Abstract:
The invention provides a diffusion function working on a multidimensional diffusion-area (plaintext/ciphertext), in which a multidimensional medium is meanwhile overlapped to the diffusion-area; accordingly, repeating the diffusion function for at least one time thus brings about the multilayer effect. FIG.  1  shows an embodiment of the present invention in flow chart diagram form, comprising of: inputting a plaintext in encryption or a ciphertext in decryption  100 ; inputting a series of password data forward in encryption or backward in decryption  200 ; further, by the password data, converting the dimensions of the plaintext  300 , and implementing with a diffusion function, repeated T E  times in encryption, T D  times in decryption  400 ; outputting the ciphertext in encryption or the plaintext in decryption  600  if completing all password data  500.

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The invention relates to a cryptographic method. More particularly, the invention relates to a diffusion function working on a multidimensional diffusion-area (plaintext/ciphertext), and further, repeating the diffusion function for at least one time to create a multilayer effect in order to perform the encryption and the decryption. 
     2. Description of the Related Art 
     The Applicant&#39;s following patent applications are related to the invention and are incorporated herein by reference: “Diffused Data Encryption/Decryption Processing Method”, application Ser. No. 12/365,160, filed Feb. 3, 2009 (CIP of application Ser. No. 10/963,014, filed Oct. 12, 2004); “Multipoint Synchronous Diffused Encryption/Decryption Method”, application Ser. No. 11/171,549, filed Jun. 30, 2005. 
     The prior art described that the coding of a 2D diffusion-area, see application Ser. No. 12/365,160, page 7, teaches the math of A(i, j)=A⊕Ac i ⊕Ar j ⊕b(i, j); further expressed the status of diffusion from inward to outward or vice versa in reverse, and implemented to multidimensional matrix A(i 1 , i 2 , . . . i n ), see application Ser. No. 11/171,549, page 4, 7. 
     The present invention emphasizes the multilayer effect of multidimensional diffusion. The diffusion function herein is notated specially by AF(p 1 , p 2 , . . . p n ) to differentiate from the traditional symbolization of matrix position. 
     SUMMARY OF THE INVENTION 
     The invention provides a diffusion function working on a multidimensional diffusion-area (plaintext/ciphertext), in which a multidimensional medium is meanwhile overlapped to the diffusion-area; accordingly, repeating the diffusion function for at least one time thus brings about the multilayer effect. In addition, the numbers of repetition, a so called diffusion-cycle, is then able to divide into two parts: one for encrypting, the other for decrypting; consequently, the original status of the diffusion-area is recovered through the diffusion-cycle. The steps are shown as follows:
         (a) Selecting a diffusion function, a multidimensional medium;   (b) Inputting a multidimensional diffusion-area (plaintext/ciphertext), for which the dimensions are the same as the medium&#39;s, and generating a diffusion-cycle;   (c) Repeating the diffusion function working on a plaintext for a first part of the diffusion-cycle to generate a ciphertext;   (d) Repeating the diffusion function working on the ciphertext for a second part of the diffusion-cycle to recover the plaintext.       

    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a summary flow chart diagram showing the main steps taken while encrypting/decrypting by repeating diffusion function in accordance with the present invention; 
         FIG. 2A  is a summary flow chart diagram of  FIG. 1 ,  410  showing the steps taken while originating the function of point-diffusion from a diffusion-center in accordance with the present invention; 
         FIG. 2B  is a two-dimension visualized diagram of  FIG. 2A  showing the point-diffusion by way of a medium anchoring to a diffusion-center in accordance with the present invention; 
         FIG. 3A  is a summarized flow chart diagram of  FIG. 1 ,  420  showing the steps taken while originating the function of block-diffusion from a block diffusion-center in accordance with the present invention; 
         FIG. 3B  is a two-dimension visualized diagram of  FIG. 3A  showing the block-diffusion by way of a medium anchoring to a diffusion-center, a block anchoring to the diffusion-center to form a block diffusion-center in accordance with the present invention. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
       FIG. 1  shows an embodiment of the present invention in flow chart diagram form. This system comprises of: inputting a plaintext in encryption or a ciphertext in decryption  100 ; inputting a series of password data forward in encryption or backward in decryption  200 ; further, by the password data, converting the dimensions of the plaintext  300 , and implementing with a function of diffusion, repeated T E  times in encryption, T D  times in decryption  400 ; outputting the ciphertext in encryption or the plaintext in decryption  600  if completing all password data  500 . 
     Function of Point-Diffusion: 
       FIG. 2A  shows an embodiment of the point-diffusion function,  FIG. 1 ,  410 , in flow chart diagram. The function comprises of: reading a diffusion-area (plaintext/ciphertext), a diffusion-center, and a medium with an anchor-point  201 ; anchoring the medium to the diffusion-center with the anchor-point  411 ; implementing the point-diffusion AF(p 1 , p 2 , . . . p n )  412 , which is further detailed in Notation of Point-Diffusion. In addition, also see  FIG. 2B  with 2D visualization for a more clear view, the diffusion effect colored from white to black generates the column segments a-g, a-b for later diffusion calculation. 
     Notation of Point-Diffusion: 
     
         
         A: a diffusion-area, wherein A expresses a d 1 ×d 2 × . . . ×d n  binary matrix, wherein A includes a diffusion-center {dot over (P)} expressed (p 1 , p 2 , . . . p n ) coordinate position. 
         S: a n-dimension medium, expresses a s 1 ×s 2 × . . . ×s n  binary matrix, wherein S includes an anchor-point {dot over (S)} expressed (s 1 , s 2 , . . . , s n ) coordinate position. 
         AF(p 1 , p 2 , . . . p n ): the diffusion-area A performs the function of point-diffusion at position {dot over (P)}, wherein S overlaps A by {dot over (S)} anchoring to {dot over (P)}; further comprising:
       AF(p 1 , p 2 , . . . , p n )=A⊕Ad 1p ⊕Ad 2p ⊕ . . . ⊕Ad np ⊕S;   Ad ip =[A d     i   (2), . . . , A d     i   (p i ), A d     i   (0), A d     i   (p i ), . . . , A d     i   (d i −1)];   Ad ip  expresses a series of n−1 dimensional binary matrix A d     i    on the axis d i . Furthermore, A d     i   (p i ) represents the original A d     i    the coordinate p i , and then, A d     i   (0) expresses a zero matrix filling at the coordinate p i .   For example: 2D point-diffusion, with rows for x, columns for y, AF(p x =3, p y =2).
           Suppose   
           
     
       
    
     
       
         
           
             
               A 
               = 
               
                 [ 
                 
                   
                     
                       
                         a 
                         11 
                       
                     
                     
                       
                         a 
                         12 
                       
                     
                     
                       
                         a 
                         13 
                       
                     
                     
                       
                         a 
                         14 
                       
                     
                   
                   
                     
                       
                         a 
                         21 
                       
                     
                     
                       
                         a 
                         22 
                       
                     
                     
                       
                         a 
                         23 
                       
                     
                     
                       
                         a 
                         24 
                       
                     
                   
                   
                     
                       
                         a 
                         31 
                       
                     
                     
                       
                         a 
                         32 
                       
                     
                     
                       
                         a 
                         33 
                       
                     
                     
                       
                         a 
                         34 
                       
                     
                   
                   
                     
                       
                         a 
                         41 
                       
                     
                     
                       
                         a 
                         42 
                       
                     
                     
                       
                         a 
                         43 
                       
                     
                     
                       
                         a 
                         44 
                       
                     
                   
                 
                 ] 
               
             
             , 
             
               
 
             
             ⁢ 
             
               S 
               = 
               
                 [ 
                 
                   
                     
                       
                         s 
                         11 
                       
                     
                     
                       
                         s 
                         12 
                       
                     
                     
                       
                         s 
                         13 
                       
                     
                     
                       
                         s 
                         14 
                       
                     
                   
                   
                     
                       
                         s 
                         21 
                       
                     
                     
                       
                         s 
                         22 
                       
                     
                     
                       
                         s 
                         23 
                       
                     
                     
                       
                         s 
                         24 
                       
                     
                   
                   
                     
                       
                         s 
                         31 
                       
                     
                     
                       
                         s 
                         32 
                       
                     
                     
                       
                         s 
                         33 
                       
                     
                     
                       
                         s 
                         34 
                       
                     
                   
                   
                     
                       
                         s 
                         41 
                       
                     
                     
                       
                         s 
                         42 
                       
                     
                     
                       
                         s 
                         43 
                       
                     
                     
                       
                         s 
                         44 
                       
                     
                   
                 
                 ] 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 S 
                 . 
               
               = 
               
                 ( 
                 
                   2 
                   , 
                   1 
                 
                 ) 
               
             
           
         
       
       
         
           thus 
         
       
       
         
           
             
               
                 
                   
                     AF 
                     ⁡ 
                     
                       ( 
                       
                         3 
                         , 
                         2 
                       
                       ) 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     A 
                     ⊕ 
                     
                       Ax 
                       3 
                     
                     ⊕ 
                     
                       Ay 
                       2 
                     
                     ⊕ 
                     S 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     A 
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             
                               a 
                               21 
                             
                           
                           
                             
                               a 
                               22 
                             
                           
                           
                             
                               a 
                               23 
                             
                           
                           
                             
                               a 
                               24 
                             
                           
                         
                         
                           
                             
                               a 
                               31 
                             
                           
                           
                             
                               a 
                               32 
                             
                           
                           
                             
                               a 
                               33 
                             
                           
                           
                             
                               a 
                               34 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             
                               a 
                               31 
                             
                           
                           
                             
                               a 
                               32 
                             
                           
                           
                             
                               a 
                               33 
                             
                           
                           
                             
                               a 
                               34 
                             
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             
                               a 
                               12 
                             
                           
                           
                             0 
                           
                           
                             
                               a 
                               12 
                             
                           
                           
                             
                               a 
                               13 
                             
                           
                         
                         
                           
                             
                               a 
                               22 
                             
                           
                           
                             0 
                           
                           
                             
                               a 
                               22 
                             
                           
                           
                             
                               a 
                               23 
                             
                           
                         
                         
                           
                             
                               a 
                               32 
                             
                           
                           
                             0 
                           
                           
                             
                               a 
                               32 
                             
                           
                           
                             
                               a 
                               33 
                             
                           
                         
                         
                           
                             
                               a 
                               42 
                             
                           
                           
                             0 
                           
                           
                             
                               a 
                               42 
                             
                           
                           
                             
                               a 
                               43 
                             
                           
                         
                       
                       ] 
                     
                     ⊕ 
                   
                 
               
             
             
               
                 
                     
                   ⁢ 
                   
                     [ 
                     
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           
                             s 
                             11 
                           
                         
                         
                           
                             s 
                             12 
                           
                         
                         
                           
                             s 
                             13 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             s 
                             21 
                           
                         
                         
                           
                             s 
                             22 
                           
                         
                         
                           
                             s 
                             23 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             s 
                             31 
                           
                         
                         
                           
                             s 
                             32 
                           
                         
                         
                           
                             s 
                             33 
                           
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     In detail, Ax 3  expresses a series of one dimensional binary matrixes A x  on the axis x; wherein Ax 3  comprises A x  (2)=[a 21  a 22  a 23  a 24 ] to position 1, A x  (3)=[a 31  a 32  a 33  a 34 ] to positions 2, 4, and A x  (0)=[0 0 0 0] at position 3. Furthermore, Ay 2  expresses a series of one dimensional binary matrixes A y  on the axis y; wherein Ay 2  comprises 
                 A   y     ⁡     (   2   )       =     [           a   12               a   22               a   32               a   42           ]           
to positions 1, 3,
 
                 A   y     ⁡     (   3   )       =     [           a   13               a   23               a   33               a   43           ]           
to position 4, and
 
                 A   y     ⁡     (   0   )       =     [         0           0           0           0         ]           
at position 2.
         Finally, the effective S comes from the overlap between S and A, while {dot over (S)}=(2,1) anchors to P=(3,2).   For example: 3D point-diffusion AF(p x =3, p y =2, p z =1). Suppose       

     
       
         
           
             
               A 
               = 
               
                 [ 
                 
                   
                     
                       
                         
                           
                             
                               a 
                               111 
                             
                           
                           
                             
                               a 
                               121 
                             
                           
                           
                             
                               a 
                               131 
                             
                           
                           
                             
                               a 
                               141 
                             
                           
                         
                         
                           
                             
                               a 
                               211 
                             
                           
                           
                             
                               a 
                               221 
                             
                           
                           
                             
                               a 
                               231 
                             
                           
                           
                             
                               a 
                               241 
                             
                           
                         
                         
                           
                             
                               a 
                               311 
                             
                           
                           
                             
                               a 
                               321 
                             
                           
                           
                             
                               a 
                               331 
                             
                           
                           
                             
                               a 
                               341 
                             
                           
                         
                         
                           
                             
                               a 
                               411 
                             
                           
                           
                             
                               a 
                               421 
                             
                           
                           
                             
                               a 
                               431 
                             
                           
                           
                             
                               a 
                               441 
                             
                           
                         
                       
                       ︷ 
                     
                     
                       z 
                       = 
                       1 
                     
                   
                   ⁢ 
                   
                      
                     
                       
                         
                           
                             
                               
                                 a 
                                 112 
                               
                             
                             
                               
                                 a 
                                 122 
                               
                             
                             
                               
                                 a 
                                 132 
                               
                             
                             
                               
                                 a 
                                 142 
                               
                             
                           
                           
                             
                               
                                 a 
                                 212 
                               
                             
                             
                               
                                 a 
                                 222 
                               
                             
                             
                               
                                 a 
                                 232 
                               
                             
                             
                               
                                 a 
                                 242 
                               
                             
                           
                           
                             
                               
                                 a 
                                 312 
                               
                             
                             
                               
                                 a 
                                 322 
                               
                             
                             
                               
                                 a 
                                 332 
                               
                             
                             
                               
                                 a 
                                 342 
                               
                             
                           
                           
                             
                               
                                 a 
                                 412 
                               
                             
                             
                               
                                 a 
                                 422 
                               
                             
                             
                               
                                 a 
                                 432 
                               
                             
                             
                               
                                 a 
                                 442 
                               
                             
                           
                         
                         ︷ 
                       
                       
                         z 
                         = 
                         2 
                       
                     
                      
                   
                   ⁢ 
                   
                     
                       
                         
                           
                             
                               a 
                               113 
                             
                           
                           
                             
                               a 
                               123 
                             
                           
                           
                             
                               a 
                               133 
                             
                           
                           
                             
                               a 
                               143 
                             
                           
                         
                         
                           
                             
                               a 
                               213 
                             
                           
                           
                             
                               a 
                               223 
                             
                           
                           
                             
                               a 
                               233 
                             
                           
                           
                             
                               a 
                               243 
                             
                           
                         
                         
                           
                             
                               a 
                               313 
                             
                           
                           
                             
                               a 
                               323 
                             
                           
                           
                             
                               a 
                               333 
                             
                           
                           
                             
                               a 
                               343 
                             
                           
                         
                         
                           
