Patent Publication Number: US-2020287704-A1

Title: Color Image Encryption Method Based on DNA Strand Displacement Analog Circuit

Description:
TECHNICAL FIELD 
     The invention relates to the field of DNA strand displacement, wherein color image encryption is performed by utilizing chaotic sequences generated by a DNA strand displacement analog chaotic circuit. 
     BACKGROUND ART 
     DNA strand displacement is a new technique and method in DNA computing, and takes full advantage of the properties of DNA molecules, for example, the composition of DNA long strands is based on the base complementary pairing principle, that is, the Wastson-Crick complementary pairing principle, so that the DNA strand displacement reaction process becomes a spontaneous reaction, which can be operated at normal temperature without annealing operation or electric field action, and the experimental operation time is short and the product yield is high, so that the DNA strand displacement reaction is more convenient and efficient compared with the traditional molecular assembly. As another example, DNA strands have powerful coding and compiling capabilities, its own DNA sequence can store and compile large amounts of information, one short strand of DNA (20 nt-80 nt) can represent a logic signal, and several strands of DNA can be assembled into a molecule operation gate or logic gate, which makes it easy to construct digital circuits and analog circuits based on DNA strands. DNA strands have great application potential in the field of encryption due to powerful compiling capability. 
     Unpredictability, ergodicity, pseudorandom and sensitivity to initial value are the main characteristics of chaotic systems, and the main characteristics can be applied to the fields of encryption and secure communication. Therefore, chaotic encryption arouses wide public concern, and people have done a lot of research work on chaotic encryption. However, some image encryption methods based on low-dimensional chaotic systems are not secure, because the low-dimensional chaotic systems often have problems of periodic orbits and lack of secret key space due to the limitation of accuracy in the process of implementation, people turn their attention to high-dimensional chaotic systems and spatiotemporal chaotic systems, but the high-dimensional chaotic systems and spatiotemporal chaotic systems are difficult to achieve. Therefore, how to ensure the security of the encryption scheme is very meaningful in current chaotic image encryption under the condition of limited accuracy. 
     SUMMARY OF THE INVENTION 
     There are a lot of uncertainties and leakage reactions in a DNA strand displacement analog circuit, and the results of the DNA strand displacement analog circuit are difficultly verified by experiments because the reaction rate is difficult to set accurately. At present, most of DNA strand displacement analog circuits are designed and implemented by Visual DSD software. However, the DNA strand displacement analog circuit is a continuous system, can dynamically react to the dynamic behavior of the system, and is conducive to the realization of the dynamic behavior of a chaotic system. 
     A Rössler chaotic system is subjected to analog by a DSD analog circuit, and the generated chaotic sequence is applied to color image encryption. Compared with the traditional chaotic encryption scheme, this encryption scheme has the following advantages: on the one hand, the secret key is unrelated to the initial value of the chaotic system, but related to the color image itself, so that the space size of the secret key is not affected by the accuracy of the chaotic sequence; on the other hand, due to the fact that the generation of the chaotic sequence relies upon biochemical tests, concentration is measured every 3 seconds to obtain a piece of data, in such a way, a large amount of time is consumed in the process of generating the chaotic sequence. The invention adopts two measures to greatly reduce the time required for measurement, first, the chaotic sequence is divided into blocks, except the first sequence block, data in the rest sequence blocks are the same; and second, a series of secret keys are generated by utilizing the information of plaintext images, and are independent of chaotic sequences, so the same-size plaintext images can utilize the same chaotic sequence, that is to say, there is no need to re-measure the concentration. 
     Compared with the prior art, the invention has the following advantages: 
     1. The invention performs color image encryption by utilizing a DNA strand displacement chaotic analog circuit for the first time.
 
2. The chaotic sequence is divided into blocks, which greatly reduces the number of data to be measured and shortens the measurement time.
 
3. The secret key is related to the information of the plaintext images and independent of the initial values of the chaotic circuit, so that the encryption effect can still be achieved in the case of low accuracy of chaotic sequences, and the secret key space can resist violence attacks.
 
