CN114915400A - Synchronous time adjustable heterogeneous fractional order hyperchaotic system image encryption and decryption method - Google Patents

Synchronous time adjustable heterogeneous fractional order hyperchaotic system image encryption and decryption method Download PDF

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CN114915400A
CN114915400A CN202210510488.7A CN202210510488A CN114915400A CN 114915400 A CN114915400 A CN 114915400A CN 202210510488 A CN202210510488 A CN 202210510488A CN 114915400 A CN114915400 A CN 114915400A
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林立雄
庄裕富
郑佳春
蔡国玮
李超鹏
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Jimei University
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    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
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    • H04L9/00—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols
    • H04L9/001—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols using chaotic signals
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L9/00—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols
    • H04L9/32—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols including means for verifying the identity or authority of a user of the system or for message authentication, e.g. authorization, entity authentication, data integrity or data verification, non-repudiation, key authentication or verification of credentials
    • H04L9/3236—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols including means for verifying the identity or authority of a user of the system or for message authentication, e.g. authorization, entity authentication, data integrity or data verification, non-repudiation, key authentication or verification of credentials using cryptographic hash functions
    • H04L9/3239—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols including means for verifying the identity or authority of a user of the system or for message authentication, e.g. authorization, entity authentication, data integrity or data verification, non-repudiation, key authentication or verification of credentials using cryptographic hash functions involving non-keyed hash functions, e.g. modification detection codes [MDCs], MD5, SHA or RIPEMD
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    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04N—PICTORIAL COMMUNICATION, e.g. TELEVISION
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Abstract

The invention relates to a heterogeneous fractional order hyperchaotic system image encryption and decryption method with adjustable synchronization time, which comprises the following steps: constructing a fractional order hyperchaotic Lorenz system with a delay characteristic based on the fractional order hyperchaotic Lorenz system, and establishing a driving system; generating an initial value of a driving system according to a plaintext image, and generating a chaotic sequence of the driving system; generating a ciphertext image for transmission by a DNA coding method, a cyclic shift scrambling method and a diffusion method; constructing a fractional order hyperchaotic Liu system with a time delay characteristic based on the fractional order hyperchaotic Liu system, establishing a response system, and establishing a time-adjustable synchronous error system; designing a time-adjustable heterogeneous fractional order hyperchaotic system synchronization method to realize time-adjustable synchronization of a synchronization error system; the receiver obtains the ciphertext image, the key and the adjustable time parameter, and the decryption of the ciphertext image is realized according to the reverse encryption process. The method can realize more parameters to expand the key space, and improves the safety of the system.

Description

Synchronous time adjustable heterogeneous fractional order hyperchaotic system image encryption and decryption method
Technical Field
The invention belongs to the field of communication safety, and particularly relates to a heterogeneous fractional order hyperchaotic system image encryption and decryption method with adjustable synchronization time.
Background
In digital image encryption and secure communication, there are three very important factors: security, attack resistance, complexity, which can increase the difficulty of hacking. The signals generated by the fractional order hyperchaotic system have the characteristics of non-periodicity, ergodicity, noise similarity and the like, so that the fractional order hyperchaotic system has natural advantages in the field of digital image encryption. In image encryption, an encryption method for scrambling image positions and pixel values by a chaotic system with a higher dimension is generally considered, or a chaotic sequence discrete cosine transform, Arnold transform and other methods are adopted, while a chaotic sequence generated by a fractional order hyperchaotic system has richer dynamic characteristics than an integer order, and is more suitable for image encryption, for example, in a document' Ni J, Liu L, Liu C, et al. However, the stable time of the fixed time synchronization is affected by the system parameters and the controller parameters, is difficult to adjust freely and is fixed, and therefore, the image decryption is unsafe. In 2020, Lin proposes a special fixed time stability theory, which can establish a direct relationship between system parameters and convergence time, and the estimation of the convergence time is closer to a true value. In fact, the complexity, the rapidity and the safety of the encryption and decryption method for researching the image are equivalent to the complexity and the unpredictability of a chaotic system. The synchronization time adjustability between the two fractional order hyperchaotic systems can also increase the complexity and the rapidity of the image encryption and decryption process.
Disclosure of Invention
The invention aims to provide a heterogeneous fractional order hyperchaotic system image encryption and decryption method with adjustable synchronization time, which can expand a key space, increase the complexity of a key and improve the safety of a system.
In order to achieve the purpose, the invention adopts the technical scheme that: a heterogeneous fractional order hyperchaotic system image encryption and decryption method with adjustable synchronization time comprises the following steps:
s1, constructing a fractional order hyperchaotic Lorenz system with a delay characteristic based on the fractional order hyperchaotic Lorenz system, and establishing a driving system;
step S2, inputting the plaintext image matrix into a hash function and carrying out XOR operation to serve as an initial value of the driving system, wherein the initial value is a secret key;
step S3, generating a chaos sequence through a driving system;
s4, generating a ciphertext image to be transmitted through a DNA coding method, a cyclic shift scrambling method and a diffusion method according to the chaotic sequence generated in the S3;
s5, constructing a fractional order hyperchaotic Liu system with a time delay characteristic based on the fractional order hyperchaotic Liu system, and establishing a response system;
step S6, establishing a time adjustable synchronous error system according to the driving system and the response system established in the step S1 and the step S5;
s7, designing a time-adjustable heterogeneous fractional order hyper-chaotic system synchronization method to realize time-adjustable synchronization of a synchronization error system;
step S8, the receiving party bases on the ciphertext image obtained in step S4, the key obtained in step S2 and the adjustable time parameter T in step S7 c Decryption of the ciphertext image is achieved according to the reverse process of step S4.
