Abstract:
The present invention applies with a diffused mechanism, as such, a variable series of diffusion functions embedded within a diffusion-medium, to generate a maximum diffusion-cycle and nonlinear complexity; additionally, it performs in a serial process for simple design, further, in a parallel process for saving time, or even in a hardware architecture, to gain greater acceleration. FIG.  1  shows an embodiment of the present invention in flow chart diagram form, comprising steps of: selecting a diffusion-area A, a diffusion-medium S, and a diffusion mechanism F t     1      100 ; initializing A by an input password  200 ; performing F t     1    to obtain a new value of A  300 ; inputting a plaintext in encryption or a ciphertext for decryption, bit by bit  400 ; XORing a plaintext bit in encryption or a ciphertext bit in decryption with a certain bit of A for stream output  500 ; continuing until completing the plaintext in encryption, or the ciphertext in decryption  600.

Description:
BACKGROUND OF THE INVENTION 
       [0001]    1. Field of the Invention 
         [0002]    The invention relates to a stream cryptographic method and apparatus therefor. More particularly, the invention relates to a diffusion mechanism for running stream cryptography bit by bit; other than the prior art with a fixed polynomial and Shift operations, the diffusion mechanism is combined flexibly with a variable series of diffusion functions embedded within a diffusion-medium control. 
         [0003]    2. Description of the Related Art 
         [0004]    The Applicant&#39;s following patent application is related to the invention and is incorporated herein by reference: “A Cryptographic Method of Multilayer Diffusion in Multidimension”, application Ser. No. 12/726,833, filed Mar. 18, 2010; the prior invention described a diffusion function for one position AF(p 1 , p 2 , . . . , p n ), notated herein as AF(i 1 , i 2 , . . . , i m )=A⊕Ad 1i ⊕A 2i ⊕ . . . ⊕Ad mi ⊕S, the encryption/decryption is completed through a diffusion-cycle T, wherein T=2 U+1 , U=┌log 2  u┐, u=max(d 1 , d 2 , . . . , d m ), based on the dimensional magnitude. 
         [0005]    The present invention emphasizes multiple diffusion functions at a series of pre-arranged positions to maximize the diffusion cycle. Other than the prior invention for one position, the notation AF(p 1 , p 2 , . . . p k ) herein is for a series of pre-arranged positions; further, it may be described more delicately as or precisely by A performs the diffusion functions F(p 1 , p 2 , . . . p k ). 
       SUMMARY OF THE INVENTION 
       [0006]    The present invention applies with a diffused mechanism, as such, a variable series of diffusion functions embedded within a diffusion-medium, to generate a maximum diffusion-cycle and nonlinear complexity; additionally, it performs in a serial process for simple design, further, in a parallel process for saving time, or even in a hardware architecture, to gain greater acceleration. Under the combination of diffusion functions, the output bits, with a wide and flexible random distribution, are thus hardly attacked during the long diffusion-cycle; the steps are shown as follows:
       (a) selecting a diffusion mechanism, a diffusion-area, and a diffusion-medium;   (b) setting an initial value of the diffusion-area by an input password;   (c) running the diffusion mechanism to obtain a new value for the diffusion-area;   (d) inputting a bit of a plaintext/ciphertext stream in sequence;   (e) outputting a bit of the ciphertext/plaintext stream, equally, XORing (exclusive-or (XOR) operation) the bit of the plaintext/ciphertext stream with an assigned bit of the diffusion-area;   (f) going back to Step (c) until the encryption/decryption is complete.       
 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0013]      FIG. 1  is a summary flow chart diagram showing the main steps taken while encrypting/decrypting by a diffusion mechanism according to the present invention; 
           [0014]      FIG. 2  is a summary flow chart diagram of  FIG. 1 ,  200  showing the steps taken while initializing by an input password according to the present invention; 
           [0015]      FIG. 3A  is a summary flow chart diagram showing the component steps taken while performing a diffusion mechanism including a medium-constant in hardware architecture according to the present invention; 
           [0016]      FIG. 3B  is a summary flow chart diagram showing the component steps taken while performing a diffusion mechanism excluding a medium-constant in hardware architecture according to the present invention; 
           [0017]      FIG. 3C  is a summary flow chart diagram showing the component steps taken while encrypting/decrypting by a diffusion mechanism in hardware architecture according to the present invention; 
           [0018]      FIG. 4  is a summary schematic diagram of preferred Embodiment I showing a diffusion-table in hardware architecture according to the present invention; 
           [0019]      FIG. 5  is a summary schematic diagram of preferred Embodiments showing a parallel connecting in hardware architecture according to the present invention; 
           [0020]      FIG. 6  is a summary schematic diagram of preferred Embodiments showing a serial connecting in hardware architecture according to the present invention. 
       
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
       [0021]      FIG. 1  shows an embodiment of the present invention in flow chart diagram form, comprising steps of: selecting a diffusion-area A, a diffusion-medium S, and a diffusion mechanism F t     1      100 ; initializing A by an input password  200 ; performing F t     1    to obtain a new value of A  300 ; inputting a plaintext in encryption or a ciphertext for decryption, bit by bit  400 ; XORing a plaintext bit in encryption or a ciphertext bit in decryption with a certain bit of A for stream output  500 ; continuing until completing the plaintext in encryption, or the ciphertext in decryption  600 . 
         [0022]    Moreover,  FIG. 2 , i.e.  FIG. 1 ,  200 , shows an initialization of the diffusion-area by an input password, comprising steps of: inputting the password, bit by bit  210 ; setting the password bit whether controlling over S  220 , if yes, then performing F t     1    whether adding S by the value (0 or 1) of the password bit, to obtain a new value of A  240 , otherwise, writing the password bit directly to A  230 ; continuing until completing the password  250 . 
       Serial Process in Detail: 
       [0023]      FIG. 1 ,  300 , obtaining a new value A t     1     ×t     2   =A t     1     ×(t     2     −1) F t     1    by a serial process, it would implement a number of diffusion functions F(p 1 , p 2 , . . . , p k ), provided with a simple design but rather time consuming. Furthermore, for the initialization of A,  FIG. 2 , a password bit selects a first value for AF(p)=A⊕Ad 1i ⊕Ad 2i ⊕ . . . ⊕Ad mi ⊕S, and a second value for AF(p)=A⊕Ad 1i ⊕Ad 2i ⊕ . . . ⊕Ad mi . The cryptographic process flow comprises steps of:
       1. choosing a diffusion-area A, with a diffusion mechanism F t     1   ;   2. setting A&#39;s initial value A 0  by a password; t 2 =1;   3. obtaining a new value A t     1     ×t     2   , wherein A t     1     ×t     2   =A t     1     ×(t     2     −1) F t     1   ;   4. inputting a bit of a plaintext/ciphertext stream in sequence;   5. outputting a bit of the ciphertext/plaintext stream, equally, XORing the bit of the plaintext/ciphertext stream with an assigned bit of A t     1     ×t     2   ;   6. letting t 2 =t 2 +1, going back to Step 3, until the encryption/decryption is complete.       
 
       Symbols and Definitions: 
       [0000]    
       
         
           
             A: a diffusion-area, wherein A is a d 1 ×d 2 × . . . ×d m  binary matrix with a combination of positions labeled sequentially from 1 to n, accordingly, the bit values from a 1  to a n ; where the positions are shown as below:
 
for example,
 
           
         
       
     
         [0000]    
       
         
           
             
               
                 
                   ( 
                   a 
                   ) 
                 
                  
                 
                     
                 
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                 m 
               
               = 
               1 
             
             , 
             
               one 
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                 - 
               
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               dimension 
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               A 
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                     1 
                   
                   
                     2 
                   
                   
                     3 
                   
                   
                     … 
                   
                   
                     
                       n 
                       - 
                       1 
                     
                   
                   
                     n 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   ( 
                   b 
                   ) 
                 
                  
                 
                     
                 
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                 m 
               
               = 
               2 
             
             , 
             
               two 
                
               
                 - 
               
                
               dimension 
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                
               A 
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                     1 
                   
                   
                     
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                       + 
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                       + 
                       1 
                     
                   
                 
                 
                   
                     2 
                   
                   
                     
                       i 
                       + 
                       2 
                     
                   
                   
                     … 
                   
                   
                     
                       n 
                       - 
                       i 
                       + 
                       2 
                     
                   
                 
                 
                   
                     ⋮ 
                   
                   
                     ⋮ 
                   
                   
                     ⋮ 
                   
                   
                     ⋮ 
                   
                 
                 
                   
                     i 
                   
                   
                     
                       2 
                        
                       
                           
                       
                        
                       i 
                     
                   
                   
                     … 
                   
                   
                     n 
                   
                 
               
             
           
         
       
       
         
           
             S: a diffusion-medium, wherein S is a m-dimensional binary matrix, and further, includes an anchor-position {dot over (S)}. 
             AF(p): A performs a diffusion function F(p), see application Ser. No. 12/726,833, Point-Diffusion, wherein p transforms into m-dimension coordinates (i 1 , i 2 , . . . , i m ), then AF(p)=A⊕Ad 1i ⊕Ad 2i ⊕ . . . ⊕Ad mi ⊕S. 
             AF(p 1 , p 2 , . . . , p k ): A performs the diffusion functions F(p 1 , p 2 , . . . , p k ), sequentially at the m-dimension positions p 1 , p 2 , . . . , p k . 
             AF t     1   : A performs a diffusion mechanism F t     1   , as abbreviated for F(p 1 , p 2   t     1   , . . . , p k ), i.e. F(p 1 , p 2 , . . . , p k ) repeated t 1  times. 
           
         
       