                             
                               a 
                               413 
                             
                           
                           
                             
                               a 
                               423 
                             
                           
                           
                             
                               a 
                               433 
                             
                           
                           
                             
                               a 
                               443 
                             
                           
                         
                       
                       ︷ 
                     
                     
                       z 
                       = 
                       3 
                     
                   
                 
                 ] 
               
             
             , 
             
               
 
             
             ⁢ 
             
               S 
               = 
               
                 [ 
                 
                   
                     
                       
                         
                           
                             
                               s 
                               111 
                             
                           
                           
                             
                               s 
                               121 
                             
                           
                           
                             
                               s 
                               131 
                             
                           
                           
                             
                               s 
                               141 
                             
                           
                         
                         
                           
                             
                               s 
                               211 
                             
                           
                           
                             
                               s 
                               221 
                             
                           
                           
                             
                               s 
                               231 
                             
                           
                           
                             
                               s 
                               241 
                             
                           
                         
                         
                           
                             
                               s 
                               311 
                             
                           
                           
                             
                               s 
                               321 
                             
                           
                           
                             
                               s 
                               331 
                             
                           
                           
                             
                               s 
                               341 
                             
                           
                         
                         
                           
                             
                               s 
                               411 
                             
                           
                           
                             
                               s 
                               421 
                             
                           
                           
                             
                               s 
                               431 
                             
                           
                           
                             
                               s 
                               441 
                             
                           
                         
                       
                       ︷ 
                     
                     
                       z 
                       = 
                       1 
                     
                   
                   ⁢ 
                   
                      
                     
                       
                         
                           
                             
                               
                                 s 
                                 112 
                               
                             
                             
                               
                                 s 
                                 122 
                               
                             
                             
                               
                                 s 
                                 132 
                               
                             
                             
                               
                                 s 
                                 142 
                               
                             
                           
                           
                             
                               
                                 s 
                                 212 
                               
                             
                             
                               
                                 s 
                                 222 
                               
                             
                             
                               
                                 s 
                                 232 
                               
                             
                             
                               
                                 s 
                                 242 
                               
                             
                           
                           
                             
                               
                                 s 
                                 312 
                               
                             
                             
                               
                                 s 
                                 322 
                               
                             
                             
                               
                                 s 
                                 332 
                               
                             
                             
                               
                                 s 
                                 342 
                               
                             
                           
                           
                             
                               
                                 s 
                                 412 
                               
                             
                             
                               
                                 s 
                                 422 
                               
                             
                             
                               
                                 s 
                                 432 
                               
                             
                             
                               
                                 s 
                                 442 
                               
                             
                           
                         
                         ︷ 
                       
                       
                         z 
                         = 
                         2 
                       
                     
                      
                   
                   ⁢ 
                   
                     
                       
                         
                           
                             
                               s 
                               113 
                             
                           
                           
                             
                               s 
                               123 
                             
                           
                           
                             
                               s 
                               133 
                             
                           
                           
                             
                               s 
                               143 
                             
                           
                         
                         
                           
                             
                               s 
                               213 
                             
                           
                           
                             
                               s 
                               223 
                             
                           
                           
                             
                               s 
                               233 
                             
                           
                           
                             
                               s 
                               243 
                             
                           
                         
                         
                           
                             
                               s 
                               313 
                             
                           
                           
                             
                               s 
                               323 
                             
                           
                           
                             
                               s 
                               333 
                             
                           
                           
                             
                               s 
                               343 
                             
                           
                         
                         
                           
                             
                               s 
                               413 
                             
                           
                           
                             
                               s 
                               423 
                             
                           
                           
                             
                               s 
                               433 
                             
                           
                           
                             
                               s 
                               443 
                             
                           
                         
                       
                       ︷ 
                     
                     
                       z 
                       = 
                       3 
                     
                   
                 
                 ] 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 
                   S 
                   . 
                 
                 = 
                 
                   ( 
                   
                     2 
                     , 
                     1 
                     , 
                     3 
                   
                   ) 
                 
               
               ; 
               thus 
             
             , 
             
               
                 AF 
                 ⁡ 
                 
                   ( 
                   
                     3 
                     , 
                     2 
                     , 
                     1 
                   
                   ) 
                 
               
               = 
               
                 
                   A 
                   ⊕ 
                   
                     Ax 
                     3 
                   
                   ⊕ 
                   
                     Ay 
                     2 
                   
                   ⊕ 
                   
                     Az 
                     1 
                   
                   ⊕ 
                   S 
                 
                 = 
                 
                   
                     A 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     … 
                   
                   ⁢ 
                   
                       
                   
                   ⊕ 
                   
                     [ 
                     
                       
                         
                           
                             
                               
                                 
                                   a 
                                   211 
                                 
                               
                               
                                 
                                   a 
                                   221 
                                 
                               
                               
                                 
                                   a 
                                   231 
                                 
                               
                               
                                 
                                   a 
                                   241 
                                 
                               
                             
                             
                               
                                 
                                   a 
                                   311 
                                 
                               
                               
                                 
                                   a 
                                   321 
                                 
                               
                               
                                 
                                   a 
                                   331 
                                 
                               
                               
                                 
                                   a 
                                   341 
                                 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 
                                   a 
                                   311 
                                 
                               
                               
                                 
                                   a 
                                   321 
                                 
                               
                               
                                 
                                   a 
                                   331 
                                 
                               
                               
                                 
                                   a 
                                   341 
                                 
                               
                             
                           
                           ︷ 
                         
                         
                           z 
                           = 
                           1 
                         
                       
                       ⁢ 
                       
                          
                         
                           
                             
                               
                                 
                                   
                                     a 
                                     212 
                                   
                                 
                                 
                                   
                                     a 
                                     222 
                                   
                                 
                                 
                                   
                                     a 
                                     232 
                                   
                                 
                                 
                                   
                                     a 
                                     242 
                                   
                                 
                               
                               
                                 
                                   
                                     a 
                                     312 
                                   
                                 
                                 
                                   
                                     a 
                                     322 
                                   
                                 
                                 
                                   
                                     a 
                                     332 
                                   
                                 
                                 
                                   
                                     a 
                                     342 
                                   
                                 
                               
                               
                                 
                                   0 
                                 
                                 
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                                     a 
                                     312 
                                   
                                 
                                 
                                   
                                     a 
                                     322 
                                   
                                 
                                 
                                   
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                                     332 
                                   
                                 
                                 
                                   
                                     a 
                                     342 
                                   
                                 
                               
                             
                             ︷ 
                           
                           
                             z 
                             = 
                             2 
                           
                         
                          
                       
                       ⁢ 
                       
                         
                           
                             
                               
                                 
                                   a 
                                   213 
                                 
                               
                               
                                 
                                   a 
                                   223 
                                 
                               
                               
                                 
                                   a 
                                   233 
                                 
                               
                               
                                 
                                   a 
                                   243 
                                 
                               
                             
                             
                               
                                 
                                   a 
                                   313 
                                 
                               
                               
                                 
                                   a 
                                   323 
                                 
                               
                               
                                 
                                   a 
                                   333 
                                 
                               
                               
                                 
                                   a 
                                   343 
                                 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 
                                   a 
                                   313 
                                 
                               
                               
                                 
                                   a 
                                   323 
                                 
                               
                               
                                 
                                   a 
                                   333 
                                 
                               
                               
                                 
                                   a 
                                   343 
                                 
                               
                             
                           
                           ︷ 
                         
                         
                           z 
                           = 
                           3 
                         
                       
                     
                     ] 
                   
                   ⊕ 
                   
                     
 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     [ 
                     
                       
                         
                           
                             
                               
                                 
                                   a 
                                   121 
                                 
                               
                               
                                 0 
                               
                               
                                 
                                   a 
                                   121 
                                 
                               
                               
                                 
                                   a 
                                   131 
                                 
                               
                             
                             
                               
                                 
                                   a 
                                   221 
                                 
                               
                               
                                 0 
                               
                               
                                 
                                   a 
                                   221 
                                 
                               
                               
                                 
                                   a 
                                   231 
                                 
                               
                             
                             
                               
                                 
                                   a 
                                   321 
                                 
                               
                               
                                 0 
                               
                               
                                 
                                   a 
                                   321 
                                 
                               
                               
                                 
                                   a 
                                   331 
                                 
                               
                             
                             
                               
                                 
                                   a 
                                   421 
                                 
                               
                               
                                 0 
                               
                               
                                 
                                   a 
                                   421 
                                 
                               
                               
                                 
                                   a 
                                   431 
                                 
                               
                             
                           
                           ︷ 
                         
                         
                           z 
                           = 
                           1 
                         
                       
                       ⁢ 
                       
                          
                         
                           
                             
                               
                                 
                                   
                                     a 
                                     122 
                                   
                                 
                                 
                                   0 
                                 
                                 
                                   
                                     a 
                                     122 
                                   
                                 
                                 
                                   
                                     a 
                                     132 
                                   
                                 
                               
                               
                                 
                                   
                                     a 
                                     222 
                                   
                                 
                                 
                                   0 
                                 
                                 
                                   
                                     a 
                                     222 
                                   
                                 
                                 
                                   
                                     a 
                                     232 
                                   
                                 
                               
                               
                                 
                                   
                                     a 
                                     322 
                                   
                                 
                                 
                                   0 
                                 
                                 
                                   
                                     a 
                                     322 
                                   
                                 
                                 
                                   
                                     a 
                                     332 
                                   
                                 
                               
                               
                                 
                                   
                                     a 
                                     422 
                                   
                                 
                                 
                                   0 
                                 
                                 
                                   
                                     a 
                                     422 
                                   
                                 
                                 
                                   
                                     a 
                                     432 
                                   
                                 
                               
                             
                             ︷ 
                           
                           
                             z 
                             = 
                             2 
                           
                         
                          
                       
                       ⁢ 
                       
                         
                           
                             
                               
                                 
                                   a 
                                   123 
                                 
                               
                               
                                 0 
                               
                               
                                 
                                   a 
                                   123 
                                 
                               
                               
                                 
                                   a 
                                   133 
                                 
                               
                             
                             
                               
                                 
                                   a 
                                   223 
                                 
                               
                               
                                 0 
                               
                               
                                 
                                   a 
                                   223 
                                 
                               
                               
                                 
                                   a 
                                   233 
                                 
                               
                             
                             
                               
                                 
                                   a 
                                   323 
                                 
                               
                               
                                 0 
                               
                               
                                 
                                   a 
                                   323 
                                 
                               
                               
                                 
                                   a 
                                   333 
                                 
                               
                             
                             
                               
                                 
                                   a 
                                   423 
                                 
                               
                               
                                 0 
                               
                               
                                 
                                   a 
                                   423 
                                 
                               
                               
                                 
                                   a 
                                   433 
                                 
                               
                             
                           
                           ︷ 
                         
                         
                           z 
                           = 
                           3 
                         
                       
                     
                     ] 
                   
                   ⊕ 
                   
                     
 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     [ 
                     
                       
                         
                           
                             
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                             
                           
                           ︷ 
                         
                         
                           z 
                           = 
                           1 
                         
                       
                       ⁢ 
                       
                          
                         
                           
                             
                               
                                 
                                   
                                     a 
                                     111 
                                   
                                 
                                 
                                   
                                     a 
                                     121 
                                   
                                 
                                 
                                   
                                     a 
                                     131 
                                   
                                 
                                 
                                   
                                     a 
                                     141 
                                   
                                 
                               
                               
                                 
                                   
                                     a 
                                     211 
                                   
                                 
                                 
                                   
                                     a 
                                     221 
                                   
                                 
                                 
                                   
                                     a 
                                     231 
                                   
                                 
                                 
                                   
                                     a 
                                     241 
                                   
                                 
                               
                               
                                 
                                   
                                     a 
                                     311 
                                   
                                 
                                 
                                   
                                     a 
                                     321 
                                   
                                 
                                 
                                   
                                     a 
                                     331 
                                   
                                 
                                 
                                   
                                     a 
                                     341 
                                   
                                 
                               
                               
                                 
                                   
                                     a 
                                     411 
                                   
                                 
                                 
                                   
                                     a 
                                     421 
                                   
                                 
                                 
                                   
                                     a 
                                     431 
                                   
                                 
                                 
                                   
                                     a 
                                     441 
                                   
                                 
                               
                             
                             ︷ 
                           
                           
                             z 
                             = 
                             2 
                           
                         
                          
                       
                       ⁢ 
                       
                         
                           
                             
                               
                                 
                                   a 
                                   112 
                                 
                               
                               
                                 
                                   a 
                                   122 
                                 
                               
                               
                                 
                                   a 
                                   132 
                                 
                               
                               
                                 
                                   a 
                                   142 
                                 
                               
                             
                             
                               
                                 
                                   a 
                                   212 
                                 
                               
                               
                                 
                                   a 
                                   222 
                                 
                               
                               
                                 
                                   a 
                                   232 
                                 
                               
                               
                                 
                                   a 
                                   242 
                                 
                               
                             
                             
                               
                                 
                                   a 
                                   312 
                                 
                               
                               
                                 
                                   a 
                                   322 
                                 
                               
                               
                                 
                                   a 
                                   332 
                                 
                               
                               
                                 
                                   a 
                                   342 
                                 
                               
                             
                             
                               
                                 
                                   a 
                                   412 
                                 
                               
                               
                                 
                                   a 
                                   422 
                                 
                               
                               
                                 
                                   a 
                                   432 
                                 
                               
                               
                                 
                                   a 
                                   442 
                                 
                               
                             
                           
                           ︷ 
                         
                         
                           z 
                           = 
                           3 
                         
                       
                     
                     ] 
                   
                   ⊕ 
                   
                     
 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   
                                     s 
                                     113 
                                   
                                 
                                 
                                   
                                     s 
                                     123 
                                   
                                 
                                 
                                   
                                     s 
                                     133 
                                   
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   
                                     s 
                                     213 
                                   
                                 
                                 
                                   
                                     s 
                                     223 
                                   
                                 
                                 
                                   
                                     s 
                                     233 
                                   
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   
                                     s 
                                     313 
                                   
                                 
                                 
                                   
                                     s 
                                     323 
                                   
                                 
                                 
                                   
                                     s 
                                     333 
                                   
                                 
                               
                             
                             ︷ 
                           
                           
                             z 
                             = 
                             1 
                           
                         
                         ⁢ 
                         
                            
                           
                             
                               
                                 
                                   
                                     0 
                                   
                                   
                                     0 
                                   
                                   
                                     0 
                                   
                                   
                                     0 
                                   
                                 
                                 
                                   
                                     0 
                                   
                                   
                                     0 
                                   
                                   
                                     0 
                                   
                                   
                                     0 
                                   
                                 
                                 
                                   
                                     0 
                                   
                                   
                                     0 
                                   
                                   
                                     0 
                                   
                                   
                                     0 
                                   
                                 
                                 
                                   
                                     0 
                                   
                                   
                                     0 
                                   
                                   
                                     0 
                                   
                                   
                                     0 
                                   
                                 
                               
                               ︷ 
                             
                             
                               z 
                               = 
                               2 
                             
                           
                            
                         
                         ⁢ 
                         
                           
                             
                               
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                               
                             
                             ︷ 
                           
                           
                             z 
                             = 
                             3 
                           
                         
                       
                       ] 
                     
                     . 
                   
