# Patent

## Title
Diffusion oriented method and apparatus for stream cryptography

## Abstract

The present invention applies with a diffused mechanism, as such, a variable series of diffusion functions embedded within a diffusion-medium, to generate a maximum diffusion-cycle and nonlinear complexity; additionally, it performs in a serial process for simple design, further, in a parallel process for saving time, or even in a hardware architecture, to gain greater acceleration. FIG. 1 shows an embodiment of the present invention in flow chart diagram form, comprising steps of: selecting a diffusion-area A, a diffusion-medium S, and a diffusion mechanism Ft1 100; initializing A by an input password 200; performing Ft1 to obtain a new value of A 300; inputting a plaintext in encryption or a ciphertext for decryption, bit by bit 400; XORing a plaintext bit in encryption or a ciphertext bit in decryption with a certain bit of A for stream output 500; continuing until completing the plaintext in encryption, or the ciphertext in decryption 600.

## Background
BACKGROUND OF THE INVENTION
1. Field of the Invention
The invention relates to a stream cryptographic method and apparatus therefor. More particularly, the invention relates to a diffusion mechanism for running stream cryptography bit by bit; other than the prior art with a fixed polynomial and Shift operations, the diffusion mechanism is combined flexibly with a variable series of diffusion functions embedded within a diffusion-medium control.
2. Description of the Related Art
The Applicant's following patent application is related to the invention and is incorporated herein by reference: “A Cryptographic Method of Multilayer Diffusion in Multidimension”, application Ser. No. 12/726,833, filed Mar. 18, 2010; the prior invention described a diffusion function for one position AF(p1, p2, . . . , pn), notated herein as AF(i1, i2, . . . , im)=A⊕Ad1i⊕A2i⊕ . . . ⊕Admi⊕S, the encryption/decryption is completed through a diffusion-cycle T, wherein T=2U+1, U=┌log2u┐, u=max(d1, d2, . . . , dm), based on the dimensional magnitude.
The present invention emphasizes multiple diffusion functions at a series of pre-arranged positions to maximize the diffusion cycle. Other than the prior invention for one position, the notation AF(p1, p2, . . . pk) herein is for a series of pre-arranged positions; further, it may be described more delicately as or precisely by A performs the diffusion functions F(p1, p2, . . . pk).
SUMMARY OF THE INVENTION
The present invention applies with a diffused mechanism, as such, a variable series of diffusion functions embedded within a diffusion-medium, to generate a maximum diffusion-cycle and nonlinear complexity; additionally, it performs in a serial process for simple design, further, in a parallel process for saving time, or even in a hardware architecture, to gain greater acceleration. Under the combination of diffusion functions, the output bits, with a wide and flexible random distribution, are thus hardly attacked during the long diffusion-cycle; the steps are shown as follows:

    
    
        (a) selecting a diffusion mechanism, a diffusion-area, and a diffusion-medium;
        (b) setting an initial value of the diffusion-area by an input password;
        (c) running the diffusion mechanism to obtain a new value for the diffusion-area;
        (d) inputting a bit of a plaintext/ciphertext stream in sequence;
        (e) outputting a bit of the ciphertext/plaintext stream, equally, XORing (exclusive-or (XOR) operation) the bit of the plaintext/ciphertext stream with an assigned bit of the diffusion-area;
        (f) going back to Step (c) until the encryption/decryption is complete.
    
    





BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a summary flow chart diagram showing the main steps taken while encrypting/decrypting by a diffusion mechanism according to the present invention;
FIG. 2 is a summary flow chart diagram of FIG. 1, 200 showing the steps taken while initializing by an input password according to the present invention;
FIG. 3A is a summary flow chart diagram showing the component steps taken while performing a diffusion mechanism including a medium-constant in hardware architecture according to the present invention;
FIG. 3B is a summary flow chart diagram showing the component steps taken while performing a diffusion mechanism excluding a medium-constant in hardware architecture according to the present invention;
FIG. 3C is a summary flow chart diagram showing the component steps taken while encrypting/decrypting by a diffusion mechanism in hardware architecture according to the present invention;
FIG. 4 is a summary schematic diagram of preferred Embodiment I showing a diffusion-table in hardware architecture according to the present invention;
FIG. 5 is a summary schematic diagram of preferred Embodiments showing a parallel connecting in hardware architecture according to the present invention;
FIG. 6 is a summary schematic diagram of preferred Embodiments showing a serial connecting in hardware architecture according to the present invention.



DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
FIG. 1 shows an embodiment of the present invention in flow chart diagram form, comprising steps of: selecting a diffusion-area A, a diffusion-medium S, and a diffusion mechanism Ft1 100; initializing A by an input password 200; performing Ft1 to obtain a new value of A 300; inputting a plaintext in encryption or a ciphertext for decryption, bit by bit 400; XORing a plaintext bit in encryption or a ciphertext bit in decryption with a certain bit of A for stream output 500; continuing until completing the plaintext in encryption, or the ciphertext in decryption 600.
Moreover, FIG. 2, i.e. FIG. 1, 200, shows an initialization of the diffusion-area by an input password, comprising steps of: inputting the password, bit by bit 210; setting the password bit whether controlling over S 220, if yes, then performing Ft1 whether adding S by the value (0 or 1) of the password bit, to obtain a new value of A 240, otherwise, writing the password bit directly to A 230; continuing until completing the password 250.
Serial Process in Detail:
FIG. 1, 300, obtaining a new value At1×t2=At1×(t2−1)Ft1 by a serial process, it would implement a number of diffusion functions F(p1, p2, . . . , pk), provided with a simple design but rather time consuming. Furthermore, for the initialization of A, FIG. 2, a password bit selects a first value for AF(p)=A⊕Ad1i⊕Ad2i⊕ . . . ⊕Admi⊕S, and a second value for AF(p)=A⊕Ad1i⊕Ad2i⊕ . . . ⊕Admi. The cryptographic process flow comprises steps of:

    
    
        1. choosing a diffusion-area A, with a diffusion mechanism Ft1;
        2. setting A's initial value A0 by a password; t2=1;
        3. obtaining a new value At1×t2, wherein At1×t2=At1×(t2−1)Ft1;
        4. inputting a bit of a plaintext/ciphertext stream in sequence;
        5. outputting a bit of the ciphertext/plaintext stream, equally, XORing the bit of the plaintext/ciphertext stream with an assigned bit of At1×t2;
        6. letting t2=t2+1, going back to Step 3, until the encryption/decryption is complete.

Symbols and Definitions:

        A: a diffusion-area, wherein A is a d1×d2× . . . ×dm binary matrix with a combination of positions labeled sequentially from 1 to n, accordingly, the bit values from a1 to an; where the positions are shown as below:

for example,

    
    





  
    
      
        
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        S: a diffusion-medium, wherein S is a m-dimensional binary matrix, and further, includes an anchor-position {dot over (S)}.
        AF(p): A performs a diffusion function F(p), see application Ser. No. 12/726,833, Point-Diffusion, wherein p transforms into m-dimension coordinates (i1, i2, . . . , im), then AF(p)=A⊕Ad1i⊕Ad2i⊕ . . . ⊕Admi⊕S.
        AF(p1, p2, . . . , pk): A performs the diffusion functions F(p1, p2, . . . , pk), sequentially at the m-dimension positions p1, p2, . . . , pk.
        AFt1: A performs a diffusion mechanism Ft1, as abbreviated for F(p1, p2t1, . . . , pk), i.e. F(p1, p2, . . . , pk) repeated t1 times.

Preferred Embodiment I: a 1×16 Diffusion-Area A, a Diffusion Mechanism F1.

    
    


Suppose that F1=F(13, 14, 15, 16, 1, . . . , 13) applied to A, further, A0=0, and S=1 placed at the anchor-position, with the encrypting value A1×1=A0F1 at the first time, t2=1. To detail the diffusion function of every position, the sequential positions are notated as no. p: no. q in abbreviation thereafter, for example 13:15 for 13,14,15, or 13:1 for 13,14,15,16,1, and so on; the steps are listed as follows (S=1 is indicated in boldface hereafter):

    A0F(p)=A0⊕A0yp⊕S;
    A0F(13)=[0000000000000000]⊕[0000000000000000]⊕[0000000000001000];
    A0F(13:14)=[0000000000001000]⊕[0000000000010000]⊕[0000000000000100];
    A0F(13:15)=[0000000000011100]⊕[0000000000111000]⊕[0000000000000010];
    A0F(13:16)=[0000000000100110]⊕[0000000001001100]⊕[0000000000000001];
    A0F(13:1)=[0000000001101011]⊕[0000000000110101]⊕[1000000000000000];
    A0F(13:2)=[1000000001011110]⊕[0000000000101111]⊕[0100000000000000];
    A0F(13:3)=[1100000001110001]⊕[1000000000111000]⊕[0010000000000000];
    A0F(13:4)=[0110000001001001]⊕[1100000000100100]⊕[0001000000000000];
    A0F(13:5)=[1011000001101101]⊕[0110000000110110]⊕[0000100000000000];
    A0F(13:6)=[1101100001011011]⊕[1011000000101101]⊕[0000010000000000];
    A0F(13:7)=[0110110001110110]⊕[1101100000111011]⊕[0000001000000000];
    A0F(13:8)=[1011011001001101]⊕[0110110000100110]⊕[0000000100000000];
    A0F(13:9)=[1101101101101011]⊕[1011011000110101]⊕[0000000010000000];
    A0F(13:10)=[0110110111011110]⊕[1101101110101111]⊕[0000000001000000];
    A0F(13:11)=[1011011000110001]⊕[0110110001011000]⊕[0000000000100000];
    A0F(13:12)=[1101101001001001]⊕[1011010010000100]⊕[0000000000010000];
    A0F(13:13)=[0110111011011101]⊕[1101110110110110]⊕[0000000000001000].


Through A0F(13:13), thus, A1×1=[1011001101100011], afterward, XORing a bit of A, assuming a16, with a plaintext bit to output an encrypted bit. Next, let t2=2, and the value A1×2 is thus derived from A1×1 which can be equivalently notated as A1 in abbreviation hereafter; the details are shown as below:

    A1F(p)=A1⊕A1yp⊕S;
    A1F(13)=[1011001101100011]⊕[0110011011000001]⊕[0000000000001000];
    A1F(13:14)=[1101010110101010]⊕[1010101101010001]⊕[0000000000000100];
    A1F(13:15)=[0111111011111111]⊕[1111110111111101]⊕[0000000000000010];
    A1F(13:16)=[1000001100000000]⊕[0000011000000000]⊕[0000000000000001];
    A1F(13:1)=[1000010100000001]⊕[0100001010000000]⊕[1000000000000000];
    A1F(13:2)=[0100011110000001]⊕[1010001111000000]⊕[0100000000000000];
    A1F(13:3)=[1010010001000001]⊕[0101001000100000]⊕[0010000000000000];
    A1F(13:4)=[1101011001100001]⊕[1010101100110000]⊕[0001000000000000];
    A1F(13:5)=[0110110101010001]⊕[1101011010101000]⊕[0000100000000000];
    A1F(13:6)=[1011001111111001]⊕[0110000111111100]⊕[0000010000000000];
    A1F(13:7)=[1101011000000101]⊕[1010110100000010]⊕[0000001000000000];
    A1F(13:8)=[0111100100000111]⊕[1111001010000011]⊕[0000000100000000];
    A1F(13:9)=[1000101010000100]⊕[0001010101000010]⊕[0000000010000000];
    A1F(13:10)=[1001111101000110]⊕[0011111010100011]⊕[0000000001000000];
    A1F(13:11)=[1010000110100101]⊕[0100001101010010]⊕[0000000000100000];
    A1F(13:12)=[1110001011010111]⊕[1100010110101011]⊕[0000000000010000];
    A1F(13:13)=[0010011101101100]⊕[0100111011010110]⊕[0000000000001000].


The second value of the diffusion-area A1×2=[0110100110110010]; then, another encrypted bit is generated. Accordingly, a diffusion-cycle of F1, implying A1×t2=A1, requires 216−1 times; thus, running the diffusion mechanism consecutively through the diffusion-cycle, the encrypted data complies with a random distribution thereby avoiding successful crypto-attack; assume a plaintext stream=0, four segments (64 bits) of the random output are appended as below, as samples:
Four Segments: (t2=1 to 64)
1011011100111011 (t2=1 to 16), 0000100100010111 (t2=17 to 32),
0100000011010100 (t2=33 to 48), 1011011111111110 (t2=49 to 64).
Preferred Embodiment II: a 1×16 Diffusion-Area A, a Diffusion Mechanism F2.
According to a diffusion mechanism arranged with the same positions in sequence, but at different times, the output is sampled out at every corresponding time. Suppose that F2=F(132:13) A0=[0000000000000000] and S=1 is placed at the anchor-position, the encrypting value A2×1 thus is obtained from A2×0F2 at the first time, t2=1, see A2 of Embodiment I, short for A2×1. The next value A2×2, notated as A4 for short, is thus obtained by running the diffusion functions of the value A3; the steps are also listed as below to let any skilled person gain more understanding: (A3=[1000010001101011])

    A3F(p)=A3⊕A3yp⊕S;
    A3F(13)=[1000010001101011]⊕[0000100011010101]⊕[0000000000001000];
    A3F(13:14)=[1000110010110110]⊕[0001100101101011]⊕[0000000000000100];
    A3F(13:15)=[1001010111011001]⊕[0010101110110000]⊕[0000000000000010];
    A3F(13:16)=[1011111001101011]⊕[0111110011010110]⊕[0000000000000001];
    A3F(13:1)=[1100001010111100]⊕[0110000101011110]⊕[1000000000000000];
    A3F(13:2)=[0010001111100010]⊕[0001000111110001]⊕[0100000000000000];
    A3F(13:3)=[0111001000010011]⊕[1101100100001001]⊕[0010000000000000];
    A3F(13:4)=[1000101100011010]⊕[0000010110001101]⊕[0001000000000000];
    A3F(13:5)=[1001111010010111]⊕[0011011101001011]⊕[0000100000000000];
    A3F(13:6)=[1010000111011100]⊕[0100000011101110]⊕[0000010000000000];
    A3F(13:7)=[1110010100110010]⊕[1100100010011001]⊕[0000001000000000];
    A3F(13:8)=[0010111110101011]⊕[0101111011010101]⊕[0000000100000000];
    A3F(13:9)=[0111000001111110]⊕[1110000000111111]⊕[0000000010000000];
    A3F(13:10)=[1001000011000001]⊕[0010000110100000]⊕[0000000001000000];
    A3F(13:11)=[1011000100100001]⊕[0110001001010000]⊕[0000000000100000];
    A3F(13:12)=[1101001101010001]⊕[1010011010101000]⊕[0000000000010000];
    A3F(13:13)=[0111010111101001]⊕[1110101111010100]⊕[0000000000001000].