4. The secret key is related to the information of the plaintext images and are independent of the initial values of the chaotic circuit. The same-size plaintext images can utilize the same chaotic sequence, so that the detection time can be omitted.
 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS AND TABLES 
         FIG. 1  Encryption flow 
         FIG. 2  Catalytic reaction module 1: X→2X 
         FIG. 3  Catalytic reaction module 2: X+Y→2Y 
         FIG. 4  Annihilation reaction module: X+Y→∅ 
         FIG. 5  Degradation reaction module 1: X+X→X 
         FIG. 6  Degradation reaction module 2: Y→∅ 
         FIG. 7  Experimental results obtained by using encryption and decryption schemes 
       Table 1 Values of parameters of a Rössler chaotic system 
       Table 2 Correlation coefficient of a “lena” ciphertext image and adjacent pixels thereof 
       Table 3 Information entropy 
       Table 4 NPCR and UACI values for encrypted images 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     The present invention will be described further below with reference to the drawings. 
     The detailed steps are as follows: 
     Step 1: determining an idealized reaction network that can describe the Rössler chaotic system as shown in Formula (1). 
     
       
         
           
             
               
                 
                   
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     Step 2: determining an idealized reaction module and a DNA strand displacement reaction module corresponding thereto, for example, a catalytic reaction module, a degradation reaction module and an annihilation reaction module according to an idealized reaction equation.
 
Step 3: constructing a DNA strand displacement analog circuit according to the idealized reaction network and the reaction modules.
 
Step 4: adopting an encryption method. The detailed steps (1)-(5) of the encryption method are as follows:
 
(1) obtaining a secret key d k  according to plaintext information, wherein K∈[4,5,6, . . . , ∞];
 
(2) obtaining chaotic sequences x 1 (i), x 2 (i) and x 3 (i) from the concentration measurement, and obtaining sequences matched with plaintext images from the extended sequences) {circumflex over (X)} 1 (i) {circumflex over (X)} 2 (i) and {circumflex over (X)} 3 (i);
 
(4) scrambling the color plaintext images at the level of the three components R, G, B in consideration of the relation between the three components R, G, B in a color image;
 
(5) obtaining scrambling sequences Ψ∈[1,N], Γ={Γ r ,Γ g ,Γ b } and Ψ={Ψ r ,Ψ g ,Ψ b } by using the extended chaotic sequences;
 
(6) scrambling the plaintext images to obtain a scrambled image P′;
 
(7) obtaining a scrambling sequence U by utilizing the extended chaotic sequences;
 
(8) obtaining a scrambling sequence V by utilizing the extended chaotic sequences; and
 
(9) performing diffusion operation on the image P′ by utilizing sequences U={U r ,U g ,U b } and V={V r ,V g ,V b } to obtain an encrypted image C={C r ,C g ,C b }.
 
Step 5: adopting a decryption method. The detailed steps (1)-(2) of the decryption method are as follows:
 
(1) removing the diffusion effect on the encrypted image from the last pixel to the first pixel; and
 
(2) removing the diffusion effect from the last column (row) to the first column (row) to obtain a plaintext image;
 
     EXAMPLE 1 
     The embodiments of the present invention are implemented on the premise of the technical proposal of the present invention, and detailed embodiments and specific operation processes are given, but the scope of protection of the present invention is not limited to the following embodiments. 
     Step 1: substituting the parameters in table 1 into the DNA strand displacement analog circuit to obtain chaotic sequences x 1 (i), x 2 (i) and x 3 (i).
 
Step 2: detecting the encryption scheme by using color images of sizes 256×256 including “Lena”, “Pepper” and “Baboon”, wherein s 1 =800, s 2 =1500.
 