                 
               
             
           
         
       
     
     In detail, Ax 3  expresses a series of two dimensional binary matrixes A x  on the axis x; wherein Ax 3  comprises 
                 A   x     ⁡     (   2   )       =     
     ⁢     [                   a   211           a   221           a   231           ⁢     a   241       ︷       z   =   1       ⁢                    a   212           a   222           a   232           a   242           ︷       z   =   2            ⁢               a   213           a   223           a   233           a   243           ︷       z   =   3         ]           
to position 1.
 
                 A   x     ⁡     (   3   )       =     
     ⁢     [                   a   311           a   321           a   331           ⁢     a   341       ︷       z   =   1       ⁢                    a   312           a   322           a   332           a   342           ︷       z   =   2            ⁢               a   313           a   323           a   333           a   343           ︷       z   =   3         ]           
to positions 2, 4, and
 
                 A   x     ⁡     (   0   )       =     [               0       0             0       0               ︷       z   =   1       ⁢                  0       0       0       0         ︷       z   =   2            ⁢             0       0       0       0         ︷       z   =   3         ]           
at position 3.
 
Furthermore, Ay 2  expresses a series of two dimensional binary matrixes A y  on the axis y; wherein Ay 2  comprises
 
                 A   y     ⁡     (   2   )       =     [             a   121       ︷     z   =   1                 a   122       ︷     z   =   2                 a   123       ︷     z   =   3                   a   221           a   222           a   223               a   321           a   322           a   323               a   421           a   422           a   423           ]           
to positions 1, 3,
 
                 A   y     ⁡     (   3   )       =     [             a   131       ︷     z   =   1                 a   132       ︷     z   =   2                 a   133       ︷     z   =   3                   a   231           a   232           a   233               a   331           a   332           a   333               a   431           a   432           a   433           ]           
to position 4, and
 
                 A   y     ⁡     (   0   )       =     [           0     ︷     z   =   1               0     ︷     z   =   2               0     ︷     z   =   3                 0       0       0           0       0       0           0       0       0         ]           
at position 2.
 
Moreover, Az 1  expresses a series of two dimensional binary matrixes A z  on the axis z; wherein Az 1  comprises
 
                 A   z     ⁡     (   1   )       =     [           a   111           a   121           a   131           a   141               a   211           a   221           a   231           a   241               a   311           a   321           a   331           a   341               a   411           a   421           a   431           a   441           ]           
to position 2,
 
                 A   z     ⁡     (   2   )       =     [           a   112           a   122           a   132           a   142               a   212           a   222           a   232           a   242               a   312           a   322           a   332           a   342               a   412           a   422           a   432           a   442           ]           
to position 3, and
 
                 A   z     ⁡     (   0   )       =     [         0       0       0       0           0       0       0       0           0       0       0       0           0       0       0       0         ]           
at position 1. Finally, the effective S comes from the overlap between S and A, while {dot over (S)}=(2,1,3) anchors to P=(3,2,1).
     AF(p 1 , p 2   t , . . . , p n ): A performs the function of point-diffusion, repeated t times.
       Example: (a) AF(p 1 , p 2   2 , . . . , p n )=AF(p 1 , p 2 , . . . , p n )F(p 1 , p 2 , . . . , p n )
           (b) AF(p 1 , p 2   1 , . . . , p n )=AF(p 1 , p 2 , . . . , p n )   (c) AF(p 1 , p 2   0 , . . . , p n )=A   
           
       T: a diffusion-cycle, expresses AF(p 1 , p 2   T , . . . , p n )=A, wherein T=2 U+1 , U=┌log 2  u┐, u=max(d 1 , d 2 , . . . , d n ).
 
Function of Block-Diffusion:
   

       FIG. 3A  shows an embodiment of the block-diffusion function,  FIG. 1 ,  420 , in flow chart diagram. The function comprises of: reading a diffusion-area (plaintext/ciphertext), a diffusion-center, a medium with an anchor-point and a block with an anchor-point  202 ; anchoring the medium and the block to the diffusion-center with the anchor-point  421 ; implementing the block-diffusion ÂF({circumflex over (p)} 1 , {circumflex over (p)} 2 , . . . {circumflex over (p)} n )  422 , further detailed in Notation of Block-Diffusion. In addition, also see  FIG. 3B  with 2D visualization for a more clear view, the diffusion effect colored from white to black generates the block-column segments a-c, a-b for later diffusion calculation. 
     Notation of Block-Diffusion: 
     
         
         A: a n-dimension plaintext, expresses a d 1 ×d 2 × . . . ×d n  binary matrix, wherein A includes a diffusion-center P expressed (p 1 , p 2 , . . . p n ) coordinate position. 
         S: a n-dimension medium, expresses a s 1 ×s 2 × . . . ×s n  binary matrix, wherein S includes an anchor-point {dot over (S)} expressed ({dot over (s)} 1 , {dot over (s)} 2 , . . . , {dot over (s)} n ) coordinate position. 
         B: a n-dimension unit-block, expresses a b 1 ×b 2 × . . . ×b n  binary matrix, wherein B includes an anchor-point {dot over (B)} expressed ({dot over (b)} 1 , {dot over (b)} 2 , . . . , {dot over (b)} n ) coordinate position. 
         ÂF({circumflex over (p)} 1 , {circumflex over (p)} 2 , . . . {circumflex over (p)} n ): Â performs the function of block-diffusion, wherein Â expresses A by B unit seeing that {dot over (B)} anchors to P, and thus, includes a block diffusion-center {circumflex over (P)} expressed ({circumflex over (p)} 1 , {circumflex over (p)} 2 , . . . {circumflex over (p)} n ) coordinate position. Therefore, A translates into a {circumflex over (d)} 1 ×{circumflex over (d)} 2 × . . . ×{circumflex over (d)} n  binary matrix, wherein {circumflex over (d)} i =┌(p i −{dot over (b)} i )/b i ┐+┌(d i −p i +{dot over (b)} i /b i ┐, and {circumflex over (p)} i =|(p i −{dot over (b)} i )/b i |+1; further comprising: 
       
    
                   A   ^     ⁢     F   ⁡     (         p   ^     1     ,       p   ^     2     ,   …   ⁢           ,       p   ^     n       )         =       A   ^     ⊕       A   ^     ⁢       d   ^       1   ⁢     p   ^           ⊕       A   ^     ⁢       d   ^       2   ⁢     p   ^           ⊕   …   ⊕       A   ^     ⁢       d   ^       n   ⁢     p   ^           ⊕   S       ;                     A   ^     ⁢       d   ^       i   ⁢     p   ^           =     [           A   ^         d   ^     i       ⁡     (   2   )       ,   …   ⁢           ,         A   ^         d   ^     i       ⁡     (       p   ^     i     )       ,         A   ^         d   ^     i       ⁡     (   0   )       ,         A   ^         d   ^     i       ⁡     (       p   ^     i     )       ,   …   ⁢           ,         A   ^         d   ^     i       ⁡     (         d   ^     i     -   1     )         ]       ;         
Â{circumflex over (d)} i{circumflex over (p)}  expresses a series of n−1 dimensional binary matrixes
 
               A   ^         d   ^     i           
on the axis {circumflex over (d)} i . Furthermore,
 
                 A   ^         d   ^     i       ⁡     (       p   ^     i     )           
represents the original
 
               A   ^         d   ^     i           
at the coordinate {circumflex over (p)} i , and then,
 
                 A   ^         d   ^     i       ⁡     (   0   )           
expresses a zero matrix tilling at the coordinate {circumflex over (p)} i .
         For example: 2D block-diffusion, with rows for x, columns for y, AF(p x =3, p y =2). Suppose       

               A   =     [           a   11           a   12           a   13           a   14               a   21           a   22           a   23           a   24               a   31           a   32           a   33           a   34               a   41           a   42           a   43           a   44           ]       ,     S   =     [           s   11           s   12           s   13           s   14               s   21           s   22           s   23           s   24               s   31           s   32           s   33           s   34               s   41           s   42           s   43           s   44           ]       ,     
     ⁢     B   =     [           b   11           b   12               b   21           b   22           ]       ,       S   .     =     (     2   ,   1     )       ,         B   .     =     (     1   ,   1     )       ;           
thus, dimensions {circumflex over (x)}=┌(3−1)/2┐+┌(4−3+1)/2┐=2 and ŷ=┌(2−1)/2┐+┌(4−2+1)/2┐=3;
 
                                
that shows the block-diffusion in 2×3 blocks, but with the data still kept in 4×4 bits. And now {circumflex over (p)} x =┌(3−1)/2┐+1=2, {circumflex over (p)} y =┌(2−1)/2┐+1=2, thus
 
     
       
                 
         
             
             
         
      
     
     In detail, Â{circumflex over (x)} 2  expresses a series of one dimensional binary matrixes Â {circumflex over (x)}  on the axis {circumflex over (x)}; wherein Â{circumflex over (x)} 2  comprises 
                   A   ^       x   ^       ⁡     (   2   )       =     [           a   31           a   32           a   33           a   34               a   41           a   42           a   43           a   44           ]           
to position 1, and
 
                   A   ^       x   ^       ⁡     (   0   )       =     [         0       0       0       0           0       0       0       0         ]           
at position 2. Furthermore, Âŷ 2  expresses a series of one dimensional binary matrixes Â ŷ  on the axis ŷ; wherein Âŷ 2  comprises
 
                   A   ^       y   ^       ⁡     (   2   )       =     [           a   12           a   13               a   22           a   23               a   32           a   33               a   42           a   43           ]           
to positions 1, 3, and
 
                   A   ^       y   ^       ⁡     (   0   )       =     [         0       0           0       0           0       0           0       0         ]           
at position 2. Finally, the effective S comes from the overlap between S and A, while {dot over (S)}=(2,1) anchors to P=(3,2).
         For example: 3D block-diffusion AF(p x =3, p y =2, p z =1). Suppose       

               A   =     [               a   111           a   121           a   131           a   141               a   211           a   221           a   231           a   241               a   311           a   321           a   331           a   341               a   411           a   421           a   431           a   441             ︷     z   =   1         ❘               a   112           a   122           a   132           a   142               a   212           a   222           a   232           a   242               a   312           a   322           a   332           a   342               a   412           a   422           a   432           a   442             ︷     z   =   2         ❘             a   113           a   123           a   133           a   143               a   213           a   223           a   233           a   243               a   313           a   323           a   333           a   343               a   413           a   423           a   433           a   443             ︷     z   =   3             ]       ,     
     ⁢     S   =     [               s   111           s   121           s   131           s   141               s   211           s   221           s   231           s   241               s   311           s   321           s   331           s   341               s   411           s   421           s   431           s   441             ︷     z   =   1         ❘               s   112           s   122           s   132           s   142               s   212           s   222           s   232           s   242               s   312           s   322           s   332           s   342               s   412           s   422           s   432           s   442             ︷     z   =   2         ❘             s   113           s   123           s   133           s   143               s   213           s   223           s   233           s   243               s   313           s   323           s   333           s   343               s   413           s   423           s   433           s   443             ︷     z   =   3             ]       ,          ⁢     B   =     [               b   111           b   121               b   211           b   221             ︷     z   =   1         |             b   112           b   122               b   212           b   222             ︷     z   =   2           ]       ,       S   .     =     (     2   ,   1   ,   3     )       ,         B   .     =     (     1   ,   1   ,   1     )       ;           
thus, dimensions {circumflex over (x)}=┌(3−1)/2┐+┌(4−3+1)/2┐=2, ŷ=┌(2−1)/2┐+┌(4−2+1)/2┐=3, and {circumflex over (z)}=┌(1−1)/2┐+┌(4−1+1)/2┐=2; further
 
     
       
                 
         
             
             
         
      
         
         
           
             that shows the block-diffusion in 2×3×2 blocks, but with the data still kept in 4×4×3 bits. And now {circumflex over (p)} x =┌(3−1)/2┐+1=2, {circumflex over (p)} y =┌(2−1)/2┐+1=2, {circumflex over (p)} z =┌(1−1)/2┐+1=1, thus, 
           
         
       
    
     
       
                 
         
             
             
         
      
     
     In detail, Â{circumflex over (x)} 2  expresses a series of two dimensional binary matrixes Â {circumflex over (x)}  on the axis {circumflex over (x)}; wherein Â{circumflex over (x)} 2  comprises 
                   A   ^       x   ^       ⁡     (   2   )       =     [                 a   311           a   321           a   331           a   341           a   312           a   322           a   332           a   342               a   411           a   421           a   431           a   441           a   412           a   422           a   432           a   442           ︷         z   ^     =   1       ⁢               a   313           a   323           a   333           a   343               a   413           a   423           a   433           a   443           ︷         z   ^     =   2         ]           
to position 1, and
 
                   A   ^       x   ^       ⁡     (   0   )       =     [               0       0       0       0       0       0       0       0           0       0       0       0       0       0       0       0         ︷         z   ^     =   1       ⁢             0       0       0       0           0       0       0       0         ︷         z   ^     =   2         ]           
at position 2.
 
Furthermore, Âŷ 2  expresses a series of two dimensional binary matrixes Â ŷ  on the axis ŷ; wherein Âŷ 2  comprises Â ŷ (2) to positions 1, 3 is equal to
 
                 [                 a   121           a   131               a   221           a   231               a   321           a   331               a   421           a   431           ❘           a   122           a   132               a   222           a   232               a   322           a   332               a   422           a   432               ︷       z   ^     =   1         ❘               a   123           a   133               a   223           a   233               a   323           a   333               a   423           a   433             ︷       z   ^     =   2         ❘       ]     ,           ⁢   and     ⁢                             A   ^       y   ^       ⁡     (   0   )       =       [               0       0           0       0           0       0           0       0         ❘         0       0           0       0           0       0           0       0             ︷       z   ^     =   1         ❘             0       0           0       0           0       0           0       0           ︷       z   ^     =   2         ❘       ]     ⁢   and           
at position 2.
 