The new value of the diffusion-area will be A2×2=[1001111000110101]. According to the diffusion mechanism using the number of times “2” is a prime number, therefore, the diffusion-cycle still keeps on 216−1 times; thus, also assume a plaintext stream=0, XORing a16 of A, the output other than Embodiment I is sampled out every two times; with four fragments of the random output appended as below, as samples:
Four Segments: (t2=1 to 64)
0111010100010111 (t2=1 to 16), 1000111001111110 (t2=17 to 32),
1000010100011110 (t2=33 to 48), 1101011100000100 (t2=49 to 64).
Preferred Embodiment III: a 4×4 Diffusion-Area A, a Diffusion Mechanism F1.
Provided that F1=F(8:8), A0=[0000000000000000] and S=1 placed at the anchor-position, the encrypting value A1×1 is obtained from A1×0 run by F1 at the first time, t=1. Now that the diffusion-area is a two-dimension matrix, a position needs to be transformed to the corresponding coordinates (i, j); the details are shown as below (Ax(0), Ay(0) also indicated in boldface hereafter):



  
    
      
        AF
        ⁡
        
          (
          p
          )
        
      
      =
      
        A
        ⊕
        
          Ax
          i
        
        ⊕
        
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          j
        
        ⊕
        S
      
    
    ;
  




  
    
      
        
          
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                    0000
                  
                
                
                  
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            ⊕
            
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          ;
        
      
    
  




  
    
      
        
          
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              [
              
                
                  
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            ⊕
            
              [
              
                
                  
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              ]
            
            ⊕
            
              [
              
                
                  
                    0000
                  
                
                
                  
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              ]
            
          
        
      
    
    
      
        
          
            =
            
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              [
              
                
                  
                    0111
                  
                
                
                  
                    0010
                  
                
                
                  
                    0000
                  
                
                
                  
                    0100
                  
                
              
              ]
            
          
          ;
        
      
    
  




  
    
      
        
          
            AF
            ⁡
            
              (
              
                8
                :
                11
              
              )
            
          
          =
          
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            ⊕
            
              Ax
              3
            
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              [
              
                
                  
                    0111
                  
                
                
                  
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                    0100
                  
                
              
              ]
            
            ⊕
            
              [
              
                
                  
                    0010
                  
                
                
                  
                    0000
                  
                
                
                  
                    0000
                  
                
                
                  
                    0000
                  
                
              
              ]
            
            ⊕
            
              [
              
                
                  
                    1101
                  
                
                
                  
                    0101
                  
                
                
                  
                    0000
                  
                
                
                  
                    1000
                  
                
              
              ]
            
            ⊕
            
              [
              
                
                  
                    0000
                  
                
                
                  
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                    0000
                  
                
              
              ]
            
          
        
      
    
    
      
        
          
            =
            
            ⁢
            
              [
              
                
                  
                    1000
                  
                
                
                  
                    0111
                  
                
                
                  
                    0010
                  
                
                
                  
                    1100
                  
                
              
              ]
            
          
          ;
        
      
    
  




  
    
      
        
          
            AF
            ⁡
            
              (
              
                8
                :
                12
              
              )
            
          
          =
          
          ⁢
          
            A
            ⊕
            
              Ax
              4
            
            ⊕
            
              Ay
              3
            
            ⊕
            S
          
        
      
    
    
      
        
          =
          
          ⁢
          
            
              [
              
                
                  
                    1000
                  
                
                
                  
                    0111
                  
                
                
                  
                    0010
                  
                
                
                  
                    1100
                  
                
              
              ]
            
            ⊕
            
              [
              
                
                  
                    0111
                  
                
                
                  
                    0010
                  
                
                
                  
                    1100
                  
                
                
                  
                    0000
                  
                
              
              ]
            
            ⊕
            
              [
              
                
                  
                    0000
                  
                
                
                  
                    1101
                  
                
                
                  
                    0101
                  
                
                
                  
                    1000
                  
                
              
              ]
            
            ⊕
            
              [
              
                
                  
                    0000
                  
                
                
                  
                    0000
                  
                
                
                  
                    0000
                  
                
                
                  
                    0010
                  
                
              
              ]
            
          
        
      
    
    
      
        
          
            =
            
            ⁢
            
              [
              
                
                  
                    1111
                  
                
                
                  
                    1000
                  
                
                
                  
                    1011
                  
                
                
                  
                    0110
                  
                
              
              ]
            
          
          ;
        
      
    
  




  
    
      
        
          
            AF
            ⁡
            
              (
              
                8
                :
                13
              
              )
            
          
          =
          
          ⁢
          
            A
            ⊕
            
              Ax
              1
            
            ⊕
            
              Ay
              4
            
            ⊕
            S
          
        
      
    
    
      
        
          =
          
          ⁢
          
            
              [
              
                
                  
                    1111
                  
                
                
                  
                    1000
                  
                
                
                  
                    1011
                  
                
                
                  
                    0110
                  
                
              
              ]
            
            ⊕
            
              [
              
                
                  
                    0000
                  
                
                
                  
                    1111
                  
                
                
                  
                    1000
                  
                
                
                  
                    1011
                  
                
              
              ]
            
            ⊕
            
              [
              
                
                  
                    1110
                  
                
                
                  
                    0000
                  
                
                
                  
                    0110
                  
                
                
                  
                    1100
                  
                
              
              ]
            
            ⊕
            
              [
              
                
                  
                    0001
                  
                
                
                  
                    0000
                  
                
                
                  
                    0000
                  
                
                
                  
                    0000
                  
                
              
              ]
            
          
        
      
    
    
      
        
          
            =
            
            ⁢
            
              [
              
                
                  
                    0000
                  
                
                
                  
                    0111
                  
                
                
                  
                    0101
                  
                
                
                  
                    0001
                  
                
              
              ]
            
          
          ;
        
      
    
  




  
    
      
        
          
            AF
            ⁡
            
              (
              
                8
                :
                14
              
              )
            
          
          =
          
          ⁢
          
            A
            ⊕
            
              Ax
              2
            
            ⊕
            
              Ay
              4
            
            ⊕
            S
          
        
      
    
    
      
        
          =
          
          ⁢
          
            
              [
              
                
                  
                    0000
                  
                
                
                  
                    0111
                  
                
                
                  
                    0101
                  
                
                
                  
                    0001
                  
                
              
              ]
            
            ⊕
            
              [
              
                
                  
                    0111
                  
                
                
                  
                    0000
                  
                
                
                  
                    0111
                  
                
                
                  
                    0101
                  
                
              
              ]
            
            ⊕
            
              [
              
                
                  
                    0000
                  
                
                
                  
                    1110
                  
                
                
                  
                    1010
                  
                
                
                  
                    0010
                  
                
              
              ]
            
            ⊕
            
              [
              
                
                  
                    0000
                  
                
                
                  
                    0001
                  
                
                
                  
                    0000
                  
                
                
                  
                    0000
                  
                
              
              ]
            
          
        
      
    
    
      
        
          
            =
            
            ⁢
            
              [
              
                
                  
                    0111
                  
                
                
                  
                    1000
                  
                
                
                  
                    1000
                  
                
                
                  
                    0110
                  
                
              
              ]
            
          
          ;
        
      
    
  




  
    
      
        
          
            AF
            ⁡
            
              (
              
                8
                :
                15
              
              )
            
          
          =
          
          ⁢
          
            A
            ⊕
            
              Ax
              3
            
            ⊕
            
              Ay
              4
            
            ⊕
            S
          
        
      
    
    
      
        
          =
          
          ⁢
          
            
              [
              
                
                  
                    0111
                  
                
                
                  
                    1000
                  
                
                
                  
                    1000
                  
                
                
                  
                    0110
                  
                
              
              ]
            
            ⊕
            
              [
              
                
                  
                    1000
                  
                
                
                  
                    1000
                  
                
                
                  
                    0000
                  
                
                
                  
                    1000
                  
                
              
              ]
            
            ⊕
            
              [
              
                
                  
                    1110
                  
                
                
                  
                    0000
                  
                
                
                  
                    0000
                  
                
                
                  
                    1100
                  
                
              
              ]
            
            ⊕
            
              [
              
                
                  
                    0000
                  
                
                
                  
                    0000
                  
                
                
                  
                    0001
                  
                
                
                  
                    0000
                  
                
              
              ]
            
          
        
      
    
    
      
        
          
            =
            
            ⁢
            
              [
              
                
                  
                    0001
                  
                
                
                  
                    0000
                  
                
                
                  
                    1001
                  
                
                
                  
                    0010
                  
                
              
              ]
            
          
          ;
        
      
    
  




  
    
      
        
          
            AF
            ⁡
            
              (
              
                8
                :
                16
              
              )
            
          
          =
          
          ⁢
          
            A
            ⊕
            
              Ax
              4
            
            ⊕
            
              Ay
              4
            
            ⊕
            S
          
        
      
    
    
      
        
          =
          
          ⁢
          
            
              [
              
                
                  
                    0001
                  
                
                
                  
                    0000
                  
                
                
                  
                    1001
                  
                
                
                  
                    0010
                  
                
              
              ]
            
            ⊕
            
              [
              
                
                  
                    0000
                  
                
                
                  
                    1001
                  
                
                
                  
                    0010
                  
                
                
                  
                    0000
                  
                
              
              ]
            
            ⊕
            
              [
              
                
                  
                    0010
                  
                
                
                  
                    0000
                  
                
                
                  
                    0010
                  
                
                
                  
                    0100
                  
                
              
              ]
            
            ⊕
            
              [
              
                
                  
                    0000
                  
                
                
                  
                    0000
                  
                
                
                  
                    0000
                  
                
                
                  
                    0001
                  
                
              
              ]
            
          
        
      
    
    
      
        
          
            =
            
            ⁢
            
              [
              
                
                  
                    0011
                  
                
                
                  
                    1001
                  
                
                
                  
                    1001
                  
                
                
                  
                    0111
                  
                
              
              ]
            
          
          ;
        
      
    
  




  
    
      
        
          
            AF
            ⁡
            
              (
              
                8
                :
                1
              
              )
            
          
          =
          
          ⁢
          
            A
            ⊕
            
              Ax
              1
            
            ⊕
            
              Ay
              1
            
            ⊕
            S
          
        
      
    
    
      
        
          =
          
          ⁢
          
            
              [
              
                
                  
                    0011
                  
                
                
                  
                    1001
                  
                
                
                  
                    1001
                  
                
                
                  
                    0111
                  
                
              
              ]
            
            ⊕
            
              [
              
                
                  
                    0000
                  
                
                
                  
                    0011
                  
                
                
                  
                    1001
                  
                
                
                  
                    1001
                  
                
              
              ]
            
            ⊕
            
              [
              
                
                  
                    0001
                  
                
                
                  
                    0100
                  
                
                
                  
                    0100
                  
                
                
                  
                    0011
                  
                
              
              ]
            
            ⊕
            
              [
              
                
                  
                    1000
                  
                
                
                  
                    0000
                  
                
                
                  
                    0000
                  
                
                
                  
                    0000
                  
                
              
              ]
            
          
        
      
    
    
      
        
          
            =
            
            ⁢
            
              [
              
                
                  
                    1010
                  
                
                
                  
                    1110
                  
                
                
                  
                    0100
                  
                
                
                  
                    1101
                  
                
              
              ]
            
          
          ;
        
      
    
  




  
    
      
        
          
            AF
            ⁡
            
              (
              
                8
                :
                2
              
              )
            
          
          =
          
          ⁢
          
            A
            ⊕
            
              Ax
              2
            
            ⊕
            
              Ay
              1
            
            ⊕
            S
          
        
      
    
    
      
        
          =
          
          ⁢
          
            
              [
              
                
                  
                    1010
                  
                
                
                  
                    1110
                  
                
                
                  
                    0100
                  
                
                
                  
                    1101
                  
                
              
              ]
            
            ⊕
            
              [
              
                
                  
                    1110
                  
                
                
                  
                    0000
                  
                
                
                  
                    1110
                  
                
                
                  
                    0100
                  
                
              
              ]
            
            ⊕
            
              [
              
                
                  
                    0101
                  
                
                
                  
                    0111
                  
                
                
                  
                    0010
                  
                
                
                  
                    0110
                  
                
              
              ]
            
            ⊕
            
              [
              
                
                  
                    0000
                  
                
                
                  
                    1000
                  
                
                
                  
                    0000
                  
                
                
                  
                    0000
                  
                
              
              ]
            
          
        
      
    
    
      
        
          
            =
            
            ⁢
            
              [
              
                
                  
                    0001
                  
                
                
                  
                    0001
                  
                
                
                  
                    1000
                  
                
                
                  
                    1111
                  
                
              
              ]
            
          
          ;
        
      
    
  




  
    
      
        
          
            AF
            ⁡
            
              (
              
                8
                :
                3
              
              )
            
