CN112543092B - Chaotic binary sequence family matrix construction method based on image compressed sensing encryption - Google Patents
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Abstract
The invention relates to a chaotic binary sequence family matrix construction method based on image compressed sensing encryption, which comprises the following steps: (1) Judging the parity of the related parameters according to the information dimension of the image block size, and selecting a corresponding trace representation function; (2) Generating a binary pseudo-random sequence set forming a corresponding binary sequence family by the trace representation function selected in the step (1), performing numerical conversion to obtain a corresponding bipolar sequence family, and selecting a part of sequences from the obtained bipolar sequence family as column vectors to be arranged to obtain a corresponding initial measurement matrix; (3) And (3) introducing a chaotic sequence, and correspondingly replacing column vectors of the initial measurement matrix obtained in the step (2) to obtain a required chaotic binary sequence family matrix. The invention can be used for constructing a compressed sensing measurement matrix with friendly hardware, high sensing performance and good encryption property, and realizing compressed encryption acquisition of image signals such as gray images, color images and the like.
Description
Technical Field
The invention belongs to the field of signal processing, and particularly relates to a chaotic binary sequence family matrix construction method in image compression perception encryption.
Background
Compressed sensing (Compressive Sensing, CS) is an effective source processing technique due to the sparsity or compressibility of the signal being considered. It is a completely new signal sampling framework that differs from Nyquist (Nyquist) sampling law. CS can realize sampling sparse signals at a sampling rate far lower than Nyquist, and is characterized in that the linear mapping of the original signals from a high-dimensional space to a low-dimensional space is realized by utilizing a measurement matrix, and then the high-probability accurate reconstruction of the original signals is realized by utilizing the sparsity/compressibility of the signals in a reconstruction algorithm. From mathematical knowledge, a good measurement matrix can maintain all information of the original signal during projection measurement, and can reconstruct the original signal by combining the measured values.
In image compressed sensing encryption, the measurement matrix can be used as an encryption key in cryptography when the encryption is hidden in the process of data sampling, if x is regarded as an image signal to be processed and A is regarded as an encryption operator, y=ax is regarded as an encrypted signal, and the measurement matrix realizes the encryption operation on the image signal without extra cost while finishing the sensing of dimension reduction information. The construction of a measurement matrix with good encryption property has important significance for popularization and application of CS theory.
Existing measurement matrices can be divided into random measurement matrices and deterministic measurement matrices. For the former, gaussian and bernoulli matrices are more common. Although the random matrix is widely applied in scientific research, the practical application has a large limitation, the uncertainty of elements consumes a large amount of storage resources, and the generation of random numbers has high requirements on hardware, so that the hardware implementation of the random matrix is difficult. While the elements of the deterministic measurement matrix are deterministic, these deficiencies can be overcome. Based on the above, from the practical popularization and application of CS theory, the research of the deterministic measurement matrix is particularly important, and becomes a future research direction. As a main part of CS theory, the design of the measurement matrix directly relates to success or failure of CS theory in practical application, and the requirements on hardware implementation simplicity and reconstruction performance of the measurement matrix are higher and higher.
Therefore, how to design a high-performance deterministic measurement matrix with good encryption properties and easy hardware implementation is a worth of research in the image CS encryption context.
Disclosure of Invention
The invention aims to provide a chaotic binary sequence family matrix construction method based on image compressed sensing encryption, which is used for solving the problems in the prior art.
The invention discloses a chaotic binary sequence family matrix construction method based on image compressed sensing encryption, which comprises the following steps: (1) Judging the parity of the related parameters according to the information dimension of the image block size, and selecting a corresponding trace representation function; (2) Generating a binary pseudo-random sequence set forming a corresponding binary sequence family by the trace representation function selected in the step (1), performing numerical conversion to obtain a corresponding bipolar sequence family, and selecting a part of sequences from the obtained bipolar sequence family as column vectors to be arranged to obtain a corresponding initial measurement matrix; (3) And (3) introducing a chaotic sequence, and correspondingly replacing column vectors of the initial measurement matrix obtained in the step (2) to obtain a required chaotic binary sequence family matrix.