     
       Preferred Embodiment I 
     A 1×16 Diffusion-Area A, a Diffusion Mechanism F 1    
       [0035]    Suppose that F 1 =F(13, 14, 15, 16, 1, . . . , 13) applied to A, further, A 0 =0, and S=1 placed at the anchor-position, with the encrypting value A 1×1 =A 0 F 1  at the first time, t 2 =1. To detail the diffusion function of every position, the sequential positions are notated as no. p: no. q in abbreviation thereafter, for example 13:15 for 13, 14, 15, or 13:1 for 13, 14, 15, 16, 1, and so on; the steps are listed as follows (S=1 is indicated in boldface hereafter): 
         [0000]    A 0 F(p)=A 0 ⊕A 0 y p ⊕S; 
       A 0 F(13)=[0000000000000000]⊕[0000000000000000]⊕[0000000000001000]; 
     A 0 F(13:14)=[0000000000001000]⊕[0000000000010000]⊕[0000000000000100]; 
     A 0 F(13:15)=[0000000000011100]⊕[0000000000111000]⊕[0000000000000010]; 
     A 0 F(13:16)=[0000000000100110]⊕[0000000001001100]⊕[0000000000000001]; 
     A 0 F(13:1)=[0000000001101011]⊕[0000000000110101]⊕[1000000000000000]; 
     A 0 F(13:2)=[1000000001011110]⊕[0000000000101111]⊕[0100000000000000]; 
     A 0 F(13:3)=[1100000001110001]⊕[1000000000111000]⊕[0010000000000000]; 
     A 0 F(13:4)=[0110000001001001]⊕[1100000000100100]⊕[0001000000000000]; 
     A 0 F(13:5)=[1011000001101101]⊕[0110000000110110]⊕[0000100000000000]; 
     A 0 F(13:6)=[1101100001011011]⊕[ 1011000000101101 ]⊕[ 0000010000000000 ]; 
     A 0 F(13:7)=[0110110001110110]⊕[1101100000111011]⊕[0000001000000000]; 
     A 0 F(13:8)=[1011011001001101]⊕[0110110000100110]⊕[0000000100000000]; 
     A 0 F(13:9)=[1101101101101011]⊕[1011011000110101]⊕[0000000010000000]; 
     A 0 F(13:10)=[0110110111011110]⊕[1101101110101111]⊕[0000000001000000]; 
     A 0 F(13:11)=[1011011000110001]⊕[ 0110110001011000 ]⊕[ 0000000000100000 ]; 
     A 0 F(13:12)=[1101101001001001]⊕[ 1011010010000100 ]⊕[ 0000000000010000 ]; 
     A 0 F(13:13)=[0110111011011101]⊕[1101110110110110]⊕[0000000000001000]. 
       [0036]    Through A 0 F(13:13), thus, A 1×1 =[1011001101100011], afterward, XORing a bit of A, assuming a 16 , with a plaintext bit to output an encrypted bit. Next, let t 2 =2, and the value A 1×2  is thus derived from A 1×1  which can be equivalently notated as A 1  in abbreviation hereafter; the details are shown as below: 
         [0000]    A 1 F(p)=A 1 ⊕A 1 y p ⊕S; 
       A 1 F(13)=[1011001101100011]⊕[0110011011000001]⊕[0000000000001000]; 
     A 1 F(13:14)=[1101010110101010]⊕[1010101101010001]⊕[0000000000000100]; 
     A 1 F(13:15)=[0111111011111111]⊕[1111110111111101]⊕[0000000000000010]; 
     A 1 F(13:16)=[1000001100000000]⊕[0000011000000000]⊕[0000000000000001]; 
     A 1 F(13:1)=[1000010100000001]⊕[0100001010000000]⊕[1000000000000000]; 
     A 1 F(13:2)=[0100011110000001]⊕[1010001111000000]⊕[0100000000000000]; 
     A 1 F(13:3)=[1010010001000001]⊕[0101001000100000]⊕[0010000000000000]; 
     A 1 F(13:4)=[1101011001100001]⊕[1010101100110000]⊕[0001000000000000]; 
     A 1 F(13:5)=[0110110101010001]⊕[1101011010101000]⊕[0000100000000000]; 
     A 1 F(13:6)=[1011001111111001]⊕[0110000111111100]⊕[0000010000000000]; 
     A 1 F(13:7)=[1101011000000101]⊕[1010110100000010]⊕[0000001000000000]; 
     A 1 F(13:8)=[0111100100000111]⊕[1111001010000011]⊕[0000000100000000]; 
     A 1 F(13:9)=[1000101010000100]⊕[0001010101000010]⊕[0000000010000000]; 
     A 1 F(13:10)=[1001111101000110]⊕[0011111010100011]⊕[0000000001000000]; 
     A 1 F(13:11)=[1010000110100101]⊕[0100001101010010]⊕[0000000000100000]; 
     A 1 F(13:12)=[1110001011010111]⊕[1100010110101011]⊕[0000000000010000]; 
     A 1 F(13:13)=[0010011101101100]⊕[0100111011010110]⊕[0000000000001000]. 
       [0037]    The second value of the diffusion-area A 1×2 =[0110100110110010]; then, another encrypted bit is generated. Accordingly, a diffusion-cycle of F 1 , implying A 1×t     2   =A 1 , requires 2 16 −1 times; thus, running the diffusion mechanism consecutively through the diffusion-cycle, the encrypted data complies with a random distribution thereby avoiding successful crypto-attack; assume a plaintext stream=0, four segments (64 bits) of the random output are appended as below, as samples: 
         [0000]    Four Segments: (t 2 =1 to 64) 
         [0038]    1011011100111011 (t 2 =1 to 16), 0000100100010111 (t 2 =17 to 32), 
         [0039]    0100000011010100 (t 2 =33 to 48), 1011011111111110 (t 2 =49 to 64). 
       Preferred Embodiment II 
     A 1×16 Diffusion-Area A, a Diffusion Mechanism F 2    
       [0040]    According to a diffusion mechanism arranged with the same positions in sequence, but at different times, the output is sampled out at every corresponding time. Suppose that F 2 =F(13 2 :13), A 0 =[0000000000000000] and S=1 is placed at the anchor-position, the encrypting value A 2×1  thus is obtained from A 2×0 F 2  at the first time, t 2 =1, see A 2  of Embodiment I, short for A 2×1 . The next value A 2×2 , notated as A 4  for short, is thus obtained by running the diffusion functions of the value A 3 ; the steps are also listed as below to let any skilled person gain more understanding: (A 3 =[1000010001101011]) 
         [0000]    A 3 F(p)=A 3 ⊕A 3 y p ⊕S; 
       A 3 F(13)=[1000010001101011]⊕[0000100011010101]⊕[0000000000001000]; 
     A 3 F(13:14)=[1000110010110110]⊕[0001100101101011]⊕[0000000000000100]; 
     A 3 F(13:15)=[1001010111011001]⊕[0010101110110000]⊕[0000000000000010]; 
     A 3 F(13:16)=[1011111001101011]⊕[0111110011010110]⊕[0000000000000001]; 
     A 3 F(13:1)=[1100001010111100]⊕[0110000101011110]⊕[1000000000000000]; 
     A 3 F(13:2)=[0010001111100010]⊕[0001000111110001]⊕[0100000000000000]; 
     A 3 F(13:3)=[0111001000010011]⊕[1101100100001001]⊕[0010000000000000]; 
     A 3 F(13:4)=[1000101100011010]⊕[0000010110001101]⊕[0001000000000000]; 
     A 3 F(13:5)=[1001111010010111]⊕[0011011101001011]⊕[0000100000000000]; 
     A 3 F(13:6)=[1010000111011100]⊕[0100000011101110]⊕[0000010000000000]; 
     A 3 F(13:7)=[1110010100110010]⊕[1100100010011001]⊕[0000001000000000]; 
     A 3 F(13:8)=[0010111110101011]⊕[0101111011010101]⊕[0000000100000000]; 
     A 3 F(13:9)=[0111000001111110]⊕[1110000000111111]⊕[0000000010000000]; 
     A 3 F(13:10)=[1001000011000001]⊕[0010000110100000]⊕[0000000001000000]; 
     A 3 F(13:11)=[1011000100100001]⊕[0110001001010000]⊕[0000000000100000]; 
     A 3 F(13:12)=[1101001101010001]⊕[1010011010101000]⊕[0000000000010000]; 
     A 3 F(13:13)=[0111010111101001]⊕[1110101111010100]⊕[0000000000001000]. 
       [0041]    The new value of the diffusion-area will be A 2×2 =[1001111000110101]. According to the diffusion mechanism using the number of times “2” is a prime number, therefore, the diffusion-cycle still keeps on 2 16 −1 times; thus, also assume a plaintext stream=0, XORing a 16  of A, the output other than Embodiment I is sampled out every two times; with four fragments of the random output appended as below, as samples: 
         [0000]    Four Segments: (t 2 =1 to 64) 
         [0042]    0111010100010111 (t 2 =1 to 16), 1000111001111110 (t 2 =17 to 32), 
         [0043]    1000010100011110 (t 2 =33 to 48), 1101011100000100 (t 2 =49 to 64). 
       Preferred Embodiment III 
     A 4×4 Diffusion-Area A, a Diffusion Mechanism F 1    
       [0044]    Provided that F 1 =F(8:8), A 0 =[0000000000000000] and S=1 placed at the anchor-position, the encrypting value A 1×1  is obtained from A 1×0  run by F 1  at the first time, t=1. Now that the diffusion-area is a two-dimension matrix, a position needs to be transformed to the corresponding coordinates (i, j); the details are shown as below (Ax(0), Ay(0) also indicated in boldface hereafter): 
         [0000]    
       
         
           
             
               
                 AF 
                  
                 
                   ( 
                   p 
                   ) 
                 
               
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                 ⊕ 
                 
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                       ( 
                       8 
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                     ⊕ 
                     
                       Ax 
                       4 
                     
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                       2 
                     
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                     ⊕ 
                     
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                     ⊕ 
                     
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                       ] 
                     
                     ⊕ 
                     
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                             1000 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             0000 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             0010 
                           
                         
                         
                           
                             0000 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                      
                     
                       [ 
                       
                         
                           
                             1000 
                           
                         
                         
                           
                             0111 
                           
                         
                         
                           
                             0010 
                           
                         
                         
                           
                             1100 
                           
                         
                       
                       ] 
                     
                   
                   ; 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     AF 
                      
                     
                       ( 
                       
                         8 
                         : 
                         12 
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     A 
                     ⊕ 
                     
                       Ax 
                       4 
                     
                     ⊕ 
                     
                       Ay 
                       3 
                     
                     ⊕ 
                     S 
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       [ 
                       
                         
                           
                             1000 
                           
                         
                         
                           
                             0111 
                           
                         
                         
                           
                             0010 
                           
                         
                         
                           
                             1100 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             0111 
                           
                         
                         
                           
                             0010 
                           
                         
                         
                           
                             1100 
                           
                         
                         
                           
                             0000 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             0000 
                           
                         
                         
                           
                             1101 
                           
                         
                         
                           
                             0101 
                           
                         
                         
                           
                             1000 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             0000 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             0010 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                      
                     
                       [ 
                       
                         
                           
                             1111 
                           
                         
                         
                           
                             1000 
                           
                         
                         
                           
                             1011 
                           
                         
                         
                           
                             0110 
                           
                         
                       
                       ] 
                     
                   
                   ; 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     AF 
                      
                     
                       ( 
                       
                         8 
                         : 
                         13 
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     A 
                     ⊕ 
                     
                       Ax 
                       1 
                     
                     ⊕ 
                     
                       Ay 
                       4 
                     
                     ⊕ 
                     S 
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       [ 
                       
                         
                           
                             1111 
                           
                         
                         
                           
                             1000 
                           
                         
                         
                           
                             1011 
                           
                         
                         
                           
                             0110 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             0000 
                           
                         
                         
                           
                             1111 
                           
                         
                         
                           
                             1000 
                           
                         
                         
                           
                             1011 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             1110 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             0110 
                           
                         
                         
                           
                             1100 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             0001 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             0000 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                      
                     
                       [ 
                       
                         
                           
                             0000 
                           
                         
                         
                           
                             0111 
                           
                         
                         
                           
                             0101 
                           
                         
                         
                           
                             0001 
                           
                         
                       
                       ] 
                     
                   
                   ; 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     AF 
                      
                     
                       ( 
                       
                         8 
                         : 
                         14 
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     A 
                     ⊕ 
                     
                       Ax 
                       2 
                     
                     ⊕ 
                     
                       Ay 
                       4 
                     
                     ⊕ 
                     S 
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       [ 
                       
                         
                           
                             0000 
                           
                         
                         
                           
                             0111 
                           
                         
                         
                           
                             0101 
                           
                         
                         
                           
                             0001 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             0111 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             0111 
                           
                         
                         
                           
                             0101 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             0000 
                           
                         
                         
                           
                             1110 
                           
                         
                         
                           
                             1010 
                           
                         
                         
                           
                             0010 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             0000 
                           
                         
                         
                           
                             0001 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             0000 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                      
                     
                       [ 
                       
                         
                           
                             0111 
                           
                         
                         
                           
                             1000 
                           
                         
                         
                           
                             1000 
                           
                         
                         
                           
                             0110 
                           
                         
                       
                       ] 
                     
                   
                   ; 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     AF 
                      
                     
                       ( 
                       
                         8 
                         : 
                         15 
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     A 
                     ⊕ 
                     
                       Ax 
                       3 
                     
                     ⊕ 
                     
                       Ay 
                       4 
                     
                     ⊕ 
                     S 
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       [ 
                       
                         
                           
                             0111 
                           
                         
                         
                           
                             1000 
                           
                         
                         
                           
                             1000 
                           
                         
                         
                           
                             0110 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             1000 
                           
                         
                         
                           
                             1000 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             1000 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             1110 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             1100 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             0000 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             0001 
                           
                         
                         
                           
                             0000 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                      
                     
                       [ 
                       
                         
                           
                             0001 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             1001 
                           
                         
                         
                           
                             0010 
                           
                         
                       
                       ] 
                     
                   
                   ; 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     AF 
                      
                     
                       ( 
                       
                         8 
                         : 
                         16 
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     A 
                     ⊕ 
                     
                       Ax 
                       4 
                     
                     ⊕ 
                     
                       Ay 
                       4 
                     
                     ⊕ 
                     S 
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       [ 
                       
                         
                           
                             0001 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             1001 
                           
                         
                         
                           
                             0010 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             0000 
                           
                         
                         
                           
                             1001 
                           
                         
                         
                           
                             0010 
                           
                         
                         
                           
                             0000 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             0010 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             0010 
                           
                         
                         
                           
                             0100 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             0000 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             0001 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                      
                     
                       [ 
                       
                         
                           
                             0011 
                           
                         
                         
                           
                             1001 
                           
                         
                         
                           
                             1001 
                           
                         
                         
                           
                             0111 
                           
                         
                       
                       ] 
                     
                   
                   ; 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     AF 
                      
                     
                       ( 
                       
                         8 
                         : 
                         1 
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     A 
                     ⊕ 
                     
                       Ax 
                       1 
                     
                     ⊕ 
                     
                       Ay 
                       1 
                     
                     ⊕ 
                     S 
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       [ 
                       
                         
                           
                             0011 
                           
                         
                         
                           
                             1001 
                           
                         
                         
                           
                             1001 
                           
                         
                         
                           
                             0111 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             0000 
                           
                         
                         
                           
                             0011 
                           
                         
                         
                           
                             1001 
                           
                         
                         
                           
                             1001 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             0001 
                           
                         
                         
                           
                             0100 
                           
                         
                         
                           
                             0100 
                           
                         
                         
                           
                             0011 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             1000 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             0000 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                      
                     