          
          =
          
          ⁢
          
            A
            ⊕
            
              Ax
              3
            
            ⊕
            
              Ay
              1
            
            ⊕
            S
          
        
      
    
    
      
        
          =
          
          ⁢
          
            
              [
              
                
                  
                    0001
                  
                
                
                  
                    0001
                  
                
                
                  
                    1000
                  
                
                
                  
                    1111
                  
                
              
              ]
            
            ⊕
            
              [
              
                
                  
                    0001
                  
                
                
                  
                    1000
                  
                
                
                  
                    0000
                  
                
                
                  
                    1000
                  
                
              
              ]
            
            ⊕
            
              [
              
                
                  
                    0000
                  
                
                
                  
                    0000
                  
                
                
                  
                    0100
                  
                
                
                  
                    0111
                  
                
              
              ]
            
            ⊕
            
              [
              
                
                  
                    0000
                  
                
                
                  
                    0000
                  
                
                
                  
                    1000
                  
                
                
                  
                    0000
                  
                
              
              ]
            
          
        
      
    
    
      
        
          
            =
            
            ⁢
            
              [
              
                
                  
                    0000
                  
                
                
                  
                    1001
                  
                
                
                  
                    0100
                  
                
                
                  
                    0000
                  
                
              
              ]
            
          
          ;
        
      
    
  




  
    
      
        
          
            AF
            ⁡
            
              (
              
                8
                :
                4
              
              )
            
          
          =
          
          ⁢
          
            A
            ⊕
            
              Ax
              4
            
            ⊕
            
              Ay
              1
            
            ⊕
            S
          
        
      
    
    
      
        
          =
          
          ⁢
          
            
              [
              
                
                  
                    0000
                  
                
                
                  
                    1001
                  
                
                
                  
                    0100
                  
                
                
                  
                    0000
                  
                
              
              ]
            
            ⊕
            
              [
              
                
                  
                    1001
                  
                
                
                  
                    0100
                  
                
                
                  
                    0000
                  
                
                
                  
                    0000
                  
                
              
              ]
            
            ⊕
            
              [
              
                
                  
                    0000
                  
                
                
                  
                    0100
                  
                
                
                  
                    0010
                  
                
                
                  
                    0000
                  
                
              
              ]
            
            ⊕
            
              [
              
                
                  
                    0000
                  
                
                
                  
                    0000
                  
                
                
                  
                    0000
                  
                
                
                  
                    1000
                  
                
              
              ]
            
          
        
      
    
    
      
        
          
            =
            
            ⁢
            
              [
              
                
                  
                    1001
                  
                
                
                  
                    1001
                  
                
                
                  
                    0110
                  
                
                
                  
                    1000
                  
                
              
              ]
            
          
          ;
        
      
    
  




  
    
      
        
          
            AF
            ⁡
            
              (
              
                8
                :
                5
              
              )
            
          
          =
          
          ⁢
          
            A
            ⊕
            
              Ax
              1
            
            ⊕
            
              Ay
              2
            
            ⊕
            S
          
        
      
    
    
      
        
          =
          
          ⁢
          
            
              [
              
                
                  
                    1001
                  
                
                
                  
                    1001
                  
                
                
                  
                    0110
                  
                
                
                  
                    1000
                  
                
              
              ]
            
            ⊕
            
              [
              
                
                  
                    0000
                  
                
                
                  
                    1001
                  
                
                
                  
                    1001
                  
                
                
                  
                    0110
                  
                
              
              ]
            
            ⊕
            
              [
              
                
                  
                    0000
                  
                
                
                  
                    0000
                  
                
                
                  
                    1011
                  
                
                
                  
                    0000
                  
                
              
              ]
            
            ⊕
            
              [
              
                
                  
                    0100
                  
                
                
                  
                    0000
                  
                
                
                  
                    0000
                  
                
                
                  
                    0000
                  
                
              
              ]
            
          
        
      
    
    
      
        
          
            =
            
            ⁢
            
              [
              
                
                  
                    1101
                  
                
                
                  
                    0000
                  
                
                
                  
                    0100
                  
                
                
                  
                    1110
                  
                
              
              ]
            
          
          ;
        
      
    
  




  
    
      
        
          
            AF
            ⁡
            
              (
              
                8
                :
                6
              
              )
            
          
          =
          
          ⁢
          
            A
            ⊕
            
              Ax
              2
            
            ⊕
            
              Ay
              2
            
            ⊕
            S
          
        
      
    
    
      
        
          =
          
          ⁢
          
            
              [
              
                
                  
                    1101
                  
                
                
                  
                    0000
                  
                
                
                  
                    0100
                  
                
                
                  
                    1110
                  
                
              
              ]
            
            ⊕
            
              [
              
                
                  
                    0000
                  
                
                
                  
                    0000
                  
                
                
                  
                    0000
                  
                
                
                  
                    0100
                  
                
              
              ]
            
            ⊕
            
              [
              
                
                  
                    1010
                  
                
                
                  
                    0000
                  
                
                
                  
                    1010
                  
                
                
                  
                    1011
                  
                
              
              ]
            
            ⊕
            
              [
              
                
                  
                    0000
                  
                
                
                  
                    0100
                  
                
                
                  
                    0000
                  
                
                
                  
                    0000
                  
                
              
              ]
            
          
        
      
    
    
      
        
          
            =
            
            ⁢
            
              [
              
                
                  
                    0111
                  
                
                
                  
                    0100
                  
                
                
                  
                    1110
                  
                
                
                  
                    0001
                  
                
              
              ]
            
          
          ;
        
      
    
  




  
    
      
        
          
            AF
            ⁡
            
              (
              
                8
                :
                7
              
              )
            
          
          =
          
          ⁢
          
            A
            ⊕
            
              Ax
              3
            
            ⊕
            
              Ay
              2
            
            ⊕
            S
          
        
      
    
    
      
        
          =
          
          ⁢
          
            
              [
              
                
                  
                    0111
                  
                
                
                  
                    0100
                  
                
                
                  
                    1110
                  
                
                
                  
                    0001
                  
                
              
              ]
            
            ⊕
            
              [
              
                
                  
                    0100
                  
                
                
                  
                    1110
                  
                
                
                  
                    0000
                  
                
                
                  
                    1110
                  
                
              
              ]
            
            ⊕
            
              [
              
                
                  
                    1011
                  
                
                
                  
                    1010
                  
                
                
                  
                    1011
                  
                
                
                  
                    0000
                  
                
              
              ]
            
            ⊕
            
              [
              
                
                  
                    0000
                  
                
                
                  
                    0000
                  
                
                
                  
                    0100
                  
                
                
                  
                    0000
                  
                
              
              ]
            
          
        
      
    
    
      
        
          
            =
            
            ⁢
            
              [
              
                
                  
                    1000
                  
                
                
                  
                    0000
                  
                
                
                  
                    0001
                  
                
                
                  
                    1111
                  
                
              
              ]
            
          
          ;
        
      
    
  




  
    
      
        
          
            AF
            ⁡
            
              (
              
                8
                :
                8
              
              )
            
          
          =
          
          ⁢
          
            A
            ⊕
            
              Ax
              4
            
            ⊕
            
              Ay
              2
            
            ⊕
            S
          
        
      
    
    
      
        
          =
          
          ⁢
          
            
              [
              
                
                  
                    1000
                  
                
                
                  
                    0000
                  
                
                
                  
                    0001
                  
                
                
                  
                    1111
                  
                
              
              ]
            
            ⊕
            
              [
              
                
                  
                    0000
                  
                
                
                  
                    0001
                  
                
                
                  
                    1111
                  
                
                
                  
                    0000
                  
                
              
              ]
            
            ⊕
            
              [
              
                
                  
                    0000
                  
                
                
                  
                    0000
                  
                
                
                  
                    0000
                  
                
                
                  
                    1011
                  
                
              
              ]
            
            ⊕
            
              [
              
                
                  
                    0000
                  
                
                
                  
                    0000
                  
                
                
                  
                    0000
                  
                
                
                  
                    0100
                  
                
              
              ]
            
          
        
      
    
    
      
        
          =
          
          ⁢
          
            
              [
              
                
                  
                    1000
                  
                
                
                  
                    0001
                  
                
                
                  
                    1110
                  
                
                
                  
                    0000
                  
                
              
              ]
            
            .
          
        
      
    
  



Assume again the last bit of A1×t2 for encrypting and a plaintext stream=0, due to a two-dimension diffusion mechanism, four fragments of the random output, which are thus very different from samples obtained for Embodiment I & II, are appended as below:
Four Segments: (t2=1 to 64)
0111000100100111 (t2=1 to 16), 0000001100001011 (t2=17 to 32),
1110101001111110 (t2=33 to 48), 0011000001101100 (t2=49 to 64).
Parallel Process in Detail:
FIG. 1, 300, obtaining a new value At1×t2=At1×(t2−1)Ft1 by a parallel process, it would transform the serial process into At1×t2=At1×(t2−1)Ft1⊕BFt1 namely, by running a diffusion-table Ft1 and a medium-constant BFt1, through a slightly complicated design but with the concomitant advantage of shortening the time required. Furthermore, for the initialization of A, FIG. 2, a password bit selects a first value for At1×t2=At1×(t2−1)Ft1⊕BFt1, and a second value for At1×t2=At1×(t2−1)Ft1.
The cryptographic process flow comprises steps of:


    
    
        1. choosing a diffusion-area A, creating a diffusion-table Ft1;
        2. choosing a medium-area B, creating a medium-constant BFt1;
        3. setting A's initial value A0 by a password; t2=1;
        4. obtaining a new value At1×t2, wherein At1×t2=At1×(t2−1)Ft1⊕BFt1;
        5. inputting a bit of a plaintext/ciphertext stream in sequence;
        6. outputting a bit of the ciphertext/plaintext stream, equally, XORing the bit of the plaintext/ciphertext stream with an assigned bit of At1×t2;
        7. letting t2=t2+1, going back to Step 4, until the encryption/decryption is complete.

Symbols and Definitions:

        A: a diffusion-area, wherein A contains an initial value A0 and further, is a d1×d2× . . . ×dm binary matrix with a combination of positions labeled sequentially from 1 to n, accordingly, the bit values from a1 to an.
        S: a diffusion-medium, wherein S is a m-dimensional binary matrix, and further, includes an anchor-position {dot over (S)}.
        B: a medium-area, wherein B is a d1×d2× . . . ×dm zero matrix with a combination of positions labeled sequentially from 1 to n.
        BFt1: a medium-constant, wherein B performs a diffusion mechanism Ft1 once, as abbreviated for F(p1, p2t1, . . . , pk), i.e. F(p1, p2, . . . , pk) repeated t1 times.
        AFt1: A performs a diffusion-table Ft1, wherein Ft1=FF(p1, p2t1, . . . , pk) performed once, comprising of:
        
            F is a zero matrix with d1×d2× . . . ×dm plus one dimension with magnitude of d1×d2× . . . ×dm, called layers, equally; in addition, 1's are at the ith position on the ith layer; for example,
        
        
    
    





  
    
      
    
    ⁢
    
      
        
          
            (
            a
            )
          
          ⁢
          
            
          
          ⁢
          a
          ⁢
          
            
          
          ⁢
          4
          ×
          4
          ⁢
          
            
          
          ⁢
          F
        
        =
        
          [
          
            
              
                1000
              
            
            
              
                0100
              
            
            
              
                0010
              
            
            
              
                0001
              
            
          
          ]
        
      
      ,
      
        

      
      ⁢
      
        
      
      ⁢
      
        
          
            
              (
              b
              )
            
            ⁢
            
              
            
            ⁢
            a
            ⁢
            
              
            
            ⁢
            2
            ×
            2
            ×
            4
            ⁢
            
              
            
            ⁢
            F
          
          =
          
            
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                  
                
                ]
              
              
                {
                
                  1
                  ⁢
                  st
                
              
            
            ⁢
            
              
                [
                
                  
                    
                      0
                    
                    
                      0
                    
                  
                  
                    
                      1
                    
                    
                      0
                    
                  
                
                ]
              
              
                {
                
                  2
                  ⁢
                  nd
                
              
            
            ⁢
            
              
                [
                
                  
                    
                      0
                    
                    
                      1
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                  
                
                ]
              
              
                {
                
                  3
                  ⁢
                  rd
                
              
            
            ⁢
            
              
                [
                
                  
                    
                      0
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                  
                
                ]
              
              
                {
                
                  4
                  ⁢
                  th
                
              
            
          
        
        ;
      
    
  




  
    
      
        
          (
          c
          )
        
        ⁢
        
          
        
        ⁢
        a
        ⁢
        
          
        
        ⁢
        4
        ×
        4
        ×
        16
        ⁢
        
          
        
        ⁢
        F
      
      =
      
        
          
            [
            
              
                
                  1000
                
              
              
                
                  0000
                
              
              
                
                  0000
                
              
              
                
                  0000
                
              
            
            ]
          
          
            {
            
              1
              ⁢
              st
            
          
        
        ⁢
        
          
            [
            
              
                
                  0000
                
              
              
                
                  1000
                
              
              
                
                  0000
                
              
              
                
                  0000
                
              
            
            ]
          
          
            {
            
              2
              ⁢
              nd
            
          
        
        ⁢
        
          
            [
            
              
                
                  0000
                
              
              
                
                  0000
                
              
              
                
                  1000
                
              
              
                
                  0000
                
              
            
            ]
          
          
            {
            
              3
              ⁢
              rd
            
          
        
        ⁢
        
          
            [
            
              
                
                  0000
                
              
              
                
                  0000
                
              
              
                
                  0000
                
              
              
                
                  1000
                
              
            
            ]
          
          
            {
            
              4
              ⁢
              th
            
          
        
        ⁢
        
          
        
        ⁢
        ...
        ⁢
        
          
        
        ⁢
        
          
            [
            
              
                
                  0001
                
              
              
                
                  0000
                
              
              
                
                  0000
                
              
              
                
                  0000
                
              
            
            ]
          
          
            {
            
              13
              ⁢
              th
            
          
        
        ⁢
        
          
            [
            
              
                
                  0000
                
              
              
                
                  0001
                
              
              
                
                  0000
                
              
              
                
                  0000
                
              
            
            ]
          
          
            {
            
              14
              ⁢
              th
            
          
        
        ⁢
        
          
            [
            
              
                
                  0000
                
              
              
                
                  0000
                
              
              
                
                  0001
                
              
              
                
                  0000
                
              
            
            ]
          
          
            {
            
              15
              ⁢
              th
            
          
        
        ⁢
        
          
            [
            
              
                
                  0000
                
              
              
                
                  0000
                
              
              
                
                  0000
                
              
              
                
                  0001
                
              
            
            ]
          
          
            {
            
              16
              ⁢
              th
            
          
        
      
    
    ;
  



    
    
         wherein a 1×4 A is applied with the 4×4 F; a 2×2 A is applied with the 2×2×4 F; further, a 4×4 A is applied with the 4×4×16 F; and so on.
        