Moreover, Â{circumflex over (z)} 1  expresses a series of two dimensional binary matrixes Â {circumflex over (z)}  on the axis {circumflex over (z)}; wherein Â{circumflex over (z)} 1  comprises
 
                   A   ^       z   ^       ⁡     (   1   )       =     [               a   111           a   121           a   131           a   141               a   211           a   221           a   231           a   241               a   311           a   321           a   331           a   341               a   411           a   421           a   431           a   441           ︷         z   ^     =   1       ]           
to position
 
                   A   ^       z   ^       ⁡     (   0   )       =     [         0       0       0       0       0       0       0       0           0       0       0       0       0       0       0       0           0       0       0       0       0       0       0       0           0       0       0       0       0       0       0       0         ]           
at position 1. Finally, the effective S comes from the overlap between S and A, while {dot over (S)}=(2,1,3) anchors to P=(3,2,1).
     ÂF({circumflex over (p)} 1 , {circumflex over (p)} 2   t  , . . . , {dot over (p)} n ): Â performs the function of block-diffusion, repeated t times.
       Example: (a) ÂF({circumflex over (p)} 1 , {circumflex over (p)} 2   2 , . . . , {circumflex over (p)} n )=ÂF({circumflex over (p)} 1 , {circumflex over (p)} 2 , . . . , {circumflex over (p)} n )F({circumflex over (p)} 1 , {circumflex over (p)} 2 , . . . {circumflex over (p)} n )
           (b) ÂF({circumflex over (p)} 1 , {circumflex over (p)} 2   1 , . . . , {circumflex over (p)} n )=ÂF({circumflex over (p)} 1 , {circumflex over (p)} 2 , . . . , {circumflex over (p)} n )   (c) ÂF({circumflex over (p)} 1 , {circumflex over (p)} 2   0 , . . . , {circumflex over (p)} n )=A   
           
       T: a diffusion-cycle, expresses {dot over (A)}F({circumflex over (p)} 1 , {circumflex over (p)} 2   T , . . . {dot over (p)} n )=A, wherein T=2 U+1 , U=┌log 2  u┌, u=max(┌d i /b i ┐, 1≦i≦n).
 
Embodiment of Cryptographic Method:
   

     To make it easier to understand the content of the present invention, examples in detail are described as follows: 
     Suppose a plaintext A: “smoother”, its ASCII code is 73 6d 6f 6f 74 68 65 72, the binary format is shown as an 8×8 two-dimensional matrix as in Table 1-1. 
     
       
         
               
             
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 1-1 
               
             
             
               
                   
               
               
                 ASCII 
               
             
          
           
               
                 73 
                 6d 
                 6f 
                 6f 
                 74 
                 68 
                 65 
                 72 
               
               
                   
               
               
                 1 
                 1 
                 1 
                 1 
                 0 
                 0 
                 1 
                 0 
               
               
                 1 
                 0 
                 1 
                 1 
                 0 
                 0 
                 0 
                 1 
               
               
                 0 
                 1 
                 1 
                 1 
                 1 
                 0 
                 1 
                 0 
               
               
                 0 
                 1 
                 1 
                 1 
                 0 
                 1 
                 0 
                 0 
               
               
                 1 
                 0 
                 0 
                 0 
                 1 
                 0 
                 0 
                 1 
               
               
                 1 
                 1 
                 1 
                 1 
                 1 
                 1 
                 1 
                 1 
               
               
                 1 
                 1 
                 1 
                 1 
                 1 
                 1 
                 1 
                 1 
               
               
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                   
               
             
          
         
       
     
     Suppose a password: “Yourlips”, its ASCII code is 59 6f 75 72 6c 69 70 73. For applying to the plaintext, the ASCII code: first, excludes the last digit 3; second, forms into octal format 26 26 75.65 34 46 61 51 34 07; third, adds 1 to each digit; Table 1-2 shows that the password includes 10 diffusion-centers. 
     
       
         
               
               
             
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 1-2 
               
             
             
               
                   
               
               
                   
                 ASCII 
               
             
          
           
               
                   
                 26 
                 26 
                 75 
                 65 
                 34 
                 46 
                 61 
                 51 
                 34 
                 07 
               
               
                   
               
               
                 Row 
                 3 
                 3  
                 8 
                 7 
                 4 
                 5  
                 7 
                 6 
                 4 
                 1 
               
               
                 Column 
                 7  
                 7 
                 6 
                 6 
                 5 
                 7 
                 2 
                 2 
                 5 
                 8 
               
               
                   
               
             
          
         
       
     
     EXAMPLE 1 
     The Function of Point-Diffusion in 2D 
     Supposes 
                 S     5   ×   5       =     [         10011           01101           10111           10010           11101         ]       ,         S   .     =     (     1   ,   1     )       ;           
reads every diffusion-center in order, if from 1 to 10 on encryption, then from 10 back to 1 on decryption; counts the diffusion-cycle T=2 3+1 =16, if 1 time on encryption, then 15 times on decryption. In math, inputs the plaintext A, then runs A 1 , A 1   1 , . . . A 9   1  and outputs A 1 , A 2 , . . . A 10  during encryption; inputs the ciphertext A 10 , then runs A 10   15 , A 9   15 , . . . A 1   15  and outputs A 9 , . . . , A 1 , A during decryption. The details on the order 1, 5, 10 are shown as below, A d     i    (0) marked in boldface.
 
Encryption at the 1 st  Diffusion-Center (3,7):
 
                     A   1     =       ⁢     AF   ⁡     (     3   ,   7     )                   =       ⁢     A   ⊕     Ax   3     ⊕     Ay   7     ⊕   S                 =       ⁢       [         11110010           10110001           01111010           01110100           10001001           11111111           11111111           00000000         ]     ⊕     [         10110001           01111010           00000000           01111010           01110100           10001001           11111111           11111111         ]     ⊕     [         11100101           01100000           11110101           11101000           00010000           11111101           11111101           00000000         ]     ⊕     [         00000000           00000000           00000010           00000001           00000010           00000010           00000011           00000000         ]                   =       ⁢       [           ⁢         10100110           10101011           10001101           11100111           11101111           10001001           11111110           11111111         ]     =       A   1     .                   
Encryption at the 5 th  Diffusion-Center (4,5):
 
     
       
         
           
             
               
                 
                   
                     A 
                     4 
                     1 
                   
                   = 
                     
                   ⁢ 
                   
                     
                       A 
                       4 
                     
                     ⁢ 
                     
                       F 
                       ⁡ 
                       
                         ( 
                         
                           4 
                           , 
                           5 
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       A 
                       4 
                     
                     ⊕ 
                     
                       
                         A 
                         4 
                       
                       ⁢ 
                       
                         x 
                         4 
                       
                     
                     ⊕ 
                     
                       
                         A 
                         4 
                       
                       ⁢ 
                       
                         y 
                         5 
                       
                     
                     ⊕ 
                     S 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       [ 
                       
                         
                           
                             11010111 
                           
                         
                         
                           
                             00010101 
                           
                         
                         
                           
                             00010001 
                           
                         
                         
                           
                             01000011 
                           
                         
                         
                           
                             00001111 
                           
                         
                         
                           
                             01011001 
                           
                         
                         
                           
                             10101011 
                           
                         
                         
                           
                             01100101 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             00010101 
                           
                         
                         
                           
                             00010001 
                           
                         
                         
                           
                             01000011 
                           
                         
                         
                           
                             00000000 
                           
                         
                         
                           
                             01000011 
                           
                         
                         
                           
                             00001111 
                           
                         
                         
                           
                             01011001 
                           
                         
                         
                           
                             10101011 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             10100011 
                           
                         
                         
                           
                             00100010 
                           
                         
                         
                           
                             00100000 
                           
                         
                         
                           
                             10000001 
                           
                         
                         
                           
                             00010111 
                           
                         
                         
                           
                             10110100 
                           
                         
                         
                           
                             01010101 
                           
                         
                         
                           
                             11000010 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             00000000 
                           
                         
                         
                           
                             00000000 
                           
                         
                         
                           
                             00000000 
                           
                         
                         
                           
                             00001001 
                           
                         
                         
                           
                             00000110 
                           
                         
                         
                           
                             00001011 
                           
                         
                         
                           
                             00001001 
                           
                         
                         
                           
                             00001110 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       [ 
                       
                         
                           
                             01100001 
                           
                         
                         
                           
                             00100110 
                           
                         
                         
                           
                             01110010 
                           
                         
                         
                           
                             11001011 
                           
                         
                         
                           
                             01011101 
                           
                         
                         
                           
                             11101001 
                           
                         
                         
                           
                             10101110 
                           
                         
                         
                           
                             00000010 
                           
                         
                       
                       ] 
                     
                     = 
                     
                       
                         A 
                         5 
                       
                       . 
                     
                   
                 
               
             
           
         
       
     
     Encryption at the 10 th  Diffusion-Center (1,8): 
     
       
         
           
             
               
                 
                   
                     A 
                     9 
                     1 
                   
                   = 
                     
                   ⁢ 
                   
                     
                       
                         A 
                         9 
                       
                       ⁢ 
                       
                         F 
                         ⁡ 
                         
                           ( 
                           
                             1 
                             , 
                             8 
                           
                           ) 
                         
                       
                     
                     = 
                     
                       
                         A 
                         9 
                       
                       ⊕ 
                       
                         
                           A 
                           9 
                         
                         ⁢ 
                         
                           x 
                           1 
                         
                       
                       ⊕ 
                       
                         
                           A 
                           9 
                         
                         ⁢ 
                         
                           y 
                           8 
                         
                       
                       ⊕ 
                       S 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       [ 
                       
                         
                           
                             01110011 
                           
                         
                         
                           
                             10000110 
                           
                         
                         
                           
                             10011100 
                           
                         
                         
                           
                             10101100 
                           
                         
                         
                           
                             01000101 
                           
                         
                         
                           
                             10001011 
                           
                         
                         
                           
                             00110101 
                           
                         
                         
                           
                             10101001 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             00000000 
                           
                         
                         
                           
                             01110011 
                           
                         
                         
                           
                             10000110 
                           
                         
                         
                           
                             10011100 
                           
                         
                         
                           
                             10101100 
                           
                         
                         
                           
                             01000101 
                           
                         
                         
                           
                             10001011 
                           
                         
                         
                           
                             00110101 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             11100110 
                           
                         
                         
                           
                             00001100 
                           
                         
                         
                           
                             00111000 
                           
                         
                         
                           
                             01011000 
                           
                         
                         
                           
                             10001010 
                           
                         
                         
                           
                             00010110 
                           
                         
                         
                           
                             01101010 
                           
                         
                         
                           
                             01010010 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             00000001 
                           
                         
                         
                           
                             00000000 
                           
                         
                         
                           
                             00000001 
                           
                         
                         
                           
                             00000001 
                           
                         
                         
                           
                             00000001 
                           
                         
                         
                           
                             00000000 
                           
                         
                         
                           
                             00000000 
                           
                         
                         
                           
                             00000000 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       [ 
                       
                         
                           
                             10010100 
                           
                         
                         
                           
                             11111001 
                           
                         
                         
                           
                             00100011 
                           
                         
                         
                           
                             01101001 
                           
                         
                         
                           
                             01100010 
                           
                         
                         
                           
                             11011000 
                           
                         
                         
                           
                             11010100 
                           
                         
                         
                           
                             11001110 
                           
                         
                       
                       ] 
                     
                     = 
                     
                       
                         A 
                         10 
                       
                       . 
                     
                   
                 
               
             
           
         
       
     
     Decryption at the 10 th  Diffusion-Center (1,8): 
     
       
         
           
             
               
                 
                   
                     A 
                     10 
                     15 
                   
                   = 
                     
                   ⁢ 
                   
                     
                       
                         A 
                         10 
                         14 
                       
                       ⁢ 
                       
                         F 
                         ⁡ 
                         
                           ( 
                           
                             1 
                             , 
                             8 
                           
                           ) 
                         
                       
                     
                     = 
                     
                       
                         A 
                         10 
                         14 
                       
                       ⊕ 
                       
                         
                           A 
                           10 
                           14 
                         
                         ⁢ 
                         
                           x 
                           1 
                         
                       
                       ⊕ 
                       
                         
                           A 
                           10 
                           14 
                         
                         ⁢ 
                         
                           y 
                           8 
                         
                       
                       ⊕ 
                       S 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       [ 
                       
                         
                           
                             00101110 
                           
                         
                         
                           
                             10011000 
                           
                         
                         
                           
                             00000011 
                           
                         
                         
                           
                             10011010 
                           
                         
                         
                           
                             01001010 
                           
                         
                         
                           
                             10111111 
                           
                         
                         
                           
                             10000110 
                           
                         
                         
                           
                             11100101 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             00000000 
                           
                         
                         
                           
                             00101110 
                           
                         
                         
                           
                             10011000 
                           
                         
                         
                           
                             00000011 
                           
                         
                         
                           
                             10011010 
                           
                         
                         
                           
                             01001010 
                           
                         
                         
                           
                             10111111 
                           
                         
                         
                           
                             10000110 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             01011100 
                           
                         
                         
                           
                             00110000 
                           
                         
                         
                           
                             00000110 
                           
                         
                         
                           
                             00110100 
                           
                         
                         
                           
                             10010100 
                           
                         
                         
                           
                             01111110 
                           
                         
                         
                           
                             00001100 
                           
                         
                         
                           
                             11001010 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             00000001 
                           
                         
                         
                           
                             00000000 
                           
                         
                         
                           
                             00000001 
                           
                         
                         
                           
                             00000001 
                           
                         
                         
                           
                             00000001 
                           
                         
                         
                           
                             00000000 
                           
                         
                         
                           
                             00000000 
                           
                         
                         
                           
                             00000000 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       [ 
                       
                         
                           
                             01110011 
                           
                         
                         
                           
                             10000110 
                           
                         
                         
                           
                             10011100 
                           
                         
                         
                           
                             10101100 
                           
                         
                         
                           
                             01000101 
                           
                         
                         
                           
                             10001011 
                           
                         
                         
                           
                             00110101 
                           
                         
                         
                           
                             10101001 
                           
                         
                       
                       ] 
                     
                     = 
                     
                       
                         A 
                         9 
                       
                       . 
                     
                   
                 
               
             
           
         
       
     
     Decryption at the 5 th  Diffusion-Center (4,5): 
     
       
         
           
             
               
                 
                   
                     A 
                     5 
                     15 
                   
                   = 
                     
                   ⁢ 
                   
                     
                       
                         A 
                         5 
                         14 
                       
                       ⁢ 
                       
                         F 
                         ⁡ 
                         
                           ( 
                           
                             4 
                             , 
                             5 
                           
                           ) 
                         
                       
                     
                     = 
                     
                       
                         A 
                         5 
                         14 
                       
                       ⊕ 
                       
                         
                           A 
                           5 
                           14 
                         
                         ⁢ 
                         
                           x 
                           4 
                         
                       
                       ⊕ 
                       
                         
                           A 
                           5 
                           14 
                         
                         ⁢ 
                         
                           y 
                           5 
                         
                       
                       ⊕ 
                       S 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       [ 
                       
                         
                           
                             00011010 
                           
                         
                         
                           
                             11111000 
                           
                         
                         
                           
                             00011001 
                           
                         
                         
                           
                             00111100 
                           
                         
                         
                           
                             00010110 
                           
                         
                         
                           
                             11000111 
                           
                         
                         
                           
                             00100110 
                           
                         
                         
                           
                             00111001 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             11111000 
                           
                         
                         
                           
                             00011001 
                           
                         
                         
                           
                             00111100 
                           
                         
                         
                           
                             00000000 
                           
                         
                         
                           
                             00111100 
                           
                         
                         
                           
                             00010110 
                           
                         
                         
                           
                             11000111 
                           
                         
                         
                           
                             00100110 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             00110101 
                           
                         
                         
                           
                             11110100 
                           
                         
                         
                           
                             00110100 
                           
                         
                         
                           
                             01110110 
                           
                         
                         
                           
                             00100011 
                           
                         
                         
                           
                             10000011 
                           
                         
                         
                           
                             01000011 
                           
                         
                         
                           
                             01110100 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             00000000 
                           
                         
                         
                           
                             00000000 
                           
                         
                         
                           
                             00000000 
                           
                         
                         
                           
                             00001001 
                           
                         
                         
                           
                             00000110 
                           
                         
                         
                           
                             00001011 
                           
                         
                         
                           
                             00001001 
                           
                         
                         
                           
                             00001110 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       [ 
                       
                         
                           
                             11010111 
                           
                         
                         
                           
                             00010101 
                           
                         
                         
                           
                             00010001 
                           
                         
                         
                           
                             01000011 
                           
                         
                         
                           
                             00001111 
                           
                         
                         
                           
                             01011001 
                           
                         
                         
                           
                             10101011 
                           
                         
                         
                           
                             01100101 
                           
                         
                       
                       ] 
                     
                     = 
                     
                       
                         A 
                         4 
                       
                       . 
                     