                       [ 
                       
                         
                           
                             1010 
                           
                         
                         
                           
                             1110 
                           
                         
                         
                           
                             0100 
                           
                         
                         
                           
                             1101 
                           
                         
                       
                       ] 
                     
                   
                   ; 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     AF 
                      
                     
                       ( 
                       
                         8 
                         : 
                         2 
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     A 
                     ⊕ 
                     
                       Ax 
                       2 
                     
                     ⊕ 
                     
                       Ay 
                       1 
                     
                     ⊕ 
                     S 
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       [ 
                       
                         
                           
                             1010 
                           
                         
                         
                           
                             1110 
                           
                         
                         
                           
                             0100 
                           
                         
                         
                           
                             1101 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             1110 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             1110 
                           
                         
                         
                           
                             0100 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             0101 
                           
                         
                         
                           
                             0111 
                           
                         
                         
                           
                             0010 
                           
                         
                         
                           
                             0110 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             0000 
                           
                         
                         
                           
                             1000 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             0000 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                      
                     
                       [ 
                       
                         
                           
                             0001 
                           
                         
                         
                           
                             0001 
                           
                         
                         
                           
                             1000 
                           
                         
                         
                           
                             1111 
                           
                         
                       
                       ] 
                     
                   
                   ; 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     AF 
                      
                     
                       ( 
                       
                         8 
                         : 
                         3 
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     A 
                     ⊕ 
                     
                       Ax 
                       3 
                     
                     ⊕ 
                     
                       Ay 
                       1 
                     
                     ⊕ 
                     S 
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       [ 
                       
                         
                           
                             0001 
                           
                         
                         
                           
                             0001 
                           
                         
                         
                           
                             1000 
                           
                         
                         
                           
                             1111 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             0001 
                           
                         
                         
                           
                             1000 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             1000 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             0000 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             0100 
                           
                         
                         
                           
                             0111 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             0000 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             1000 
                           
                         
                         
                           
                             0000 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                      
                     
                       [ 
                       
                         
                           
                             0000 
                           
                         
                         
                           
                             1001 
                           
                         
                         
                           
                             0100 
                           
                         
                         
                           
                             0000 
                           
                         
                       
                       ] 
                     
                   
                   ; 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     AF 
                      
                     
                       ( 
                       
                         8 
                         : 
                         4 
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     A 
                     ⊕ 
                     
                       Ax 
                       4 
                     
                     ⊕ 
                     
                       Ay 
                       1 
                     
                     ⊕ 
                     S 
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       [ 
                       
                         
                           
                             0000 
                           
                         
                         
                           
                             1001 
                           
                         
                         
                           
                             0100 
                           
                         
                         
                           
                             0000 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             1001 
                           
                         
                         
                           
                             0100 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             0000 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             0000 
                           
                         
                         
                           
                             0100 
                           
                         
                         
                           
                             0010 
                           
                         
                         
                           
                             0000 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             0000 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             1000 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                      
                     
                       [ 
                       
                         
                           
                             1001 
                           
                         
                         
                           
                             1001 
                           
                         
                         
                           
                             0110 
                           
                         
                         
                           
                             1000 
                           
                         
                       
                       ] 
                     
                   
                   ; 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     AF 
                      
                     
                       ( 
                       
                         8 
                         : 
                         5 
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     A 
                     ⊕ 
                     
                       Ax 
                       1 
                     
                     ⊕ 
                     
                       Ay 
                       2 
                     
                     ⊕ 
                     S 
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       [ 
                       
                         
                           
                             1001 
                           
                         
                         
                           
                             1001 
                           
                         
                         
                           
                             0110 
                           
                         
                         
                           
                             1000 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             0000 
                           
                         
                         
                           
                             1001 
                           
                         
                         
                           
                             1001 
                           
                         
                         
                           
                             0110 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             0000 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             1011 
                           
                         
                         
                           
                             0000 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             0100 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             0000 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                      
                     
                       [ 
                       
                         
                           
                             1101 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             0100 
                           
                         
                         
                           
                             1110 
                           
                         
                       
                       ] 
                     
                   
                   ; 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     AF 
                      
                     
                       ( 
                       
                         8 
                         : 
                         6 
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     A 
                     ⊕ 
                     
                       Ax 
                       2 
                     
                     ⊕ 
                     
                       Ay 
                       2 
                     
                     ⊕ 
                     S 
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       [ 
                       
                         
                           
                             1101 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             0100 
                           
                         
                         
                           
                             1110 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             0000 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             0100 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             1010 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             1010 
                           
                         
                         
                           
                             1011 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             0000 
                           
                         
                         
                           
                             0100 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             0000 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                      
                     
                       [ 
                       
                         
                           
                             0111 
                           
                         
                         
                           
                             0100 
                           
                         
                         
                           
                             1110 
                           
                         
                         
                           
                             0001 
                           
                         
                       
                       ] 
                     
                   
                   ; 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     AF 
                      
                     
                       ( 
                       
                         8 
                         : 
                         7 
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     A 
                     ⊕ 
                     
                       Ax 
                       3 
                     
                     ⊕ 
                     
                       Ay 
                       2 
                     
                     ⊕ 
                     S 
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       [ 
                       
                         
                           
                             0111 
                           
                         
                         
                           
                             0100 
                           
                         
                         
                           
                             1110 
                           
                         
                         
                           
                             0001 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             0100 
                           
                         
                         
                           
                             1110 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             1110 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             1011 
                           
                         
                         
                           
                             1010 
                           
                         
                         
                           
                             1011 
                           
                         
                         
                           
                             0000 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             0000 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             0100 
                           
                         
                         
                           
                             0000 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                      
                     
                       [ 
                       
                         
                           
                             1000 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             0001 
                           
                         
                         
                           
                             1111 
                           
                         
                       
                       ] 
                     
                   
                   ; 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     AF 
                      
                     
                       ( 
                       
                         8 
                         : 
                         8 
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     A 
                     ⊕ 
                     
                       Ax 
                       4 
                     
                     ⊕ 
                     
                       Ay 
                       2 
                     
                     ⊕ 
                     S 
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       [ 
                       
                         
                           
                             1000 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             0001 
                           
                         
                         
                           
                             1111 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             0000 
                           
                         
                         
                           
                             0001 
                           
                         
                         
                           
                             1111 
                           
                         
                         
                           
                             0000 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             0000 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             1011 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             0000 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             0000 
                           
                         
                         
                           
                             0100 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       [ 
                       
                         
                           
                             1000 
                           
                         
                         
                           
                             0001 
                           
                         
                         
                           
                             1110 
                           
                         
                         
                           
                             0000 
                           
                         
                       
                       ] 
                     
                     . 
                   
                 
               
             
           
         
       
     
         [0045]    Assume again the last bit of A 1×t     2    for encrypting and a plaintext stream=0, due to a two-dimension diffusion mechanism, four fragments of the random output, which are thus very different from samples obtained for Embodiment I &amp; II, are appended as below: 
         [0000]    Four Segments: (t 2 =1 to 64) 
         [0046]    0111000100100111 (t 2 =1 to 16), 0000001100001011 (t 2 =17 to 32), 
         [0047]    1110101001111110 (t 2 =33 to 48), 0011000001101100 (t 2 =49 to 64). 
       Parallel Process in Detail: 
       [0048]      FIG. 1 ,  300 , obtaining a new value A t     1     ×t     2   =A t     1     ×(t     2     −1) F t     1    by a parallel process, it would transform the serial process into A t     1     ×t     2   =A t     1     ×(t     2     −1) F t     1   ⊕BF t     1    namely, by running a diffusion-table F t     1    and a medium-constant BF t     1   , through a slightly complicated design but with the concomitant advantage of shortening the time required. Furthermore, for the initialization of A,  FIG. 2 , a password bit selects a first value for A t     1     ×t     2   =A t     1     ×(t     2     −1) F t     1   ⊕BF t     1   , and a second value for A t     1     ×t     2   =A t     1     ×(t     2     −1) F t     1   . 
         [0000]    The cryptographic process flow comprises steps of:
       1. choosing a diffusion-area A, creating a diffusion-table F t     1   ;   2. choosing a medium-area B, creating a medium-constant BF t     1   ;   3. setting A&#39;s initial value A 0  by a password; t 2 =1;   4. obtaining a new value A t     1     ×t     2   , wherein A t     1     ×t     2   =A t     1     ×(t     2     −1) F t     1   ⊕BF t     1   ;   5. inputting a bit of a plaintext/ciphertext stream in sequence;   6. outputting a bit of the ciphertext/plaintext stream, equally, XORing the bit of the plaintext/ciphertext stream with an assigned bit of A t     1     ×t     2   ;   7. letting t 2 =t 2 +1, going back to Step 4, until the encryption/decryption is complete.       
 
       Symbols and Definitions: 
       [0000]    
       
         
           
             A: a diffusion-area, wherein A contains an initial value A 0  and further, is a d 1 ×d 2 × . . . ×d m  binary matrix with a combination of positions labeled sequentially from 1 to n, accordingly, the bit values from a 1  to a n . 
             S: a diffusion-medium, wherein S is a m-dimensional binary matrix, and further, includes an anchor-position {dot over (S)}. 
             B: a medium-area, wherein B is a d 1 ×d 2 × . . . ×d m  zero matrix with a combination of positions labeled sequentially from 1 to n. 
             BF t     1   : a medium-constant, wherein B performs a diffusion mechanism F t     1    once, as abbreviated for F(p 1 , p 2   t     1   , . . . , p k ), i.e. F(p 1 , p 2 , . . . , p k ) repeated t 1  times. 
             AF t     1   : A performs a diffusion-table F t     1   , wherein F t     1   =FF(p 1 , p 2   t     1   , . . . , p k ) performed once, comprising of:
           F is a zero matrix with d 1 ×d 2 × . . . ×d m  plus one dimension with magnitude of d 1 ×d 2 × . . . ×d m , called layers, equally; in addition, 1&#39;s are at the i th  position on the i th  layer; for example,   
         
           
         
       
     
         [0000]    
       
         
           
             
                 
             
              
             
               
                 
                   
                     ( 
                     a 
                     ) 
                   
                    
                   
                       
                   
                    
                   a 
                    
                   
                       
                   
                    
                   4 
                   × 
                   4 
                    
                   
                       
                   
                    
                   F 
                 
                 = 
                 
                   [ 
                   
                     
                       
                         1000 
                       
                     
                     
                       
                         0100 
                       
                     
                     
                       
                         0010 
                       
                     
                     
                       
                         0001 
                       
                     
                   
                   ] 
                 
               
               , 
               
                 
 
               
                
               
                   
               
                
               
                 
                   
                     
                       ( 
                       b 
                       ) 
                     
                      
                     
                         
                     
                      
                     a 
                      
                     
                         
                     
                      
                     2 
                     × 
                     2 
                     × 
                     4 
                      
                     
                         
                     
                      
                     F 
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               1 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                           
                         
                         ] 
                       
                       
                          
                         
                           1 
                            
                           st 
                         
                       
                     
                      
                     
                       
                         [ 
                         
                           
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               1 
                             
                             
                               0 
                             
                           
                         
                         ] 
                       
                       
                          
                         
                           2 
                            
                           nd 
                         
                       
                     
                      
                     
                       
                         [ 
                         
                           
                             
                               0 
                             
                             
                               1 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                           
                         
                         ] 
                       
                       
                          
                         
                           3 
                            
                           rd 
                         
                       
                     
                      
                     
                       
                         [ 
                         
                           
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               1 
                             
                           
                         
                         ] 
                       
                       
                          
                         
                           4 
                            
                           th 
                         
                       
                     
                   
                 
                 ; 
               
             
           
         
       
       
         
           
             
               
                 
                   ( 
                   c 
                   ) 
                 
                  
                 
                     
                 
                  
                 a 
                  
                 
                     
                 
                  
                 4 
                 × 
                 4 
                 × 
                 16 
                  
                 
                     
                 
                  
                 F 
               
               = 
               
                 
                   
                     [ 
                     
                       
                         
                           1000 
                         
                       
                       
                         
                           0000 
                         
                       
                       
                         
                           0000 
                         
                       
                       
                         
                           0000 
                         
                       
                     
                     ] 
                   
                   
                      
                     
                       1 
                        
                       st 
                     
                   
                 
                  
                 
                   
                     [ 
                     
                       
                         
                           0000 
                         
                       
                       
                         
                           1000 
                         
                       
                       
                         
                           0000 
                         
                       
                       
                         
                           0000 
                         
                       
                     
                     ] 
                   
                   
                      
                     
                       2 
                        
                       nd 
                     
                   
                 
                  
                 
                   
                     [ 
                     
                       
                         