            FF(p1, p2, . . . , pk): F performs the diffusion functions F(p1, p2, . . . , pk), sequentially at the m-dimension positions p1, p2, . . . , pk; for example, a 1×4 A applied with the 4×4 F, FF(p)=F⊕Fyp, FF(1:4) as follows (Fy(0) indicated in boldface hereafter):
        
        
    
    





  
    
      
        FF
        ⁡
        
          (
          1
          )
        
      
      =
      
        
          
            [
            
              
                
                  1000
                
              
              
                
                  0100
                
              
              
                
                  0010
                
              
              
                
                  0001
                
              
            
            ]
          
          ⊕
          
            [
            
              
                
                  0100
                
              
              
                
                  0010
                
              
              
                
                  0001
                
              
              
                
                  0000
                
              
            
            ]
          
        
        =
        
          [
          
            
              
                1100
              
            
            
              
                0110
              
            
            
              
                0011
              
            
            
              
                0001
              
            
          
          ]
        
      
    
    ;
  




  
    
      
        FF
        ⁡
        
          (
          
            1
            :
            2
          
          )
        
      
      =
      
        
          
            [
            
              
                
                  1100
                
              
              
                
                  0110
                
              
              
                
                  0011
                
              
              
                
                  0001
                
              
            
            ]
          
          ⊕
          
            [
            
              
                
                  1010
                
              
              
                
                  1011
                
              
              
                
                  0001
                
              
              
                
                  0000
                
              
            
            ]
          
        
        =
        
          [
          
            
              
                0110
              
            
            
              
                1101
              
            
            
              
                0010
              
            
            
              
                0001
              
            
          
          ]
        
      
    
    ;
  




  
    
      
        FF
        ⁡
        
          (
          
            1
            :
            3
          
          )
        
      
      =
      
        
          
            [
            
              
                
                  0110
                
              
              
                
                  1101
                
              
              
                
                  0010
                
              
              
                
                  0001
                
              
            
            ]
          
          ⊕
          
            [
            
              
                
                  1101
                
              
              
                
                  1000
                
              
              
                
                  0101
                
              
              
                
                  0000
                
              
            
            ]
          
        
        =
        
          [
          
            
              
                1011
              
            
            
              
                0101
              
            
            
              
                0111
              
            
            
              
                0001
              
            
          
          ]
        
      
    
    ;
  




  
    
      FF
      ⁡
      
        (
        
          1
          :
          4
        
        )
      
    
    =
    
      
        
          [
          
            
              
                1011
              
            
            
              
                0101
              
            
            
              
                0111
              
            
            
              
                0001
              
            
          
          ]
        
        ⊕
        
          [
          
            
              
                0110
              
            
            
              
                1010
              
            
            
              
                1110
              
            
            
              
                0010
              
            
          
          ]
        
      
      =
      
        
          [
          
            
              
                1101
              
            
            
              
                1111
              
            
            
              
                1001
              
            
            
              
                0011
              
            
          
          ]
        
        .
      
    
  



Suppose AF1, with the diffusion-table F1=FF(1:4), so according to the above derivation, the next a1=a1⊕a2⊕a3, see 1's at the first position on every layer, the ith layer represented for current ai; thus referring the second position on every layer, the next a2=a1⊕a2; further, a3=a2⊕a4; finally, a4=a1⊕a2⊕a3⊕a4.
Moreover, a 2×2 A applied with the 2×2×4 F, FF(p)=F⊕Fxi⊕Fyj, FF(1:4) as follows (Fx(0) also indicated in boldface hereafter):



  
    
      
        
          
            FF
            ⁡
            
              (
              1
              )
            
          
          =
          
          ⁢
          
            F
            ⊕
            
              Fx
              1
            
            ⊕
            
              Fy
              1
            
          
        
      
    
    
      
        
          =
          
          ⁢
          
            
              
                
                  [
                  
                    
                      
                        1
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                    
                  
                  ]
                
                
                  {
                  
                    1
                    ⁢
                    st
                  
                
              
              ⁢
              
                
                  [
                  
                    
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        1
                      
                      
                        0
                      
                    
                  
                  ]
                
                
                  {
                  
                    2
                    ⁢
                    nd
                  
                
              
              ⁢
              
                
                  [
                  
                    
                      
                        0
                      
                      
                        1
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                    
                  
                  ]
                
                
                  {
                  
                    3
                    ⁢
                    rd
                  
                
              
              ⁢
              
                
                  [
                  
                    
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        1
                      
                    
                  
                  ]
                
                
                  {
                  
                    4
                    ⁢
                    th
                  
                
              
            
            ⊕
          
        
      
    
    
      
        
          
          ⁢
          
            
              
                
                  [
                  
                    
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        1
                      
                      
                        0
                      
                    
                  
                  ]
                
                
                  {
                  
                    1
                    ⁢
                    st
                  
                
              
              ⁢
              
                
                  [
                  
                    
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                    
                  
                  ]
                
                
                  {
                  
                    2
                    ⁢
                    nd
                  
                
              
              ⁢
              
                
                  [
                  
                    
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        1
                      
                    
                  
                  ]
                
                
                  {
                  
                    3
                    ⁢
                    rd
                  
                
              
              ⁢
              
                
                  [
                  
                    
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                    
                  
                  ]
                
                
                  {
                  
                    4
                    ⁢
                    th
                  
                
              
            
            ⊕
          
        
      
    
    
      
        
          
          ⁢
          
            
              
                
                  [
                  
                    
                      
                        0
                      
                      
                        1
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                    
                  
                  ]
                
                
                  {
                  
                    1
                    ⁢
                    st
                  
                
              
              ⁢
              
                
                  [
                  
                    
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        1
                      
                    
                  
                  ]
                
                
                  {
                  
                    2
                    ⁢
                    nd
                  
                
              
              ⁢
              
                
                  [
                  
                    
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                    
                  
                  ]
                
                
                  {
                  
                    3
                    ⁢
                    rd
                  
                
              
              ⁢
              
                
                  [
                  
                    
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                    
                  
                  ]
                
                
                  {
                  
                    4
                    ⁢
                    th
                  
                
              
            
            ;
          
        
      
    
  




  
    
      
        
          
            FF
            ⁡
            
              (
              
                1
                :
                2
              
              )
            
          
          =
          
          ⁢
          
            
              [
              
                FF
                ⁡
                
                  (
                  1
                  )
                
              
              ]
            
            ⊕
            
              
                [
                
                  FF
                  (
                  1
                  ]
                
                )
              
              ⁢
              
                x
                2
              
            
            ⊕
            
              
                [
                
                  FF
                  ⁡
                  
                    (
                    1
                    )
                  
                
                ]
              
              ⁢
              
                y
                1
              
            
          
        
      
    
    
      
        
          =
          
          ⁢
          
            
              
                
                  [
                  
                    
                      
                        1
                      
                      
                        1
                      
                    
                    
                      
                        1
                      
                      
                        0
                      
                    
                  
                  ]
                
                
                  {
                  
                    1
                    ⁢
                    st
                  
                
              
              ⁢
              
                
                  [
                  
                    
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        1
                      
                      
                        1
                      
                    
                  
                  ]
                
                
                  {
                  
                    2
                    ⁢
                    nd
                  
                
              
              ⁢
              
                
                  [
                  
                    
                      
                        0
                      
                      
                        1
                      
                    
                    
                      
                        0
                      
                      
                        1
                      
                    
                  
                  ]
                
                
                  {
                  
                    3
                    ⁢
                    rd
                  
                
              
              ⁢
              
                
                  [
                  
                    
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        1
                      
                    
                  
                  ]
                
                
                  {
                  
                    4
                    ⁢
                    th
                  
                
              
            
            ⊕
          
        
      
    
    
      
        
          
          ⁢
          
            
              
                
                  [
                  
                    
                      
                        1
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                    
                  
                  ]
                
                
                  {
                  
                    1
                    ⁢
                    st
                  
                
              
              ⁢
              
                
                  [
                  
                    
                      
                        1
                      
                      
                        1
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                    
                  
                  ]
                
                
                  {
                  
                    2
                    ⁢
                    nd
                  
                
              
              ⁢
              
                
                  [
                  
                    
                      
                        0
                      
                      
                        1
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                    
                  
                  ]
                
                
                  {
                  
                    3
                    ⁢
                    rd
                  
                
              
              ⁢
              
                
                  [
                  
                    
                      
                        0
                      
                      
                        1
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                    
                  
                  ]
                
                
                  {
                  
                    4
                    ⁢
                    th
                  
                
              
            
            ⊕
          
        
      
    
    
      
        
          
          ⁢
          
            
              
                
                  [
                  
                    
                      
                        0
                      
                      
                        1
                      
                    
                    
                      
                        0
                      
                      
                        1
                      
                    
                  
                  ]
                
                
                  {
                  
                    1
                    ⁢
                    st
                  
                
              
              ⁢
              
                
                  [
                  
                    
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        1
                      
                    
                  
                  ]
                
                
                  {
                  
                    2
                    ⁢
                    nd
                  
                
              
              ⁢
              
                
                  [
                  
                    
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                    
                  
                  ]
                
                
                  {
                  
                    3
                    ⁢
                    rd
                  
                
              
              ⁢
              
                
                  [
                  
                    
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                    
                  
                  ]
                
                
                  {
                  
                    4
                    ⁢
                    th
                  
                
              
            
            ;
          
        
      
    
  




  
    
      
        
          
            FF
            ⁡
            
              (
              
                1
                :
                3
              
              )
            
          
          =
          
          ⁢
          
            
              [
              
                FF
                ⁡
                
                  (
                  
                    1
                    :
                    2
                  
                  )
                
              
              ]
            
            ⊕
            
              
                [
                
                  FF
                  (
                  
                    1
                    :
                    2
                  
                  ]
                
                )
              
              ⁢
              
                x
                1
              
            
            ⊕
            
              
                [
                
                  FF
                  ⁡
                  
                    (
                    
                      1
                      :
                      2
                    
                    )
                  
                
                ]
              
              ⁢
              
                y
                2
              
            
          
        
      
    
    
      
        
          =
          
          ⁢
          
            
              
                
                  [
                  
                    
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        1
                      
                      
                        1
                      
                    
                  
                  ]
                
                
                  {
                  
                    1
                    ⁢
                    st
                  
                
              
              ⁢
              
                
                  [
                  
                    
                      
                        1
                      
                      
                        1
                      
                    
                    
                      
                        1
                      
                      
                        0
                      
                    
                  
                  ]
                
                
                  {
                  
                    2
                    ⁢
                    nd
                  
                
              
              ⁢
              
                
                  [
                  
                    
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        1
                      
                    
                  
                  ]
                
                
                  {
                  
                    3
                    ⁢
                    rd
                  
                
              
              ⁢
              
                
                  [
                  
                    
                      
                        0
                      
                      
                        1
                      
                    
                    
                      
                        0
                      
                      
                        1
                      
                    
                  
                  ]
                
                
                  {
                  
                    4
                    ⁢
                    th
                  
                
              
            
            ⊕
          
        
      
    
    
      
        
          
          ⁢
          
            
              
                
                  [
                  
                    
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                    
                  
                  ]
                
                
                  {
                  
                    1
                    ⁢
                    st
                  
                
              
              ⁢
              
                
                  [
                  
                    
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        1
                      
                      
                        1
                      
                    
                  
                  ]
                
                
                  {
                  
                    2
                    ⁢
                    nd
                  
                
              
              ⁢
              
                
                  [
                  
                    
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                    
                  
                  ]
                
                
                  {
                  
                    3
                    ⁢
                    rd
                  
                
              
              ⁢
              
                
                  [
                  
                    
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        1
                      
                    
                  
                  ]
                
                
                  {
                  
                    4
                    ⁢
                    th
                  
                
              
            
            ⊕
          
        
      
    
    
      
        
          
          ⁢
          
            
              
                
                  [
                  
                    
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        1
                      
                      
                        0
                      
                    
                  
                  ]
                
                
                  {
                  
                    1
                    ⁢
                    st
                  
                
              
              ⁢
              
                
                  [
                  
                    
                      
                        1
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                    
                  
                  ]
                
                
                  {
                  
                    2
                    ⁢
                    nd
                  
                
              
              ⁢
              
                
                  [
                  
                    
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        1
                      
                      
                        0
                      
                    
                  
                  ]
                
                
                  {
                  
                    3
                    ⁢
                    rd
                  
                
              
              ⁢
              
                
                  [
                  
                    
                      
                        1
                      
                      
                        0
                      
                    
                    
                      
                        1
                      
                      
                        0
                      
                    
                  
                  ]
                
                
                  {
                  
                    4
                    ⁢
                    th
                  
                
              