                   
                 
               
             
           
         
       
     
     Decryption at the 1 st  Diffusion-Center (3,7): 
     
       
         
           
             
               
                 
                   
                     A 
                     1 
                     15 
                   
                   = 
                     
                   ⁢ 
                   
                     
                       
                         A 
                         1 
                         14 
                       
                       ⁢ 
                       
                         F 
                         ⁡ 
                         
                           ( 
                           
                             3 
                             , 
                             7 
                           
                           ) 
                         
                       
                     
                     = 
                     
                       
                         A 
                         1 
                         14 
                       
                       ⊕ 
                       
                         
                           A 
                           1 
                           14 
                         
                         ⁢ 
                         
                           x 
                           3 
                         
                       
                       ⊕ 
                       
                         
                           A 
                           1 
                           14 
                         
                         ⁢ 
                         
                           y 
                           7 
                         
                       
                       ⊕ 
                       S 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       [ 
                       
                         
                           
                             11010110 
                           
                         
                         
                           
                             10001001 
                           
                         
                         
                           
                             00101000 
                           
                         
                         
                           
                             00110101 
                           
                         
                         
                           
                             01101011 
                           
                         
                         
                           
                             01110011 
                           
                         
                         
                           
                             01111010 
                           
                         
                         
                           
                             11010111 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             10001001 
                           
                         
                         
                           
                             00101000 
                           
                         
                         
                           
                             00000000 
                           
                         
                         
                           
                             00101000 
                           
                         
                         
                           
                             00110101 
                           
                         
                         
                           
                             01101011 
                           
                         
                         
                           
                             01110011 
                           
                         
                         
                           
                             01111010 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             10101101 
                           
                         
                         
                           
                             00010000 
                           
                         
                         
                           
                             01010000 
                           
                         
                         
                           
                             01101000 
                           
                         
                         
                           
                             11010101 
                           
                         
                         
                           
                             11100101 
                           
                         
                         
                           
                             11110101 
                           
                         
                         
                           
                             10101101 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             00000000 
                           
                         
                         
                           
                             00000000 
                           
                         
                         
                           
                             00000010 
                           
                         
                         
                           
                             00000001 
                           
                         
                         
                           
                             00000010 
                           
                         
                         
                           
                             00000010 
                           
                         
                         
                           
                             00000011 
                           
                         
                         
                           
                             00000000 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       [ 
                       
                         
                           
                             11110010 
                           
                         
                         
                           
                             10110001 
                           
                         
                         
                           
                             01111010 
                           
                         
                         
                           
                             01110100 
                           
                         
                         
                           
                             10001001 
                           
                         
                         
                           
                             11111111 
                           
                         
                         
                           
                             11111111 
                           
                         
                         
                           
                             00000000 
                           
                         
                       
                       ] 
                     
                     = 
                     
                       A 
                       . 
                     
                   
                 
               
             
           
         
       
     
     EXAMPLE 2 
     The Function of Block-Diffusion in 2D 
     Supposes that 
                 S     5   ×   5       =     [         10011           01101           10111           10010           11101         ]       ,       S   .     =     (     1   ,   1     )       ,     
     ⁢       B     2   ×   2       =     [           b   11           b   12               b   21           b   22           ]       ,         B   .     =     (     1   ,   1     )       ;           
reads
 
every diffusion-center in order, if from 1 to 10 on encryption, then from 10 back to 1 on decryption; counts the diffusion-cycle T=2 2+1 =8, since d i /b i =4=2 2 , and if 1 time on encryption, then 7 times on decryption. In math, inputs the plaintext A, then runs Â 1 , Â 1   1 , . . . Â 9   1 , and outputs A 1 , A 2 , . . . A 10  during encryption; inputs the ciphertext A 10 , then runs Â 10   7 , Â 9   7 , . . . Â 1   7  and outputs A 9 , . . . , A 1 , A during decryption. The details on the order 1, 5, 10 are shown as below,
 
                 A   ^         d   ^     i       ⁡     (   0   )           
marked in boldface.
 
Encryption at the 1 st  Diffusion-Center (3,7):
 
                       A   ^     1     =       ⁢         A   ^     ⁢     F   ⁡     (     2   ,   4     )         =                 =       ⁢       A   ^     ⊕       A   ^     ⁢       x   ^     2       ⊕       A   ^     ⁢       y   ^     4       ⊕     S   (         ∵       p   ^     x       =       ⌈       (     3   -   1     )     /   2     ⌉     +   1       ,       p   ^     y                         =       ⁢       ⌈       (     7   -   1     )     /   2     ⌉     +   1       )               =       ⁢       [         11110010           10110001           01111010           01110100           10001001           11111111           11111111           00000000         ]     ⊕     [         01111010           01110100           00000000           00000000           01111010           01110100           10001001           11111111         ]     ⊕     [         11001000           11000100           11101000           11010000           00100100           11111100           11111100           00000000         ]     ⊕     [         00000000           00000000           00000010           00000001           00000010           00000010           00000011           00000000         ]                   =       ⁢       [         01000000           00000001           10010000           10100101           11010101           01110101           10001001           11111111         ]     =       A   1     .                   
Encryption at the 5 th  Diffusion-Center (4,5):
 
                       A   ^     4   1     =       ⁢         A   ^     4     ⁢     F   ⁡     (     3   ,   3     )                     =       ⁢         A   ^     4     ⊕         A   ^     4     ⁢       x   ^     3       ⊕         A   ^     4     ⁢       y   ^     3       ⊕     S   ⁡     (         ∵       p   ^     x       =       ⌈       (     4   -   1     )     /   2     ⌉     +   1       ,         p   ^     y     =       ⌈       (     5   -   1     )     /   2     ⌉     +   1         )                     =       ⁢       [         11000011           10100110           10001001           01000110           00110011           01100010           11011111           00000000         ]     ⊕     [         10001001           01000110           00110011           00000000           00000000           01000110           00110011           01100010         ]     ⊕     [           0000   ⁢   00   ⁢   00               1001   ⁢   00   ⁢   01               0010   ⁢   00   ⁢   10               0001   ⁢   00   ⁢   01               1100   ⁢   00   ⁢   00               1000   ⁢   00   ⁢   00               0111   ⁢   00   ⁢   11               0000   ⁢   00   ⁢   00           ]     ⊕     [         00000000           00000000           00000000           00001001           00000110           00001011           00001001           00001110         ]                   =       ⁢     [         01001010           01110001           10011000           01011110           11110101           10101111           10010110           01101100         ]                 =       ⁢       A   5     .                 
Encryption at the 10 th  Diffusion-Center (1,8): (□, Zero in A ŷ , (5), 2 nd  Col.)
 
                       A   ^     9   1     =       ⁢         A   ^     9     ⁢     F   ⁡     (     1   ,   5     )                     =       ⁢         A   ^     9     ⊕         A   ^     9     ⁢       x   ^     1       ⊕         A   ^     9     ⁢       y   ^     5       ⊕     S   ⁡     (         ∵       p   ^     x       =       ⌈       (     1   -   1     )     /   2     ⌉     +   1       ,         p   ^     y     =       ⌈       (     8   -   1     )     /   2     ⌉     +   1         )                     =       ⁢       [         10011001           11110100           10001001           10001000           11011000           10000001           01110101           10011001         ]     ⊕     [         00000000           00000000           10011001           11110100           10001000           00010011           11011000           10000001         ]     ⊕     [           011001   ⁢     •   ⁢   0                 110100   ⁢     •   ⁢   0                 001000   ⁢     •   ⁢   0                 010011   ⁢     •   ⁢   0                 011000   ⁢     •   ⁢   0                 000001   ⁢     •   ⁢   0                 110101   ⁢     •   ⁢   0                 011001   ⁢     •   ⁢   0             ]     ⊕     [         00000001           00000000           00000001           00000001           00000001           00000000           00000000           00000000         ]                   =       ⁢     [         11111100           00100100           00110000           10101010           00110001           10010110           01111001           01111100         ]                 =       ⁢       A   10     .                 
Decryption at the 10 th  Diffusion-Center (1,8):
 
                       A   ^     10   7     =       ⁢         A   ^     10   6     ⁢     F   ⁡     (     1   ,   5     )                     =       ⁢         A   ^     10   6     ⊕         A   ^     10   6     ⁢       x   ^     1       ⊕         A   ^     10   6     ⁢       y   ^     5       ⊕                     ⁢     S   ⁡     (         ∵       p   ^     x       =       ⌈       (     1   -   1     )     /   2     ⌉     +   1       ,         p   ^     y     =       ⌈       (     8   -   1     )     /   2     ⌉     +   1         )                   =       ⁢       [         01111000           01100100           01100101           01001110           10001100           11000011           11001101           00010010         ]     ⊕     [         00000000           00000000           01111000           01100100           01100101           01001110           10001100           11000011         ]     ⊕     [           111000   ⁢     •   ⁢   0                 100100   ⁢     •   ⁢   0                 100101   ⁢     •   ⁢   0                 001110   ⁢     •   ⁢   0                 001100   ⁢     •   ⁢   0                 000011   ⁢     •   ⁢   0                 001101   ⁢     •   ⁢   0                 010010   ⁢     •   ⁢   0             ]     ⊕     [         00000001           00000000           00000001           00000001           00000001           00000000           00000000           00000000         ]                   =       ⁢     [         10011001           11110100           10001000           00010011           11011000           10000001           01110101           10011001         ]                 =       ⁢       A   9     .                 
Decryption at the 5 th  diffusion-center (4,5):
 
                       A   ^     5   7     =       ⁢         A   ^     5   6     ⁢     F   ⁡     (     3   ,   3     )                     =       ⁢         A   ^     5   6     ⊕         A   ^     5   6     ⁢       x   ^     3       ⊕         A   ^     5   6     ⁢       y   ^     3       ⊕     S   ⁡     (         ∵       p   ^     x       =       ⌈       (     4   -   1     )     /   2     ⌉     +   1       ,         p   ^     y     =       ⌈       (     5   -   1     )     /   2     ⌉     +   1         )                     =       ⁢       [         00101110           11111000           01011110           10111100           10100100           11000100           10110010           01101000         ]     ⊕     [         01011110           10111100           10100100           00000000           00000000           10111100           10100100           11000100         ]     ⊕     [           1011   ⁢   00   ⁢   11               1110   ⁢   00   ⁢   10               0111   ⁢   00   ⁢   11               1111   ⁢   00   ⁢   11               1001   ⁢   00   ⁢   01               0001   ⁢   00   ⁢   01               1100   ⁢   00   ⁢   00               1010   ⁢   00   ⁢   10           ]     ⊕     [         00000000           00000000           00000000           00001001           00000110           00001011           00001001           00001110         ]                   =       ⁢     [         11000011           10100110           10001001           01000110           00110011           01100010           11011111           00000000         ]                 =       ⁢       A   4     .                 
Decryption at the 1 st  Diffusion-Center (3,7):
 
     
       
         
           
             
               
                 
                   
                     
                       A 
                       ^ 
                     
                     1 
                     7 
                   
                   = 
                     
                   ⁢ 
                   
                     
                       
                         A 
                         ^ 
                       
                       1 
                       6 
                     
                     ⁢ 
                     
                       F 
                       ⁡ 
                       
                         ( 
                         
                           2 
                           , 
                           4 
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       
                         A 
                         ^ 
                       
                       1 
                       6 
                     
                     ⊕ 
                     
                       
                         
                           A 
                           ^ 
                         
                         1 
                         6 
                       
                       ⁢ 
                       
                         
                           x 
                           ^ 
                         
                         2 
                       
                     
                     ⊕ 
                     
                       
                         
                           A 
                           ^ 
                         
                         1 
                         6 
                       
                       ⁢ 
                       
                         
                           y 
                           ^ 
                         
                         4 
                       
                     
                     ⊕ 
                     
                       S 
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               ∵ 
                               
                                 
                                   p 
                                   ^ 
                                 
                                 x 
                               
                             
                             = 
                             
                               
                                 ⌈ 
                                 
                                   
                                     ( 
                                     
                                       3 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                   / 
                                   2 
                                 
                                 ⌉ 
                               
                               + 
                               1 
                             
                           
                           , 
                           
                             
                               
                                 p 
                                 ^ 
                               
                               y 
                             
                             = 
                             
                               
                                 ⌈ 
                                 
                                   
                                     ( 
                                     
                                       7 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                   / 
                                   2 
                                 
                                 ⌉ 
                               
                               + 
                               1 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       [ 
                       
                         
                           
                             01100010 
                           
                         
                         
                           
                             00000000 
                           
                         
                         
                           
                             00011000 
                           
                         
                         
                           
                             10110001 
                           
                         
                         
                           
                             00101111 
                           
                         
                         
                           
                             10111100 
                           
                         
                         
                           
                             01101111 
                           
                         
                         
                           
                             10001100 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             00011000 
                           
                         
                         
                           
                             10110001 
                           
                         
                         
                           
                             00000000 
                           
                         
                         
                           
                             00000000 
                           
                         
                         
                           
                             00011000 
                           
                         
                         
                           
                             10110001 
                           
                         
                         
                           
                             00101111 
                           
                         
                         
                           
                             10111100 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             
                               100010 
                               ⁢ 
                               00 
                             
                           
                         
                         
                           
                             
                               000000 
                               ⁢ 
                               00 
                             
                           
                         
                         
                           
                             
                               011000 
                               ⁢ 
                               00 
                             
                           
                         
                         
                           
                             
                               110001 
                               ⁢ 
                               00 
                             
                           
                         
                         
                           
                             
                               101111 
                               ⁢ 
                               00 
                             
                           
                         
                         
                           
                             
                               111100 
                               ⁢ 
                               00 
                             
                           
                         
                         
                           
                             
                               101111 
                               ⁢ 
                               00 
                             
                           
                         
                         
                           
                             
                               001100 
                               ⁢ 
                               00 
                             
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             00000000 
                           
                         
                         
                           
                             00000000 
                           
                         
                         
                           
                             00000010 
                           
                         
                         
                           
                             00000001 
                           
                         
                         
                           
                             00000010 
                           
                         
                         
                           
                             00000010 
                           
                         
                         
                           
                             00000011 
                           
                         
                         
                           
                             00000000 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     [ 
                     
                       
                         
                           11110010 
                         
                       
                       
                         
                           10110001 
                         
                       
                       
                         
                           01111010 
                         
                       
                       
                         
                           01110100 
                         
                       
                       
                         
                           10001001 
                         
                       
                       
                         
                           11111111 
                         
                       
                       
                         
                           11111111 
                         
                       
                       
                         
                           00000000 
                         
                       
                     
                     ] 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     A 
                     . 
                   