Further, in step S1, the state equation of the driving system is established as follows:
Figure BDA0003639238210000021
wherein D is α Alpha order derivative, 0, of the drive system<α<1 represents the order, x i (t), i ═ 1,2,3,4 denotes a state variable of the drive system, σ i >0, i-1, 2,3,4 is a self-suppression parameter, τ>0 is a time-invariant delay term, and a, b, h, r, l, c and b are constants.
Further, in step S2, the plaintext image matrix is input into the SHA-256 function to form a 64-bit hexadecimal digest character string with a length of 256 bits; the 64-bit hexadecimal digest is converted to a decimal number and divided into 8 sequences: h ═ H 1 ,H 2 ,H 3 ,H 4 ,H 5 ,H 6 ,H 7 ,H 8 }; and carrying out XOR operation on the 8 numbers to obtain 4 numbers: x ═ x 1 ,x 2 ,x 3 ,x 4 }; these four numbers are used as initial values for the drive system.
Further, in step S3, the initial value x obtained in step S2 is set to { x ═ x } 1 ,x 2 ,x 3 ,x 4 Inputting the chaos sequences X into a driving system, and generating a set number of chaos sequences X in a given time.
Further, step S4 specifically includes the following steps:
step S41: the DNA sequence has four nucleobases, adenine (A), thymine (T), cytosine (C) and guanine (G), wherein A and T are complementary and G and C are complementary; for binary numbers, 0 and 1 are complementary, so 00 and 11, 01 and 10 are also complementary; by using four bases A, T, C, G to code 00, 01, 10, 11, there are 24 coding combinations, but only eight satisfy the base complementary pairing principle, according to the DNA coding rule, the image pixel is coded;
step S42: the color image has three channels: r, G and B, both of size M N, convert the image into a matrix I of M3N m (i, j); in the cyclic shift scrambling, only the rows and columns are operated, and therefore, the chaos generated in step S3 is arbitrarily takenObtaining a chaotic sequence by M +3N groups of data in the sequence X: s 1 ＝{X 1 ,Y 1 ,Z 1 }; to S 1 Each sequence of (a) performs the following operations:
Figure BDA0003639238210000031
wherein m is derived from S 1 A number of the sequence; m, N denotes the size of the image, T m ，W m Represents a new sequence value obtained by formula calculation, and the sequence T is { T ═ T x ,T y ,T z }、W＝{W x ,W y ,W z Is S 1 ＝{X 1 ,Y 1 ,Z 1 Calculating each sequence in the sequence list by a formula to obtain a new sequence; the scrambling sequence is obtained according to the following formula:
Figure BDA0003639238210000032
wherein S is r (i) Is the sequence required for the cyclic shift of the rows, S c (i) Is the sequence required for the column cyclic shift;
scrambling the image matrix according to the following formula:
Figure BDA0003639238210000033
wherein i is 1,2, … …, M; j ═ 1,2, … …, 3N; i is p (I,: represents I) p Values of all columns in the ith row of the matrix, I p (j) represents I p Values of all rows of the jth column in the matrix, I m (I,: represents I) m Values of all columns in the ith row of the matrix, I m (j) represents I m The values of all rows in the jth column in the matrix, and the function Circhift (a, b) circularly shifts the matrix a by b bits; finally, a scrambling matrix I is obtained p (i，j)；
Step S43: diffusing pixels of all components of the color image; from the chaotic sequence XRandomly extracting M × N groups of data to obtain a sequence: s 2 ＝{X 2 ,Y 2 ,Z 2 }; the obtained sequences are then merged into a matrix S of size M x 3N 2 ', and processing S according to the following formula d ：
Figure BDA0003639238210000034
The diffusion operation is performed according to the following formula:
Figure BDA0003639238210000041
Figure BDA0003639238210000042
wherein, I c (1,: represents I) c Values of all columns in row 1 in the matrix, I c (1) represents I c The values of all rows in column 1 of the matrix, where I c (I,: represents I) c Values of all columns in the ith row of the matrix, I c (j) represents I c The values of all rows in the jth column in the matrix;
obtaining a matrix I after diffusion c Is shown by c Obtaining matrix I 'after Arnold scrambling' c Prepared from l' c Reconstructing a matrix I of MXNX 3 e The matrix I e Namely the obtained ciphertext image; and packaging the obtained ciphertext, the initial driving signal and the predefined time, and transmitting the ciphertext, the initial driving signal and the predefined time to a receiving party through signals.
Further, in step S5, the state equation of the response system is established as follows:
Figure BDA0003639238210000043
wherein, y i (t), i ═ 1,2,3,4 denotes a state variable of the drive system, σ i >0, i-1, 2,3,4 is a self-suppression parameter, τ>0 is not time-varyingA delay term; a is 1 、b 1 、c 1 、r 1 Is a constant; u. of j (t), j ═ 1,2,3,4, which denotes a predefined time synchronization controller.