            
            ;
          
        
      
    
  




  
    
      
        
          
            FF
            ⁡
            
              (
              
                1
                :
                4
              
              )
            
          
          =
          
          ⁢
          
            
              [
              
                FF
                ⁡
                
                  (
                  
                    1
                    :
                    3
                  
                  )
                
              
              ]
            
            ⊕
            
              
                [
                
                  FF
                  (
                  
                    1
                    :
                    3
                  
                  ]
                
                )
              
              ⁢
              
                x
                2
              
            
            ⊕
            
              
                [
                
                  FF
                  ⁡
                  
                    (
                    
                      1
                      :
                      3
                    
                    )
                  
                
                ]
              
              ⁢
              
                y
                2
              
            
          
        
      
    
    
      
        
          =
          
          ⁢
          
            
              
                
                  [
                  
                    
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        1
                      
                    
                  
                  ]
                
                
                  {
                  
                    1
                    ⁢
                    st
                  
                
              
              ⁢
              
                
                  [
                  
                    
                      
                        0
                      
                      
                        1
                      
                    
                    
                      
                        0
                      
                      
                        1
                      
                    
                  
                  ]
                
                
                  {
                  
                    2
                    ⁢
                    nd
                  
                
              
              ⁢
              
                
                  [
                  
                    
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        1
                      
                      
                        1
                      
                    
                  
                  ]
                
                
                  {
                  
                    3
                    ⁢
                    rd
                  
                
              
              ⁢
              
                
                  [
                  
                    
                      
                        0
                      
                      
                        1
                      
                    
                    
                      
                        1
                      
                      
                        0
                      
                    
                  
                  ]
                
                
                  {
                  
                    4
                    ⁢
                    th
                  
                
              
            
            ⊕
          
        
      
    
    
      
        
          
          ⁢
          
            
              
                
                  [
                  
                    
                      
                        0
                      
                      
                        1
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                    
                  
                  ]
                
                
                  {
                  
                    1
                    ⁢
                    st
                  
                
              
              ⁢
              
                
                  [
                  
                    
                      
                        0
                      
                      
                        1
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                    
                  
                  ]
                
                
                  {
                  
                    2
                    ⁢
                    nd
                  
                
              
              ⁢
              
                
                  [
                  
                    
                      
                        1
                      
                      
                        1
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                    
                  
                  ]
                
                
                  {
                  
                    3
                    ⁢
                    rd
                  
                
              
              ⁢
              
                
                  [
                  
                    
                      
                        1
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                    
                  
                  ]
                
                
                  {
                  
                    4
                    ⁢
                    th
                  
                
              
            
            ⊕
          
        
      
    
    
      
        
          
          ⁢
          
            
              
                [
                
                  
                    
                      0
                    
                    
                      0
                    
                  
                  
                    
                      1
                    
                    
                      0
                    
                  
                
                ]
              
              
                {
                
                  1
                  ⁢
                  st
                
              
            
            ⁢
            
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      1
                    
                    
                      0
                    
                  
                
                ]
              
              
                {
                
                  2
                  ⁢
                  nd
                
              
            
            ⁢
            
              
                [
                
                  
                    
                      0
                    
                    
                      0
                    
                  
                  
                    
                      1
                    
                    
                      0
                    
                  
                
                ]
              
              
                {
                
                  3
                  ⁢
                  rd
                
              
            
            ⁢
            
              
                
                  [
                  
                    
                      
                        1
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                    
                  
                  ]
                
                
                  {
                  
                    4
                    ⁢
                    th
                  
                
              
              .
            
          
        
      
    
  



Suppose AF1, wherein





  
    
      F
      1
    
    =
    
      
        FF
        ⁡
        
          (
          
            1
            :
            4
          
          )
        
      
      =
      
        
          
            [
            
              
                
                  0
                
                
                  1
                
              
              
                
                  1
                
                
                  1
                
              
            
            ]
          
          
            {
            
              1
              ⁢
              st
            
          
        
        ⁢
        
          
            [
            
              
                
                  1
                
                
                  0
                
              
              
                
                  1
                
                
                  1
                
              
            
            ]
          
          
            {
            
              2
              ⁢
              nd
            
          
        
        ⁢
        
          
            [
            
              
                
                  1
                
                
                  1
                
              
              
                
                  0
                
                
                  1
                
              
            
            ]
          
          
            {
            
              3
              ⁢
              rd
            
          
        
        ⁢
        
          
            [
            
              
                
                  1
                
                
                  1
                
              
              
                
                  1
                
                
                  0
                
              
            
            ]
          
          
            {
            
              4
              ⁢
              th
            
          
        
      
    
  
  ;




four layers represent the current a1 to a4, through by the first position, the next a1=a2⊕a3⊕a4; by the second position, the next a2=a1⊕a2⊕a4; further, by the third position, the next a3=a1⊕a3⊕a4; and finally, by the fourth position, the next a4=a1⊕a2⊕a3.

Preferred Embodiment IV: Diffusion Mechanisms in a Parallel Process.

Firstly, in Embodiment I, now that the diffusion mechanism F1=F(13:13), A0=[0000000000000000], and S=1 placed at the anchor-position, the sequential value of diffusion-area A1×t2=A1×(t2−1)F1⊕BF1, wherein a medium-constant BF1 and a diffusion-table F1 are performed only once at the beginning; BF1=BF(13:13)=[1011001101100011], derived as A1, in Embodiment I, and further, F1=FF(13:13), wherein FF(p)=F⊕Fyp, may refer to the whole derivation of two-dimensional FF(1:4), and F is such as a 16×16 unit matrix, with a one-dimension diffusion function in a two-dimension matrix; positions are orderly ordered from 13 to 16 then from 1 to 13, each position's diffusion function is shown in Table 1, and for greater clarity, the final result is shown in Table 2. Furthermore, reading from the ith position of all layers, namely, referring to the ith column to obtain the next ai, all combinations are shown in Table 3.
Therefore, according to Table 3, new values in sequence as below:



  
    
      
        for
        ⁢
        
          
        
        ⁢
        
          t
          2
        
      
      =
      1
    
    ,
    
      

    
    ⁢
    
      
        
          
            
              A
              
                1
                ×
                1
              
            
            =
            
            ⁢
            
              
                
                  A
                  0
                
                ⁢
                
                  F
                  1
                
              
              ⊕
              
                BF
                1
              
            
          
        
      
      
        
          
            =
            
            ⁢
            
              
                [
                
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  0
                
                ]
              
              ⊕
            
          
        
      
      
        
          
            
            ⁢
            
              
                [
                
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  1
                
                ]
              
              ;
            
          
        
      
    
  




  
    
      
        the
        ⁢
        
          
        
        ⁢
        results
        ⁢
        
          
        
        ⁢
        is
        ⁢
        
          
        
        ⁢
        the
        ⁢
        
          
        
        ⁢
        same
        ⁢
        
          
        
        ⁢
        as
        ⁢
        
          
        
        ⁢
        
          A
          1
        
        ⁢
        
          
        
        ⁢
        in
        ⁢
        
          
        
        ⁢
        Embodiment
        ⁢
        
          
        
        ⁢
        
          I
          .
          
            

          
          ⁢
          for
        
        ⁢
        
          
        
        ⁢
        
          t
          2
        
      
      =
      2
    
    ,
    
      

    
    ⁢
    
      
        
          
            
              A
              
                1
                ×
                2
              
            
            =
            
            ⁢
            
              
                
                  A
                  
                    1
                    ×
                    1
                  
                
                ⁢
                
                  F
                  1
                
              
              ⊕
              
                BF
                1
              
            
          
        
      
      
        
          
            =
            
            ⁢
            
              
                [
                
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  1
                
                ]
              
              ⊕
              
                [
                
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  1
                
                ]
              
            
          
        
      
      
        
          
            
              =
              
              ⁢
              
                0
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                0
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                0
                ⁢
                
                  
                
                ⁢
                0
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                0
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                0
                ⁢
                
                  
                
                ⁢
                0
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                0
              
            
            ;
          
        
      
    
  




  
    the
    ⁢
    
      
    
    ⁢
    results
    ⁢
    
      
    
    ⁢
    is
    ⁢
    
      
    
    ⁢
    the
    ⁢
    
      
    
    ⁢
    same
    ⁢
    
      
    
    ⁢
    as
    ⁢
    
      
    
    ⁢
    
      A
      2
    
    ⁢
    
      
    
    ⁢
    in
    ⁢
    
      
    
    ⁢
    Embodiment
    ⁢
    
      
    
    ⁢
    
      I
      .
      
        

      
      ⁢
      Derivation
    
    ⁢
    
      
    
    ⁢
    of
    ⁢
    
      
    
    ⁢
    
      A
      
        1
        ×
        1
      
    
    ⁢
    
      F
      1
    
    ⁢
    
      :
    
  




  
    Now
    ⁢
    
      
    
    ⁢
    that
  




  
    
      
        
          
            A
            
              1
              ×
              1
            
          
          =
          
          ⁢
          
            [
            
              
                a
                1
              
              ,
              
                a
                2
              
              ,
              ...
              ⁢
              
                
              
              ,
              
                a
                16
              
            
            ]
          
        
      
    
    
      
        
          
            =
            
            ⁢
            
              [
              
                
                  
                    
                      1
                      1
                    
                  
                  
                    
                      0
                      2
                    
                  
                  
                    
                      1
                      3
                    
                  
                  
                    
                      1
                      4
                    
                  
                  
                    
                      0
                      5
                    
                  
                  
                    
                      0
                      6
                    
                  
                  
                    
                      1
                      7
                    
                  
                  
                    
                      1
                      8
                    
                  
                  
                    
                      0
                      9
                    
                  
                  
                    
                      1
                      10
                    
                  
                  
                    
                      1
                      11
                    
                  
                  
                    
                      0
                      12
                    
                  
                  
                    
                      0
                      13
                    
                  
                  
                    
                      0
                      14
                    
                  
                  
                    
                      1
                      15
                    
                  
                  
                    
                      1
                      16
                    
                  
                
              
              ]
            
          
          ;
        
      
    
  




  
    
      
        the
        ⁢
        
          
        
        ⁢
        next
        ⁢
        
          
        
        ⁢
        
          a
          1
        
      
      =
      
        
          
            a
            1
          
          ⊕
          
            a
            7
          
          ⊕
          
            a
            9
          
          ⊕
          
            a
            11
          
        
        =
        
          
            1
            ⊕
            1
            ⊕
            0
            ⊕
            1
          
          =
          1
        
      
    
    ;
  




  
    the
    ⁢
    
      
    
    ⁢
    next
  




  
    
      
        
          
            a
            2
          
          =
          
          ⁢
          
            
              a
              1
            
            ⊕
            
              a
              2
            
            ⊕
            
              a
              5
            
            ⊕
            
              a
              8
            
            ⊕
            
              a
              10
            
            ⊕
            
              a
              12
            
          
        
      
    
    
      
        
          =
          
          ⁢
          
            
              1
              ⊕
              0
              ⊕
              0
              ⊕
              1
              ⊕
              1
              ⊕
              0
            
            =
            
              
            
            ⁢
            
               
              
                1
                ;
                ...
                ⁢
                
                  
                
                ;
              
            
          
        
      
    
  




  
    
      the
      ⁢
      
        
      
      ⁢
      next
      ⁢
      
        
      
      ⁢
      
        a
        16
      
    
    =
    
      
        
          a
          3
        
        ⊕
        
          a
          4
        
        ⊕
        
          a
          13
        
        ⊕
        
          a
          15
        
      
      =
      
        
          1
          ⊕
          1
          ⊕
          0
          ⊕
          1
        
        =
        1.
      
    
  




  
    
      Prove
      ⁢
      
        
      
      ⁢
      likewise
    
    ,
    
      

    
    ⁢
    
      
        for
        ⁢
        
          
        
        ⁢
        
          t
          2
        
      
      =
      3
    
    ,
    
      

    
    ⁢
    
      
        
          
            
              A
              
                1
                ×
                2
              
            
            =
            
            ⁢
            
              [
              
                
                  a
                  1
                
                ,
                
                  a
                  2
                
                ,
                ...
                ⁢
                
                  
                
                ,
                
                  a
                  16
                
              
              ]
            
          
        
      
      
        
          
            
              =
              
              ⁢
              
                [
                
                  
                    
                      
                        0
                        1
                      
                    
                    
                      
                        1
                        2
                      
                    
                    
                      
                        1
                        3
                      
                    
                    
                      
                        0
                        4
                      
                    
                    
                      
                        1
                        5
                      
                    
                    
                      
                        0
                        6
                      
                    
                    
                      
                        0
                        7
                      
                    
                    
                      
                        1
                        8
                      
                    
                    
                      
                        1
                        9
                      
                    
                    
                      
                        0
                        10
                      
                    
                    
                      
                        1
                        11
                      
                    
                    
                      
                        1
                        12
                      
                    
                    
                      
                        0
                        13
                      
                    
                    
                      
                        0
                        14
                      
                    
                    
                      
                        1
                        15
                      
                    
                    
                      
                        0
                        16
                      
                    
                  
                
                ]
              
            
            ,
          
        
      
    
  




  
    
      
        
          
            A
            
              1
              ×
              3
            
          
          =
          
          ⁢
          
            
              
                A
                
                  1
                  ×
                  2
                
              
              ⁢
              
                F
                1
              
            
            ⊕
            
              BF
              1
            
          
        
      
    
    
      
        
          =
          
          ⁢
          
            
              [
              
                0
                ⁢
                
                  
                
                ⁢
                0
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                0
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                0
                ⁢
                
                  
                
                ⁢
                0
                ⁢
                
                  
                
                ⁢
                0
                ⁢
                
                  
                
                ⁢
                0
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                0
                ⁢
                
                  
                
                ⁢
                0
                ⁢
                
                  
                
                ⁢
                0
              
              ]
            
            ⊕
            
              [
              
                1
                ⁢
                
                  
                
                ⁢
                0
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                0
                ⁢
                
                  
                
                ⁢
                0
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                0
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                0
                ⁢
                
                  
                
                ⁢
                0
                ⁢
                
                  
                
                ⁢
                0
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                1
              
              ]
            
          
        
      
    
    
      
        
          
            =
            
            ⁢
            
              1
              ⁢
              
                
              
              ⁢
              0
              ⁢
              
                
              
              ⁢
              0
              ⁢
              
                
              
              ⁢
              0
              ⁢
              
                
              
              ⁢
              0
              ⁢
              
                
              
              ⁢
              1
              ⁢
              
                
              
              ⁢
              0
              ⁢
              
                
              
              ⁢
              0
              ⁢
              
                
              
              ⁢
              0
              ⁢
              
                
              
              ⁢
              1
              ⁢
              
                
              
              ⁢
              1
              ⁢
              
                
              
              ⁢
              0
              ⁢
              
                
              
              ⁢
              1
              ⁢
              
                
              
              ⁢
              0
              ⁢
              
                
              
              ⁢
              1
              ⁢
              
                
              
              ⁢
              1
            
          
          ;
        
      
    
  




  
    
      
        the
        ⁢
        
          
        
        ⁢
        result
        ⁢
        
          
        
        ⁢
        is
        ⁢
        
          
        
        ⁢
        the
        ⁢
        
          
        
        ⁢
        same
        ⁢
        
          
        
        ⁢
        as
        ⁢
        
          
        
        ⁢
        
          A
          3
        
        ⁢
        
          
        
        ⁢
        in
        ⁢
        
          
        
        ⁢
        Embodiment
        ⁢
        
          
        
        ⁢
        
          II
          .
          