An embodiment of the chaotic binary sequence family matrix construction method based on image compressed sensing encryption according to the present invention, wherein the trace function related to the trace representing function in the step (1) is defined as follows:
Let n, m be a positive integer, and m divide n, the trace function from the finite field GF (2 n) to GF (2 m) is:
When m=1, GF (2 m) =gf (2) = {0,1}, Abbreviated as Tr (x);
Wherein, for a finite field GF (q) of size q, let β be the primitive field element of GF (q), then all field elements of GF (q) can be generated from powers of 0 and β: GF (q) = {0, β 0＝1,β,…,βq-2 }; wherein, the latter q-1 nonzero elements form a multiplication group GF (q) \ {0}, denoted GF (q) *; all elements of GF (q) are denoted as {0,1, and q-1.
According to an embodiment of the chaotic binary sequence family matrix construction method based on image compressed sensing encryption, the step (1) specifically comprises the following steps:
1a) Judging the parity of a related parameter N according to the information dimension N=2 n+1 (N is larger than or equal to 5) of the image block size, and executing the step 1b if N is an odd number; if n is an even number, executing the step 1 c);
1b) Selection trace representation function Wherein x ε GF (2 n)*,λ0,λ1∈GF(2n);
1c) Selection trace representation function Wherein x ε GF (2 n)*,λ0,λ1∈GF(2n).
According to an embodiment of the chaotic binary sequence family matrix construction method based on image compressed sensing encryption, the step (2) specifically includes:
2a) Selecting primitive domain element beta on GF (2 n), letting At this time, t is {0,1, ··2 n-2},λ0,λ1∈GF(2n), letTraversing and taking value of parameter (lambda 0,λ1) to obtain binary pseudo-random sequence set forming corresponding binary sequence familyExecuting step 2 b);
2b) Binary sequence All elements are sequentially input into the numerical conversion functionObtaining corresponding bipolar pseudo-random sequenceFor the corresponding bipolar sequence familyExecuting step 2 c);
2c) The elements of (lambda 0,λ1) are ordered by dictionary as (0, 0), (0, 1), …, (0, 2 n-1),(1,0),(1,1),…,(1,2n-1),(2,0),…,(2n-1,2n -1), and Is used as a column vector to be orderly arranged to obtain an initial measurement matrix A with the size of (2 n-1)×2n+1;
according to an embodiment of the chaotic binary sequence family matrix construction method based on image compressed sensing encryption, the step (3) specifically includes:
3a) According to chebyshev mapping R j+1＝cos(w·arccos(rj)) to obtain a corresponding chebyshev chaotic sequence R (R 0,s,l)＝{r0,rs,r2s,…,r(l-1)s), for a given R 0, recording each value of R (R 0, s, l) when w= 5,s =5 and l=n=2 n+1, arranging the corresponding chaotic sequences { R 0,rs,r2s,…,r(l-1)s } in descending order to obtain a corresponding index set χ, wherein the set becomes a chaotic set due to the pseudo-randomness of R (R 0, s, l), and executing step 3 b);
In the chebyshev mapping formula, j=0, 1,2, …, r 0 e [ -1,1] is an initial state, and w is a positive integer greater than 1, which is also called mapping degree;
3b) The column vector of matrix A is adjusted by utilizing the sequence of the X set element to obtain a new matrix which is a chaotic binary sequence family matrix The chaotic binary sequence family matrix is expressed as a matrix formPermutation operator based on chaosIs an identity matrix determined by the element sequence of the aggregation χIs a deterministic column arrangement of (c).
The invention can be used for constructing a compressed sensing measurement matrix with friendly hardware, high sensing performance and good encryption property, and realizing compressed encryption acquisition of image signals such as gray images, color images and the like.