Further, the specific implementation method of step S6 is as follows:
error e for synchronizing the drive system with the response system at a predetermined time i The setting is as follows:
Figure BDA0003639238210000044
wherein i is 1,2,3,4,
Figure BDA0003639238210000045
representing the response system output value, x i Expressing the output value of the driving system, and obtaining a preset time synchronization error system e by derivation according to the formula i (t)：
e i (t)＝y i (t)-x i (t),i＝1,2,…,n (9)
Let E (t) ═ e 1 (t),e 2 (t),…,e n (t)) T ∈R n Is the state vector of the error system, and the initial value is E (0) ═ Y (0) -X (0); the synchronization error system is described as:
Figure BDA0003639238210000046
substituting formulae (1) and (7) into formula (10) yields:
D α e i (t)＝-σ i e i (t)+h i (e i (t))+H i (e i (t-τ))+u i (t) (11)
in the formula, h i (e i (t))＝f i (Y,t)-f i (X,t)；H i (e i (t-τ))＝F i (Y,t-τ)-F i (X,t-τ)；f i ，F i Representing some functional relationship.
Further, the specific implementation method of step S7 is:
the adaptive predefined time synchronization controller function is described as follows:
Figure BDA0003639238210000051
wherein Tc represents an adjustable predefined time; c v Is a normal number determined by other parameters; beta is a 2 ，λ 2 Q, k, μ, ω are custom normality; c v The calculation formula of (a) is as follows:
Figure BDA0003639238210000052
the following continuous functions were chosen as the lyapunov function:
Figure BDA0003639238210000053
V 2 (t) denotes the Lyapunov function, e (t) is the error function,
Figure BDA0003639238210000054
denotes e i Adding the absolute values of (A) to (B), taking V 2 (t) in combination with equation (11) in step S6, the following is obtained:
Figure BDA0003639238210000061
further derivation of equation (14) yields:
Figure BDA0003639238210000062
wherein the content of the first and second substances,
Figure BDA0003639238210000063
for the
Figure BDA0003639238210000064
To obtain V 2 (t) ═ 0; the synchronization error formula (8) in step S6 is at the predetermined time T c And (4) internal synchronization.
Further, in step S8, a synchronization system is constructed by using the synchronization method in step S7, a synchronization chaotic sequence is generated to perform the reverse operation in step S4, and the ciphertext information is sequentially subjected to reverse Arnold scrambling, reverse diffusion, reverse circular scrambling, and DNA decoding to obtain a decrypted image.
Compared with the prior art, the invention has the following beneficial effects: the synchronization time between the drive-response systems is adjustable, which is equivalent to more synchronization time keys, the key space is expanded, and the complexity of the keys is increased. Meanwhile, the invention has more parameters to expand the key space, and uses the chaos sequence to carry out cyclic shift, DNA coding, position disorder and XOR diffusion, and SHA-256 hash function to provide the system with the guarantee of 'one-time pad', and improve the system security. In addition, the method has higher key sensitivity and plaintext sensitivity, and can resist various attacks such as statistical attack, differential attack and the like. Therefore, the invention has good encryption effect and good application prospect.
Drawings
FIG. 1 is a schematic flow chart of a method according to an embodiment of the present invention.
Fig. 2 is a chaotic behavior curve of the driving system according to the embodiment of the present invention.
FIG. 3 is a digital image after encryption and decryption according to an embodiment of the present invention; wherein, (a) is the original digital color image, (b) is the cipher text image, (c) is the digital image after deciphering, and (d) is that the image after deciphering is compared with the original image to obtain a completely black image.
FIG. 4 is a histogram of a plaintext image according to an embodiment of the invention; wherein, (a) is a histogram of a clear text image R channel, (B) is a histogram of a clear text image G channel, and (c) is a histogram of a clear text image B channel.
FIG. 5 is a histogram of a ciphertext image in an embodiment of the present invention; wherein (a) is a histogram of a ciphertext image R channel, (B) is a histogram of a ciphertext image G channel, and (c) is a histogram of a ciphertext image B channel.
FIG. 6 is a correlation scatter plot in an embodiment of the present invention; wherein (a) is a correlation scatter diagram of adjacent pixels of the original image, and (b) is a correlation scatter diagram of adjacent pixels of the encrypted image.
FIG. 7 is a drive-response error system response curve for an embodiment of the present invention; wherein (a) is e 1 Is (b) is e 2 Is (c) is e 3 Is (d) is e 4 The response curve of (c).
Detailed Description
The invention is further explained below with reference to the drawings and the embodiments.
It should be noted that the following detailed description is exemplary and is intended to provide further explanation of the disclosure. Unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this application belongs.
It is noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of example embodiments according to the present application. As used herein, the singular forms "a", "an", and "the" are intended to include the plural forms as well, and it should be understood that when the terms "comprises" and/or "comprising" are used in this specification, they specify the presence of stated features, steps, operations, devices, components, and/or combinations thereof, unless the context clearly indicates otherwise.
As shown in fig. 1, the embodiment provides a heterogeneous fractional order hyper-chaotic system image encryption and decryption method with adjustable synchronization time, which is characterized by comprising the following steps:
and S1, constructing a fractional order hyperchaotic Lorenz system with a delay characteristic based on the fractional order hyperchaotic Lorenz system, and establishing a driving system.
And step S2, inputting the plaintext image matrix into a hash function and carrying out XOR operation to serve as an initial value of the driving system, wherein the initial value is a secret key.