            

          
          ⁢
          for
        
        ⁢
        
          
        
        ⁢
        
          t
          2
        
      
      =
      4
    
    ,
    
      

    
    ⁢
    
      
        
          
            
              A
              
                1
                ×
                3
              
            
            =
            
            ⁢
            
              [
              
                
                  a
                  1
                
                ,
                
                  a
                  2
                
                ,
                ...
                ⁢
                
                  
                
                ,
                
                  a
                  16
                
              
              ]
            
          
        
      
      
        
          
            
              =
              
              ⁢
              
                [
                
                  
                    
                      
                        1
                        1
                      
                    
                    
                      
                        0
                        2
                      
                    
                    
                      
                        0
                        3
                      
                    
                    
                      
                        0
                        4
                      
                    
                    
                      
                        0
                        5
                      
                    
                    
                      
                        1
                        6
                      
                    
                    
                      
                        0
                        7
                      
                    
                    
                      
                        0
                        8
                      
                    
                    
                      
                        0
                        9
                      
                    
                    
                      
                        1
                        10
                      
                    
                    
                      
                        1
                        11
                      
                    
                    
                      
                        0
                        12
                      
                    
                    
                      
                        1
                        13
                      
                    
                    
                      
                        0
                        14
                      
                    
                    
                      
                        1
                        15
                      
                    
                    
                      
                        1
                        16
                      
                    
                  
                
                ]
              
            
            ,
          
        
      
    
  




  
    
      
        
          
            A
            
              1
              ×
              4
            
          
          =
          
          ⁢
          
            
              
                A
                
                  1
                  ×
                  3
                
              
              ⁢
              
                F
                1
              
            
            ⊕
            
              BF
              1
            
          
        
      
    
    
      
        
          =
          
          ⁢
          
            
              [
              
                0
                ⁢
                
                  
                
                ⁢
                0
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                0
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                0
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                0
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                0
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                0
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                0
              
              ]
            
            ⊕
            
              [
              
                1
                ⁢
                
                  
                
                ⁢
                0
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                0
                ⁢
                
                  
                
                ⁢
                0
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                0
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                0
                ⁢
                
                  
                
                ⁢
                0
                ⁢
                
                  
                
                ⁢
                0
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                1
              
              ]
            
          
        
      
    
    
      
        
          
            =
            
            ⁢
            
              1
              ⁢
              
                
              
              ⁢
              0
              ⁢
              
                
              
              ⁢
              0
              ⁢
              
                
              
              ⁢
              1
              ⁢
              
                
              
              ⁢
              1
              ⁢
              
                
              
              ⁢
              1
              ⁢
              
                
              
              ⁢
              1
              ⁢
              
                
              
              ⁢
              0
              ⁢
              
                
              
              ⁢
              0
              ⁢
              
                
              
              ⁢
              0
              ⁢
              
                
              
              ⁢
              1
              ⁢
              
                
              
              ⁢
              1
              ⁢
              
                
              
              ⁢
              0
              ⁢
              
                
              
              ⁢
              1
              ⁢
              
                
              
              ⁢
              0
              ⁢
              
                
              
              ⁢
              1
            
          
          ;
        
      
    
  




  
    the
    ⁢
    
      
    
    ⁢
    result
    ⁢
    
      
    
    ⁢
    is
    ⁢
    
      
    
    ⁢
    the
    ⁢
    
      
    
    ⁢
    same
    ⁢
    
      
    
    ⁢
    as
    ⁢
    
      
    
    ⁢
    
      A
      4
    
    ⁢
    
      
    
    ⁢
    in
    ⁢
    
      
    
    ⁢
    Embodiment
    ⁢
    
      
    
    ⁢
    
      II
      .
    
  










TABLE 1







    



  
    
      FF
      ⁡
      
        (
        13
        )
      
    
    =
    
      
        
          [
          
            
              
                
                  1000000000000000
                
              
            
            
              
                ⋮
              
            
            
              
                
                  0000000000010000
                
              
            
            
              
                
                  0000000000001000
                
              
            
            
              
                
                  0000000000000100
                
              
            
            
              
                
                  0000000000000010
                
              
            
            
              
                
                  0000000000000001
                
              
            
          
          ]
        
        ⊕
        
          [
          
            
              
                
                  000000000000
                  0
                  000
                
              
            
            
              
                ⋮
              
            
            
              
                
                  000000000010
                  0
                  000
                
              
            
            
              
                
                  000000000001
                  0
                  100
                
              
            
            
              
                
                  000000000000
                  0
                  010
                
              
            
            
              
                
                  000000000000
                  0
                  001
                
              
            
            
              
                
                  000000000000
                  0
                  000
                
              
            
          
          ]
        
      
      =
      
        [
        
          
            
              
                1000000000000000
              
            
          
          
            
              ⋮
            
          
          
            
              
                0000000000110000
              
            
          
          
            
              
                0000000000011100
              
            
          
          
            
              
                0000000000000110
              
            
          
          
            
              
                0000000000000011
              
            
          
          
            
              
                0000000000000001
              
            
          
        
        ]
      
    
  
  ;






 






  
    
      FF
      ⁡
      
        (
        
          13
          :
          14
        
        )
      
    
    =
    
      
        
          [
          
            
              
                
                  1000000000000000
                
              
            
            
              
                ⋮
              
            
            
              
                
                  0000000000110000
                
              
            
            
              
                
                  0000000000011100
                
              
            
            
              
                
                  0000000000000110
                
              
            
            
              
                
                  0000000000000011
                
              
            
            
              
                
                  0000000000000001
                
              
            
          
          ]
        
        ⊕
        
          [
          
            
              
                
                  000000000000
                  0
                  0
                  00
                
              
            
            
              
                ⋮
              
            
            
              
                
                  000000000110
                  0
                  0
                  00
                
              
            
            
              
                
                  000000000011
                  1
                  0
                  10
                
              
            
            
              
                
                  000000000000
                  1
                  0
                  11
                
              
            
            
              
                
                  000000000000
                  0
                  0
                  01
                
              
            
            
              
                
                  000000000000
                  0
                  0
                  00
                
              
            
          
          ]
        
      
      =
      
        [
        
          
            
              
                1000000000000000
              
            
          
          
            
              ⋮
            
          
          
            
              
                0000000001010000
              
            
          
          
            
              
                0000000000100110
              
            
          
          
            
              
                0000000000001101
              
            
          
          
            
              
                0000000000000010
              
            
          
          
            
              
                0000000000000001
              
            
          
        
        ]
      
    
  
  ;






 






  
    FF
    ⁡
    
      (
      
        13
        :
        13
      
      )
    
  
  =
  
    
      
        [
        
          
            
              
                1011011011011000
              
            
          
          
            
              ⋮
            
          
          
            
              
                0011001100110000
              
            
          
          
            
              
                0000101101110110
              
            
          
          
            
              
                0000000010101100
              
            
          
          
            
              
                0000000000001001
              
            
          
          
            
              
                0000000000000011
              
            
          
        
        ]
      
      ⊕
      
        [
        
          
            
              
                011011011011
                0
                100
              
            
          
          
            
              ⋮
            
          
          
            
              
                011001100110
                0
                000
              
            
          
          
            
              
                000101101110
                0
                011
              
            
          
          
            
              
                000000010101
                0
                110
              
            
          
          
            
              
                000000000001
                0
                100
              
            
          
          
            
              
                000000000000
                0
                0
                01
              
            
          
        
        ]
      
    
    =
    
      
        [
        
          
            
              
                1101101101101100
              
            
          
          
            
              ⋮
            
          
          
            
              
                0101010101010000
              
            
          
          
            
              
                0001110110010101
              
            
          
          
            
              
                0000000111111010
              
            
          
          
            
              
                0000000000011101
              
            
          
          
            
              
                0000000000000010
              
            
          
        
        ]
      
      .
    
  




















TABLE 2











  
    F
    1
  
  =
  
    
      FF
      ⁡
      
        (
        
          13
          ⁢
          
            :
          
          ⁢
          13
        
        )
      
    
    =
    
      [
      
        
          
            1
          
          
            1
          
          
            0
          
          
            1
          
          
            1
          
          
            0
          
          
            1
          
          
            1
          
          
            0
          
          
            1
          
          
            1
          
          
            0
          
          
            1
          
          
            1
          
          
            0
          
          
            0
          
        
        
          
            0
          
          
            1
          
          
            0
          
          
            0
          
          
            0
          
          
            1
          
          
            0
          
          
            1
          
          
            0
          
          
            0
          
          
            0
          
          
            1
          
          
            0
          
          
            1
          
          
            1
          
          
            0
          
        
        
          
            0
          
          
            0
          
          
            0
          
          
            1
          
          
            1
          
          
            0
          
          
            1
          
          
            0
          
          
            1
          
          
            0
          
          
            1
          
          
            1
          
          
            0
          
          
            0
          
          
            1
          
          
            1
          
        
        
          
            0
          
          
            0
          
          
            0
          
          
            0
          
          
            0
          
          
            1
          
          
            0
          
          
            1
          
          
            0
          
          
            1
          
          
            0
          
          
            1
          
          
            0
          
          
            0
          
          
            0
          
          
            1
          
        
        
          
            0
          
          
            1
          
          
            1
          
          
            0
          
          
            1
          
          
            1
          
          
            0
          
          
            0
          
          
            0
          
          
            1
          
          
            1
          
          
            1
          
          
            0
          
          
            0
          
          
            0
          
          
            0
          
        
        
          
            0
          
          
            0
          
          
            0
          
          
            1
          
          
            0
          
          
            1
          
          
            0
          
          
            0
          
          
            0
          
          
            0
          
          
            0
          
          
            1
          
          
            0
          
          
            0
          
          
            0
          
          
            0
          
        
        
          
            1
          
          
            0
          
          
            1
          
          
            1
          
          
            0
          
          
            0
          
          
            0
          
          
            1
          
          
            1
          
          
            0
          
          
            1
          
          
            1
          
          
            0
          
          
            0
          
          
            0
          
          
            0
          
        
        
          
            0
          
          
            1
          
          
            0
          
          
            1
          
          
            0
          
          
            0
          
          
            0
          
          
            0
          
          
            0
          
          
            1
          
          
            0
          
          
            1
          
          
            0
          
          
            0
          
          
            0
          
          
            0
          
        
        
          
            1
          
          
            0
          
          
            1
          
          
            0
          
          
            1
          
          
            0
          
          
            1
          
          
            1
          
          
            0
          
          
            1
          
          
            1
          
          
            1
          
          
            0
          
          
            0
          
          
            0
          
          
            0
          
        
        
          
            0
          
          
            1
          
          
            0
          
          
            1
          
          
            0
          
          
            1
          
          
            0
          
          
            1
          
          
            0
          
          
            0
          
          
            0
          
          
            1
          
          
            0
          
          
            0
          
          
            0
          
          
            0
          
        
        
          
            1
          
          
            0
          
          
            1
          
          
            0
          
          
            1
          
          
            0
          
          
            1
          
          
            0
          
          
            1
          
          
            0
          
          
            1
          
          
            1
          
          
            0
          
          
            0
          
          
            0
          
          
            0
          
        
        
          
            0
          
          
            1
          
          
            0
          
          
            1
          
          
            0
          
          
            1
          
          
            0
          
          
            1
          
          
            0
          
          
            1
          
          
            0
          
          
            1
          
          
            0
          
          
            0
          
          
            0
          
          
            0
          
        
        
          
            0
          
          
            0
          
          
            0
          
          
            1
          
          
            1
          
          
            1
          
          
            0
          
          
            1
          
          
            1
          
          
            0
          
          
            0
          
          
            1
          
          
            0
          
          
            1
          
          
            0
          
          
            1
          
        
        
          
            0
          
          
            0
          
          
            0
          
          
            0
          
          
            0
          
          
            0
          
          
            0
          
          
            1
          
          
            1
          
          
            1
          
          
            1
          
          
            1
          
          
            1
          
          
            0
          
          
            1
          
          
            0
          
        
        