Step 3: arranging the pixel values of the plaintext image according to the size, then representing the pixel values by p r , p g  and p b , and recording the positions of the pixel values respectively by q r ×{q 1   r ,q 2   r , . . . ,q M×N   r }, q q ={q 1   g ,q 2   g , . . . ,q M×N   g } and q b ={q 1   b ,q 2   b , . . . ,q M×N   b , then
 
     
       
         
           
             
               
                 
                   
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     wherein, 0&lt;α k ≤10, K∈[4,5,6, . . . , ∞]. 
     Step 4: extending x 1 (i), x 2 (i), and x 3 (i) into three sets of sequences and {circumflex over (X)} 1 (i), {circumflex over (X)} 2 (i) and {circumflex over (X)} 3 (i) matched with the plaintext images as follows, 
     
       
         
           
             
               
                 
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     the first m data are discarded, and the detection time can be obtained by the following equation: 
         T= (π+ m )×3(seconds)   (30)
 
     Step 5: scrambling the color plaintext images at the level of the three components R, G, B as follows in consideration of the relation among the three components R, G, B in a color image: 
     
       
         
           
             
               
                 
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     wherein 1≤θ≤M×N
 
Step 6: repetitively iterating equations (34) and (35) until 6 one-dimensional arrays are obtained, wherein Γ∈[1,M], Ψ∈[1,N], Γ={Γ r ,Γ g ,Γ b } and Ψ={Ψ r ,Ψ g ,Ψ b }.
 
     
       
         
           
             
               
                 
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                                       ( 
                                       
                                         
                                           
                                             
                                               X 
                                               ^ 
                                             
                                             2 
                                           
                                           d 
                                         
                                         × 
                                         
                                           10 
                                           14 
                                         
                                       
                                       ) 
                                     
                                   
                                   , 
                                   N 
                                 
                                 ) 
                               
                             
                             + 
                             1 
                           
                         
                       
                     
                     
                       
                         
                           
                             Ψ 
                             b 
                           
                           = 
                           
                             
                               mod 
                                
                               
                                 ( 
                                 
                                   
                                     floor 
                                      
                                     
                                       ( 
                                       
                                         
                                           
                                             
                                               X 
                                               ^ 
                                             
                                             3 
                                           
                                           d 
                                         
                                         × 
                                         
                                           10 
                                           14 
                                         
                                       
                                       ) 
                                     
                                   
                                   , 
                                   N 
                                 
                                 ) 
                               
                             
                             + 
                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   35 
                   ) 
                 
               
             
           
         
       
     
     step 7: obtaining a scrambled image P′ according to the following operation: 
       f(i,j)⇄f(Γ(i+s 1 ),Ψ(j+s 2 ))   (36)
 
     Step 8, generating a diffusion sequence U by using the chaotic sequences; 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             U 
                             r 
                           
                           = 
                           
                             mod 
                              
                             
                               ( 
                               
                                 
                                   floor 
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           
                                             X 
                                             ^ 
                                           
                                           1 
                                         
                                         d 
                                       
                                       × 
                                       
                                         10 
                                         14 
                                       
                                     
                                     ) 
                                   
                                 
                                 , 
                                 256 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             U 
                             g 
                           
                           = 
                           
                             mod 
                              
                             
                               ( 
                               
                                 
                                   floor 
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           
                                             X 
                                             ^ 
                                           
                                           1 
                                         
                                         d 
                                       
                                       × 
                                       
                                         10 
                                         14 
                                       
                                     
                                     ) 
                                   
                                 
                                 , 
                                 256 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             U 
                             b 
                           
                           = 
                           
                             mod 
                              
                             
                               ( 
                               
                                 
                                   floor 
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           
                                             X 
                                             ^ 
                                           
                                           3 
                                         
                                         d 
                                       
                                       × 
                                       
                                         10 
                                         14 
                                       
                                     
                                     ) 
                                   
                                 
                                 , 
                                 256 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   37 
                   ) 
                 
               
             
           
         
       
     
     Step 9: generating a diffusion sequence V by using the chaotic sequences; 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             V 
                             r 
                           
                           = 
                           255 
                         
                       
                       
                         
                           