                 
               
             
           
         
       
     
     EXAMPLE 3 
     The Functions of Point-Diffusion and Block-Diffusion in 2D 
     Supposes 
                 S     5   ×   5       =     [         10011           01101           10111           10010           11101         ]       ,     
     ⁢       S   .     =     (     1   ,   1     )       ,     
     ⁢       B     2   ×   2       =     [           b   11           b   12               b   21           b   22           ]       ,     
     ⁢         B   .     =     (     1   ,   1     )       ;           
selects a switch set Y=[1011011101]; reads every diffusion-center and Y element in order, if from 1 to 10 on encryption, then from 10 back to 1 on decryption; counts the diffusion-cycle, if Y element is 1, then T=2 3+1 =16 with point-diffusion, otherwise, T=2 2+1 =8 with block-diffusion, and if 1 time on encryption, then 15 or 7 times on decryption.
 
     In math, inputs the plaintext A, then runs A 1 , Â 1   1 , A 2   1 , A 3   1 , Â 4   1 , A 5   1 , A 6   1 , A 7   1 , Â 8   1 , A 9   1  and outputs A 1 , A 2 , . . . , A 9 , A 10  during encryption; inputs the ciphertext A 10 , then runs A 10   15 , Â 9   7 , A 8   15 , A 7   15 , A 6   15 , Â 5   7 , A 4   15 , A 3   15 , Â 2   7 , A 1   15  and outputs A 9 , A 8 , . . . , A 1 , A during decryption. The details on the order 1, 5, 10 are shown as below, A d     i    (0) and 
                 A   ^         d   ^     i       ⁡     (   0   )           
marked in boldface.
 
Encryption at the 1 st  Diffusion-Center (3,7): Y(1)=1, Point-Diffusion.
 
                     A   1     =       ⁢     AF   ⁡     (     3   ,   7     )                   =       ⁢     A   ⊕     Ax   3     ⊕     Ay   7     ⊕   S                 =       ⁢       [         11110010           10110001           01111010           01110100           10001001           11111111           11111111           00000000         ]     ⊕     [         10110001           01111010           00000000           01111010           01110100           10001001           11111111           11111111         ]     ⊕     [           111001   ⁢   0   ⁢   1               011000   ⁢   0   ⁢   0               111101   ⁢   0   ⁢   1               111010   ⁢   0   ⁢   0               000100   ⁢   0   ⁢   0               111111   ⁢   0   ⁢   1               111111   ⁢   0   ⁢   1               000000   ⁢   0   ⁢   0           ]     ⊕     [         00000000           00000000           00000010           00000001           00000010           00000010           00000011           00000000         ]                   =       ⁢     [         10100110           10101011           10001101           11100111           11101111           10001001           11111110           11111111         ]                 =       ⁢     A   1                 
Encryption at the 5 th  Diffusion-Center (4,5): Y(5)=0, Block-Diffusion.
 
                       A   ^     4   1     =       ⁢         A   ^     4     ⁢     F   ⁡     (     3   ,   3     )                     =       ⁢         A   ^     4     ⊕       A   4     ⁢       x   ^     3       ⊕         A   ^     4     ⁢       y   ^     3       ⊕     S   ⁡     (         ∵       p   ^     x       =       ⌈       (     4   -   1     )     /   2     ⌉     +   1       ,         p   ^     y     =       ⌈       (     5   -   1     )     /   2     ⌉     +   1         )                     =       ⁢       [         11001000           00011010           10000111           11010010           01000111           11100010           11010101           00010110         ]     ⊕     [         10000111           11010010           01000111           00000000           00000000           11010010           01000111           11100010         ]     ⊕     [           0010   ⁢   00   ⁢   10               0110   ⁢   00   ⁢   10               0001   ⁢   00   ⁢   01               0100   ⁢   00   ⁢   00               0001   ⁢   00   ⁢   01               1000   ⁢   00   ⁢   00               0101   ⁢   00   ⁢   01               0101   ⁢   00   ⁢   01           ]     ⊕     [         00000000           00000000           00000000           00001001           00000110           00001011           00001001           00001110         ]                   =       ⁢     [         01101101           10101010           11010001           10011011           01010000           10111011           11001010           10101011         ]                 =       ⁢     A   5                 
Encryption at the 10 th  Diffusion-Center (1,8): Y(10)=1, Point-Diffusion.
 
                     A   9   1     =       ⁢       A   9     ⁢     F   ⁡     (     1   ,   8     )                     =       ⁢       A   9     ⊕       A   9     ⁢     x   1       ⊕       A   9     ⁢     y   8       ⊕   S                 =       ⁢       [         00110000           11000111           00001010           10000100           00101100           11110100           00000111           10011011         ]     ⊕     [         00000000           00110000           11000111           00001010           10000100           00101100           11110100           00000111         ]     ⊕     [           0110000   ⁢   0               1000111   ⁢   0               0001010   ⁢   0               0000100   ⁢   0               0101100   ⁢   0               1110100   ⁢   0               0000111   ⁢   0               0011011   ⁢   0           ]     ⊕     [         00000001           00000000           00000001           00000001           00000001           00000000           00000000           00000000         ]                   =       ⁢     [         01010001           01111001           11011000           10000111           11110001           00110000           11111101           10101010         ]                 =       ⁢     A   10                 
Decryption at the 10 th  Diffusion-Center (1,8): Y(10)=1, Point-Diffusion.
 
                     A   10   15     =       ⁢       A   10   14     ⁢     F   ⁡     (     1   ,   8     )                     =       ⁢       A   10   14     ⊕       A   10   14     ⁢     x   1       ⊕       A   10   14     ⁢     y   8       ⊕   S                 =       ⁢       [         11101111           00011000           11110001           00101100           11111111           11111001           10101010           11101111         ]     ⊕     [         00000000           11101111           00011000           11110001           00101100           11111111           11111001           10101010         ]     ⊕     [           1101111   ⁢   0               0011000   ⁢   0               1110001   ⁢   0               0101100   ⁢   0               1111111   ⁢   0               1111001   ⁢   0               0101010   ⁢   0               1101111   ⁢   0           ]     ⊕     [         00000001           00000000           00000001           00000001           00000001           00000000           00000000           00000000         ]                   =       ⁢     [         00110000           11000111           00001010           10000100           00101100           11110100           00000111           10011011         ]                 =       ⁢     A   9                 
Decryption at the 5 th  Diffusion-Center (4,5): Y(5)=0, Block-Diffusion.
 
                       A   ^     5   7     =       ⁢         A   ^     5   6     ⁢     F   ⁡     (     3   ,   3     )                     =       ⁢         A   ^     5   6     ⊕         A   ^     5   6     ⁢       x   ^     3       ⊕         A   ^     5   6     ⁢       y   ^     3       ⊕     S   ⁡     (         ∵       p   ^     x       =       ⌈       (     4   -   1     )     /   2     ⌉     +   1       ,         p   ^     y     =       ⌈       (     5   -   1     )     /   2     ⌉     +   1         )                     =       ⁢       [         11101100           10100011           10010111           00111001           01000111           10010000           00101110           00101010         ]     ⊕     [         10010111           00111001           01000001           00000000           00000000           00111001           01000001           10010000         ]     ⊕     [           1011   ⁢   00   ⁢   11               1000   ⁢   00   ⁢   0   ⁢   0               0101   ⁢   00   ⁢   01               1110   ⁢   00   ⁢   1   ⁢   0               0000   ⁢   00   ⁢   00               0100   ⁢   00   ⁢   00               1011   ⁢   00   ⁢   11               1010   ⁢   00   ⁢   10           ]     ⊕     [         00000000           00000000           00000000           00001001           00000110           00001011           00001001           00001110         ]                   =       ⁢     [         11001000           00011010           10000111           11010010           01000111           11100010           11010101           00010110         ]                 =       ⁢     A   4                 
Decryption at the 1 st  Diffusion-Center (3,7): Y(1)=1, Point-Diffusion.
 
     
       
         
           
             
               
                 
                   
                     A 
                     1 
                     15 
                   
                   = 
                     
                   ⁢ 
                   
                     
                       A 
                       1 
                       14 
                     
                     ⁢ 
                     
                       F 
                       ⁡ 
                       
                         ( 
                         
                           3 
                           , 
                           7 
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       A 
                       1 
                       14 
                     
                     ⊕ 
                     
                       
                         A 
                         1 
                         14 
                       
                       ⁢ 
                       
                         x 
                         3 
                       
                     
                     ⊕ 
                     
                       
                         A 
                         1 
                         14 
                       
                       ⁢ 
                       
                         y 
                         7 
                       
                     
                     ⊕ 
                     S 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       [ 
                       
                         
                           
                             11010110 
                           
                         
                         
                           
                             10001001 
                           
                         
                         
                           
                             00101000 
                           
                         
                         
                           
                             00110101 
                           
                         
                         
                           
                             01101011 
                           
                         
                         
                           
                             01110011 
                           
                         
                         
                           
                             01111010 
                           
                         
                         
                           
                             11010111 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             10001001 
                           
                         
                         
                           
                             00101000 
                           
                         
                         
                           
                             00000000 
                           
                         
                         
                           
                             00101000 
                           
                         
                         
                           
                             00110101 
                           
                         
                         
                           
                             01101011 
                           
                         
                         
                           
                             01110011 
                           
                         
                         
                           
                             01111010 
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             
                               101011 
                               ⁢ 
                               0 
                               ⁢ 
                               1 
                             
                           
                         
                         
                           
                             
                               000100 
                               ⁢ 
                               0 
                               ⁢ 
                               0 
                             
                           
                         
                         
                           
                             
                               010100 
                               ⁢ 
                               0 
                               ⁢ 
                               0 
                             
                           
                         
                         
                           
                             
                               011010 
                               ⁢ 
                               0 
                               ⁢ 
                               0 
                             
                           
                         
                         
                           
                             
                               110101 
                               ⁢ 
                               0 
                               ⁢ 
                               1 
                             
                           
                         
                         
                           
                             
                               111001 
                               ⁢ 
                               0 
                               ⁢ 
                               1 
                             
                           
                         
                         
                           
                             
                               111101 
                               ⁢ 
                               0 
                               ⁢ 
                               1 
                             
                           
                         
                         
                           
                             
                               101011 
                               ⁢ 
                               0 
                               ⁢ 
                               1 
                             
                           
                         
                       
                       ] 
                     
                     ⊕ 
                     
                       [ 
                       
                         
                           
                             00000000 
                           
                         
                         
                           
                             00000000 
                           
                         
                         
                           
                             00000010 
                           
                         
                         
                           
                             00000001 
                           
                         
                         
                           
                             00000010 
                           
                         
                         
                           
                             00000010 
                           
                         
                         
                           
                             00000011 
                           
                         
                         
                           
                             00000000 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     [ 
                     
                       
                         
                           11110010 
                         
                       
                       
                         
                           10110001 
                         
                       
                       
                         
                           01111010 
                         
                       
                       
                         
                           01110100 
                         
                       
                       
                         
                           10001001 
                         
                       
                       
                         
                           11111111 
                         
                       
                       
                         
                           11111111 
                         
                       
                       
                         
                           00000000 
                         
                       
                     
                     ] 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   A 
                 
               
             
           
         
       
     
     EXAMPLE 4 
     The Function of Point-Diffusion in 3D 
     Supposes a plaintext A: let Table 1-1 overlap for 8 times to shape a 8×8×8 binary matrix, shown as a 8×8 matrix in ASCII code format as in Table 2-1. To figure out the later 3D calculation clearly with Table 2-1, the row stands for a x-y plane, namely Table 1-1, and all rows resolve as the axis z. 
     
       
         
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 2-1 
               
               
                   
               
             
             
               
                 73 
                 6d 
                 6f 
                 6f 
                 74 
                 68 
                 65 
                 72 
               
               
                 73 
                 6d 
                 6f 
                 6f 
                 74 
                 68 
                 65 
                 72 
               
               
                 73 
                 6d 
                 6f 
                 6f 
                 74 
                 68 
                 65 
                 72 
               
               
                 73 
                 6d 
                 6f 
                 6f 
                 74 
                 68 
                 65 
                 72 
               
               
                 73 
                 6d 
                 6f 
                 6f 
                 74 
                 68 
                 65 
                 72 
               
               
                 73 
                 6d 
                 6f 
                 6f 
                 74 
                 68 
                 65 
                 72 
               
               
                 73 
                 6d 
                 6f 
                 6f 
                 74 
                 68 
                 65 
                 72 
               
               
                 73 
                 6d 
                 6f 
                 6f 
                 74 
                 68 
                 65 
                 72 
               
               
                   
               
             
          
         
       
     
     Suppose a password: “YourlipsY”, its ASCII code is 59 6f 75 72 6c 69 70 73 59. For applying to the plaintext, the ASCII code: first, subtracts 8 if a digit &gt;7 and leaves 51 67 75 72 64 61 70 73 51; second, every three-digit forms a division; third, adds 1 to each digit; Table 2-2 shows that the password includes 6 diffusion-centers. 
     
       
         
               
               
               
             
               
               
               
               
               
               
               
               
             
           
               
                   
                 TABLE 2-2 
               
             
             
               
                   
                   
               
               
                   
                   
                 ASCII 
               
             
          
           
               
                   
                   
                 516 
                 775 
                 726 
                 461 
                 707 
                 351 
               
               
                   
                   
               
               
                   
                 First Dimension 
                 6 
                 8 
                 8 
                 5 
                 8 
                 4 
               
               
                   
                 Second Dimension 
                 2 
                 8 
                 3 
                 7 
                 1 
                 6 
               
               
                   
                 Third Dimension 
                 7 
                 6 
                 7 
                 2 
                 8 
                 2 
               
               
                   
                   
               
             
          
         
       
     
     Supposes S 1×1×1 =1, {dot over (S)}=(1,1,1); reads every diffusion-center in order, if from 1 to 6 on encryption, then from 6 back to 1 on decryption; counts the diffusion-cycle T=2 3+1 =16, if 1 time on encryption, then 15 times on decryption. In math, inputs the plaintext A, then runs A 1 , A 1   1 , . . . A 5   1  and outputs A 1 , A 2 , . . . A 6  during encryption; inputs the ciphertext A 6 , then runs A 6   15 , A 5   15 , . . . A 1   15  and outputs A 5 , . . . , A 1 , A during decryption. The details on the order 1, 6 are shown as below, A d     i    (0) marked in boldface. 
     Encryption at the 1 st  Diffusion-Center (6,2,7):
 
 A   1   =AF (6,2,7)= A⊕Ax   6   ⊕Ay   2   ⊕Az   7   ⊕S=A   1 ;
 
     Considering that the row of Table 2-1 means a x-y plane, it can be figured out by the 3D scheme through rearranging every plane then placing to the corresponding row of 2D table, as Ax 6  and Ay 2  as follows. 
     