Drawings
FIG. 1 is a general flow chart of an implementation of the present invention;
FIGS. 2a and 2b are graphs comparing performance results of successful reconstruction probabilities in different initial states;
FIGS. 3 a-3 d are graphs comparing the results of the reconstruction of the BSFDBC matrix image "liftingbody" under different keys;
FIG. 4 is a graph comparing PSNR performance results of image reconstruction decrypted by different keys;
FIGS. 5a and 5b are graphs comparing performance results of successful reconstruction probabilities under different sparsities in a noise-free scenario;
FIGS. 6a and 6b are graphs comparing reconstructed SNR performance results at different sparsities in noisy scenarios;
FIGS. 7a and 7b are graphs comparing reconstructed SNR performance results under different input noise conditions;
fig. 8a to 8f are six test images including three gray images and three color images.
Detailed Description
For the purposes of clarity, content, and advantages of the present invention, a detailed description of the embodiments of the present invention will be described in detail below with reference to the drawings and examples.
The invention adopts a chaotic binary sequence group matrix construction method based on image compression perception encryption, which comprises the following steps:
(1) Judging the parity of the related parameters according to the information dimension of the image block size, and selecting a corresponding trace representation function;
the trace function related to the trace representing function in the step (1) is defined as follows:
Let n, m be a positive integer, and m divide n, the trace function from the finite field GF (2 n) to GF (2 m) is:
When m=1, GF (2 m) =gf (2) = {0,1}. Here the number of the elements is the number, Abbreviated as Tr (x).
Wherein, for a finite field GF (q) of size q, let β be the primitive field element of GF (q), then all field elements of GF (q) can be generated from powers of 0 and β, i.e.: GF (q) = {0, β 0＝1,β,…,βq-2 }. The latter q-1 nonzero elements constitute a multiplication group GF (q) \ {0} denoted GF (q) *. For the sake of convenience of description, all elements of GF (q) may also be represented as {0,1, and q-1.
(2) Generating a binary pseudo-random sequence set forming a corresponding binary sequence family by the trace representation function selected in the step (1), performing numerical conversion on the binary pseudo-random sequence set to obtain a corresponding bipolar sequence family, and selecting a part of sequences from the obtained bipolar sequence family as column vectors to be arranged to obtain a corresponding initial measurement matrix;
(3) And (3) introducing a chaotic sequence to correspondingly replace the column vector of the initial measurement matrix obtained in the step (2) to obtain the required chaotic binary sequence family matrix.
Compared with the prior art, the invention has the following advantages:
The correlation of the matrix BSFDBC is smaller than the Gaussian random matrix and the Bernoulli random matrix with the same size, the correlation is an important criterion for describing the matrix property, the correlation is reduced, the sparseness of a reconstructable signal is increased, the reconstruction precision is improved, and theoretical guarantee is provided for the high perception performance of the matrix BSFDBC;
The inventive matrix has good potential encryption properties because of the high complexity of the exhaustive search permutation operator. The initial state determines the order of their column vectors. When the initial states are different, the corresponding column vector arrangement order is different, which will further generate a different BSFDBC matrix, so the initial states can be considered as keys to construct BSFDBC matrix, which is beneficial for the actual CS application;
The matrix BSFDBC is derived from BSF and Chebyshev chaos sequences, the reconstruction precision is superior to that of Gaussian random matrixes and Bernoulli random matrixes with the same size, the method has the practical characteristics of easiness in hardware implementation, high perceptive performance and good encryption property, the negative influence of unfriendly implementation of random matrix hardware and large storage resources on practical application is overcome, and the practicality of CS theory is facilitated;
The matrix of the invention is sensitive to the initial state, under different initial states, BSFDBC matrix has limited influence on corresponding reconstruction precision, the initial states of the seed functions of the chaotic sequence are different, the obtained measurement matrix is also different, the difference of the initial states corresponds to different arrangements of the vector of the measurement matrix array, but the correlation of the measurement matrix is not changed, and theoretical guarantee is provided for good encryption property of the measurement matrix.
Referring to fig. 1, the chaotic binary sequence family matrix construction method based on image compressed sensing encryption of the present invention comprises the following steps:
step 1: and judging the parity of the related parameters according to the information dimension of the image block size, and selecting a corresponding trace representation function.
