely transmitting the encrypted file2.jpg and file3.jpg files.
7. The method of claim 1, wherein the real encrypted file and the imaginary encrypted file are received at a remote terminal and stored as a real receiving matrix and an imaginary receiving matrix, respectively. Then, the row matrix transformation and the normalization transformation of the real part receiving matrix and the imaginary part receiving matrix comprise:
first, the real part encrypted file2.jpg is received at the remote terminal and stored as a real part reception matrix P1(m × n); the imaginary encrypted file3.jpg is received and stored as the imaginary reception matrix Q1(m × n). Next, P1(m × n) is column-transformed to obtain a real-part reception column matrix R1(mn × 1), and Q1(m × n) is column-transformed to obtain an imaginary-part reception matrix S1(mn × 1). Finally, the column matrices R1(mn × 1), S1(m × n) are simply normalized as follows:
T1(i)＝R1(i)/255；
U1(i)＝S1(i)/255；
where T1(m × n) is the real-part receive normalized column matrix and U1(m × n) is the imaginary-part receive normalized column matrix, T1(i) is the i-th element of matrix T1(mn × 1), R1(i) is the i-th element of matrix R1(mn × 1), U1(i) is the i-th element of matrix U1(mn × 1), S1(i) is the i-th element of matrix S1(mn × 1), and 1 ≦ i ≦ mn.
8. The method of claim 1, wherein fast normalized inverse transformation is performed on the real part receiving normalized column matrix and the imaginary part receiving normalized column matrix according to the normalized information factors to obtain a real part inverse transformation matrix and an imaginary part inverse transformation matrix, and then inverse proportional-differential transformation is performed on the real part inverse transformation matrix to obtain a real part inverse proportional-differential transformation matrix, and the method further comprises:
V1(i)＝M(2)+(M(1)-M(2))*T1(i)；
W1(i)＝N(2)+(N(1)-N(2))*U1(i)；
X1(1)＝V1(1)；
D(i)＝(V1(i)-k1X(i))/k2
X1(i+1)＝X1(i)+D(i)*T1；
where V1(i) is the ith element of matrix V1(mn × 1), W1(i) is the ith element of matrix W1(mn × 1), and 1 ≦ i ≦ mn. V1(mn × 1) is an inverse real part transform matrix, and W1(mn × 1) is an inverse imaginary part transform matrix. X1(mn X1) is a real part proportional differential inverse transformation matrix, i is more than or equal to 1 and less than or equal to mn-1, k1、k2For the gain factor, T, used in encryption1The time interval parameter between data used for encryption is described in detail in the following examples.
9. The method according to claim 1, wherein the combining of the fourier transform and the differential transform to encrypt the image comprises combining the real part inverse proportional-differential transform matrix and the imaginary part inverse transform matrix to obtain a decrypted complex matrix, performing array recombination, performing two-dimensional inverse fast fourier transform, performing fast normalization to obtain a decrypted image matrix, and storing the decrypted image matrix as a decrypted image file, wherein the method comprises:
first, matrix compounding is performed as follows:
xj＝X1(j)；
wj＝W1(j)；
Y1(j)＝xj+wj*i；
wherein X1(mn × 1) is the real part inverse proportional differential transform matrix, W1(mn × 1) is the imaginary part inverse transform matrix, Y1(mn × 1) is the decrypted complex matrix, and i is the imaginary part unit. Y1(j) is the jth variable of matrix Y1(mn × 1), X1(j) is the jth variable of matrix X1(mn × 1), W1(j) is the jth variable of matrix W1(mn × 1), and j is more than or equal to 1 and less than or equal to mn. Next, the decrypted complex matrix Y1(mn × 1) is recombined to obtain a two-dimensional complex matrix Z1 (mn × n). Then, two-dimensional inverse fast fourier transform is performed on the two-dimensional complex matrix Z1(m × n) to obtain a matrix a2(m × n), and the matrix a2(m × n) is normalized and converted into data between (0,1), thereby obtaining a normalized matrix B2(m × n). And finally storing B2 (mxn) as a data file, and recording the data file as file4.jpg, thereby obtaining a final decrypted image.
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