       
         
           
             
               
                 a 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 plane 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 of 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   Ax 
                   6 
                 
               
               = 
               
                 [ 
                 
                   
                     
                       10110001 
                     
                   
                   
                     
                       01111010 
                     
                   
                   
                     
                       01110100 
                     
                   
                   
                     
                       10001001 
                     
                   
                   
                     
                       11111111 
                     
                   
                   
                     
                       00000000 
                     
                   
                   
                     
                       11111111 
                     
                   
                   
                     
                       11111111 
                     
                   
                 
                 ] 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 
                   Ax 
                   6 
                 
                 = 
                 
                   [ 
                   
                     
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           9 
                         
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           6 
                         
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           7 
                         
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           7 
                         
                       
                       
                         da 
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           4 
                         
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                         
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           9 
                         
                       
                     
                     
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           9 
                         
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           6 
                         
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           7 
                         
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           7 
                         
                       
                       
                         da 
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           4 
                         
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                         
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           9 
                         
                       
                     
                     
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           9 
                         
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           6 
                         
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           7 
                         
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           7 
                         
                       
                       
                         da 
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           4 
                         
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                         
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           9 
                         
                       
                     
                     
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           9 
                         
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           6 
                         
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           7 
                         
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           7 
                         
                       
                       
                         da 
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           4 
                         
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                         
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           9 
                         
                       
                     
                     
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           9 
                         
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           6 
                         
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           7 
                         
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           7 
                         
                       
                       
                         da 
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           4 
                         
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                         
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           9 
                         
                       
                     
                     
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           9 
                         
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           6 
                         
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           7 
                         
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           7 
                         
                       
                       
                         da 
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           4 
                         
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                         
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           9 
                         
                       
                     
                     
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           9 
                         
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           6 
                         
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           7 
                         
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           7 
                         
                       
                       
                         da 
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           4 
                         
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                         
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           9 
                         
                       
                     
                     
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           9 
                         
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           6 
                         
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           7 
                         
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           7 
                         
                       
                       
                         da 
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           4 
                         
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                         
                       
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           9 
                         
                       
                     
                   
                   ] 
                 
               
               ; 
             
           
         
       
       
         
           
             
               
                 a 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 plane 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 of 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   Ay 
                   2 
                 
               
               = 
               
                 [ 
                 
                   
                     
                       10111001 
                     
                   
                   
                     
                       00011000 
                     
                   
                   
                     
                       10111101 
                     
                   
                   
                     
                       10111010 
                     
                   
                   
                     
                       00000000 
                     
                   
                   
                     
                       10111111 
                     
                   
                   
                     
                       10111111 
                     
                   
                   
                     
                       00000000 
                     
                   
                 
                 ] 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 
                   Ay 
                   2 
                 
                 = 
                 
                   [ 
                   
                     
                       
                         
                           6 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           d 
                         
                       
                       
                         00 
                       
                       
                         
                           6 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           d 
                         
                       
                       
                         
                           6 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           f 
                         
                       
                       
                         
                           6 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           f 
                         
                       
                       
                         74 
                       
                       
                         68 
                       
                       
                         65 
                       
                     
                     
                       
                         
                           6 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           d 
                         
                       
                       
                         00 
                       
                       
                         
                           6 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           d 
                         
                       
                       
                         
                           6 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           f 
                         
                       
                       
                         
                           6 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           f 
                         
                       
                       
                         74 
                       
                       
                         68 
                       
                       
                         65 
                       
                     
                     
                       
                         
                           6 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           d 
                         
                       
                       
                         00 
                       
                       
                         
                           6 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           d 
                         
                       
                       
                         
                           6 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           f 
                         
                       
                       
                         
                           6 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           f 
                         
                       
                       
                         74 
                       
                       
                         68 
                       
                       
                         65 
                       
                     
                     
                       
                         
                           6 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           d 
                         
                       
                       
                         00 
                       
                       
                         
                           6 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           d 
                         
                       
                       
                         
                           6 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           f 
                         
                       
                       
                         
                           6 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           f 
                         
                       
                       
                         74 
                       
                       
                         68 
                       
                       
                         65 
                       
                     
                     
                       
                         
                           6 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           d 
                         
                       
                       
                         00 
                       
                       
                         
                           6 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           d 
                         
                       
                       
                         
                           6 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           f 
                         
                       
                       
                         
                           6 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           f 
                         
                       
                       
                         74 
                       
                       
                         68 
                       
                       
                         65 
                       
                     
                     
                       
                         
                           6 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           d 
                         
                       
                       
                         00 
                       
                       
                         
                           6 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           d 
                         
                       
                       
                         
                           6 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           f 
                         
                       
                       
                         
                           6 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           f 
                         
                       
                       
                         74 
                       
                       
                         68 
                       
                       
                         65 
                       
                     
                     
                       
                         
                           6 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           d 
                         
                       
                       
                         00 
                       
                       
                         
                           6 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           d 
                         
                       
                       
                         
                           6 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           f 
                         
                       
                       
                         
                           6 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           f 
                         
                       
                       
                         74 
                       
                       
                         68 
                       
                       
                         65 
                       
                     
                     
                       
                         
                           6 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           d 
                         
                       
                       
                         00 
                       
                       
                         
                           6 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           d 
                         
                       
                       
                         
                           6 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           f 
                         
                       
                       
                         
                           6 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           f 
                         
                       
                       
                         74 
                       
                       
                         68 
                       
                       
                         65 
                       
                     
                   
                   ] 
                 
               
               ; 
             
           
         
       
     
     In addition, S is anchored to position (6,2,7), see below, value 1 found at p x =6, p y =2 on the 7 th  plane (p z =7). 
                 the   ⁢           ⁢     7   th     ⁢           ⁢   plane   ⁢           ⁢   of   ⁢           ⁢   S     =     [         00000000           00000000           00000000           00000000           00000000           01000000           00000000           00000000         ]       ,     
     ⁢     S   =       [         00       00       00       00       00       00       00       00           00       00       00       00       00       00       00       00           00       00       00       00       00       00       00       00           00       00       00       00       00       00       00       00           00       00       00       00       00       00       00       00           00       00       00       00       00       00       00       00           00       20       00       00       00       00       00       00           00       00       00       00       00       00       00       00         ]     .             
Encryption at the 6 th  Diffusion-Center (4,6,2):
 
                   A   5     ⁢     F   ⁡     (     4   ,   6   ,   2     )         =         A   5     ⊕       A   5     ⁢     x   4       ⊕       A   5     ⁢     y   6       ⊕       A   5     ⁢     z   2       ⊕   S     =     A   6         ;                   A   5     =     [           c   ⁢           ⁢   4         11         3   ⁢   a           b   ⁢           ⁢   7           7   ⁢   a         64       01       ed             8   ⁢   f           c   ⁢           ⁢   0           8   ⁢   e           8   ⁢   d           a   ⁢           ⁢   1           b   ⁢           ⁢   7         cb       00           53         9   ⁢   c         48       26       ee       eb         8   ⁢   b         23             5   ⁢   b           5   ⁢   b         27       47       91         b   ⁢           ⁢   1           5   ⁢   c         cb             b   ⁢           ⁢   8         34       67         4   ⁢   a           9   ⁢   f           b   ⁢           ⁢   3         74         4   ⁢   c             92       17         5   ⁢   d           5   ⁢   a           7   ⁢   e           7   ⁢   d         84         0   ⁢   f               a   ⁢           ⁢   0         19         5   ⁢   d           e   ⁢           ⁢   6         48         7   ⁢   a         82         c   ⁢           ⁢   3             bf       ac       38       df       13         2   ⁢   f           d   ⁢           ⁢   8           1   ⁢   a           ]       ;                     A   5     ⁢     z   2       =     [           8   ⁢   f           c   ⁢           ⁢   0           8   ⁢   e           8   ⁢   d           a   ⁢           ⁢   1           b   ⁢           ⁢   7         cb       00           00       00       00       00       00       00       00       00             8   ⁢   f           c   ⁢           ⁢   0           8   ⁢   e           8   ⁢   d           a   ⁢           ⁢   1           b   ⁢           ⁢   7         cb       00           53         9   ⁢   c         48       26       ee       eb         8   ⁢   b         23             5   ⁢   b           5   ⁢   b         27       47       91         b   ⁢           ⁢   1           5   ⁢   c         cb             b   ⁢           ⁢   8         34       67         4   ⁢   a           9   ⁢   f           b   ⁢           ⁢   3         74         4   ⁢   c             92       17         5   ⁢   d           5   ⁢   a           7   ⁢   e           7   ⁢   d         84         0   ⁢   f               a   ⁢           ⁢   0         19         5   ⁢   d           e   ⁢           ⁢   6         48         7   ⁢   a         82         c   ⁢           ⁢   3           ]       ;                     A   5     ⁢     x   4       =     [         82       20       75       63         f   ⁢           ⁢   5           c   ⁢           ⁢   2         00         d   ⁢           ⁢   6             17       80       17       16       40       63       95       00             a   ⁢           ⁢   1         36       94       43         d   ⁢           ⁢   7           d   ⁢           ⁢   5         15       41             b   ⁢           ⁢   5           b   ⁢           ⁢   5         43       83       20       60         b   ⁢           ⁢   6         95           74       62         c   ⁢           ⁢   3         95       37       61         e   ⁢           ⁢   2         96           21       23         b   ⁢           ⁢   6           b   ⁢           ⁢   5           f   ⁢           ⁢   7           f   ⁢           ⁢   6         02       17           40       34         b   ⁢           ⁢   6           c   ⁢           ⁢   3         94         f   ⁢           ⁢   5         01       81           77       56       74         b   ⁢           ⁢   7         21       57         b   ⁢           ⁢   4         35         ]       ;                 S   =     [         00       00       00       00       00       00       00       00           00       00       00       00       00       08       00       00           00       00       00       00       00       00       00       00           00       00       00       00       00       00       00       00           00       00       00       00       00       00       00       00           00       00       00       00       00       00       00       00           00       00       00       00       00       00       00       00           00       00       00       00       00       00       00       00         ]       ;                     A   5     ⁢     y   6       =     [         11         3   ⁢   a           b   ⁢           ⁢   7           7   ⁢   a         64       00       64       01             c   ⁢           ⁢   0           8   ⁢   e           8   ⁢   d           a   ⁢           ⁢   1           b   ⁢           ⁢   7         00         b   ⁢           ⁢   7         cb             9   ⁢   c         48       26       ee       eb       00       eb         8   ⁢   b               5   ⁢   b         27       47       91         b   ⁢           ⁢   1         00         b   ⁢           ⁢   1           5   ⁢   c             34       67         4   ⁢   a           9   ⁢   f           b   ⁢           ⁢   3         00         b   ⁢           ⁢   3         74           17         5   ⁢   d           5   ⁢   a           7   ⁢   e           7   ⁢   d         00         7   ⁢   d         84           19         5   ⁢   d           e   ⁢           ⁢   6         48         7   ⁢   a         00         7   ⁢   a         82           ac       38       df       13         2   ⁢   f         00         2   ⁢   f           d   ⁢           ⁢   8           ]       ;                 A   6     =       [           d   ⁢           ⁢   8         cb       76       23         4   ⁢   a         11       ae         3   ⁢   a             58       ce       14         3   ⁢   a         56       dc         e   ⁢           ⁢   9         cb             e   ⁢           ⁢   1         22       74       06       73       89       be         e   ⁢           ⁢   9               e   ⁢           ⁢   6         55         6   ⁢   b         73       ee         3   ⁢   a           d   ⁢           ⁢   0         21             a   ⁢           ⁢   3           6   ⁢   a           c   ⁢           ⁢   9         07         8   ⁢   a         63       79       65             1   ⁢   c           5   ⁢   d           d   ⁢           ⁢   6         db         6   ⁢   b         38         8   ⁢   f           d   ⁢           ⁢   0               6   ⁢   b         67       50       37         d   ⁢           ⁢   8           f   ⁢           ⁢   2           7   ⁢   d         cf             c   ⁢           ⁢   4         db       ce         9   ⁢   d         55       02         c   ⁢           ⁢   1         34         ]     .           
Decryption at the 6 th  Diffusion-Center (4,6,2):
 
                 A   6   15     =         A   6   14     ⁢     F   ⁡     (     4   ,   6   ,   2     )         =         A   6   14     ⊕       A   6   14     ⁢     x   4       ⊕       A   6   14     ⁢     y   6       ⊕       A   6   14     ⁢     z   2       ⊕   S     =     A   5           ;                   A   6   14     =     [         24       31         7   ⁢   d           3   ⁢   a           e   ⁢           ⁢   8         fb         8   ⁢   d           c   ⁢           ⁢   1             93         3   ⁢   d           8   ⁢   b         10         a   ⁢           ⁢   1           6   ⁢   a         61       21             a   ⁢           ⁢   6         25         c   ⁢           ⁢   6         86       ee       81         d   ⁢           ⁢   0           4   ⁢   a             47       39       33         b   ⁢           ⁢   3         91       10       71       50                     ⁢   50           0   ⁢   f         15       63         9   ⁢   f         62       25       ee             c   ⁢           ⁢   6         87         9   ⁢   c           e   ⁢           ⁢   2           7   ⁢   e           0   ⁢   a           9   ⁢   d         28             9   ⁢   f         ce         c   ⁢           ⁢   7         85       48         d   ⁢           ⁢   0         ba       31             3   ⁢   c           6   ⁢   a         dd       94       13       aa       bf       77         ]       ;                     A   6   14     ⁢     x   4       =     [         42       60         f   ⁢           ⁢   6         75         d   ⁢           ⁢   4           f   ⁢           ⁢   5         16       80           21       76       15       20       76         d   ⁢           ⁢   5           c   ⁢           ⁢   0         40           43       42       83       03       61       00         b   ⁢           ⁢   6         95           83       74       61       61       21       20         e   ⁢           ⁢   0           a   ⁢           ⁢   0               a   ⁢           ⁢   0         17       22         c   ⁢           ⁢   1           b   ⁢           ⁢   5           c   ⁢           ⁢   1         42         d   ⁢           ⁢   7             83       03       36         c   ⁢           ⁢   1         35       15       36       54           37       97       83       02       01         a   ⁢           ⁢   0         75       60           76         d   ⁢           ⁢   5           b   ⁢           ⁢   6         22         d   ⁢           ⁢   6         55       77         e   ⁢           ⁢   3           ]       ;                     A   6   14     ⁢     y   6       =     [         31         7   ⁢           ⁢   d           3   ⁢   a           e   ⁢           ⁢   8         fb       00       fb         8   ⁢   d               3   ⁢   d           8   ⁢   b         10       bd         6   ⁢   a         00         6   ⁢   a         61           25         c   ⁢           ⁢   6         86         b   ⁢           ⁢   3         81       00       81       dd           39       33         b   ⁢           ⁢   3         13       10       00       10       71             0   ⁢   f         15       63         5   ⁢   b         62       00       62       25           87         9   ⁢   c           e   ⁢           ⁢   2           1   ⁢   a           0   ⁢   a         00         0   ⁢   a           9   ⁢   d             ce         c   ⁢           ⁢   7         85       83         d   ⁢           ⁢   0         00         d   ⁢           ⁢   0         ba             6   ⁢   a         dd       94       ec       aa       00       aa       bf         ]       ;                     A   6   14     ⁢     z   2       =     [         93         3   ⁢   d           8   ⁢   b         10       bd         6   ⁢   a         61       21           00       00       00       00       00       00       00       00           93         3   ⁢   d           8   ⁢   b         10       bd         6   ⁢   a         61       21             a   ⁢           ⁢   6         25         c   ⁢           ⁢   6         86         b   ⁢           ⁢   3         81       dd         4   ⁢   a             47       39       33         b   ⁢           ⁢   3         13       10       71       50           50         0   ⁢   f         15       63         5   ⁢   b         62       25       ee             c   ⁢           ⁢   6         87         9   ⁢   c           e   ⁢           ⁢   2           1   ⁢   a           0   ⁢   a           9   ⁢   d         28             9   ⁢   f         ce         c   ⁢           ⁢   7         85       83         d   ⁢           ⁢   0         ba       31         ]       ;         
Decryption at the 1 st  Diffusion-Center (6,2,7):
 