          
            0
          
          
            0
          
          
            0
          
          
            0
          
          
            0
          
          
            0
          
          
            0
          
          
            0
          
          
            0
          
          
            0
          
          
            0
          
          
            1
          
          
            1
          
          
            1
          
          
            0
          
          
            1
          
        
        
          
            0
          
          
            0
          
          
            0
          
          
            0
          
          
            0
          
          
            0
          
          
            0
          
          
            0
          
          
            0
          
          
            0
          
          
            0
          
          
            0
          
          
            0
          
          
            0
          
          
            1
          
          
            0
          
        
      
      ]
    
  























TABLE 3







at




time t2
at time t2 − 1









a1
a1 ⊕ a7 ⊕ a9 ⊕ a11



a2
a1 ⊕ a2 ⊕ a5 ⊕ a8 ⊕ a10 ⊕ a12



a3
a5 ⊕ a7 ⊕ a9 ⊕ a11



a4
a1 ⊕ a3 ⊕ a6 ⊕ a7 ⊕ a8 ⊕ a10 ⊕ a12 ⊕ a13



a5
a1 ⊕ a3 ⊕ a5 ⊕ a9 ⊕ a11 ⊕ a13



a6
a2 ⊕ a4 ⊕ a5 ⊕ a6 ⊕ a10 ⊕ a12 ⊕ a13



a7
a1 ⊕ a3 ⊕ a9 ⊕ a11



a8
a1 ⊕ a2 ⊕ a4 ⊕ a7 ⊕ a9 ⊕ a10 ⊕ a12 ⊕ a13 ⊕ a14



a9
a3 ⊕ a7 ⊕ a11 ⊕ a13 ⊕ a14



a10
a1 ⊕ a4 ⊕ a5 ⊕ a8 ⊕ a9 ⊕ a12 ⊕ a14



a11
a1 ⊕ a3 ⊕ a5 ⊕ a7 ⊕ a9 ⊕ a11 ⊕ a14



a12
a2 ⊕ a3 ⊕ a4 ⊕ a5 ⊕ a6 ⊕ a7 ⊕ a8 ⊕




a9 ⊕ a10 ⊕ a11 ⊕ a12 ⊕ a13 ⊕ a14 ⊕ a15



a13
a1 ⊕ a14 ⊕ a15



a14
a1 ⊕ a2 ⊕ a13 ⊕ a15



a15
a2 ⊕ a3 ⊕ a14 ⊕ a16



a16
a3 ⊕ a4 ⊕ a13 ⊕ a15










Secondly, with Embodiment II, to comply with the diffusion mechanism F2=F(132:13), the sequential value A2×t2=A2×(t2−1)F2⊕BF2, wherein BF2=[BF(13:13)]F(13:13), is derived as A2, in Embodiment I, equals to [0110100110110010], and further, F2=FF(132:13), wherein a value FF(13:13), see Table 2, performs the diffusion functions F(13:13), derived as Table 1; the diffusion-table F2 is shown in Table 4.








TABLE 4





at



time t2
at time t2 − 1







a1
a1 ⊕ a3 ⊕ a5 ⊕ a7 ⊕ a9 ⊕ a13


a2
a1 ⊕ a2 ⊕ a4 ⊕ a6 ⊕ a7 ⊕ a8 ⊕



a9 ⊕ a10 ⊕ a11 ⊕ a13 ⊕ a14 ⊕ a15


a3
a1 ⊕ a9


a4
a1 ⊕ a2 ⊕ a10 ⊕ a13


a5
a3 ⊕ a5 ⊕ a11 ⊕ a14 ⊕ a15


a6
a1 ⊕ a2 ⊕ a3 ⊕ a4 ⊕ a5 ⊕ a6 ⊕ a9 ⊕ a12 ⊕ a14


a7
a9 ⊕ a13


a8
a1 ⊕ a2 ⊕ a5 ⊕ a10 ⊕ a15


a9
a2 ⊕ a9 ⊕ a11 ⊕ a13


a10
a1 ⊕ a2 ⊕ a7 ⊕ a10 ⊕ a12 ⊕ a14


a11
a1 ⊕ a2 ⊕ a5 ⊕ a9 ⊕ a13 ⊕ a15


a12
a1 ⊕ a3 ⊕ a5 ⊕ a6 ⊕ a9 ⊕ a10 ⊕ a13 ⊕ a14 ⊕ a15 ⊕ a16


a13
a3 ⊕ a7 ⊕ a9 ⊕ a11 ⊕ a13 ⊕ a14 ⊕ a15 ⊕ a16


a14
a1⊕ a3 ⊕ a5 ⊕ a7 ⊕ a8 ⊕ a9 ⊕ a10 ⊕ a11 ⊕ a12 ⊕ a15 ⊕ a16


a15
a3 ⊕ a4 ⊕ a7 ⊕ a8 ⊕ a9 ⊕ a10 ⊕ a11 ⊕ a12


a16
a2 ⊕ a5 ⊕ a6 ⊕ a8 ⊕ a9 ⊕ a10 ⊕ a11 ⊕ a12 ⊕ a13 ⊕ a15 ⊕ a16









Therefore, according to Table 4, new values in sequence as below:



  
    
      
        for
        ⁢
        
          
        
        ⁢
        
          t
          2
        
      
      =
      1
    
    ,
    
      

    
    ⁢
    
      
        
          
            
              A
              
                2
                ×
                1
              
            
            =
            
            ⁢
            
              
                
                  A
                  0
                
                ⁢
                
                  F
                  2
                
              
              ⊕
              
                BF
                2
              
            
          
        
      
      
        
          
            
              =
              
              ⁢
              
                
                  [
                  
                    0
                    ⁢
                    
                      
                    
                    ⁢
                    0
                    ⁢
                    
                      
                    
                    ⁢
                    0
                    ⁢
                    
                      
                    
                    ⁢
                    0
                    ⁢
                    
                      
                    
                    ⁢
                    0
                    ⁢
                    
                      
                    
                    ⁢
                    0
                    ⁢
                    
                      
                    
                    ⁢
                    0
                    ⁢
                    
                      
                    
                    ⁢
                    0
                    ⁢
                    
                      
                    
                    ⁢
                    0
                    ⁢
                    
                      
                    
                    ⁢
                    0
                    ⁢
                    
                      
                    
                    ⁢
                    0
                    ⁢
                    
                      
                    
                    ⁢
                    0
                    ⁢
                    
                      
                    
                    ⁢
                    0
                    ⁢
                    
                      
                    
                    ⁢
                    0
                    ⁢
                    
                      
                    
                    ⁢
                    0
                    ⁢
                    
                      
                    
                    ⁢
                    0
                  
                  ]
                
                ⊕
                
                  [
                  
                    0
                    ⁢
                    
                      
                    
                    ⁢
                    1
                    ⁢
                    
                      
                    
                    ⁢
                    1
                    ⁢
                    
                      
                    
                    ⁢
                    0
                    ⁢
                    
                      
                    
                    ⁢
                    1
                    ⁢
                    
                      
                    
                    ⁢
                    0
                    ⁢
                    
                      
                    
                    ⁢
                    0
                    ⁢
                    
                      
                    
                    ⁢
                    1
                    ⁢
                    
                      
                    
                    ⁢
                    1
                    ⁢
                    
                      
                    
                    ⁢
                    0
                    ⁢
                    
                      
                    
                    ⁢
                    1
                    ⁢
                    
                      
                    
                    ⁢
                    1
                    ⁢
                    
                      
                    
                    ⁢
                    0
                    ⁢
                    
                      
                    
                    ⁢
                    0
                    ⁢
                    
                      
                    
                    ⁢
                    1
                    ⁢
                    
                      
                    
                    ⁢
                    0
                  
                  ]
                
              
            
            ;
          
        
      
    
  




  
    
      
        the
        ⁢
        
          
        
        ⁢
        results
        ⁢
        
          
        
        ⁢
        is
        ⁢
        
          
        
        ⁢
        the
        ⁢
        
          
        
        ⁢
        same
        ⁢
        
          
        
        ⁢
        as
        ⁢
        
          
        
        ⁢
        
          A
          1
        
        ⁢
        
          
        
        ⁢
        in
        ⁢
        
          
        
        ⁢
        Embodiment
        ⁢
        
          
        
        ⁢
        
          I
          .
          
            

          
          ⁢
          for
        
        ⁢
        
          
        
        ⁢
        
          t
          2
        
      
      =
      2
    
    ,
    
      

    
    ⁢
    
      
        
          
            
              A
              
                2
                ×
                2
              
            
            =
            
            ⁢
            
              
                
                  A
                  
                    2
                    ×
                    1
                  
                
                ⁢
                
                  F
                  2
                
              
              ⊕
              
                BF
                2
              
            
          
        
      
      
        
          
            =
            
            ⁢
            
              
                [
                
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  1
                
                ⁢
                
                  
                
                ]
              
              ⊕
              
                [
                
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  0
                
                ]
              
            
          
        
      
      
        
          
            
              =
              
              ⁢
              
                1
                ⁢
                
                  
                
                ⁢
                0
                ⁢
                
                  
                
                ⁢
                0
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                0
                ⁢
                
                  
                
                ⁢
                0
                ⁢
                
                  
                
                ⁢
                0
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                0
                ⁢
                
                  
                
                ⁢
                1
                ⁢
                
                  
                
                ⁢
                0
                ⁢
                
                  
                
                ⁢
                1
              
            
            ;
          
        
      
    
  




  
    the
    ⁢
    
      
    
    ⁢
    results
    ⁢
    
      
    
    ⁢
    is
    ⁢
    
      
    
    ⁢
    the
    ⁢
    
      
    
    ⁢
    same
    ⁢
    
      
    
    ⁢
    as
    ⁢
    
      
    
    ⁢
    
      A
      4
    
    ⁢
    
      
    
    ⁢
    in
    ⁢
    
      
    
    ⁢
    Embodiment
    ⁢
    
      
    
    ⁢
    
      II
      .
      
        

      
      ⁢
      Derivation
    
    ⁢
    
      
    
    ⁢
    of
    ⁢
    
      
    
    ⁢
    
      A
      
        2
        ×
        1
      
    
    ⁢
    
      F
      2
    
    ⁢
    
      :
    
  




  
    Now
    ⁢
    
      
    
    ⁢
    that
  




  
    
      
        
          
            A
            
              2
              ×
              1
            
          
          =
          
          ⁢
          
            [
            
              
                a
                1
              
              ,
              
                a
                2
              
              ,
              ...
              ⁢
              
                
              
              ,
              
                a
                16
              
            
            ]
          
        
      
    
    
      
        
          
            =
            
            ⁢
            
              [
              
                
                  
                    
                      0
                      1
                    
                  
                  
                    
                      1
                      2
                    
                  
                  
                    
                      1
                      3
                    
                  
                  
                    
                      0
                      4
                    
                  
                  
                    
                      1
                      5
                    
                  
                  
                    
                      0
                      6
                    
                  
                  
                    
                      0
                      7
                    
                  
                  
                    
                      1
                      8
                    
                  
                  
                    
                      1
                      9
                    
                  
                  
                    
                      0
                      10
                    
                  
                  
                    
                      1
                      11
                    
                  
                  
                    
                      1
                      12
                    
                  
                  
                    
                      0
                      13
                    
                  
                  
                    
                      0
                      14
                    
                  
                  
                    
                      1
                      15
                    
                  
                  
                    
                      0
                      16
                    
                  
                
              
              ]
            
          
          ;
        
      
    
  




  
    the
    ⁢
    
      
    
    ⁢
    next
  




  
    
      
        a
        1
      
      =
      
        
          
            a
            1
          
          ⊕
          
            a
            3
          
          ⊕
          
            a
            5
          
          ⊕
          
            a
            7
          
          ⊕
          
            a
            9
          
          ⊕
          
            a
            13
          
        
        =
        
          
            0
            ⊕
            1
            ⊕
            1
            ⊕
            0
            ⊕
            1
            ⊕
            0
          
          =
          1
        
      
    
    ;
  




  
    the
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Thirdly, in Embodiment III, now that A is a 4×4 matrix, thus, the F will be a 4×4×16 zero matrix, further, 1's at the ith position on the ith layer; the diffusion function F(p)=F⊕Fxi⊕Fyj, may refer to the whole derivation of three-dimensional FF(1:4). According to the diffusion mechanism F1=F(8:8), the sequential value of diffusion-area A1×t2=A1×(t2−1)F1⊕BF1, wherein BF1=BF(8:8)=[1010001000100100], derived as A1, Embodiment III, and F1=FF(8:8) is shown in Table 5.