                             if 
                              
                             
                                 
                             
                              
                             
                               
                                 X 
                                 ^ 
                               
                               1 
                             
                           
                           &gt; 
                           
                             10 
                              
                             
                                 
                             
                              
                             nM 
                           
                         
                       
                     
                     
                       
                         
                           
                             V 
                             r 
                           
                           = 
                           0 
                         
                       
                       
                         
                           
                             if 
                              
                             
                                 
                             
                              
                             
                               
                                 X 
                                 ^ 
                               
                               1 
                             
                           
                           ≤ 
                           
                             10 
                              
                             
                                 
                             
                              
                             nM 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   38 
                   ) 
                 
               
             
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             V 
                             g 
                           
                           = 
                           255 
                         
                       
                       
                         
                           
                             if 
                              
                             
                                 
                             
                              
                             
                               
                                 X 
                                 ^ 
                               
                               2 
                             
                           
                           &gt; 
                           
                             10 
                              
                             
                                 
                             
                              
                             nM 
                           
                         
                       
                     
                     
                       
                         
                           
                             V 
                             g 
                           
                           = 
                           0 
                         
                       
                       
                         
                           
                             if 
                              
                             
                                 
                             
                              
                             
                               
                                 X 
                                 ^ 
                               
                               2 
                             
                           
                           ≤ 
                           
                             10 
                              
                             
                                 
                             
                              
                             nM 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   39 
                   ) 
                 
               
             
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             V 
                             b 
                           
                           = 
                           255 
                         
                       
                       
                         
                           
                             if 
                              
                             
                                 
                             
                              
                             
                               
                                 X 
                                 ^ 
                               
                               3 
                             
                           
                           &gt; 
                           
                             10 
                              
                             
                                 
                             
                              
                             nM 
                           
                         
                       
                     
                     
                       
                         
                           
                             V 
                             b 
                           
                           = 
                           0 
                         
                       
                       
                         
                           
                             if 
                              
                             
                                 
                             
                              
                             
                               
                                 X 
                                 ^ 
                               
                               3 
                             
                           
                           ≤ 
                           
                             10 
                              
                             
                                 
                             
                              
                             nM 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   40 
                   ) 
                 
               
             
           
         
       
     
     step 10: performing diffusion operation on the image p′ by using the matrix U={U r ,U g ,U b } and V={V r ,V g ,V b } to obtain an encrypted image C={C r ,C g ,C b }. 
     
       
         
           
             
               
                 
                   
                     C 
                     1 
                   
                   = 
                   
                     
                       P 
                       1 
                       ′ 
                     
                     ⊕ 
                     
                       U 
                       i 
                     
                     ⊕ 
                     
                       V 
                       1 
                     
                   
                 
               
               
                 
                   ( 
                   41 
                   ) 
                 
               
             
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               
                                 C 
                                 i 
                                 r 
                               
                               = 
                               
                                 
                                   P 
                                   i 
                                   
                                     ′ 
                                      
                                     
                                         
                                     
                                      
                                     r 
                                   
                                 
                                 ⊕ 
                                 
                                   C 
                                   
                                     i 
                                     - 
                                     1 
                                   
                                   r 
                                 
                                 ⊕ 
                                 
                                   U 
                                   i 
                                   r 
                                 
                                 ⊕ 
                                 
                                   V 
                                   i 
                                   r 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 C 
                                 i 
                                 g 
                               
                               = 
                               
                                 
                                   P 
                                   i 
                                   
                                     ′ 
                                      
                                     
                                         
                                     
                                      
                                     g 
                                   
                                 
                                 ⊕ 
                                 
                                   C 
                                   
                                     i 
                                     - 
                                     1 
                                   
                                   g 
                                 
                                 ⊕ 
                                 
                                   U 
                                   i 
                                   g 
                                 
                                 ⊕ 
                                 
                                   V 
                                   i 
                                   g 
                                 
                                 ⊕ 
                                 
                                   C 
                                   i 
                                   r 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 C 
                                 i 
                                 b 
                               