                           0000 
                         
                       
                       
                         
                           0000 
                         
                       
                       
                         
                           1000 
                         
                       
                       
                         
                           0000 
                         
                       
                     
                     ] 
                   
                   
                      
                     
                       3 
                        
                       rd 
                     
                   
                 
                  
                 
                   
                     [ 
                     
                       
                         
                           0000 
                         
                       
                       
                         
                           0000 
                         
                       
                       
                         
                           0000 
                         
                       
                       
                         
                           1000 
                         
                       
                     
                     ] 
                   
                   
                      
                     
                       4 
                        
                       th 
                     
                   
                 
                  
                 
                     
                 
                  
                 … 
                  
                 
                     
                 
                  
                 
                   
                     [ 
                     
                       
                         
                           0001 
                         
                       
                       
                         
                           0000 
                         
                       
                       
                         
                           0000 
                         
                       
                       
                         
                           0000 
                         
                       
                     
                     ] 
                   
                   
                      
                     
                       13 
                        
                       th 
                     
                   
                 
                  
                 
                   
                     [ 
                     
                       
                         
                           0000 
                         
                       
                       
                         
                           0001 
                         
                       
                       
                         
                           0000 
                         
                       
                       
                         
                           0000 
                         
                       
                     
                     ] 
                   
                   
                      
                     
                       14 
                        
                       th 
                     
                   
                 
                  
                 
                   
                     [ 
                     
                       
                         
                           0000 
                         
                       
                       
                         
                           0000 
                         
                       
                       
                         
                           0001 
                         
                       
                       
                         
                           0000 
                         
                       
                     
                     ] 
                   
                   
                      
                     
                       15 
                        
                       th 
                     
                   
                 
                  
                 
                   
                     [ 
                     
                       
                         
                           0000 
                         
                       
                       
                         
                           0000 
                         
                       
                       
                         
                           0000 
                         
                       
                       
                         
                           0001 
                         
                       
                     
                     ] 
                   
                   
                      
                     
                       16 
                        
                       th 
                     
                   
                 
               
             
             ; 
           
         
       
       
         
           
              wherein a 1×4 A is applied with the 4×4 F; a 2×2 A is applied with the 2×2×4 F; further, a 4×4 A is applied with the 4×4×16 F; and so on.
           FF(p 1 , p 2 , . . . , p k ): F performs the diffusion functions F(p 1 , p 2 , . . . , p k ), sequentially at the m-dimension positions p 1 , p 2 , . . . , p k ; for example, a 1×4 A applied with the 4×4 F, FF(p)=F⊕Fy p , FF(1:4) as follows (Fy(0) indicated in boldface hereafter):   
         
           
         
       
     
         [0000]    
       
         
           
             
               
                 FF 
                  
                 
                   ( 
                   1 
                   ) 
                 
               
               = 
               
                 
                   
                     [ 
                     
                       
                         
                           1000 
                         
                       
                       
                         
                           0100 
                         
                       
                       
                         
                           0010 
                         
                       
                       
                         
                           0001 
                         
                       
                     
                     ] 
                   
                   ⊕ 
                   
                     [ 
                     
                       
                         
                           0100 
                         
                       
                       
                         
                           0010 
                         
                       
                       
                         
                           0001 
                         
                       
                       
                         
                           0000 
                         
                       
                     
                     ] 
                   
                 
                 = 
                 
                   [ 
                   
                     
                       
                         1100 
                       
                     
                     
                       
                         0110 
                       
                     
                     
                       
                         0011 
                       
                     
                     
                       
                         0001 
                       
                     
                   
                   ] 
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 FF 
                  
                 
                   ( 
                   
                     1 
                     : 
                     2 
                   
                   ) 
                 
               
               = 
               
                 
                   
                     [ 
                     
                       
                         
                           1100 
                         
                       
                       
                         
                           0110 
                         
                       
                       
                         
                           0011 
                         
                       
                       
                         
                           0001 
                         
                       
                     
                     ] 
                   
                   ⊕ 
                   
                     [ 
                     
                       
                         
                           1010 
                         
                       
                       
                         
                           1011 
                         
                       
                       
                         
                           0001 
                         
                       
                       
                         
                           0000 
                         
                       
                     
                     ] 
                   
                 
                 = 
                 
                   [ 
                   
                     
                       
                         0110 
                       
                     
                     
                       
                         1101 
                       
                     
                     
                       
                         0010 
                       
                     
                     
                       
                         0001 
                       
                     
                   
                   ] 
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 FF 
                  
                 
                   ( 
                   
                     1 
                     : 
                     3 
                   
                   ) 
                 
               
               = 
               
                 
                   
                     [ 
                     
                       
                         
                           0110 
                         
                       
                       
                         
                           1101 
                         
                       
                       
                         
                           0010 
                         
                       
                       
                         
                           0001 
                         
                       
                     
                     ] 
                   
                   ⊕ 
                   
                     [ 
                     
                       
                         
                           1101 
                         
                       
                       
                         
                           1000 
                         
                       
                       
                         
                           0101 
                         
                       
                       
                         
                           0000 
                         
                       
                     
                     ] 
                   
                 
                 = 
                 
                   [ 
                   
                     
                       
                         1011 
                       
                     
                     
                       
                         0101 
                       
                     
                     
                       
                         0111 
                       
                     
                     
                       
                         0001 
                       
                     
                   
                   ] 
                 
               
             
             ; 
           
         
       
       
         
           
             
               FF 
                
               
                 ( 
                 
                   1 
                   : 
                   4 
                 
                 ) 
               
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         1011 
                       
                     
                     
                       
                         0101 
                       
                     
                     
                       
                         0111 
                       
                     
                     
                       
                         0001 
                       
                     
                   
                   ] 
                 
                 ⊕ 
                 
                   [ 
                   
                     
                       
                         0110 
                       
                     
                     
                       
                         1010 
                       
                     
                     
                       
                         1110 
                       
                     
                     
                       
                         0010 
                       
                     
                   
                   ] 
                 
               
               = 
               
                 
                   [ 
                   
                     
                       
                         1101 
                       
                     
                     
                       
                         1111 
                       
                     
                     
                       
                         1001 
                       
                     
                     
                       
                         0011 
                       
                     
                   
                   ] 
                 
                 . 
               
             
           
         
       
     
         [0063]    Suppose AF 1 , with the diffusion-table F 1 =FF(1:4), so according to the above derivation, the next a 1 =a 1 ⊕a 2 ⊕a 3 , see 1&#39;s at the first position on every layer, the i th  layer represented for current a i ; thus referring the second position on every layer, the next a 2 =a 1 ⊕a 2 ; further, a 3 =a 2 ⊕a 4 ; finally, a 4 =a 1 ⊕a 2 ⊕a 3 ⊕a 4 . 
         [0064]    Moreover, a 2×2 A applied with the 2×2×4 F, FF(p)=F⊕Fx i ⊕Fy j , FF(1:4) as follows (Fx(0) also indicated in boldface hereafter): 
         [0000]    
       
         
           
             
               
                 
                   
                     FF 
                      
                     
                       ( 
                       1 
                       ) 
                     
                   
                   = 
                     
                    
                   
                     F 
                     ⊕ 
                     
                       Fx 
                       1 
                     
                     ⊕ 
                     
                       Fy 
                       1 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             1 
                              
                             st 
                           
                         
                       
                        
                       
                         
                           [ 
                           
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 1 
                               
                               
                                 0 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             2 
                              
                             nd 
                           
                         
                       
                        
                       
                         
                           [ 
                           
                             
                               
                                 0 
                               
                               
                                 1 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             3 
                              
                             rd 
                           
                         
                       
                        
                       
                         
                           [ 
                           
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             4 
                              
                             th 
                           
                         
                       
                     
                     ⊕ 
                   
                 
               
             
             
               
                 
                     
                    
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 1 
                               
                               
                                 0 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             1 
                              
                             st 
                           
                         
                       
                        
                       
                         
                           [ 
                           
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             2 
                              
                             nd 
                           
                         
                       
                        
                       
                         
                           [ 
                           
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             3 
                              
                             rd 
                           
                         
                       
                        
                       
                         
                           [ 
                           
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             4 
                              
                             th 
                           
                         
                       
                     
                     ⊕ 
                   
                 
               
             
             
               
                 
                     
                    
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 0 
                               
                               
                                 1 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             1 
                              
                             st 
                           
                         
                       
                        
                       
                         
                           [ 
                           
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             2 
                              
                             nd 
                           
                         
                       
                        
                       
                         
                           [ 
                           
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             3 
                              
                             rd 
                           
                         
                       
                        
                       
                         
                           [ 
                           
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             4 
                              
                             th 
                           
                         
                       
                     
                     ; 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     FF 
                      
                     
                       ( 
                       
                         1 
                         : 
                         2 
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     
                       [ 
                       
                         FF 
                          
                         
                           ( 
                           1 
                           ) 
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       
                         [ 
                         
                           FF 
                           ( 
                           1 
                           ] 
                         
                         ) 
                       
                        
                       
                         x 
                         2 
                       
                     
                     ⊕ 
                     
                       
                         [ 
                         
                           FF 
                            
                           
                             ( 
                             1 
                             ) 
                           
                         
                         ] 
                       
                        
                       
                         y 
                         1 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 1 
                               
                             
                             
                               
                                 1 
                               
                               
                                 0 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             1 
                              
                             st 
                           
                         
                       
                        
                       
                         
                           [ 
                           
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 1 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             2 
                              
                             nd 
                           
                         
                       
                        
                       
                         
                           [ 
                           
                             
                               
                                 0 
                               
                               
                                 1 
                               
                             
                             
                               
                                 0 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             3 
                              
                             rd 
                           
                         
                       
                        
                       
                         
                           [ 
                           
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             4 
                              
                             th 
                           
                         
                       
                     
                     ⊕ 
                   
                 
               
             
             
               
                 
                     
                    
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             1 
                              
                             st 
                           
                         
                       
                        
                       
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 1 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             2 
                              
                             nd 
                           
                         
                       
                        
                       
                         
                           [ 
                           
                             
                               
                                 0 
                               
                               
                                 1 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             3 
                              
                             rd 
                           
                         
                       
                        
                       
                         
                           [ 
                           
                             
                               
                                 0 
                               
                               
                                 1 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             4 
                              
                             th 
                           
                         
                       
                     
                     ⊕ 
                   
                 
               
             
             
               
                 
                     
                    
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 0 
                               
                               
                                 1 
                               
                             
                             
                               
                                 0 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             1 
                              
                             st 
                           
                         
                       
                        
                       
                         
                           [ 
                           
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             2 
                              
                             nd 
                           
                         
                       
                        
                       
                         
                           [ 
                           
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             3 
                              
                             rd 
                           
                         
                       
                        
                       
                         
                           [ 
                           
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             4 
                              
                             th 
                           
                         
                       
                     
                     ; 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     FF 
                      
                     
                       ( 
                       
                         1 
                         : 
                         3 
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     
                       [ 
                       
                         FF 
                          
                         
                           ( 
                           
                             1 
                             : 
                             2 
                           
                           ) 
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       
                         [ 
                         
                           FF 
                           ( 
                           
                             1 
                             : 
                             2 
                           
                           ] 
                         
                         ) 
                       
                        
                       
                         x 
                         1 
                       
                     
                     ⊕ 
                     
                       
                         [ 
                         
                           FF 
                            
                           
                             ( 
                             
                               1 
                               : 
                               2 
                             
                             ) 
                           
                         
                         ] 
                       
                        
                       
                         y 
                         2 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 1 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             1 
                              
                             st 
                           
                         
                       
                        
                       
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 1 
                               
                             
                             
                               
                                 1 
                               
                               
                                 0 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             2 
                              
                             nd 
                           
                         
                       
                        
                       
                         
                           [ 
                           
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             3 
                              
                             rd 
                           
                         
                       
                        
                       
                         
                           [ 
                           
                             
                               
                                 0 
                               
                               
                                 1 
                               
                             
                             
                               
                                 0 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             4 
                              
                             th 
                           
                         
                       
                     
                     ⊕ 
                   
                 
               
             
             
               
                 
                     
                    
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             1 
                              
                             st 
                           
                         
                       
                        
                       
                         
                           [ 
                           
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 1 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             2 
                              
                             nd 
                           
                         
                       
                        
                       
                         
                           [ 
                           
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             3 
                              
                             rd 
                           
                         
                       
                        
                       
                         
                           [ 
                           
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             4 
                              
                             th 
                           
                         
                       
                     
                     ⊕ 
                   
                 
               
             
             
               
                 
                     
                    
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 1 
                               
                               
                                 0 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             1 
                              
                             st 
                           
                         
                       
                        
                       