And step S3, generating a chaotic sequence of the driving system.
And S4, generating a ciphertext image for transmission by a DNA coding method, a cyclic shift scrambling method and a diffusion method according to the chaotic sequence generated in the step S3.
And S5, constructing a fractional order hyperchaotic Liu system with a time delay characteristic based on the fractional order hyperchaotic Liu system, and establishing a response system.
And step S6, establishing a time adjustable synchronous error system according to the driving system and the response system established in the steps S1 and S5.
And step S7, designing a time-adjustable heterogeneous fractional order hyper-chaotic system synchronization method to realize time-adjustable synchronization of a synchronization error system.
Step S8, the receiving side obtains the ciphertext image of step S4, the key obtained in step S2 and the adjustable time parameter T in step S7 c Decryption of the ciphertext image is achieved according to the reverse process of step S4.
In step S1, the state equation of the drive system is established as follows:
Figure BDA0003639238210000081
wherein D is α Alpha order derivative, 0, of the drive system<α<1 represents the order, x i (t), i ═ 1,2,3,4 denotes a state variable of the drive system, σ i >0, i-1, 2,3,4 is a self-suppression parameter, τ>0 is a time-invariant delay term, and a, b, h, r, l, c and b are constants.
In step S2, inputting the plaintext image matrix into the SHA-256 function to form a 64-bit hexadecimal digest character string with a length of 256 bits; the 64-bit hexadecimal digest is converted to a decimal number and divided into 8 sequences: h ═ H 1 ,H 2 ,H 3 ,H 4 ,H 5 ,H 6 ,H 7 ,H 8 }; and carrying out XOR operation on the 8 numbers to obtain 4 numbers: x ═ x 1 ,x 2 ,x 3 ,x 4 }; these four numbers are used as initial values for the drive system.
In step S3, step S2 is performedX ═ x 1 ,x 2 ,x 3 ,x 4 Inputting the chaos sequences X into a driving system, and generating a set number of chaos sequences X in a given time.
Step S4 specifically includes the following steps:
step S41: DNA coding
The DNA sequence has four nucleobases, adenine (A), thymine (T), cytosine (C) and guanine (G), wherein A and T are complementary and G and C are complementary; for binary numbers, 0 and 1 are complementary, so 00 and 11, 01 and 10 are also complementary; by using four bases A, T, C, G to encode 00, 01, 10, 11, there are 24 coding combinations, but only eight satisfy the base complementary pairing rules, as shown in Table 1.
TABLE 1 DNA coding rules
Figure BDA0003639238210000082
Figure BDA0003639238210000091
The image pixels may be encoded according to the DNA coding rules. For example, if the pixel value of the normal image is 100, it is converted into a binary sequence [01100100 ]. We can obtain the DNA sequence [ CGCA ] by encoding it using DNA coding rule 1.
Step S42: cyclic shift scrambling
The color image has three channels: r, G and B, both of size M N, convert the image into a matrix I of M3N m (i, j); in the cyclic shift scrambling, only rows and columns are operated, and therefore, M +3N groups of data in the chaotic sequence X generated in step S3 are arbitrarily taken to obtain a chaotic sequence: s 1 ＝{X 1 ,Y 1 ,Z 1 }; to S 1 Each sequence of (a) performs the following operations:
Figure BDA0003639238210000092
wherein m is derived from S 1 The number of the sequence; m, N denotes the size of the image, T m ，W m Represents a new sequence value obtained by formula calculation, and the sequence T is { T ═ T x ,T y ,T z }、W＝{W x ,W y ,W z Is S 1 ＝{X 1 ,Y 1 ,Z 1 Calculating each sequence in the sequence list by a formula to obtain a new sequence; the scrambling sequence is obtained according to the following formula:
Figure BDA0003639238210000093
wherein S is r (i) Is the sequence required for the cyclic shift of the rows, S c (i) Is the sequence required for the column cyclic shift.
In this context it is explained whether we are working on a matrix or a sequence, e.g. the sequence S r Can be represented as a sequence S r (i) (ii) a Matrix I p Can be represented as I p (i, j), the parentheses here are not calculation methods but representation methods.
Scrambling the image matrix according to the following formula:
Figure BDA0003639238210000094
wherein i is 1,2, … …, M; j ═ 1,2, … …, 3N; i is p (I,: represents I) p Values of all columns in the ith row of the matrix, I p (j) represents I p Values of all rows of the jth column in the matrix, I m (I,: represents I) m Values of all columns in the ith row of the matrix, I m (j) represents I m The values of all rows in the jth column in the matrix, and the function Circhift (a, b) circularly shifts the matrix a by b bits; finally, a scrambling matrix I is obtained p (i，j)。
Step S43: diffusion
The pixels of all components of the color image need to be spread. Randomly extracting M multiplied by N groups of numbers from chaos sequence XAccording to this, the sequence was obtained: s 2 ＝{X 2 ,Y 2 ,Z 2 }. The obtained sequences are then merged into a matrix S of size M x 3N 2 ', and processing S according to the following formula d ：
Figure BDA0003639238210000101
The diffusion operation is then performed according to the following formula:
Figure BDA0003639238210000102
Figure BDA0003639238210000103
wherein, I c (1,: represents I) c Values of all columns in row 1 in the matrix, I c (1) represents I c The values of all rows in column 1 of the matrix, where I c (I,: represents I) c Values of all columns in the ith row of the matrix, I c (j) represents I c The values of all rows in the jth column of the matrix, matrix S d (i, j) the same.