TABLE 5





at



time t2
at time t2 − 1







a1
a1 ⊕ a4 ⊕ a5 ⊕ a6 ⊕ a9 ⊕ a11 ⊕ a12 ⊕ a16


a2
a4 ⊕ a5 ⊕ a6 ⊕ a11 ⊕ a12 ⊕ a13 ⊕ a15


a3
a1 ⊕ a2 ⊕ a7 ⊕ a8 ⊕ a10 ⊕ a12 ⊕ a13 ⊕ a14


a4
a1 ⊕ a8 ⊕ a9 ⊕ a11 ⊕ a12 ⊕ a15


a5
a1 ⊕ a5 ⊕ a6 ⊕ a9 ⊕ a10 ⊕ a12 ⊕ a13 ⊕ a14 ⊕ a15 ⊕ a16


a6
a2 ⊕ a3 ⊕ a4 ⊕ a5 ⊕ a7 ⊕ a8 ⊕ a9 ⊕ a11 ⊕ a14 ⊕ a16


a7
a3 ⊕ a8 ⊕ a9 ⊕ a12 ⊕ a15 ⊕ a16


a8
a1 ⊕ a2 ⊕ a3 ⊕ a5 ⊕ a7 ⊕ a12 ⊕ a13 ⊕ a15 ⊕ a16


a9
a1 ⊕ a2 ⊕ a3 ⊕ a4 ⊕ a5 ⊕ a8 ⊕ a9 ⊕ a13 ⊕ a14 ⊕ a15 ⊕ a16


a10
a3 ⊕ a5 ⊕ a8 ⊕ a10 ⊕ a12 ⊕ a13 ⊕ a14


a11
a2 ⊕ a3 ⊕ a4 ⊕ a7 ⊕ a9 ⊕ a11 ⊕ a12 ⊕ a15


a12
a2 ⊕ a3 ⊕ a5 ⊕ a6 ⊕ a8 ⊕ a10 ⊕ a11 ⊕ a13 ⊕ a14


a13
a2 ⊕ a3 ⊕ a4 ⊕ a5 ⊕ a8 ⊕ a10 ⊕ a12 ⊕ a13 ⊕ a14 ⊕ a15 ⊕ a16


a14
a4 ⊕ a6 ⊕ a7 ⊕ a8 ⊕ a9 ⊕ a11 ⊕ a13 ⊕ a14


a15
a1 ⊕ a4 ⊕ a8 ⊕ a9 ⊕ a10 ⊕ a13


a16
a1 ⊕ a2 ⊕ a4 ⊕ a5 ⊕ a6 ⊕ a10 ⊕ a12 ⊕ a15 ⊕ a16









Hardware Architecture in Detail:

FIG. 1, 300, obtaining a new value At1×t2=At1×(t2−1)Ft1 in a hardware architecture, would transform At1×t2=At1×(t2−1)Ft1⊕BFt1 in a parallel process into a first circuit design shown as FIG. 3A, comprising of: a first register R1 provided for storing an input value At1×(t2−1) 310; a diffuser DF provided for receiving At1×(t2−1) from R1, further, recombining At1×(t2−1) with a function-table Ft1 and transmitting an output value At1×(t2−1)Ft1 320; a first memory M1 provided for writing a medium-constant BFt1 330; an XOR Unit provided for receiving At1×(t2−1)Ft1 from DF as a first operand, BFt1 from M1 as a second operand, then, performing a bitwise XOR operation and transmitting an output value At1×t2 340; a second register R2 provided for buffering At1×t2 from said XOR Unit, and feeding back said At1×t2 to R1 350.
In addition, for the initialization of A, FIG. 2, a password bit selects a first value for At1×t2=At1×(t2−1)Ft1⊕BFt1 as by the above architecture, and a second value for At1×t2=At1×(t2−1)Ft1 shown as FIG. 3B as a second design of FIG. 1, 300, comprising of: a first register R1 provided for storing an input value At1×(t2−1) 310; a diffuser DF provided for receiving At1×(t2−1) from R1, further, recombining At1×(t2−1) with a function-table Ft1 and transmitting an output value At1×t2 320; a second register R2 provided for buffering At1×t2 from DF, and feeding back said At1×t2 to R1 350.
The whole scheme is shown as FIG. 3C, a main component, provided for performing a diffusion mechanism 300, as FIG. 1, 300, i.e. FIGS. 3A and 3B; in addition, a switch SW, provided for receiving an input password for initialization 200, as FIG. 1, 200; further comprising of: a multiplexer MUX provided for receiving At1×t2 from R2, further, selecting a bit of At1×t2 and transmitting the assigned bit 360; a third register R3 provided for storing a stream plaintext/ciphertext in sequence and transmitting a bit each time 400, as FIG. 1, 400; an XOR gate provided for receiving said assigned bit from MUX as a first operand, said bit from R3 as a second operand, then, performing an XOR operation and transmitting an output bit 500, as FIG. 1, 500.
Preferred Embodiment V: Diffusion Mechanisms in Hardware Architecture.
According to Embodiment I in a parallel process, a medium-constant BF1=BF(13:13)=[1011001101100011] and a diffusion-table F1=FF(13:13), see Table 3, are generated in advance; therefore, to gain more acceleration, it would write BF1 to M1, and transform the diffusion-table F1 into a diffuser DF combined with a number of XOR gates shown as FIG. 4; further, in Embodiment II, write BF2=BF(132:13)=[0110100110110010] to M1, and transform F2=FF(132:13), see Table 4, into a diffuser DF; in Embodiment III, BF1=BF(8:8)=[1010001000100100] to M1, and F1=FF(8:8), see Table 5, into a diffuser DF.
The above preferred embodiments are not only able to be applied separately, but also, can be taken as a whole. For example, as a number of parallel connections as FIG. 5, every SW, provided for receiving a password in sequence; every IN, provided for receiving an assigned part of a plaintext/ciphertext in sequence; and every OUT, provided for transmitting the assigned part of the ciphertext/plaintext in sequence. In addition, as a number of serial connections as FIG. 6, comprises of: the first SW, provided for receiving the password in sequence, further, the former OUT, connecting to the latter SW; every IN, provided for receiving an assigned part of a plaintext/ciphertext in sequence, and every OUT, provided for transmitting the assigned part of the ciphertext/plaintext in sequence; or for more security, only encrypting or decrypting a plaintext/ciphertext stream in Embodiment III.
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.

## Claims

1. A stream cryptographic method comprising:
a communication channel between a first computer and a second computer;
a diffusion-area A, wherein A contains an initial value A0 and further, is a d1×d2× . . . ×dm binary matrix with a combination of positions labeled sequentially from 1 to n;
a diffusion mechanism Ft1 applied to A, notated AFt1 in abbreviation, wherein AFt1=AF(p1, p2t1, . . . , pk);
a diffusion-medium S, wherein S is a m-dimensional binary matrix which includes an anchor positon {dot over (S)};
and further, comprising steps of:
(I) generating, by the first and the second computers, and assigned bit for steam encryption/decryption by:
(a) setting t2=1, At1×(t2−1)=A0;
(b) obtaining a new value At1×t2, wherein At1×t2−1)Ft1,
further comprising steps of:
(i) setting t=1;
(ii) setting s=1;
(iii) letting At2−1=A, and performing A with a diffusion function F(ps), whrein S overlaps A by {dot over (S)} anchoring to the position ps transforming into m-dimension coordinates (i1, i2, . . . , im), then AF(ps)=A⊕Ad1i⊕ . . . ⊕Admi⊕S, 1≦ps≦n;
(iv) if s=k, then going to next step; otherwise, s=s+1, going back to Step(iii);
(v) if t≢t1, then t=t+1, going to Step(ii);

(c) outputting the assigned bit of At1×t2;

(II) encrypting, by the first computer, a bit of a plaintext stream in sequence, wherein the bit of the plaintext stream XORs with the assigned bit of At1×t2;
(III) decrypting, by the second computer, a bit of the ciphertext stream in sequence, received from the first computer, wherein the bit of the ciphertext stream XORs with the assigned bit of At1×t2;
(IV) going back to Step (b), and further, letting t2=t2+1.

2. The stream cryptographic method according to claim 1, Step (a), wherein A0 is generated by a password written directly to A, or by reading every password bit, such that At1×t2=At133(t2−1)Ft1 in sequence for A0, if a first value, then including S, otherwise, if a second value, then excluding S wherein S=0.

3. The stream cryptographic method according to claim 1, further providing a signal to another said method, wherein the former method, Step (c) sends the signal to the latter method, Step (b), if a first value, then including S, otherwise, if a second value, then excluding S, wherein S=0.

4. A stream cryptographic method comprising:
a communications channel between a first computer and a second computer;
a diffusion-area A, wherein A contains an initial value A0 and further, is a d1×d2× . . . ×dm binary matrix with a combination of positions labeled sequentially from 1 to n, accordingly, the bit values from a1 to an;
a function-table Ft1 applied to A, notated AFt1 in abbreviation, wherein Ft1 has d1×d2× . . . ×dm layer, and every layer is a d1×d2× . . . ×dm, binary matrix;
a diffusion-medium S, wherein S is a m-dimensional binary matrix which includes an anchor-position {dot over (S)};
a medium-constant BFt1, wherein BFt1=BF(p1, p2t1pk) and B is a D1×d2=. . . =dm zero matrix;
and further, comprising steps of;
(I) generating, by the first and second computers, an assigned bit for stream encryption/decryption by;
(a) setting t2=1, At1×(t2−1)=A0;
(b) obtaining the medium-constant BFt1, further comprising steps of:
(i) setting t=1;
(ii) setting s=1;
(iii) performing B with a diffusion function F(ps), namely BF(ps), wherein S overlaps B by {dot over (S)} anchoring to the position ps transforming into m-dimension coordinates (i1, i2, . . . , im), then BF(ps)=B⊕Bd1i⊕Bd2i⊕. . . ⊕ Bdmi⊕S, 1≦ps≦n;
(iv) if s=k, then going to next step; otherwise, s=s+1, going back to Step (iii)
(v) if t≢t1, then t=t+1, going back to Step (ii),

(c) obtaining a new value At1×t2, wherein At1×t2=At1×(t2−1)Ft1⊕BFt1, wherein At1×(t2−1)Ft1 obtains new ai, 1≦n, by XORing with said layers having value 1 at position i, wherein the ith layer represent ai;
(d) outputting an assigned bit of At1×t2;

(II) encrypting, by the first computer, a bit of a plaintext stream in sequence, wherein in the bit of the plaintext stream XORs with the assigned bit of At1×t2;
(III) decrypting, by the second computer, a bit of the ciphertext stream in sequence, received from the first computer, wherein the bit of the ciphertext steam XORs with the assigned bit of At1×t2;
(IV) going to Step (c), and further, ; letting t2=2+1.

5. The stream cryptographic method according to claim 4, Step (c) wherein Ft1=FF(p1, p2t1, . . . , pk) generated once, comprising steps of:
(i) setting F is a zero matrix and 1's at the ith position on the ith layer;
(ii) setting t=1;
(iii) setting s=1;
(iv) performing F with a diffusion function F(ps), namely FF(ps), wherein ps is transformed into m-dimension coordinates (i1, i2, . . . , im), then FF(ps)=F⊕Fd1i⊕Fd2i⊕ . . . ⊕Fdmi, 1≦ps≦n;
(v) if s=k, then going to next step; otherwise, s=s+1, going back to Step (iv);
(vi) if t≠t1, then t=t+1, going to back Step (iii).

6. The stream cryptographic method according to claim 4, Step (a), wherein A0 is generated by a password written directly to A, or by reading every password bit, such that At1×t2=At1×(t2−1)Ft1⊕BFt1 in sequence for A0, if a first value, then including BFt1, otherwise, if a second value, then excluding BFt1 wherein BFt1=0.

7. The stream cryptographic method according to claim 4, further providing a signal to another said method, wherein the former method, Step (d) sends the signal to the latter method, Step(c), if a first value, then including BFt1, otherwise, if a second value, then excluding BFt1 wherein BFt1=0.

8. A stream cryptographic apparatus comprising of:
a first register R1 provided for storing an input value At1×(t2−1);
a diffuser DF provided for receiving At1×(t2−1) from the first register R1, further,
recombining At1×(t2−1) with a function-table Ft1 and transmitting an output value At1×t2=At1×(t2−1)Ft1;
a second register R2 provided for receiving and buffering At1×t2 from the diffuser DF, then, feeding back said At1×t2 to the first register R1;
a multiplexer MUX provided for receiving At1×t2 from the second register R2, further, selecting an assigned bit of At1×t2 and transmitting the assigned bit;
a third register R3 provided for storing a plaintext/ciphertext stream in sequence, the third register R3 input marked IN, and transmitting a bit each time;
an XOR gate provided for receiving said assigned bit from the multiplexer MUX as a first operand, said bit from the third register R3 as a second operand, then, performing an exclusive or operation and transmitting an output bit; the output marked OUT.

9. The stream cryptographic apparatus according to claim 8, the first register R1 further provided for storing an initial value A0 written directly by a password.

10. The stream cryptographic apparatus according to claim 8, further comprising of:
a first memory M1 provided for storing a medium-constant BFt1;
an XOR Unit provided for receiving At1×t2 from the diffuser DF as a first operand, the medium-constant BFt1 from the first memory M1 as a second operand, then, performing an bitwise exclusive or operation and transmitting an output value At1×t2=At1×t2⊕BFt1;
the second register R2 provided for receiving and buffering At1×t2 from the XOR Unit, then, feeding back said At1×t2 to the first register R1;
a switch SW provided for receiving a password in sequence and performing said Ft1 recombination, further checking the bit if a first value, then the second register R2 receiving from the diffuser DF, if a second value, then the second register R2 receiving from the XOR Unit.

11. The stream cryptographic apparatus according to claim 10, the first register R1 further provided for storing an initial value A0 by the switch SW receiving the password in sequence.

12. The stream cryptographic apparatus according to claim 10, further provided for a parallel connecting to another said apparatuses, comprising of:
every apparatus, the switch SW, provided for receiving a password in sequence;
every apparatus, the third register R3 input IN, provided for receiving an assigned part of a plaintext/ciphertext in sequence;
every apparatus, the XOR gate's output OUT, provided for transmitting the assigned part of the ciphertext/plaintext in sequence.

13. The stream cryptographic apparatus according to claim 10, further provided for a serial connecting to another said apparatuses, comprising of:
the first apparatus, the switch SW, provided for receiving a password in sequence;
the last apparatus, the third register R3 input IN, provided for receiving a plaintext/ciphertext in sequence;
the last apparatus, the XOR gate's output OUT, provided for transmitting the ciphertext/plaintext in sequence;
every apparatus except the first one, the switch SW, provided for receiving the output from the former apparatus, the XOR gate's output OUT;
every apparatus except the last one, the XOR gate's output OUT, provided for transmitting the output to the latter apparatus, the switch SW.

14. The stream cryptographic apparatus according to claim 13, further comprising of:
every apparatus, the third register R3 input IN, provided for receiving an assigned part of a plaintext/ciphertext in sequence;
every apparatus, the XOR gate's output OUT, provided for transmitting the assigned part of the ciphertext/plaintext in sequence.