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             2 
                              
                             nd 
                           
                         
                       
                        
                       
                         
                           [ 
                           
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 1 
                               
                               
                                 0 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             3 
                              
                             rd 
                           
                         
                       
                        
                       
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 0 
                               
                             
                             
                               
                                 1 
                               
                               
                                 0 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             4 
                              
                             th 
                           
                         
                       
                     
                     ; 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     FF 
                      
                     
                       ( 
                       
                         1 
                         : 
                         4 
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     
                       [ 
                       
                         FF 
                          
                         
                           ( 
                           
                             1 
                             : 
                             3 
                           
                           ) 
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       
                         [ 
                         
                           FF 
                           ( 
                           
                             1 
                             : 
                             3 
                           
                           ] 
                         
                         ) 
                       
                        
                       
                         x 
                         2 
                       
                     
                     ⊕ 
                     
                       
                         [ 
                         
                           FF 
                            
                           
                             ( 
                             
                               1 
                               : 
                               3 
                             
                             ) 
                           
                         
                         ] 
                       
                        
                       
                         y 
                         2 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             1 
                              
                             st 
                           
                         
                       
                        
                       
                         
                           [ 
                           
                             
                               
                                 0 
                               
                               
                                 1 
                               
                             
                             
                               
                                 0 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             2 
                              
                             nd 
                           
                         
                       
                        
                       
                         
                           [ 
                           
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 1 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             3 
                              
                             rd 
                           
                         
                       
                        
                       
                         
                           [ 
                           
                             
                               
                                 0 
                               
                               
                                 1 
                               
                             
                             
                               
                                 1 
                               
                               
                                 0 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             4 
                              
                             th 
                           
                         
                       
                     
                     ⊕ 
                   
                 
               
             
             
               
                 
                     
                    
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 0 
                               
                               
                                 1 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             1 
                              
                             st 
                           
                         
                       
                        
                       
                         
                           [ 
                           
                             
                               
                                 0 
                               
                               
                                 1 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             2 
                              
                             nd 
                           
                         
                       
                        
                       
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 1 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             3 
                              
                             rd 
                           
                         
                       
                        
                       
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             4 
                              
                             th 
                           
                         
                       
                     
                     ⊕ 
                   
                 
               
             
             
               
                 
                     
                    
                   
                     
                       
                         [ 
                         
                           
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               1 
                             
                             
                               0 
                             
                           
                         
                         ] 
                       
                       
                          
                         
                           1 
                            
                           st 
                         
                       
                     
                      
                     
                       
                         [ 
                         
                           
                             
                               1 
                             
                             
                               0 
                             
                           
                           
                             
                               1 
                             
                             
                               0 
                             
                           
                         
                         ] 
                       
                       
                          
                         
                           2 
                            
                           nd 
                         
                       
                     
                      
                     
                       
                         [ 
                         
                           
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               1 
                             
                             
                               0 
                             
                           
                         
                         ] 
                       
                       
                          
                         
                           3 
                            
                           rd 
                         
                       
                     
                      
                     
                       
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                           
                           ] 
                         
                         
                            
                           
                             4 
                              
                             th 
                           
                         
                       
                       . 
                     
                   
                 
               
             
           
         
       
     
         [0000]    Suppose AF 1 , wherein 
         [0000]    
       
         
           
             
               
                 F 
                 1 
               
               = 
               
                 
                   FF 
                    
                   
                     ( 
                     
                       1 
                       : 
                       4 
                     
                     ) 
                   
                 
                 = 
                 
                   
                     
                       [ 
                       
                         
                           
                             0 
                           
                           
                             1 
                           
                         
                         
                           
                             1 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                     
                        
                       
                         1 
                          
                         st 
                       
                     
                   
                    
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                         
                         
                           
                             1 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                     
                        
                       
                         2 
                          
                         nd 
                       
                     
                   
                    
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             1 
                           
                         
                         
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                     
                        
                       
                         3 
                          
                         rd 
                       
                     
                   
                    
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             1 
                           
                         
                         
                           
                             1 
                           
                           
                             0 
                           
                         
                       
                       ] 
                     
                     
                        
                       
                         4 
                          
                         th 
                       
                     
                   
                 
               
             
             ; 
           
         
       
     
         [0000]    four layers represent the current a 1  to a 4 , through by the first position, the next a 1 =a 2 ⊕a 3 ⊕a 4 ; by the second position, the next a 2 =a 1 ⊕a 2 ⊕a 4 ; further, by the third position, the next a 3 =a 1 ⊕a 3 ⊕a 4 ; and finally, by the fourth position, the next a 4 =a 1 ⊕a 2 ⊕a 3 . 
       Preferred Embodiment IV 
     Diffusion Mechanisms in a Parallel Process 
       [0065]    Firstly, in Embodiment I, now that the diffusion mechanism F 1 =F(13:13), A 0 =[0000000000000000], and S=1 placed at the anchor-position, the sequential value of diffusion-area A 1×t     2   =A 1×(t     2     −1) F 1 ⊕BF 1 , wherein a medium-constant BF 1  and a diffusion-table F 1  are performed only once at the beginning; BF 1 =BF(13:13)=[1011001101100011], derived as A 1 , in Embodiment I, and further, F 1 =FF(13:13), wherein FF(p)=F⊕Fy p , may refer to the whole derivation of two-dimensional FF(1:4), and F is such as a 16×16 unit matrix, with a one-dimension diffusion function in a two-dimension matrix; positions are orderly ordered from 13 to 16 then from 1 to 13, each position&#39;s diffusion function is shown in Table 1, and for greater clarity, the final result is shown in Table 2. Furthermore, reading from the i th  position of all layers, namely, referring to the i th  column to obtain the next a i , all combinations are shown in Table 3. 
         [0066]    Therefore, according to Table 3, new values in sequence as below: 
         [0000]    
       
         
           
             
               
                 for 
                  
                 
                     
                 
                  
                 
                   t 
                   2 
                 
               
               = 
               1 
             
             , 
             
               
 
             
              
             
               
                 
                   
                     
                       A 
                       
                         1 
                         × 
                         1 
                       
                     
                     = 
                       
                      
                     
                       
                         
                           A 
                           0 
                         
                          
                         
                           F 
                           1 
                         
                       
                       ⊕ 
                       
                         BF 
                         1 
                       
                     
                   
                 
               
               
                 
                   
                     = 
                       
                      
                     
                       
                         [ 
                         
                           0 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           0 
                         
                         ] 
                       
                       ⊕ 
                     
                   
                 
               
               
                 
                   
                       
                      
                     
                       
                         [ 
                         
                           1 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           1 
                            
                           
                               
                           
                            
                           1 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           1 
                            
                           
                               
                           
                            
                           1 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           1 
                            
                           
                               
                           
                            
                           1 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           1 
                            
                           
                               
                           
                            
                           1 
                         
                         ] 
                       
                       ; 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 the 
                  
                 
                     
                 
                  
                 results 
                  
                 
                     
                 
                  
                 is 
                  
                 
                     
                 
                  
                 the 
                  
                 
                     
                 
                  
                 same 
                  
                 
                     
                 
                  
                 as 
                  
                 
                     
                 
                  
                 
                   A 
                   1 
                 
                  
                 
                     
                 
                  
                 in 
                  
                 
                     
                 
                  
                 Embodiment 
                  
                 
                     
                 
                  
                 
                   I 
                   . 
                   
                     
 
                   
                    
                   for 
                 
                  
                 
                     
                 
                  
                 
                   t 
                   2 
                 
               
               = 
               2 
             
             , 
             
               
 
             
              
             
               
                 
                   
                     
                       A 
                       
                         1 
                         × 
                         2 
                       
                     
                     = 
                       
                      
                     
                       
                         
                           A 
                           
                             1 
                             × 
                             1 
                           
                         
                          
                         
                           F 
                           1 
                         
                       
                       ⊕ 
                       
                         BF 
                         1 
                       
                     
                   
                 
               
               
                 
                   
                     = 
                       
                      
                     
                       
                         [ 
                         
                           1 
                            
                           
                               
                           
                            
                           1 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           1 
                            
                           
                               
                           
                            
                           1 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           1 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           1 
                            
                           
                               
                           
                            
                           1 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           1 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           1 
                         
                         ] 
                       
                       ⊕ 
                       
                         [ 
                         
                           1 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           1 
                            
                           
                               
                           
                            
                           1 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           1 
                            
                           
                               
                           
                            
                           1 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           1 
                            
                           
                               
                           
                            
                           1 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           1 
                            
                           
                               
                           
                            
                           1 
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   
                     
                       = 
                         
                        
                       
                         0 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         0 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         0 
                          
                         
                             
                         
                          
                         0 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         0 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         0 
                          
                         
                             
                         
                          
                         0 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         0 
                       
                     
                     ; 
                   
                 
               
             
           
         
       
       
         
           
             the 
              
             
                 
             
              
             results 
              
             
                 
             
              
             is 
              
             
                 
             
              
             the 
              
             
                 
             
              
             same 
              
             
                 
             
              
             as 
              
             
                 
             
              
             
               A 
               2 
             
              
             
                 
             
              
             in 
              
             
                 
             
              
             Embodiment 
              
             
                 
             
              
             
               I 
               . 
               
                 
 
               
                
               Derivation 
             
              
             
                 
             
              
             of 
              
             
                 
             
              
             
               A 
               
                 1 
                 × 
                 1 
               
             
              
             
               F 
               1 
             
              
             
               : 
             
           
         
       
       
         
           
             Now 
              
             
                 
             
              
             that 
           
         
       
       
         
           
             
               
                 
                   
                     A 
                     
                       1 
                       × 
                       1 
                     
                   
                   = 
                     
                    
                   
                     [ 
                     
                       
                         a 
                         1 
                       
                       , 
                       
                         a 
                         2 
                       
                       , 
                       … 
                        
                       
                           
                       
                       , 
                       
                         a 
                         16 
                       
                     
                     ] 
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                      
                     
                       [ 
                       
                         
                           
                             
                               1 
                               1 
                             
                           
                           
                             
                               0 
                               2 
                             
                           
                           
                             
                               1 
                               3 
                             
                           
                           
                             
                               1 
                               4 
                             
                           
                           
                             
                               0 
                               5 
                             
                           
                           
                             
                               0 
                               6 
                             
                           
                           
                             
                               1 
                               7 
                             
                           
                           
                             
                               1 
                               8 
                             
                           
                           
                             
                               0 
                               9 
                             
                           
                           
                             
                               1 
                               10 
                             
                           
                           
                             
                               1 
                               11 
                             
                           
                           
                             
                               0 
                               12 
                             
                           
                           
                             
                               0 
                               13 
                             
                           
                           
                             
                               0 
                               14 
                             
                           
                           
                             
                               1 
                               15 
                             
                           
                           
                             
                               1 
                               16 
                             
                           
                         
                       
                       ] 
                     
                   
                   ; 
                 
               
             
           
         
       
       
         
           
             
               
                 the 
                  
                 
                     
                 
                  
                 next 
                  
                 
                     
                 
                  
                 
                   a 
                   1 
                 
               
               = 
               
                 
                   
                     a 
                     1 
                   
                   ⊕ 
                   
                     a 
                     7 
                   
                   ⊕ 
                   
                     a 
                     9 
                   
                   ⊕ 
                   
                     a 
                     11 
                   
                 
                 = 
                 
                   
                     1 
                     ⊕ 
                     1 
                     ⊕ 
                     0 
                     ⊕ 
                     1 
                   
                   = 
                   1 
                 
               
             
             ; 
           
         
       
       
         
           
             the 
              
             
                 
             
              
             next 
           
         
       
       
         
           
             
               
                 
                   
                     a 
                     2 
                   
                   = 
                     
                    
                   
                     
                       a 
                       1 
                     
                     ⊕ 
                     
                       a 
                       2 
                     
                     ⊕ 
                     
                       a 
                       5 
                     
                     ⊕ 
                     
                       a 
                       8 
                     
                     ⊕ 
                     
                       a 
                       10 
                     
                     ⊕ 
                     
                       a 
                       12 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       1 
                       ⊕ 
                       0 
                       ⊕ 
                       0 
                       ⊕ 
                       1 
                       ⊕ 
                       1 
                       ⊕ 
                       0 
                     
                     = 
                     
                         
                     
                      
                     
                         
                       
                         1 
                         ; 
                         … 
                          
                         
                             
                         
                         ; 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
               the 
                
               
                   
               
                
               next 
                
               
                   
               
                
               
                 a 
                 16 
               
             
             = 
             
               
                 
                   a 
                   3 
                 
                 ⊕ 
                 
                   a 
                   4 
                 
                 ⊕ 
                 
                   a 
                   13 
                 
                 ⊕ 
                 
                   a 
                   15 
                 
               
               = 
               
                 
                   1 
                   ⊕ 
                   1 
                   ⊕ 
                   0 
                   ⊕ 
                   1 
                 
                 = 
                 1. 
               