Obtaining a matrix I after diffusion c Is shown by c Obtaining matrix I 'after Arnold scrambling' c Prepared from l' c Reconstructing a matrix I of MXNX 3 e The matrix I e Namely the obtained ciphertext image; and packaging the obtained ciphertext, the initial driving signal and the predefined time, and transmitting the ciphertext, the initial driving signal and the predefined time to a receiving party through signals.
In step S5, the state equation of the response system is established as follows:
Figure BDA0003639238210000104
wherein, y i (t), i ═ 1,2,3,4 denotes a state variable of the drive system, σ i >0, i-1, 2,3,4 is a self-suppression parameter, τ>0 is not timeA variable delay term; a is 1 、b 1 、c 1 、r 1 Is a constant; u. of j (t), j ═ 1,2,3,4, which denotes a predefined time synchronization controller.
The specific implementation method of step S6 is:
error e for synchronizing the drive system with the response system at a predetermined time i The following settings are set:
Figure BDA0003639238210000105
wherein i is 1,2,3,4,
Figure BDA0003639238210000106
representing the response system output value, x i Expressing the output value of the driving system, and obtaining a preset time synchronization error system e by derivation according to the formula i (t)：
e i (t)＝y i (t)-x i (t),i＝1,2,…,n (9)
Suppose e (t) ═ e 1 (t),e 2 (t),…,e n (t)) T ∈R n Is the state vector of the error system, and the initial value is E (0) ═ Y (0) -X (0); the synchronization error system is described as:
Figure BDA0003639238210000111
substituting formulae (1) and (7) into formula (10) yields:
D α e i (t)＝-σ i e i (t)+h i (e i (t))+H i (e i (t-τ))+u i (t) (11)
in the formula, h i (e i (t))＝f i (Y,t)-f i (X,t)；H i (e i (t-τ))＝F i (Y,t-τ)-F i (X,t-τ)；f i ，F i Representing some functional relationship.
The specific implementation method of step S7 is:
the adaptive predefined time synchronization controller function is described as follows:
Figure BDA0003639238210000112
wherein Tc represents an adjustable predefined time; c v Is a normal number determined by other parameters; beta is a beta 2 ，λ 2 Q, k, μ, ω are custom normality; c v The calculation formula of (a) is as follows:
Figure BDA0003639238210000113
the following continuous functions were chosen as the lyapunov function:
Figure BDA0003639238210000121
V 2 (t) denotes the Lyapunov function, e (t) is the error function,
Figure BDA0003639238210000122
denotes e i Adding the absolute values of (A) to (B), taking V 2 (t) in combination with equation (11) in step S6, the following is obtained:
Figure BDA0003639238210000123
further derivation of equation (14) yields:
Figure BDA0003639238210000124
wherein the content of the first and second substances,
Figure BDA0003639238210000125
for the
Figure BDA0003639238210000126
To obtain V 2 (t) ═ 0; the synchronization error formula (8) in step S6 is at the predetermined time T c And (4) internal synchronization.
In step S8, a synchronization system is constructed by using the synchronization method in step S7, a synchronization chaotic sequence is generated to perform the reverse operation in steps S2 to S4, and the ciphertext information is sequentially subjected to reverse Arnold scrambling, reverse diffusion, and reverse circular scrambling; and decoding the DNA to obtain a decrypted image. Wherein, the DNA decoding process is as follows: DNA decoding is the reverse of DNA encoding. By decoding the DNA sequence, the pixel values can be recovered. If the DNA sequence is [ CGCA ]]Then it is decoded to [01100100] according to step S41 table 1 rule 1]Decimal 100. If decoded by step S41 Table 1 rule 2, it will be translated as [ 10011000%]And represents a decimal number 152. In view of this property, the rule R can be passed 1 Ordinary image pixels are encoded and then passed through rule R 2 It is decoded to modify the image pixel values and hide the image original information.
In order to more intuitively illustrate the effectiveness and feasibility of the synchronization time adjustable heterogeneous fractional order hyperchaotic system image encryption and decryption method provided by the invention, the embodiment uses MATLAB software to perform a computer simulation experiment on the method.
The state equation of the driving system is as follows:
Figure BDA0003639238210000131
the driving system is a 4-dimensional fractional order hyperchaotic system, and the chaotic behavior curve is shown in fig. 2. The state equation of the response system is:
Figure BDA0003639238210000132
wherein the response system is a 4-dimensional fractional order hyperchaotic system, and when the parameter is sigma i ＝0.1(i＝1,2,3,4),τ＝0.8,a＝10,b＝40,c＝2.5,l＝10,h＝4,r＝2.5,a 1 ＝10,b 1 ＝28,c 1 ＝8/3,r 1 1, and the order α is 0.9. The error of the drive-response system is set to:
Figure BDA0003639238210000133
obtaining a predetermined time synchronization error system:
D α e i (t)＝-σ i e i (t)+h i (e i (t))+H i (e i (t-τ))+u i (t) (19) the active controller is designed to:
Figure BDA0003639238210000134
wherein α is 0.9, i is 1,2,3,4, σ i 0.1, 5.2, 12.9, 11. We chose L ═ 0.2. At this time, the predetermined synchronization time of its drive-response system is: t is C 1. The above results were applied to digital image encryption, and the test image was Lena (256 × 256). In this chapter, we simulate the synchronization of a predetermined time fractional order hyper-chaotic system and evaluate the encryption effect.