The trace function related to the trace representing function in the step 1 is defined as follows:
Let n, m be a positive integer, and m divide n, the trace function from the finite field GF (2 n) to GF (2 m) is:
When m=1, GF (2 m) =gf (2) = {0,1}. Here the number of the elements is the number, Abbreviated as Tr (x).
Wherein, for a finite field GF (q) of size q, let β be the primitive field element of GF (q), then all field elements of GF (q) can be generated from powers of 0 and β, i.e.: GF (q) = {0, β 0＝1,β,…,βq-2 }. The latter q-1 nonzero elements constitute a multiplication group GF (q) \ {0} denoted GF (q) *. For the sake of convenience of description, all elements of GF (q) may also be represented as {0,1, and q-1.
The step 1 specifically comprises the following steps:
1a) Judging the parity of a related parameter N according to the information dimension N=2 n+1 (N is larger than or equal to 5) of the image block size, and executing the step 1b if N is an odd number; if n is an even number, executing the step 1 c);
1b) Selection trace representation function Wherein x ε GF (2 n)*,λ0,λ1∈GF(2n);
1c) Selection trace representation function Wherein x ε GF (2 n)*,λ0,λ1∈GF(2n).
Step 2: and (2) generating a binary pseudo-random sequence set forming a corresponding binary sequence family for the trace representation function selected in the step (1), performing numerical conversion on the binary pseudo-random sequence set to obtain a corresponding bipolar sequence family, and selecting a part of sequences from the obtained bipolar sequence family as column vectors to be arranged to obtain a corresponding initial measurement matrix.
2A) Selecting primitive domain element beta on GF (2 n), lettingAt this point t.epsilon. {0,1,. Cndot.2 n-2},λ0,λ1∈GF(2n). Order theTraversing and taking value of parameter (lambda 0,λ1) to obtain binary pseudo-random sequence set forming corresponding binary sequence familyExecuting step 2 b);
2b) Binary sequence All elements of (a) are sequentially input into a numerical conversion function to obtain a corresponding bipolar pseudo-random sequenceFor the corresponding bipolar sequence familyExecuting step 2 c);
2c) The elements of (lambda 0,λ1) are ordered by dictionary as (0, 0), (0, 1), …, (0, 2 n-1),(1,0),(1,1),…,(1,2n-1),(2,0),…,(2n-1,2n -1). Will be Is used as a column vector to be orderly arranged to obtain an initial measurement matrix A with the size of (2 n-1)×2n+1;
step 3: and (3) introducing a chaotic sequence to correspondingly replace the column vector of the initial measurement matrix obtained in the step (2) to obtain the required chaotic binary sequence family matrix.
3A) According to chebyshev mapping R j+1＝cos(w·arccos(rj)) to obtain a corresponding chebyshev chaotic sequence R (R 0,s,l)＝{r0,rs,r2s,…,r(l-1)s), for a given R 0, recording each value of R (R 0, s, l) when w= 5,s =5 and l=n=2 n+1, arranging the corresponding chaotic sequences { R 0,rs,r2s,…,r(l-1)s } in descending order to obtain a corresponding index set χ, wherein the set becomes a chaotic set due to the pseudo-randomness of R (R 0, s, l), and executing step 3 b);
In the chebyshev mapping formula, j=0, 1,2, …, r 0 e [ -1,1] is the initial state, and w is a positive integer greater than 1, which is also called the mapping degree.
3B) The column vector of the matrix A is adjusted by utilizing the sequence of the X set elements to obtain a new matrix, namely the chaotic binary sequence family matrixThe chaotic binary sequence family matrix is expressed as a matrix formHere, the permutation operator based on chaosIs an identity matrix determined by the element sequence of the aggregation χIs a deterministic column arrangement of (c).
The effect of the invention can be further illustrated by the following experiments:
1. Experimental conditions and content:
Matlab simulation platform is adopted in the experiment, and the performance of the BSFDBC matrix, gaussian random matrix Gaussian and Bernoulli random matrix Bernoulli is compared through simulation experiments on sparse signals and image signals. In the Gaussian matrix, each element value follows a standard normal distribution of independent same distribution. In the Bernoulli matrix, each element value is subjected to Bernoulli distribution with independent same distribution, the elements consist of +1 and-1, and the signal reconstruction algorithm selects an orthogonal matching pursuit (Orthogonal Matching Pursuit, OMP) algorithm. Without loss of generality, the initial state r 0 of the BSFDBC matrix is set to 0.8.