             
           
         
       
       
         
           
             
               Prove 
                
               
                   
               
                
               likewise 
             
             , 
             
               
 
             
              
             
               
                 for 
                  
                 
                     
                 
                  
                 
                   t 
                   2 
                 
               
               = 
               3 
             
             , 
             
               
 
             
              
             
               
                 
                   
                     
                       A 
                       
                         1 
                         × 
                         2 
                       
                     
                     = 
                       
                      
                     
                       [ 
                       
                         
                           a 
                           1 
                         
                         , 
                         
                           a 
                           2 
                         
                         , 
                         … 
                          
                         
                             
                         
                         , 
                         
                           a 
                           16 
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   
                     
                       = 
                         
                        
                       
                         [ 
                         
                           
                             
                               
                                 0 
                                 1 
                               
                             
                             
                               
                                 1 
                                 2 
                               
                             
                             
                               
                                 1 
                                 3 
                               
                             
                             
                               
                                 0 
                                 4 
                               
                             
                             
                               
                                 1 
                                 5 
                               
                             
                             
                               
                                 0 
                                 6 
                               
                             
                             
                               
                                 0 
                                 7 
                               
                             
                             
                               
                                 1 
                                 8 
                               
                             
                             
                               
                                 1 
                                 9 
                               
                             
                             
                               
                                 0 
                                 10 
                               
                             
                             
                               
                                 1 
                                 11 
                               
                             
                             
                               
                                 1 
                                 12 
                               
                             
                             
                               
                                 0 
                                 13 
                               
                             
                             
                               
                                 0 
                                 14 
                               
                             
                             
                               
                                 1 
                                 15 
                               
                             
                             
                               
                                 0 
                                 16 
                               
                             
                           
                         
                         ] 
                       
                     
                     , 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     A 
                     
                       1 
                       × 
                       3 
                     
                   
                   = 
                     
                    
                   
                     
                       
                         A 
                         
                           1 
                           × 
                           2 
                         
                       
                        
                       
                         F 
                         1 
                       
                     
                     ⊕ 
                     
                       BF 
                       1 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       [ 
                       
                         0 
                          
                         
                             
                         
                          
                         0 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         0 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         0 
                          
                         
                             
                         
                          
                         0 
                          
                         
                             
                         
                          
                         0 
                          
                         
                             
                         
                          
                         0 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         0 
                          
                         
                             
                         
                          
                         0 
                          
                         
                             
                         
                          
                         0 
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         1 
                          
                         
                             
                         
                          
                         0 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         0 
                          
                         
                             
                         
                          
                         0 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         0 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         0 
                          
                         
                             
                         
                          
                         0 
                          
                         
                             
                         
                          
                         0 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         1 
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                      
                     
                       1 
                        
                       
                           
                       
                        
                       0 
                        
                       
                           
                       
                        
                       0 
                        
                       
                           
                       
                        
                       0 
                        
                       
                           
                       
                        
                       0 
                        
                       
                           
                       
                        
                       1 
                        
                       
                           
                       
                        
                       0 
                        
                       
                           
                       
                        
                       0 
                        
                       
                           
                       
                        
                       0 
                        
                       
                           
                       
                        
                       1 
                        
                       
                           
                       
                        
                       1 
                        
                       
                           
                       
                        
                       0 
                        
                       
                           
                       
                        
                       1 
                        
                       
                           
                       
                        
                       0 
                        
                       
                           
                       
                        
                       1 
                        
                       
                           
                       
                        
                       1 
                     
                   
                   ; 
                 
               
             
           
         
       
       
         
           
             
               
                 the 
                  
                 
                     
                 
                  
                 result 
                  
                 
                     
                 
                  
                 is 
                  
                 
                     
                 
                  
                 the 
                  
                 
                     
                 
                  
                 same 
                  
                 
                     
                 
                  
                 as 
                  
                 
                     
                 
                  
                 
                   A 
                   3 
                 
                  
                 
                     
                 
                  
                 in 
                  
                 
                     
                 
                  
                 Embodiment 
                  
                 
                     
                 
                  
                 
                   II 
                   . 
                   
                     
 
                   
                    
                   for 
                 
                  
                 
                     
                 
                  
                 
                   t 
                   2 
                 
               
               = 
               4 
             
             , 
             
               
 
             
              
             
               
                 
                   
                     
                       A 
                       
                         1 
                         × 
                         3 
                       
                     
                     = 
                       
                      
                     
                       [ 
                       
                         
                           a 
                           1 
                         
                         , 
                         
                           a 
                           2 
                         
                         , 
                         … 
                          
                         
                             
                         
                         , 
                         
                           a 
                           16 
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   
                     
                       = 
                         
                        
                       
                         [ 
                         
                           
                             
                               
                                 1 
                                 1 
                               
                             
                             
                               
                                 0 
                                 2 
                               
                             
                             
                               
                                 0 
                                 3 
                               
                             
                             
                               
                                 0 
                                 4 
                               
                             
                             
                               
                                 0 
                                 5 
                               
                             
                             
                               
                                 1 
                                 6 
                               
                             
                             
                               
                                 0 
                                 7 
                               
                             
                             
                               
                                 0 
                                 8 
                               
                             
                             
                               
                                 0 
                                 9 
                               
                             
                             
                               
                                 1 
                                 10 
                               
                             
                             
                               
                                 1 
                                 11 
                               
                             
                             
                               
                                 0 
                                 12 
                               
                             
                             
                               
                                 1 
                                 13 
                               
                             
                             
                               
                                 0 
                                 14 
                               
                             
                             
                               
                                 1 
                                 15 
                               
                             
                             
                               
                                 1 
                                 16 
                               
                             
                           
                         
                         ] 
                       
                     
                     , 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     A 
                     
                       1 
                       × 
                       4 
                     
                   
                   = 
                     
                    
                   
                     
                       
                         A 
                         
                           1 
                           × 
                           3 
                         
                       
                        
                       
                         F 
                         1 
                       
                     
                     ⊕ 
                     
                       BF 
                       1 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       [ 
                       
                         0 
                          
                         
                             
                         
                          
                         0 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         0 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         0 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         0 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         0 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         0 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         0 
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         1 
                          
                         
                             
                         
                          
                         0 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         0 
                          
                         
                             
                         
                          
                         0 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         0 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         0 
                          
                         
                             
                         
                          
                         0 
                          
                         
                             
                         
                          
                         0 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         1 
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                      
                     
                       1 
                        
                       
                           
                       
                        
                       0 
                        
                       
                           
                       
                        
                       0 
                        
                       
                           
                       
                        
                       1 
                        
                       
                           
                       
                        
                       1 
                        
                       
                           
                       
                        
                       1 
                        
                       
                           
                       
                        
                       1 
                        
                       
                           
                       
                        
                       0 
                        
                       
                           
                       
                        
                       0 
                        
                       
                           
                       
                        
                       0 
                        
                       
                           
                       
                        
                       1 
                        
                       
                           
                       
                        
                       1 
                        
                       
                           
                       
                        
                       0 
                        
                       
                           
                       
                        
                       1 
                        
                       
                           
                       
                        
                       0 
                        
                       
                           
                       
                        
                       1 
                     
                   
                   ; 
                 
               
             
           
         
       
       
         
           
             the 
              
             
                 
             
              
             result 
              
             
                 
             
              
             is 
              
             
                 
             
              
             the 
              
             
                 
             
              
             same 
              
             
                 
             
              
             as 
              
             
                 
             
              
             
               A 
               4 
             
              
             
                 
             
              
             in 
              
             
                 
             
              
             Embodiment 
              
             
                 
             
              
             
               II 
               . 
             
           
         
       
     
         [0000]    
       
         
               
             
           
               
                 TABLE 1 
               
               
                   
               
             
             
               
                       
           FF        (   13   )       =         [           1000000000000000             ⋮             0000000000010000               0000000000001000               0000000000000100               0000000000000010               0000000000000001           ]     ⊕     [           000000000000   0   000             ⋮             000000000010   0   000               000000000001   0   100               000000000000   0   010               000000000000   0   001               000000000000   0   000           ]       =     [           1000000000000000             ⋮             0000000000110000               0000000000011100               0000000000000110               0000000000000011               0000000000000001           ]         ;       
 
               
               
                   
               
               
                 
                   
                     
                       
                         
                           
                             FF 
                              
                             
                               ( 
                               
                                 13 
                                 : 
                                 14 
                               
                               ) 
                             
                           
                           = 
                           
                             
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         1000000000000000 
                                       
                                     
                                   
                                   
                                     
                                       ⋮ 
                                     
                                   
                                   
                                     
                                       
                                         0000000000110000 
                                       
                                     
                                   
                                   
                                     
                                       
                                         0000000000011100 
                                       
                                     
                                   
                                   
                                     
                                       
                                         0000000000000110 
                                       
                                     
                                   
                                   
                                     
                                       
                                         0000000000000011 
                                       
                                     
                                   
                                   
                                     
                                       
                                         0000000000000001 
                                       
                                     
                                   
                                 
                                 ] 
                               
                               ⊕ 
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         000000000000 
                                         0 
                                         0 
                                         00 
                                       
                                     
                                   
                                   
                                     
                                       ⋮ 
                                     
                                   
                                   
                                     
                                       
                                         000000000110 
                                         0 
                                         0 
                                         00 
                                       
                                     
                                   
                                   
                                     
                                       
                                         000000000011 
                                         1 
                                         0 
                                         10 
                                       
                                     
                                   
                                   
                                     
                                       
                                         000000000000 
                                         1 
                                         0 
                                         11 
                                       
                                     
                                   
                                   
                                     
                                       
                                         000000000000 
                                         0 
                                         0 
                                         01 
                                       
                                     
                                   
                                   
                                     
                                       
                                         000000000000 
                                         0 
                                         0 
                                         00 
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                             = 
                             
                               [ 
                               
                                 
                                   
                                     
                                       1000000000000000 
                                     
                                   
                                 
                                 
                                   
                                     ⋮ 
                                   
                                 
                                 
                                   
                                     
                                       0000000001010000 
                                     
                                   
                                 
                                 
                                   
                                     
                                       0000000000100110 
                                     
                                   
                                 
                                 
                                   
                                     
                                       0000000000001101 
                                     
                                   
                                 
                                 
                                   
                                     
                                       0000000000000010 
                                     
                                   
                                 
                                 
                                   
                                     
                                       0000000000000001 
                                     
                                   
                                 
                               
                               ] 
                             
                           
                         
                         ; 
                       
                     
                   
                 
               
               
                   
               
               
                 
                   
                     
                       
                         
                           FF 
                            
                           
                             ( 
                             
                               13 
                               : 
                               13 
                             
                             ) 
                           
                         
                         = 
                         
                           
                             
                               [ 
                               
                                 
                                   
                                     
                                       1011011011011000 
                                     
                                   
                                 
                                 
                                   
                                     ⋮ 
                                   
                                 
                                 
                                   
                                     
                                       0011001100110000 
                                     
                                   
                                 
                                 
                                   
                                     
                                       0000101101110110 
                                     
                                   
                                 
                                 
                                   
                                     
                                       0000000010101100 
                                     
                                   
                                 
                                 
                                   
                                     
                                       0000000000001001 
                                     
                                   
                                 
                                 
                                   
                                     
                                       0000000000000011 
                                     
                                   
                                 
                               
                               ] 
                             
                             ⊕ 
                             
                               [ 
                               
                                 
                                   
                                     
                                       011011011011 
                                       0 
                                       100 
                                     
                                   
                                 
                                 
                                   
                                     ⋮ 
                                   
                                 
                                 
                                   
                                     
                                       011001100110 
                                       0 
                                       000 
                                     
                                   
                                 
                                 
                                   
                                     
                                       000101101110 
                                       0 
                                       011 
                                     
                                   
                                 
                                 
                                   
                                     
                                       000000010101 
                                       0 
                                       110 
                                     
                                   
                                 
                                 
                                   
                                     
                                       000000000001 
                                       0 
                                       100 
                                     
                                   
                                 
                                 
                                   
                                     
                                       000000000000 
                                       0 
                                       0 
                                       01 
                                     
                                   
                                 
                               
                               ] 
                             
                           
                           = 
                           
                             
                               [ 
                               
                                 
                                   
                                     
                                       1101101101101100 
                                     
                                   
                                 
                                 
                                   
                                     ⋮ 
                                   
                                 
                                 
                                   
                                     
                                       0101010101010000 
                                     
                                   
                                 
                                 
                                   
                                     
                                       0001110110010101 
                                     
                                   
                                 
                                 
                                   
                                     
                                       0000000111111010 
                                     
                                   
                                 
                                 
                                   
                                     
                                       0000000000011101 
                                     
                                   
                                 
                                 
                                   
                                     
                                       0000000000000010 
                                     
                                   
                                 
                               
                               ] 
                             
                             . 
                           