                               = 
                               
                                 
                                   P 
                                   i 
                                   
                                     ′ 
                                      
                                     
                                         
                                     
                                      
                                     b 
                                   
                                 
                                 ⊕ 
                                 
                                   C 
                                   
                                     i 
                                     - 
                                     1 
                                   
                                   b 
                                 
                                 ⊕ 
                                 
                                   U 
                                   i 
                                   b 
                                 
                                 ⊕ 
                                 
                                   V 
                                   i 
                                   b 
                                 
                                 ⊕ 
                                 
                                   C 
                                   i 
                                   r 
                                 
                                 ⊕ 
                                 
                                   C 
                                   i 
                                   g 
                                 
                               
                             
                           
                         
                       
                        
                       
                         
 
                       
                        
                       M 
                       × 
                       N 
                     
                     ≥ 
                     i 
                     &gt; 
                     1 
                   
                 
               
               
                 
                   ( 
                   42 
                   ) 
                 
               
             
           
         
       
     
     step 11: adopting a decryption method. The detailed steps: (1)-(2) of the decryption method are as follows: 
     (1) removing the diffusion effect on the encrypted image from the last pixel to the first pixel; 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               
                                 E 
                                 i 
                                 r 
                               
                               = 
                               
                                 
                                   C 
                                   i 
                                   r 
                                 
                                 ⊕ 
                                 
                                   C 
                                   
                                     i 
                                     - 
                                     1 
                                   
                                   r 
                                 
                                 ⊕ 
                                 
                                   U 
                                   i 
                                   r 
                                 
                                 ⊕ 
                                 
                                   V 
                                   i 
                                   r 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 E 
                                 i 
                                 g 
                               
                               = 
                               
                                 
                                   C 
                                   i 
                                   g 
                                 
                                 ⊕ 
                                 
                                   C 
                                   
                                     i 
                                     - 
                                     1 
                                   
                                   g 
                                 
                                 ⊕ 
                                 
                                   U 
                                   i 
                                   g 
                                 
                                 ⊕ 
                                 
                                   V 
                                   i 
                                   g 
                                 
                                 ⊕ 
                                 
                                   C 
                                   i 
                                   r 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 E 
                                 i 
                                 b 
                               
                               = 
                               
                                 
                                   C 
                                   i 
                                   b 
                                 
                                 ⊕ 
                                 
                                   C 
                                   
                                     i 
                                     - 
                                     1 
                                   
                                   b 
                                 
                                 ⊕ 
                                 
                                   U 
                                   i 
                                   b 
                                 
                                 ⊕ 
                                 
                                   V 
                                   i 
                                   b 
                                 
                                 ⊕ 
                                 
                                   C 
                                   i 
                                   r 
                                 
                                 ⊕ 
                                 
                                   C 
                                   i 
                                   g 
                                 
                               
                             
                           
                         
                       
                        
                       
                         
 
                       
                        
                       i 
                     
                     &gt; 
                     1 
                   
                 
               
               
                 
                   ( 
                   43 
                   ) 
                 
               
             
             
               
                 
                   
                     E 
                     1 
                   
                   = 
                   
                     
                       C 
                       1 
                     
                     ⊕ 
                     
                       U 
                       1 
                     
                     ⊕ 
                     
                       V 
                       1 
                     
                   
                 
               
               
                 
                   ( 
                   44 
                   ) 
                 
               
             
           
         
       
     
     (2) removing the scrambling effect from the last column (row) to the first column (row) to obtain a plaintext image; 
     The effects of encryption and decryption of “Lena”, “Pepper” and “Babook” images are shown in  FIG. 7 , and the indices of the encrypted images are shown in tables 2-4. 
     The above is only the specific embodiment of the present invention, but the protection scope of the present invention is not limited thereto, and any changes or substitutions without creative efforts shall fall within the protection scope of the present invention. Therefore, the claimed protection extent of the invention shall be determined with reference to the appended claims.