And (3) encryption effect analysis: classical Lena images were selected for simulation experiments. We first perform the encryption algorithm presented in step S4 to encrypt the digital color image shown in fig. 3 (a). After the plaintext image is encrypted, a ciphertext image is formed as shown in fig. 3 (b). The ciphertext image is in a noise state and the original image information cannot be identified.
Histogram analysis: the histogram of the digital image is a frequency statistic of the gray levels. The histogram analysis reflects the statistical characteristics of the image. An intruder can attack the encrypted image by histogram analysis. The histogram of a random image with uniform gray distribution should be smooth and uniform, so a good encryption strategy should make the gray distribution uniform. Fig. 4 shows histograms of the plaintext image, where (a) is a histogram of the R channel of the plaintext image, (B) is a histogram of the G channel of the plaintext image, and (c) is a histogram of the B channel of the plaintext image. Intuitively, the histogram of the plaintext image obviously fluctuates, while the histogram of the ciphertext is shown in fig. 5, wherein (a) is the histogram of the R channel of the ciphertext image, (B) is the histogram of the G channel of the ciphertext image, and (c) is the histogram of the B channel of the ciphertext image, so that the distribution is uniform and stable, the gray value is approximately uniformly distributed, the plaintext information is effectively hidden, and an attacker is difficult to crack the ciphertext image through statistical analysis.
And (3) correlation analysis: the correlation coefficient of two data sets X and Y of length n is calculated according to a formula.
Figure BDA0003639238210000141
Figure BDA0003639238210000142
Figure BDA0003639238210000143
In the experiment, 8000 pairs of adjacent pixels were randomly selected in the horizontal, vertical, and diagonal directions of the plaintext image and the ciphertext image, and the correlation coefficient of the pixel values thereof was calculated. As can be seen from fig. 6, the correlation of the plaintext image is shown in (a), the scattered points are concentrated on a straight line, and have obvious linear correlation, while the correlation of the ciphertext image is shown in (b), and the scattered points are distributed uniformly without obvious aggregation. The invention can reduce the correlation of adjacent pixels to a low level and has good encryption performance.
Information entropy analysis: the information entropy is the embodiment of the chaos degree of the system. The higher the entropy of the information, the more chaotic the system. The information entropy of the image with the gray level L is:
Figure BDA0003639238210000151
wherein P(s) i ) Is that the pixel belongs to a grey level s i The probability of (c).
TABLE 2 information entropy of plaintext image and ciphertext image
Figure BDA0003639238210000152
The theoretical maximum value of the information entropy of the 256-gray-scale image is 8. Table 2 calculates the information entropy of each component of the plaintext image and the ciphertext image. It can be seen that the information entropy of the plaintext image is very low, and the information entropy of the ciphertext image is very high, which is close to 8.
Clear text sensitivity analysis: the differential attack resistance of encryption methods is usually tested by the pixel count rate of change (NPCR) and the uniform average variation intensity (UACI). The NPCR and UACI values for the red, green, and blue components of a digital color image can be calculated by the following equations:
Figure BDA0003639238210000153
Figure BDA0003639238210000154
the present invention uses the SHA-256 hash function to generate an initial key sequence associated with the plaintext that is highly sensitive to the input image data. One pixel value of the input image is changed randomly, so that the key is changed greatly, and the generated chaotic sequence is also changed greatly. The NPCR and UACI (%) of the two encrypted digital images are shown in table 3.
TABLE 3 NPCR and UACI (%) between two ciphertext images after plaintext change
Index (I) R G B
NPCR 99.5422 99.6124 99.6613
UACI 33.4426 33.5336 33.5158
The theoretical values for NPCR and UACI are 99.6094% and 33.4635%, respectively. The result of the encryption method is very close to the theoretical value, and the method is proved to have good differential attack resistance.
Key sensitivity analysis: NPCR and UACI (%) of the two encrypted digital images are shown in table 4.
Table 4 NPCR and UACI (%) (between two ciphertext images after key change)
Figure BDA0003639238210000155
Figure BDA0003639238210000161
As can be seen from table 4, the experimental values of NPCR and UACI are very close to the theoretical values, which demonstrates the good key sensitivity of the present invention. Receiver reception key sigma i (i ═ 1,2,3,4), τ, a, b, c, l, h, r and the encrypted image information, and then the encrypted image information is decrypted by the response system (17), the active controller (20), the reception key and the encrypted image, and the same of the response system is drivenThe results of step (a) are shown in FIG. 7, where (a) is e 1 Is (b) is e 2 Is (c) is e 3 Is (d) is e 4 The response curve of (c). The decrypted restored image looks little different from the original image as shown in fig. 3 (c). To be accurate, we compare the decrypted image with the original image to obtain a completely black image, as shown in fig. 3 (d). This shows that the decrypted image is completely consistent with the plaintext image, proving that the encryption/decryption method provided by the invention is effective and can realize lossless encryption and decryption.