                         
                       
                     
                   
                 
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
           
               
                 TABLE 2 
               
               
                   
               
             
             
               
                 
                   
                     
                       
                         
                           F 
                           1 
                         
                         = 
                         
                           
                             FF 
                              
                             
                               ( 
                               
                                 13 
                                  
                                 
                                   : 
                                 
                                  
                                 13 
                               
                               ) 
                             
                           
                           = 
                           
                             [ 
                             
                               
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                 
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
               
               
             
           
               
                   
                 TABLE 3 
               
               
                   
                   
               
               
                   
                 at 
                   
               
               
                   
                 time t 2   
                 at time t 2  − 1 
               
               
                   
                   
               
             
             
               
                   
                 a 1   
                 a 1  ⊕ a 7  ⊕ a 9  ⊕ a 11   
               
               
                   
                 a 2   
                 a 1  ⊕ a 2  ⊕ a 5  ⊕ a 8  ⊕ a 10  ⊕ a 12   
               
               
                   
                 a 3   
                 a 5  ⊕ a 7  ⊕ a 9  ⊕ a 11   
               
               
                   
                 a 4   
                 a 1  ⊕ a 3  ⊕ a 6  ⊕ a 7  ⊕ a 8  ⊕ a 10  ⊕ a 12  ⊕ a 13   
               
               
                   
                 a 5   
                 a 1  ⊕ a 3  ⊕ a 5  ⊕ a 9  ⊕ a 11  ⊕ a 13   
               
               
                   
                 a 6   
                 a 2  ⊕ a 4  ⊕ a 5  ⊕ a 6  ⊕ a 10  ⊕ a 12  ⊕ a 13   
               
               
                   
                 a 7   
                 a 1  ⊕ a 3  ⊕ a 9  ⊕ a 11   
               
               
                   
                 a 8   
                 a 1  ⊕ a 2  ⊕ a 4  ⊕ a 7  ⊕ a 9  ⊕ a 10  ⊕ a 12  ⊕ a 13  ⊕ a 14   
               
               
                   
                 a 9   
                 a 3  ⊕ a 7  ⊕ a 11  ⊕ a 13  ⊕ a 14   
               
               
                   
                 a 10   
                 a 1  ⊕ a 4  ⊕ a 5  ⊕ a 8  ⊕ a 9  ⊕ a 12  ⊕ a 14   
               
               
                   
                 a 11   
                 a 1  ⊕ a 3  ⊕ a 5  ⊕ a 7  ⊕ a 9  ⊕ a 11  ⊕ a 14   
               
               
                   
                 a 12   
                 a 2  ⊕ a 3  ⊕ a 4  ⊕ a 5  ⊕ a 6  ⊕ a 7  ⊕ a 8  ⊕ 
               
               
                   
                   
                 a 9  ⊕ a 10  ⊕ a 11  ⊕ a 12  ⊕ a 13  ⊕ a 14  ⊕ a 15   
               
               
                   
                 a 13   
                 a 1  ⊕ a 14  ⊕ a 15   
               
               
                   
                 a 14   
                 a 1  ⊕ a 2  ⊕ a 13  ⊕ a 15   
               
               
                   
                 a 15   
                 a 2  ⊕ a 3  ⊕ a 14  ⊕ a 16   
               
               
                   
                 a 16   
                 a 3  ⊕ a 4  ⊕ a 13  ⊕ a 15   
               
               
                   
                   
               
             
          
         
       
     
         [0067]    Secondly, with Embodiment II, to comply with the diffusion mechanism F 2 =F(13 2 :13), the sequential value A 2×t     2   =A 2×(t     2     −1) F 2 ⊕BF 2 , wherein BF 2 =[BF(13:13)]F(13:13), is derived as A 2 , in Embodiment I, equals to [0110100110110010], and further, F 2 =FF(13 2 :13), wherein a value FF(13:13), see Table 2, performs the diffusion functions F(13:13), derived as Table 1; the diffusion-table F 2  is shown in Table 4. 
         [0000]    
       
         
               
               
             
           
               
                 TABLE 4 
               
               
                   
               
               
                 at 
                   
               
               
                 time t 2   
                 at time t 2  − 1 
               
               
                   
               
             
             
               
                 a 1   
                 a 1  ⊕ a 3  ⊕ a 5  ⊕ a 7  ⊕ a 9  ⊕ a 13   
               
               
                 a 2   
                 a 1  ⊕ a 2  ⊕ a 4  ⊕ a 6  ⊕ a 7  ⊕ a 8  ⊕ 
               
               
                   
                 a 9  ⊕ a 10  ⊕ a 11  ⊕ a 13  ⊕ a 14  ⊕ a 15   
               
               
                 a 3   
                 a 1  ⊕ a 9   
               
               
                 a 4   
                 a 1  ⊕ a 2  ⊕ a 10  ⊕ a 13   
               
               
                 a 5   
                 a 3  ⊕ a 5  ⊕ a 11  ⊕ a 14  ⊕ a 15   
               
               
                 a 6   
                 a 1  ⊕ a 2  ⊕ a 3  ⊕ a 4  ⊕ a 5  ⊕ a 6  ⊕ a 9  ⊕ a 12  ⊕ a 14   
               
               
                 a 7   
                 a 9  ⊕ a 13   
               
               
                 a 8   
                 a 1  ⊕ a 2  ⊕ a 5  ⊕ a 10  ⊕ a 15   
               
               
                 a 9   
                 a 2  ⊕ a 9  ⊕ a 11  ⊕ a 13   
               
               
                 a 10   
                 a 1  ⊕ a 2  ⊕ a 7  ⊕ a 10  ⊕ a 12  ⊕ a 14   
               
               
                 a 11   
                 a 1  ⊕ a 2  ⊕ a 5  ⊕ a 9  ⊕ a 13  ⊕ a 15   
               
               
                 a 12   
                 a 1  ⊕ a 3  ⊕ a 5  ⊕ a 6  ⊕ a 9  ⊕ a 10  ⊕ a 13  ⊕ a 14  ⊕ a 15  ⊕ a 16   
               
               
                 a 13   
                 a 3  ⊕ a 7  ⊕ a 9  ⊕ a 11  ⊕ a 13  ⊕ a 14  ⊕ a 15  ⊕ a 16   
               
               
                 a 14   
                 a 1 ⊕ a 3  ⊕ a 5  ⊕ a 7  ⊕ a 8  ⊕ a 9  ⊕ a 10  ⊕ a 11  ⊕ a 12  ⊕ a 15  ⊕ a 16   
               
               
                 a 15   
                 a 3  ⊕ a 4  ⊕ a 7  ⊕ a 8  ⊕ a 9  ⊕ a 10  ⊕ a 11  ⊕ a 12   
               
               
                 a 16   
                 a 2  ⊕ a 5  ⊕ a 6  ⊕ a 8  ⊕ a 9  ⊕ a 10  ⊕ a 11  ⊕ a 12  ⊕ a 13  ⊕ a 15  ⊕ a 16   
               
               
                   
               
             
          
         
       
     
         [0068]    Therefore, according to Table 4, new values in sequence as below: 
         [0000]    
       
         
           
             
               
                 for 
                  
                 
                     
                 
                  
                 
                   t 
                   2 
                 
               
               = 
               1 
             
             , 
             
               
 
             
              
             
               
                 
                   
                     
                       A 
                       
                         2 
                         × 
                         1 
                       
                     
                     = 
                       
                      
                     
                       
                         
                           A 
                           0 
                         
                          
                         
                           F 
                           2 
                         
                       
                       ⊕ 
                       
                         BF 
                         2 
                       
                     
                   
                 
               
               
                 
                   
                     
                       = 
                         
                        
                       
                         
                           [ 
                           
                             0 
                              
                             
                                 
                             
                              
                             0 
                              
                             
                                 
                             
                              
                             0 
                              
                             
                                 
                             
                              
                             0 
                              
                             
                                 
                             
                              
                             0 
                              
                             
                                 
                             
                              
                             0 
                              
                             
                                 
                             
                              
                             0 
                              
                             
                                 
                             
                              
                             0 
                              
                             
                                 
                             
                              
                             0 
                              
                             
                                 
                             
                              
                             0 
                              
                             
                                 
                             
                              
                             0 
                              
                             
                                 
                             
                              
                             0 
                              
                             
                                 
                             
                              
                             0 
                              
                             
                                 
                             
                              
                             0 
                              
                             
                                 
                             
                              
                             0 
                              
                             
                                 
                             
                              
                             0 
                           
                           ] 
                         
                         ⊕ 
                         
                           [ 
                           
                             0 
                              
                             
                                 
                             
                              
                             1 
                              
                             
                                 
                             
                              
                             1 
                              
                             
                                 
                             
                              
                             0 
                              
                             
                                 
                             
                              
                             1 
                              
                             
                                 
                             
                              
                             0 
                              
                             
                                 
                             
                              
                             0 
                              
                             
                                 
                             
                              
                             1 
                              
                             
                                 
                             
                              
                             1 
                              
                             
                                 
                             
                              
                             0 
                              
                             
                                 
                             
                              
                             1 
                              
                             
                                 
                             
                              
                             1 
                              
                             
                                 
                             
                              
                             0 
                              
                             
                                 
                             
                              
                             0 
                              
                             
                                 
                             
                              
                             1 
                              
                             
                                 
                             
                              
                             0 
                           
                           ] 
                         
                       
                     
                     ; 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 the 
                  
                 
                     
                 
                  
                 results 
                  
                 
                     
                 
                  
                 is 
                  
                 
                     
                 
                  
                 the 
                  
                 
                     
                 
                  
                 same 
                  
                 
                     
                 
                  
                 as 
                  
                 
                     
                 
                  
                 
                   A 
                   1 
                 
                  
                 
                     
                 
                  
                 in 
                  
                 
                     
                 
                  
                 Embodiment 
                  
                 
                     
                 
                  
                 
                   I 
                   . 
                   
                     
 
                   
                    
                   for 
                 
                  
                 
                     
                 
                  
                 
                   t 
                   2 
                 
               
               = 
               2 
             
             , 
             
               
 
             
              
             
               
                 
                   
                     
                       A 
                       
                         2 
                         × 
                         2 
                       
                     
                     = 
                       
                      
                     
                       
                         
                           A 
                           
                             2 
                             × 
                             1 
                           
                         
                          
                         
                           F 
                           2 
                         
                       
                       ⊕ 
                       
                         BF 
                         2 
                       
                     
                   
                 
               
               
                 
                   
                     = 
                       
                      
                     
                       
                         [ 
                         
                           1 
                            
                           
                               
                           
                            
                           1 
                            
                           
                               
                           
                            
                           1 
                            
                           
                               
                           
                            
                           1 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           1 
                            
                           
                               
                           
                            
                           1 
                            
                           
                               
                           
                            
                           1 
                            
                           
                               
                           
                            
                           1 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           1 
                            
                           
                               
                           
                            
                           1 
                            
                           
                               
                           
                            
                           1 
                         
                          
                         
                             
                         
                         ] 
                       
                       ⊕ 
                       
                         [ 
                         
                           0 
                            
                           
                               
                           
                            
                           1 
                            
                           
                               
                           
                            
                           1 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           1 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           1 
                            
                           
                               
                           
                            
                           1 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           1 
                            
                           
                               
                           
                            
                           1 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           1 
                            
                           
                               
                           
                            
                           0 
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   
                     
                       = 
                         
                        
                       
                         1 
                          
                         
                             
                         
                          
                         0 
                          
                         
                             
                         
                          
                         0 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         0 
                          
                         
                             
                         
                          
                         0 
                          
                         
                             
                         
                          
                         0 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         0 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         0 
                          
                         
                             
                         
                          
                         1 
                       
                     
                     ; 
                   
                 
               
             
           
         
       
       
         
           
             the 
              
             
                 
             
              
             results 
              
             
                 
             
              
             is 
              
             
                 
             
              
             the 
              
             
                 
             
              
             same 
              
             
                 
             
              
             as 
              
             
                 
             
              
             
               A 
               4 
             
              
             
                 
             
              
             in 
              
             
                 
             
              
             Embodiment 
              
             
                 
             
              
             
               II 
               . 
               
