Abstract:
The present invention relates to information security and discloses a method of establishing public key cryptographic protocols against the quantum computational attack. The method includes the following steps: definition of an infinite non-abelian group G; choosing two private keys in G by two entities; a second entity computing y, and sending y to a first entity; the first entity computing x and z, and sending (x, z) to the second entity; the second entity computing w and v, and sending (w, v) to the first entity; the first entity computing u, and sending u to the second entity; and the first entity computing K A , and the second entity computing K B , thereby reaching a shared key K=K A =K B . The security guarantee of a public key cryptographic algorithm created by the present invention relies on unsolvability of a problem, and has an advantage of free of the quantum computational attack.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application claims priority of Chinese patent application No. 201310382299.7 filed on Aug. 21, 2013, the entire content of which are hereby incorporated by reference. 
     
    
     BACKGROUND 
       [0002]    1. Technical Field 
         [0003]    The present invention relates to the field of information security, and in particular, to a cryptogram technology for establishing public key cryptographic protocols against the quantum computational attack. 
         [0004]    2. Related Art 
         [0005]    The verification for a real identity of a person who sends and receives information, and the non-repudiation of the sentreceived information after the information is sent or received and the guarantee of the integrity of data are two important issues about the theme of modern cryptography. 
         [0006]    Disclosure of a key cryptogram system presents excellent answers to the issues of the two aspects, and more new ideas and solutions are being generated continually. In a public key system, an encryption key is different from a decryption key. People bring an encryption key to public, so that anyone can use the encryption key; but a decryption key is only known by a person performing decryption. In modern periods, the security of a public key cryptosystem is almost based on two categories of mathematic problems that are considered to be difficult to compute, a first category being a decomposition problem of a big prime number, for example, an RSA algorithm; and a second category being a discrete logarithm problem, for example, a key exchange algorithm of Diffie-Hellman, an El Gamal algorithm, an elliptic curve public key cryptographic algorithm (ECC for short), and the like. 
       SUMMARY 
       [0007]    In order to solve a problem that a hidden trouble exists in identity verification and security of data guarantee based on an existing public key cryptographic protocol, an objective of the present invention is to establish public key cryptographic protocols technology capable of resisting various known attacks, and provide various application protocols on this basis. 
         [0008]    One manner for implementing the objective of the present invention is: a method of establishing public key cryptographic protocols against the quantum computational attack, which includes a method for generating a shared key. The method for generating a shared key is also referred to as generating a shared key protocol, and the method for generating a shared key includes the following steps: 
         [0009]    (11) establishing an infinite non-abelian group G and two subgroups A and B of G, so that for any a∈A and any b∈B, the equation ab=ba is true; 
         [0010]    (12) choosing, by a first entity of a protocol, an element g in G, where the first entity of the protocol chooses two elements b 1 , b 2 ∈A as private keys, and a second entity of the protocol chooses two elements d 1 , d 2 ∈B as private keys; 
         [0011]    (13) choosing, by the second entity of the protocol, two elements c 1 , c 2 ∈B, computing y=d 1 c 1 gc 2 d 2 , and sending y to the first entity of the protocol; 
         [0012]    (14) choosing, by the first entity of the protocol, four elements a 1 , a 2 , b 3 , b 4 ∈A, computing 
         [0000]      x=b 1 a 1 ga 2 b 2  and z=b 3 a 1 ya 2 b 4 =b 3 a 1 d 1 c 1 gc 2 d 2 a 2 b 4 , 
         [0013]    and sending (x, z) to the second entity of the protocol; 
         [0014]    (15) choosing, by the second entity of the protocol, two elements d 3 , d 2 ∈B, computing 
         [0000]      w=d 3 c 1 xc 2 d 4 =d 3 c 1 b 1 a 1 ga 2 b 2 c 2 d 4    
         [0000]      and 
         [0000]      v=d 1   1 zd 2   1 =d 1   1 b 3 a 1 d 1 c 1 gc 2 d 2 a 2 b 4 d 2   1 =b 3 a 1 c 1gc   2 a 2 b 4    
         [0015]    and sending (w, v) to the first entity of the protocol; 
         [0016]    (16) computing, by the first entity of the protocol, 
         [0000]      u=b 1   −1 wb 2   −1 =b 1   −1 d 3 c 1 a 1 ga 2 b 2 c 2 d 4 b 2   −1 =d 3 c 1 a 1 ga 2 c 2 d 4 , 
         [0017]    and sending u to the second entity of the protocol; and 
         [0018]    (17) computing, by the first entity of the protocol, K A =b 3   −1 vb 4   −1 =a 1 c 1 gc 2 a 2 , and computing, by the second entity of the protocol, K B =d 3   −1 ud 4   −1 =c 1 a 1 ga 2 c 2 ; 
         [0019]    because a 1 , a 2 ∈A, and c 1 , c 2 ∈B, a 1  and c 1  are separately commute with a 2  and c 2  in multiplication, so that the first entity of the protocol and the second entity of the protocol reach a shared key K=K A =K B . 
         [0020]    In the present invention, an algebra system in which an unsolvable problem exists is first established theoretically, and second, the unsolvability of the problem is used as security guarantee to establish a public key cryptographic algorithm. The security of the algorithm and the equivalence of the unsolvable problem of the present invention prove that the present invention is immune to the quantum computational attack and the like. Because the method of establishing public key cryptographic protocols of the present invention uses an unsolvable decision problem as the security guarantee, the method is powerfully guaranteed both theoretically and in an actual application aspect, and compared with the prior art, has the following advantages: 
         [0021]    1. The security guarantee of a built public key cryptographic algorithm relies on the unsolvability of the problem rather than the computation difficulty of the problem, (a classic public key cryptographic algorithm is based on the computation difficulty); 
         [0022]    2. That the security of the public key cryptographic algorithm of the present invention is equivalent to the unsolvability of the problem on which the public key cryptographic algorithm relies has been proved mathematically; 
         [0023]    3. The public key cryptographic algorithm of the present invention resists the quantum computational attack. 
     