     
       
         
           
             
               
                 A 
                 1 
                 15 
               
               = 
               
                 
                   
                     A 
                     1 
                     14 
                   
                   ⁢ 
                   
                     F 
                     ⁡ 
                     
                       ( 
                       
                         6 
                         , 
                         2 
                         , 
                         7 
                       
                       ) 
                     
                   
                 
                 = 
                 
                   
                     
                       A 
                       1 
                       14 
                     
                     ⊕ 
                     
                       
                         A 
                         1 
                         14 
                       
                       ⁢ 
                       
                         x 
                         6 
                       
                     
                     ⊕ 
                     
                       
                         A 
                         1 
                         14 
                       
                       ⁢ 
                       
                         y 
                         2 
                       
                     
                     ⊕ 
                     
                       
                         A 
                         1 
                         14 
                       
                       ⁢ 
                       
                         z 
                         7 
                       
                     
                     ⊕ 
                     S 
                   
                   = 
                   
                     
                       A 
                       . 
                       
                         
 
                       
                       ⁢ 
                       
                         A 
                         1 
                         14 
                       
                     
                     = 
                     
                       [ 
                       
                         
                           
                             53 
                           
                           
                             
                               1 
                               ⁢ 
                               d 
                             
                           
                           
                             
                               4 
                               ⁢ 
                               f 
                             
                           
                           
                             
                               6 
                               ⁢ 
                               e 
                             
                           
                           
                             77 
                           
                           
                             
                               6 
                               ⁢ 
                               e 
                             
                           
                           
                             
                               e 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               8 
                             
                           
                           
                             28 
                           
                         
                         
                           
                             
                               b 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               4 
                             
                           
                           
                             
                               7 
                               ⁢ 
                               e 
                             
                           
                           
                             ba 
                           
                           
                             99 
                           
                           
                             
                               b 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               6 
                             
                           
                           
                             
                               a 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               6 
                             
                           
                           
                             37 
                           
                           
                             
                               e 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               6 
                             
                           
                         
                         
                           
                             
                               e 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               3 
                             
                           
                           
                             cc 
                           
                           
                             
                               f 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               6 
                             
                           
                           
                             40 
                           
                           
                             00 
                           
                           
                             
                               2 
                               ⁢ 
                               b 
                             
                           
                           
                             af 
                           
                           
                             70 
                           
                         
                         
                           
                             
                               8 
                               ⁢ 
                               d 
                             
                           
                           
                             27 
                           
                           
                             
                               8 
                               ⁢ 
                               e 
                             
                           
                           
                             59 
                           
                           
                             34 
                           
                           
                             16 
                           
                           
                             
                               b 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               6 
                             
                           
                           
                             75 
                           
                         
                         
                           
                             df 
                           
                           
                             19 
                           
                           
                             
                               c 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               1 
                             
                           
                           
                             34 
                           
                           
                             43 
                           
                           
                             41 
                           
                           
                             
                               9 
                               ⁢ 
                               e 
                             
                           
                           
                             
                               6 
                               ⁢ 
                               b 
                             
                           
                         
                         
                           
                             
                               3 
                               ⁢ 
                               a 
                             
                           
                           
                             78 
                           
                           
                             37 
                           
                           
                             
                               c 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               0 
                             
                           
                           
                             82 
                           
                           
                             ea 
                           
                           
                             
                               b 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               5 
                             
                           
                           
                             52 
                           
                         
                         
                           
                             
                               6 
                               ⁢ 
                               c 
                             
                           
                           
                             
                               c 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               9 
                             
                           
                           
                             
                               7 
                               ⁢ 
                               b 
                             
                           
                           
                             18 
                           
                           
                             37 
                           
                           
                             
                               d 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               5 
                             
                           
                           
                             60 
                           
                           
                             
                               1 
                               ⁢ 
                               c 
                             
                           
                         
                         
                           
                             
                               3 
                               ⁢ 
                               a 
                             
                           
                           
                             78 
                           
                           
                             37 
                           
                           
                             
                               c 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               0 
                             
                           
                           
                             82 
                           
                           
                             ea 
                           
                           
                             
                               b 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               5 
                             
                           
                           
                             52 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   A 
                   1 
                   14 
                 
                 ⁢ 
                 
                   x 
                   6 
                 
               
               = 
               
                 [ 
                 
                   
                     
                       89 
                     
                     
                       
                         0 
                         ⁢ 
                         e 
                       
                     
                     
                       87 
                     
                     
                       
                         d 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         7 
                       
                     
                     
                       db 
                     
                     
                       
                         d 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         7 
                       
                     
                     
                       
                         d 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         4 
                       
                     
                     
                       54 
                     
                   
                   
                     
                       
                         5 
                         ⁢ 
                         a 
                       
                     
                     
                       df 
                     
                     
                       
                         5 
                         ⁢ 
                         d 
                       
                     
                     
                       
                         0 
                         ⁢ 
                         c 
                       
                     
                     
                       
                         5 
                         ⁢ 
                         b 
                       
                     
                     
                       53 
                     
                     
                       
                         5 
                         ⁢ 
                         b 
                       
                     
                     
                       
                         d 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         3 
                       
                     
                   
                   
                     
                       
                         d 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         1 
                       
                     
                     
                       86 
                     
                     
                       db 
                     
                     
                       80 
                     
                     
                       00 
                     
                     
                       55 
                     
                     
                       57 
                     
                     
                       
                         d 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         8 
                       
                     
                   
                   
                     
                       06 
                     
                     
                       53 
                     
                     
                       07 
                     
                     
                       
                         8 
                         ⁢ 
                         c 
                       
                     
                     
                       
                         5 
                         ⁢ 
                         a 
                       
                     
                     
                       
                         0 
                         ⁢ 
                         b 
                       
                     
                     
                       
                         5 
                         ⁢ 
                         b 
                       
                     
                     
                       da 
                     
                   
                   
                     
                       
                         8 
                         ⁢ 
                         f 
                       
                     
                     
                       
                         0 
                         ⁢ 
                         c 
                       
                     
                     
                       80 
                     
                     
                       
                         5 
                         ⁢ 
                         a 
                       
                     
                     
                       81 
                     
                     
                       80 
                     
                     
                       
                         0 
                         ⁢ 
                         f 
                       
                     
                     
                       
                         d 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         5 
                       
                     
                   
                   
                     
                       
                         5 
                         ⁢ 
                         d 
                       
                     
                     
                       dc 
                     
                     
                       
                         5 
                         ⁢ 
                         b 
                       
                     
                     
                       80 
                     
                     
                       01 
                     
                     
                       
                         d 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         5 
                       
                     
                     
                       
                         5 
                         ⁢ 
                         a 
                       
                     
                     
                       89 
                     
                   
                   
                     
                       
                         d 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         6 
                       
                     
                     
                       84 
                     
                     
                       dd 
                     
                     
                       
                         0 
                         ⁢ 
                         c 
                       
                     
                     
                       
                         5 
                         ⁢ 
                         b 
                       
                     
                     
                       
                         8 
                         ⁢ 
                         a 
                       
                     
                     
                       
                         d 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         0 
                       
                     
                     
                       
                         0 
                         ⁢ 
                         e 
                       
                     
                   
                   
                     
                       
                         5 
                         ⁢ 
                         d 
                       
                     
                     
                       dc 
                     
                     
                       
                         5 
                         ⁢ 
                         b 
                       
                     
                     
                       80 
                     
                     
                       01 
                     
                     
                       
                         d 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         5 
                       
                     
                     
                       
                         5 
                         ⁢ 
                         a 
                       
                     
                     
                       89 
                     
                   
                 
                 ] 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   A 
                   1 
                   14 
                 
                 ⁢ 
                 
                   y 
                   2 
                 
               
               = 
               
                 [ 
                 
                   
                     
                       
                         1 
                         ⁢ 
                         d 
                       
                     
                     
                       00 
                     
                     
                       
                         1 
                         ⁢ 
                         d 
                       
                     
                     
                       
                         4 
                         ⁢ 
                         f 
                       
                     
                     
                       
                         6 
                         ⁢ 
                         e 
                       
                     
                     
                       77 
                     
                     
                       
                         6 
                         ⁢ 
                         e 
                       
                     
                     
                       
                         e 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         8 
                       
                     
                   
                   
                     
                       
                         7 
                         ⁢ 
                         e 
                       
                     
                     
                       00 
                     
                     
                       
                         7 
                         ⁢ 
                         e 
                       
                     
                     
                       ba 
                     
                     
                       99 
                     
                     
                       
                         b 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         6 
                       
                     
                     
                       
                         a 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         6 
                       
                     
                     
                       37 
                     
                   
                   
                     
                       cc 
                     
                     
                       00 
                     
                     
                       cc 
                     
                     
                       
                         f 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         6 
                       
                     
                     
                       40 
                     
                     
                       00 
                     
                     
                       
                         2 
                         ⁢ 
                         b 
                       
                     
                     
                       af 
                     
                   
                   
                     
                       27 
                     
                     
                       00 
                     
                     
                       27 
                     
                     
                       
                         8 
                         ⁢ 
                         e 
                       
                     
                     
                       59 
                     
                     
                       34 
                     
                     
                       16 
                     
                     
                       
                         b 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         6 
                       
                     
                   
                   
                     
                       19 
                     
                     
                       00 
                     
                     
                       19 
                     
                     
                       
                         c 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         1 
                       
                     
                     
                       34 
                     
                     
                       43 
                     
                     
                       41 
                     
                     
                       
                         9 
                         ⁢ 
                         e 
                       
                     
                   
                   
                     
                       78 
                     
                     
                       00 
                     
                     
                       78 
                     
                     
                       37 
                     
                     
                       
                         c 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         0 
                       
                     
                     
                       82 
                     
                     
                       ea 
                     
                     
                       
                         b 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         5 
                       
                     
                   
                   
                     
                       
                         c 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         9 
                       
                     
                     
                       00 
                     
                     
                       
                         c 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         9 
                       
                     
                     
                       
                         7 
                         ⁢ 
                         b 
                       
                     
                     
                       18 
                     
                     
                       37 
                     
                     
                       
                         d 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         5 
                       
                     
                     
                       60 
                     
                   
                   
                     
                       78 
                     
                     
                       00 
                     
                     
                       78 
                     
                     
                       
                         
                             
                         
                         ⁢ 
                         37 
                       
                     
                     
                       
                         c 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         0 
                       
                     
                     
                       82 
                     
                     
                       ea 
                     
                     
                       
                         b 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         5 
                       
                     
                   
                 
                 ] 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   A 
                   1 
                   14 
                 
                 ⁢ 
                 
                   z 
                   7 
                 
               
               = 
               
                 [ 
                 
                   
                     
                       
                         b 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         4 
                       
                     
                     
                       
                         7 
                         ⁢ 
                         e 
                       
                     
                     
                       ba 
                     
                     
                       99 
                     
                     
                       
                         b 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         6 
                       
                     
                     
                       
                         a 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         6 
                       
                     
                     
                       37 
                     
                     
                       
                         e 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         6 
                       
                     
                   
                   
                     
                       
                         e 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         3 
                       
                     
                     
                       cc 
                     
                     
                       
                         f 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         6 
                       
                     
                     
                       40 
                     
                     
                       00 
                     
                     
                       
                         2 
                         ⁢ 
                         b 
                       
                     
                     
                       af 
                     
                     
                       70 
                     
                   
                   
                     
                       
                         8 
                         ⁢ 
                         d 
                       
                     
                     
                       27 
                     
                     
                       
                         8 
                         ⁢ 
                         e 
                       
                     
                     
                       59 
                     
                     
                       34 
                     
                     
                       16 
                     
                     
                       
                         b 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         6 
                       
                     
                     
                       75 
                     
                   
                   
                     
                       df 
                     
                     
                       19 
                     
                     
                       
                         c 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         1 
                       
                     
                     
                       34 
                     
                     
                       43 
                     
                     
                       41 
                     
                     
                       
                         9 
                         ⁢ 
                         e 
                       
                     
                     
                       
                         6 
                         ⁢ 
                         b 
                       
                     
                   
                   
                     
                       
                         3 
                         ⁢ 
                         a 
                       
                     
                     
                       78 
                     
                     
                       37 
                     
                     
                       
                         c 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         0 
                       
                     
                     
                       82 
                     
                     
                       ea 
                     
                     
                       
                         b 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         5 
                       
                     
                     
                       52 
                     
                   
                   
                     
                       
                         6 
                         ⁢ 
                         c 
                       
                     
                     
                       
                         c 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         9 
                       
                     
                     
                       
                         7 
                         ⁢ 
                         b 
                       
                     
                     
                       18 
                     
                     
                       37 
                     
                     
                       
                         d 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         5 
                       
                     
                     
                       60 
                     
                     
                       
                         1 
                         ⁢ 
                         c 
                       
                     
                   
                   
                     
                       00 
                     
                     
                       00 
                     
                     
                       00 
                     
                     
                       00 
                     
                     
                       00 
                     
                     
                       00 
                     
                     
                       00 
                     
                     
                       00 
                     
                   
                   
                     
                       
                         6 
                         ⁢ 
                         c 
                       
                     
                     
                       
                         c 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         9 
                       
                     
                     
                       
                         7 
                         ⁢ 
                         b 
                       
                     
                     
                       18 
                     
                     
                       37 
                     
                     
                       
                         d 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         5 
                       
                     
                     
                       60 
                     
                     
                       
                         1 
                         ⁢ 
                         c 
                       
                     
                   
                 
                 ] 
               
             
             ; 
           
         
       
     
     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.