                 
 
               
                
               Derivation 
             
              
             
                 
             
              
             of 
              
             
                 
             
              
             
               A 
               
                 2 
                 × 
                 1 
               
             
              
             
               F 
               2 
             
              
             
               : 
             
           
         
       
       
         
           
             Now 
              
             
                 
             
              
             that 
           
         
       
       
         
           
             
               
                 
                   
                     A 
                     
                       2 
                       × 
                       1 
                     
                   
                   = 
                     
                    
                   
                     [ 
                     
                       
                         a 
                         1 
                       
                       , 
                       
                         a 
                         2 
                       
                       , 
                       … 
                        
                       
                           
                       
                       , 
                       
                         a 
                         16 
                       
                     
                     ] 
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                      
                     
                       [ 
                       
                         
                           
                             
                               0 
                               1 
                             
                           
                           
                             
                               1 
                               2 
                             
                           
                           
                             
                               1 
                               3 
                             
                           
                           
                             
                               0 
                               4 
                             
                           
                           
                             
                               1 
                               5 
                             
                           
                           
                             
                               0 
                               6 
                             
                           
                           
                             
                               0 
                               7 
                             
                           
                           
                             
                               1 
                               8 
                             
                           
                           
                             
                               1 
                               9 
                             
                           
                           
                             
                               0 
                               10 
                             
                           
                           
                             
                               1 
                               11 
                             
                           
                           
                             
                               1 
                               12 
                             
                           
                           
                             
                               0 
                               13 
                             
                           
                           
                             
                               0 
                               14 
                             
                           
                           
                             
                               1 
                               15 
                             
                           
                           
                             
                               0 
                               16 
                             
                           
                         
                       
                       ] 
                     
                   
                   ; 
                 
               
             
           
         
       
       
         
           
             the 
              
             
                 
             
              
             next 
           
         
       
       
         
           
             
               
                 a 
                 1 
               
               = 
               
                 
                   
                     a 
                     1 
                   
                   ⊕ 
                   
                     a 
                     3 
                   
                   ⊕ 
                   
                     a 
                     5 
                   
                   ⊕ 
                   
                     a 
                     7 
                   
                   ⊕ 
                   
                     a 
                     9 
                   
                   ⊕ 
                   
                     a 
                     13 
                   
                 
                 = 
                 
                   
                     0 
                     ⊕ 
                     1 
                     ⊕ 
                     1 
                     ⊕ 
                     0 
                     ⊕ 
                     1 
                     ⊕ 
                     0 
                   
                   = 
                   1 
                 
               
             
             ; 
           
         
       
       
         
           
             the 
              
             
                 
             
              
             next 
           
         
       
       
         
           
             
               
                 
                   
                     a 
                     2 
                   
                   = 
                     
                    
                   
                     
                       a 
                       1 
                     
                     ⊕ 
                     
                       a 
                       2 
                     
                     ⊕ 
                     
                       a 
                       4 
                     
                     ⊕ 
                     
                       a 
                       6 
                     
                     ⊕ 
                     
                       a 
                       7 
                     
                     ⊕ 
                     
                       a 
                       8 
                     
                     ⊕ 
                     
                       a 
                       9 
                     
                     ⊕ 
                     
                       a 
                       10 
                     
                     ⊕ 
                   
                 
               
             
             
               
                 
                     
                    
                   
                     
                       a 
                       11 
                     
                     ⊕ 
                     
                       a 
                       13 
                     
                     ⊕ 
                     
                       a 
                       14 
                     
                     ⊕ 
                     
                       a 
                       15 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     0 
                     ⊕ 
                     1 
                     ⊕ 
                     0 
                     ⊕ 
                     0 
                     ⊕ 
                     0 
                     ⊕ 
                     1 
                     ⊕ 
                     1 
                     ⊕ 
                     0 
                     ⊕ 
                     1 
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         [0069]    Thirdly, in Embodiment III, now that A is a 4×4 matrix, thus, the F will be a 4×4×16 zero matrix, further, 1&#39;s at the i th  position on the i th  layer; the diffusion function F(p)=F⊕Fx i ⊕Fy j , may refer to the whole derivation of three-dimensional FF(1:4). According to the diffusion mechanism F 1 =F(8:8), the sequential value of diffusion-area A 1×t     2   =A 1×(t     2     −1) F 1 ⊕BF 1 , wherein BF 1 =BF(8:8)=[1010001000100100], derived as A 1 , Embodiment III, and F 1 =FF(8:8) is shown in Table 5. 
         [0000]    
       
         
               
               
             
           
               
                 TABLE 5 
               
               
                   
               
               
                 at 
                   
               
               
                 time t 2   
                 at time t 2  − 1 
               
               
                   
               
             
             
               
                 a 1   
                 a 1  ⊕ a 4  ⊕ a 5  ⊕ a 6  ⊕ a 9  ⊕ a 11  ⊕ a 12  ⊕ a 16   
               
               
                 a 2   
                 a 4  ⊕ a 5  ⊕ a 6  ⊕ a 11  ⊕ a 12  ⊕ a 13  ⊕ a 15   
               
               
                 a 3   
                 a 1  ⊕ a 2  ⊕ a 7  ⊕ a 8  ⊕ a 10  ⊕ a 12  ⊕ a 13  ⊕ a 14   
               
               
                 a 4   
                 a 1  ⊕ a 8  ⊕ a 9  ⊕ a 11  ⊕ a 12  ⊕ a 15   
               
               
                 a 5   
                 a 1  ⊕ a 5  ⊕ a 6  ⊕ a 9  ⊕ a 10  ⊕ a 12  ⊕ a 13  ⊕ a 14  ⊕ a 15  ⊕ a 16   
               
               
                 a 6   
                 a 2  ⊕ a 3  ⊕ a 4  ⊕ a 5  ⊕ a 7  ⊕ a 8  ⊕ a 9  ⊕ a 11  ⊕ a 14  ⊕ a 16   
               
               
                 a 7   
                 a 3  ⊕ a 8  ⊕ a 9  ⊕ a 12  ⊕ a 15  ⊕ a 16   
               
               
                 a 8   
                 a 1  ⊕ a 2  ⊕ a 3  ⊕ a 5  ⊕ a 7  ⊕ a 12  ⊕ a 13  ⊕ a 15  ⊕ a 16   
               
               
                 a 9   
                 a 1  ⊕ a 2  ⊕ a 3  ⊕ a 4  ⊕ a 5  ⊕ a 8  ⊕ a 9  ⊕ a 13  ⊕ a 14  ⊕ a 15  ⊕ a 16   
               
               
                 a 10   
                 a 3  ⊕ a 5  ⊕ a 8  ⊕ a 10  ⊕ a 12  ⊕ a 13  ⊕ a 14   
               
               
                 a 11   
                 a 2  ⊕ a 3  ⊕ a 4  ⊕ a 7  ⊕ a 9  ⊕ a 11  ⊕ a 12  ⊕ a 15   
               
               
                 a 12   
                 a 2  ⊕ a 3  ⊕ a 5  ⊕ a 6  ⊕ a 8  ⊕ a 10  ⊕ a 11  ⊕ a 13  ⊕ a 14   
               
               
                 a 13   
                 a 2  ⊕ a 3  ⊕ a 4  ⊕ a 5  ⊕ a 8  ⊕ a 10  ⊕ a 12  ⊕ a 13  ⊕ a 14  ⊕ a 15  ⊕ a 16   
               
               
                 a 14   
                 a 4  ⊕ a 6  ⊕ a 7  ⊕ a 8  ⊕ a 9  ⊕ a 11  ⊕ a 13  ⊕ a 14   
               
               
                 a 15   
                 a 1  ⊕ a 4  ⊕ a 8  ⊕ a 9  ⊕ a 10  ⊕ a 13   
               
               
                 a 16   
                 a 1  ⊕ a 2  ⊕ a 4  ⊕ a 5  ⊕ a 6  ⊕ a 10  ⊕ a 12  ⊕ a 15  ⊕ a 16   
               
               
                   
               
             
          
         
       
     
       Hardware Architecture in Detail: 
       [0070]      FIG. 1 ,  300 , obtaining a new value A t     1     ×t     2   =A t     1     ×(t     2     −1) F t     1    in a hardware architecture, would transform A t     1     ×t     2   =A t     1     ×(t     2     −1) F t     1   ⊕BF t     1    in a parallel process into a first circuit design shown as  FIG. 3A , comprising of: a first register R 1  provided for storing an input value A t     1     ×(t     2     −1)    310 ; a diffuser DF provided for receiving A t     1     ×(t     2     −1)  from R 1 , further, recombining A t     1     ×(t     2     −1)  with a function-table F t     1    and transmitting an output value A t   1   ×(t     2     −1) F t     1      320 ; a first memory M 1  provided for writing a medium-constant BF t     1      330 ; an XOR Unit provided for receiving A t     1     ×(t     2     −1) F t     1    from DF as a first operand, BF t     1    from M 1  as a second operand, then, performing a bitwise XOR operation and transmitting an output value A t     1     ×t     2      340 ; a second register R 2  provided for buffering A t     1     ×t     2    from said XOR Unit, and feeding back said A t     1     ×t     2    to R 1   350 . 
         [0071]    In addition, for the initialization of A,  FIG. 2 , a password bit selects a first value for A t     1     ×t     2   =A t     1     ×(t     2     −1) F t     1   ⊕BF t     1    as by the above architecture, and a second value for A t     1     ×t     2   =A t     1     ×(t     2     −1) F t     1    shown as  FIG. 3B  as a second design of  FIG. 1 ,  300 , comprising of: a first register R 1  provided for storing an input value A t     1     ×(t     2     −1)    310 ; a diffuser DF provided for receiving A t     1     ×(t     2     −1)  from R 1 , further, recombining A t     1     ×(t     2     −1)  with a function-table F t     1    and transmitting an output value A t     1     ×t     2      320 ; a second register R 2  provided for buffering A t     1     ×t     2    from DF, and feeding back said A t     1     ×t     2    to R 1   350 . 
         [0072]    The whole scheme is shown as  FIG. 3C , a main component, provided for performing a diffusion mechanism  300 , as  FIG. 1 ,  300 , i.e.  FIGS. 3A and 3B ; in addition, a switch SW, provided for receiving an input password for initialization  200 , as  FIG. 1 ,  200 ; further comprising of: a multiplexer MUX provided for receiving A t     1     ×t     2    from R 2 , further, selecting a bit of A t     1     ×t     2    and transmitting the assigned bit  360 ; a third register R 3  provided for storing a stream plaintext/ciphertext in sequence and transmitting a bit each time  400 , as  FIG. 1 ,  400 ; an XOR gate provided for receiving said assigned bit from MUX as a first operand, said bit from R 3  as a second operand, then, performing an XOR operation and transmitting an output bit  500 , as  FIG. 1 ,  500 . 
       Preferred Embodiment V 
     Diffusion Mechanisms in Hardware Architecture 
       [0073]    According to Embodiment I in a parallel process, a medium-constant BF 1 =BF(13:13)=[1011001101100011] and a diffusion-table F 1 =FF(13:13), see Table 3, are generated in advance; therefore, to gain more acceleration, it would write BF 1  to M 1 , and transform the diffusion-table F 1  into a diffuser DF combined with a number of XOR gates shown as  FIG. 4 ; further, in Embodiment II, write BF 2 =BF(13 2 :13)=[0110100110110010] to M 1 , and transform F 2 =FF(13 2 :13), see Table 4, into a diffuser DF; in Embodiment III, BF 1 =BF(8:8)=[1010001000100100] to M 1 , and F 1 =FF(8:8), see Table 5, into a diffuser DF. 
         [0074]    The above preferred embodiments are not only able to be applied separately, but also, can be taken as a whole. For example, as a number of parallel connections as  FIG. 5 , every SW, provided for receiving a password in sequence; every IN, provided for receiving an assigned part of a plaintext/ciphertext in sequence; and every OUT, provided for transmitting the assigned part of the ciphertext/plaintext in sequence. In addition, as a number of serial connections as  FIG. 6 , comprises of: the first SW, provided for receiving the password in sequence, further, the former OUT, connecting to the latter SW; every IN, provided for receiving an assigned part of a plaintext/ciphertext in sequence, and every OUT, provided for transmitting the assigned part of the ciphertext/plaintext in sequence; or for more security, only encrypting or decrypting a plaintext/ciphertext stream in Embodiment III. 
         [0075]    In summation of the above description, the present invention herein complies with the constitutional, statutory, regulatory and treaty, patent application requirements and is herewith submitted for patent application. However, the description and its accompanied drawings are used for describing preferred embodiments of the present invention, and it is to be understood that the invention is not limited thereto. To the contrary, it is intended to cover various modifications and similar arrangements and procedures, and the scope of the appended claims therefore should be accorded the broadest interpretation so as to encompass all such modifications and similar arrangements and procedures.