In the simulation experiment of sparse signals, two types of BSFDBC matrices were generated, the size of which was (2 n-1)×2n+1: one is when n is even and n=8, the matrix size was 255×512 at this time; the other is when n is odd and n=7, the matrix size is 127×256, in a Signal with length of 2 n+1, k positions are randomly selected, the values of which follow the standard gaussian distribution, the values of other positions are 0, thus obtaining a k sparse Signal x.
In the simulation experiment of the image signal, BSFDBC matrix is compared by a block CS algorithmImage reconstruction performance with Gaussian and Bernoulli matrices of the same size. Considering the trade-off between reconstruction quality, hardware implementation and reconstruction time, the block sizes are chosen to be 32×16 and 32×32, which correspond to one type of BSFDBC matrix, respectively. To characterize the image reconstruction performance, a peak signal-to-Noise Ratio (PSNR) was chosen as the evaluation criterion. For a two-dimensional image signal x of size mxn, let x R be the corresponding reconstructed image signal, PSNR is defined as follows
If the original image x is a three-dimensional color image, the signal x is first converted into a two-dimensional gray image signal x F by concatenating its R, G, B components in column-wise spread form, and then the resultant signal x F and the corresponding reconstructed signal are usedPSNR (x F) was calculated.
2. Simulation experiment and results:
Experiment one: for a measurement matrix BSFDBC with the size of 255 multiplied by 512, the sparsity k epsilon {60,95,105,115}, and a comparison curve of the successful reconstruction probabilities of k sparse noise-free signals under different initial states r 0 is shown in fig. 2 (a), wherein r 0 is more than or equal to-1 and less than or equal to-1; for a measurement matrix BSFDBC with the size of 127 multiplied by 256, the sparsity k epsilon {20,45,50,55}, and a comparison curve of the successful reconstruction probabilities of k sparse noise-free signals under different initial states r 0 is shown in fig. 2 (b), wherein r 0 is more than or equal to-1 and less than or equal to-1.
Fig. 2 shows that the different initial states of BSFDBC have limited impact on reconstruction accuracy for all sparsity values. This is because the correlation size of BSFDBC matrix is insensitive to the value of its initial state.
Experiment II: in the simulation experiment, matrixIs used as an encryption key, meaning that the encryption is hidden during the data sampling process, where r 0 =0.8. In the phase of signal reconstruction, the matrix generated by the correct key r 0 =0.8 is consideredAnd the matrix generated by the error key r 1 The test image is "liftingbody" of size 512×512 as shown in fig. 3 (a), where the block size is chosen to be 32×16. Fig. 3 (b) and 3 (c) are decrypted images of the wrong keys r 1 =0.3 and r 1 = -0.8, respectively, and fig. 3 (d) is a decrypted image of the correct key r 0 =0.8.
The experimental results show that: the corresponding reconstructed PSNR of the three decrypted images of FIG. 3 (b), FIG. 3 (c) and FIG. 3 (d) are 2.01dB, 2.14dB and 36.48dB, respectively. It is apparent that the encrypted image cannot be correctly decrypted by the wrong key r 1.
Experiment III: in the simulation environment of experiment two, the reconstructed PSNR (reconstruction PSNR) contrast curve of the decrypted image under different keys r 1 is shown in FIG. 4 for "liftingbody", where-1.ltoreq.r 1.ltoreq.1.
Fig. 4 shows that the encrypted image signal cannot be correctly decrypted by the wrong key r 1≠r0. Thus BSFDBC matrixSensitive to the key r 0, the data security can be effectively ensured.