The foregoing is directed to preferred embodiments of the present invention, other and further embodiments of the invention may be devised without departing from the basic scope thereof, and the scope thereof is determined by the claims that follow. However, any simple modification, equivalent change and modification of the above embodiments according to the technical essence of the present invention are within the protection scope of the technical solution of the present invention.

Claims (9)

1. A heterogeneous fractional order hyper-chaotic system image encryption and decryption method with adjustable synchronization time is characterized by comprising the following steps:
s1, constructing a fractional order hyperchaotic Lorenz system with a delay characteristic based on the fractional order hyperchaotic Lorenz system, and establishing a driving system;
step S2, inputting the plaintext image matrix into a hash function and carrying out XOR operation to serve as an initial value of the driving system, wherein the initial value is a secret key;
step S3, generating a chaos sequence through a driving system;
s4, generating a ciphertext image to be transmitted through a DNA coding method, a cyclic shift scrambling method and a diffusion method according to the chaotic sequence generated in the S3;
s5, constructing a fractional order hyperchaotic Liu system with a time delay characteristic based on the fractional order hyperchaotic Liu system, and establishing a response system;
step S6, establishing a time adjustable synchronous error system according to the driving system and the response system established in the step S1 and the step S5;
s7, designing a time-adjustable heterogeneous fractional order hyper-chaotic system synchronization method to realize time-adjustable synchronization of a synchronization error system;
step S8, the receiving party bases on the ciphertext image obtained in step S4, the key obtained in step S2 and the adjustable time parameter T in step S7 c Decryption of the ciphertext image is achieved according to the reverse process of step S4.
2. The image encryption and decryption method of the synchronous time adjustable heterogeneous fractional order hyperchaotic system as claimed in claim 1, wherein in step S1, the state equation of the driving system is established as follows:
Figure FDA0003639238200000011
wherein D is α Alpha order derivative, 0, of the drive system<α<1 represents the order, x i (t), i ═ 1,2,3,4 denotes a state variable of the drive system, σ i >0, i-1, 2,3,4 is a self-suppression parameter, τ>0 is a time-invariant delay term, and a, b, h, r, l, c and b are constants.
3. The image encryption and decryption method of the synchronous time adjustable heterogeneous fractional order hyperchaotic system as claimed in claim 2, wherein in step S2, the plaintext image matrix is input into SHA-256 function to form a 64-bit hexadecimal digest character string with length of 256 bits; the 64-bit hexadecimal digest is converted to a decimal number and divided into 8 sequences: h ═ H 1 ,H 2 ,H 3 ,H 4 ,H 5 ,H 6 ,H 7 ,H 8 }; and carrying out XOR operation on the 8 numbers to obtain 4 numbers: x ═ x 1 ,x 2 ,x 3 ,x 4 }; these four numbers are used as initial values for the drive system.
4. According toThe image encryption and decryption method of the synchronous time adjustable heterogeneous fractional order hyperchaotic system as claimed in claim 3, wherein in step S3, the initial value x ═ x obtained in step S2 is set as 1 ,x 2 ,x 3 ,x 4 Inputting the chaos sequences X into a driving system, and generating a set number of chaos sequences X in a given time.
5. The method for encrypting and decrypting the images of the heterogeneous fractional order hyperchaotic system with the adjustable synchronization time according to claim 4, wherein the step S4 specifically comprises the following steps:
step S41: the DNA sequence has four nucleobases, adenine (A), thymine (T), cytosine (C) and guanine (G), wherein A and T are complementary and G and C are complementary; for binary numbers, 0 and 1 are complementary, so 00 and 11, 01 and 10 are also complementary; by using four bases A, T, C, G to code 00, 01, 10, 11, there are 24 coding combinations, but only eight satisfy the base complementary pairing principle, according to the DNA coding rule, the image pixel is coded;
step S42: the color image has three channels: r, G and B, both of size M N, convert the image into a matrix I of M3N m (i, j); in the cyclic shift scrambling, only rows and columns are operated, and therefore, M +3N groups of data in the chaotic sequence X generated in step S3 are arbitrarily taken to obtain a chaotic sequence: s 1 ＝{X 1 ,Y 1 ,Z 1 }; to S 1 Performs the following operations:
Figure FDA0003639238200000021
wherein m is derived from S 1 The number of the sequence; m, N denotes the size of the image, T m ，W m Represents a new sequence value obtained by formula calculation, and the sequence T is { T ═ T x ,T y ,T z }、W＝{W x ,W y ,W z Is S 1 ＝{X 1 ,Y 1 ,Z 1 Obtaining a new sequence after each sequence in the sequence is calculated by a formula(ii) a The scrambling sequence is obtained according to the following formula:
Figure FDA0003639238200000022
wherein S is r (i) Is the sequence required for the cyclic shift of the rows, S c (i) Is the sequence required for column cyclic shift;
scrambling the image matrix according to the following formula:
Figure FDA0003639238200000023
wherein i is 1,2, … …, M; j ═ 1,2, … …, 3N; i is p (I,: represents I) p Values of all columns in the ith row of the matrix, I p (j) represents I p Values of all rows in the jth column of the matrix, I m (I,: represents I) m Values of all columns in the ith row of the matrix, I m (j) represents I m The values of all rows in the jth column in the matrix, and the function Circhift (a, b) circularly shifts the matrix a by b bits; finally, a scrambling matrix I is obtained p (i，j)；
Step S43: diffusing pixels of all components of the color image; randomly extracting M multiplied by N groups of data from the chaotic sequence X to obtain a sequence: s. the 2 ＝{X 2 ,Y 2 ,Z 2 }; the obtained sequences are then merged into a matrix S of size M x 3N 2 ', and process S according to the following formula d ：
Figure FDA0003639238200000031
The diffusion operation is performed according to the following formula:
Figure FDA0003639238200000032
Figure FDA0003639238200000033
wherein, I c (1,: represents I) c Values of all columns in row 1 in the matrix, I c (1) represents I c The values of all rows in column 1 of the matrix, where I c (I,: represents I) c Values of all columns in the ith row of the matrix, I c (j) represents I c The values of all rows in the jth column in the matrix;
obtaining a matrix I after diffusion c Is shown by c Obtaining matrix I 'after Arnold scrambling' c Prepared from l' c Reconstructing a matrix I of MXNX 3 e The matrix I e Namely the obtained ciphertext image; and packaging the obtained ciphertext, the initial driving signal and the predefined time, and transmitting the ciphertext, the initial driving signal and the predefined time to a receiving party through signals.