Experiment IV: for a measuring matrix with the size of 255 multiplied by 512, the k sparse noise-free signal successful reconstruction probability comparison curve under different sparsity is shown as a graph (a) of fig. 5, wherein k is more than or equal to 30 and less than or equal to 150; for a measurement matrix with the size of 127 multiplied by 256, the comparison curve of the successful reconstruction probability of k sparse noise-free signals under different sparsity is shown in fig. 5 (b), wherein k is more than or equal to 10 and less than or equal to 80.
Fig. 5 shows that the reconstruction accuracy of the BSFDBC matrix of the present invention is better than the Gaussian matrix and Bernoulli matrix of the same size.
Experiment five: selecting a sampled signal with a signal-to-noise ratio of 30dB, and reconstructing an SNR contrast curve of k sparse noisy signals with different sparsity for a measuring matrix with a size of 255 multiplied by 512, wherein k is more than or equal to 30 and less than or equal to 150 as shown in a figure 6 (a); for a measurement matrix with the size of 127 multiplied by 256, k sparse noisy signal reconstruction SNR contrast curves at different sparsities are shown in FIG. 6 (b), wherein k is more than or equal to 10 and less than or equal to 80.
Fig. 6 shows that the matrix BSFDMC of the present invention reconstructs higher SNR than the Gaussian and Bernoulli matrices of the same size for all sparsity values.
Experiment six: under different noise conditions, selecting a sampled signal with the sparsity of 70 for a measuring matrix with the size of 255 multiplied by 512, wherein a reconstructed SNR (signal to noise ratio) comparison curve of the obtained noisy signal is shown in fig. 7 (a); for a measurement matrix of size 127×256, a sampled signal with a sparsity of 35 is selected, and the reconstructed SNR contrast curve of the resulting noisy signal is shown in fig. 7 (b).
Fig. 7 shows that the inventive matrix BSFDBC has a higher reconstructed SNR than the Gaussian and Bernoulli matrices of the same size under different input noise conditions.
The three simulation experiments show that the reconstruction accuracy of the BSFDBC matrix provided by the invention is better than that of a Gaussian random matrix and a Bernoulli random matrix under the same condition for a noiseless scene and a noisy scene.
Experiment seven: as shown in fig. 8, the test image includes three gray images and three color images. Wherein, figures (a), (b) and (c) are grey test images "lena", "peppers" and "air", of sizes 256× 256,256 ×256 and 1024×1024, respectively; graphs (d), (e) and (f) are color test images "Earth", "air" and "bone", the sizes are 512 multiplied by 512 respectively 3,512X 512X 3 and 675X 653X 3. The following table shows the reconstructed PSNR for different test images for block sizes of 32 x 16 and 32 x 32, respectively.
Table 1 shows that the BSFDBC matrix of the present invention has the highest reconstructed PSNR for all gray images and color images, among the three different matrices. Also, for the same matrix, the reconstructed PSNR increases with increasing block size.
From the above simulation experiments on sparse signals and image signals, it can be seen that the reconstruction accuracy of the BSFDBC matrix of the present invention is better than that of the Gaussian matrix and the Bernoulli matrix under the same conditions.
The invention provides a chaotic binary sequence family matrix construction method based on image compressed sensing encryption, which constructs a deterministic bipolar measurement matrix-BSFDBC (Binary Sequence Family based Deterministic Bipolar Chaotic) on the basis of binary sequence families (Binary Sequence Family, BSF) and Chebyshev chaotic sequences, firstly selects trace expression functions to generate binary pseudo-random sequence sets forming corresponding BSF, then carries out numerical conversion on the binary pseudo-random sequence sets to obtain corresponding bipolar sequence families, selects a part of sequences from the obtained bipolar sequence families as column vectors to be arranged, and introduces the chaotic sequences to carry out corresponding replacement on the column vectors of coefficient matrixes to form the measurement matrix in the invention. The matrix constructed by the method has practical characteristics of easy hardware realization, high perceptibility and good encryption property, and can be used for efficient compression encryption acquisition of image signals such as gray images, color images and the like.
The invention can be used for constructing a compressed sensing measurement matrix with friendly hardware, high sensing performance and good encryption property, and realizing compressed encryption acquisition of image signals such as gray images, color images and the like.