6. The image encryption and decryption method of the synchronous time adjustable heterogeneous fractional order hyper-chaotic system according to claim 5, wherein in step S5, the state equation of the established response system is:
Figure FDA0003639238200000034
wherein, y i (t), i ═ 1,2,3,4 denotes a state variable of the drive system, σ i >0, i-1, 2,3,4 is a self-suppression parameter, τ>0 is a time-invariant delay term; a is 1 、b 1 、c 1 、r 1 Is a constant; u. of j (t), j ═ 1,2,3,4, which denotes a predefined time synchronization controller.
7. The method for encrypting and decrypting the images of the heterogeneous fractional order hyperchaotic system with the adjustable synchronization time according to claim 6, wherein the step S6 is implemented by the following steps:
error e for synchronizing the drive system with the response system at a predetermined time i The following settings are set:
Figure FDA0003639238200000041
wherein i is 1,2,3,4,
Figure FDA0003639238200000042
representing the response system output value, x i Expressing the output value of the driving system, and obtaining a preset time synchronization error system e by derivation according to the formula i (t)：
e i (t)＝y i (t)-x i (t),i＝1,2,…,n (9)
Let E (t) ═ e 1 (t),e 2 (t),…,e n (t)) T ∈R n Is the state vector of the error system, and the initial value is E (0) ═ Y (0) -X (0); the synchronization error system is described as:
Figure FDA0003639238200000043
substituting formulae (1) and (7) into formula (10) yields:
D α e i (t)＝-σ i e i (t)+h i (e i (t))+H i (e i (t-τ))+u i (t) (11)
in the formula, h i (e i (t))＝f i (Y,t)-f i (X,t)；H i (e i (t-τ))＝F i (Y,t-τ)-F i (X,t-τ)；f i ，F i Representing some functional relationship.
8. The method for encrypting and decrypting the images of the heterogeneous fractional order hyperchaotic system with the adjustable synchronization time according to claim 7, wherein the step S7 is implemented by the following steps:
the adaptive predefined time synchronization controller function is described as follows:
Figure FDA0003639238200000044
where Tc represents an adjustable predefined time; c v Is a normal number determined by other parameters; beta is a 2 ，λ 2 Q, k, μ, ω are custom normality; c v The calculation formula of (a) is as follows:
Figure FDA0003639238200000051
the following continuous functions were chosen as the lyapunov function:
Figure FDA0003639238200000052
V 2 (t) denotes the Lyapunov function, e (t) is the error function,
Figure FDA0003639238200000053
denotes e i Adding the absolute values of (A) to (B), taking V 2 (t) in combination with equation (11) in step S6, the following is obtained:
Figure FDA0003639238200000054
further derivation of equation (14) yields:
Figure FDA0003639238200000055
wherein the content of the first and second substances,
Figure FDA0003639238200000056
for the
Figure FDA0003639238200000057
To obtain V 2 (t) ═ 0; the synchronization error formula (8) in step S6 is at the predetermined time T c And (4) internal synchronization.
9. The image encryption and decryption method of the heterogeneous fractional order hyperchaotic system with adjustable synchronization time as claimed in claim 8, wherein in step S8, a synchronization system is constructed by using the synchronization method of step S7, a synchronization chaotic sequence is generated to perform the reverse operation of step S4, and the ciphertext information is sequentially subjected to reverse Arnold scrambling, reverse diffusion, reverse cycle scrambling and DNA decoding to obtain the decrypted image.
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* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN115499557A (en) * 2022-09-13 2022-12-20 洛阳师范学院 Delay chaotic image encryption method based on Arnold mapping and multi-shift mapping function
CN116073982A (en) * 2023-02-07 2023-05-05 中国人民解放军陆军工程大学 Secret communication method and system for resisting DoS attack in limited time

Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN115499557A (en) * 2022-09-13 2022-12-20 洛阳师范学院 Delay chaotic image encryption method based on Arnold mapping and multi-shift mapping function
CN116073982A (en) * 2023-02-07 2023-05-05 中国人民解放军陆军工程大学 Secret communication method and system for resisting DoS attack in limited time
CN116073982B (en) * 2023-02-07 2024-01-19 中国人民解放军陆军工程大学 Secret communication method and system for resisting DoS attack in limited time

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