The foregoing is merely a preferred embodiment of the present invention, and it should be noted that modifications and variations could be made by those skilled in the art without departing from the technical principles of the present invention, and such modifications and variations should also be regarded as being within the scope of the invention.
Claims (1)
1. The chaotic binary sequence family matrix construction method based on image compression perception encryption is characterized by comprising the following steps of:
(1) Judging the parity of the related parameters according to the information dimension of the image block size, and selecting a corresponding trace representation function;
(2) Generating a binary pseudo-random sequence set forming a corresponding binary sequence family by the trace representation function selected in the step (1), performing numerical conversion to obtain a corresponding bipolar sequence family, and selecting a part of sequences from the obtained bipolar sequence family as column vectors to be arranged to obtain a corresponding initial measurement matrix;
(3) Introducing a chaotic sequence to correspondingly replace column vectors of the initial measurement matrix obtained in the step (2) to obtain a required chaotic binary sequence family matrix;
Wherein,
The trace function related to the trace representing function in the step (1) is defined as follows:
Let n, m be a positive integer, and m divide n, the trace function from the finite field GF (2 n) to GF (2 m) is:
GF (2 m)＝GF(2)＝{0,1},Tr1 n (x) is abbreviated Tr (x) when m=1;
Wherein, for a finite field GF (q) of size q, let β be the primitive field element of GF (q), then all field elements of GF (q) can be generated from powers of 0 and β: GF (q) = {0, β 0＝1,β,…,βq-2 }; wherein, the latter q-1 nonzero elements form a multiplication group GF (q) \ {0}, denoted GF (q) *; all elements of GF (q) are denoted {0,1, … q-1};
the step (1) specifically comprises:
1a) Judging the parity of a related parameter N according to the information dimension N=2 n+1 (N is larger than or equal to 5) of the image block size, and executing the step 1b if N is an odd number; if n is an even number, executing the step 1 c);
1b) Selection trace representation function Wherein x ε GF (2 n)*,λ0,λ1∈GF(2n);
1c) Selection trace representation function Wherein x ε GF (2 n)*,λ0,λ1∈GF(2n);
The step (2) specifically comprises:
2a) Selecting primitive domain element beta on GF (2 n), letting At this time, t is {0,1, …, n-2},λ0,λ1∈GF(2n), letTraversing and taking value of parameter (lambda 0,λ1) to obtain binary pseudo-random sequence set forming corresponding binary sequence familyExecuting step 2 b);
2b) Binary sequence All elements are sequentially input into the numerical conversion functionObtaining corresponding bipolar pseudo-random sequenceFor the corresponding bipolar sequence familyExecuting step 2 c);
2c) The elements of (lambda 0,λ1) are ordered by dictionary as (0, 0), (0, 1), …, (0, 2 n-1),(1,0),(1,1),…,(1,2n-1),(2,0),…,(2n-1,2n -1), and Is used as a column vector to be orderly arranged to obtain an initial measurement matrix A with the size of (2 n-1)×2n+1;
The step (3) specifically comprises:
3a) According to chebyshev mapping R j+1＝cos(w·arccos(rj)) to obtain a corresponding chebyshev chaotic sequence R (R 0,s,l)＝{r0,rs,r2s,…,r(l-1)s), for a given R 0, recording each value of R (R 0, s, l) when w= 5,s =5 and l=n=2 n+1, arranging the corresponding chaotic sequences { R 0,rs,r2s,…,r(l-1)s } in descending order to obtain a corresponding index set χ, wherein the set becomes a chaotic set due to the pseudo-randomness of R (R 0, s, l), and executing step 3 b);
In the chebyshev mapping formula, j=0, 1,2, …, r 0 e [ -1,1] is an initial state, and w is a positive integer greater than 1, which is also called mapping degree;
3b) The column vector of matrix A is adjusted by utilizing the sequence of the X set element to obtain a new matrix which is a chaotic binary sequence family matrix The chaotic binary sequence family matrix is expressed as a matrix formPermutation operator based on chaosIs an identity matrix determined by the element sequence of the aggregation χIs a deterministic column arrangement of (c).
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