    
     DETAILED DESCRIPTION 
       [0024]    The following further describes in detail establishment of public key cryptographic protocols against the quantum computational attack according to the present invention with reference to embodiments. 
         [0025]    1. A Platform for Establishing Public Key Cryptographic Protocols 
         [0026]    A platform for establishing all public key cryptographic protocols is an infinite non-abelian group G and two subgroups A and B of G, so that for any a∈A and any b∈B, the equation ab=ba is true. In addition, because of demands of encoding and key generating, G must further satisfy the following conditions: 
         [0027]    1) Any word in terms of generators of G representing an element of G has an unique computable normal form; 
         [0028]    2) G at least is in exponential growth, that is, the number of elements whose word length is a positive integer n in G is confined to an exponential function about n; 
         [0029]    3) Multiplication and inversion of a group based on the normal form is computable. 
         [0030]    Therefore, a braid group B n  with n≧12 is taken as the infinite non-abelian group G, where B n  has the foregoing properties and is a group defined by the following presentation: 
         [0000]        B   n =         σ 1 , σ 2 , . . . , σ n−1 |σ 1 σ j =σ j σ 1   , |i−j|≧ 2, σ 1 σ 1+1 σ 1 =σ 1+1 σ 1 σ 1+1 , 1≦ i≦n− 2         ,
 
         [0031]    the braid group B n  contains the following two subgroups: 
         [0032]    let m=└n/2┘ be a maximum integer not greater than n/2, and a left braid LB n  and a right braid RB n  of the braid group B n  separately are 
         [0000]      LB n =         σ 1 , σ 2 , . . . , σ m−1           and RB n =         σ m+1 , σ m+2 , . . . , σ n−1           
 
         [0033]    that is, separately are subgroups generated by σ 1 , σ 2 , . . . , σ m−1  and σ m+1 , σ m+2 , . . . , σ n−1 , and for any a∈ LB n  and any be RB n , ab=ba is true. 
         [0034]    When n≧12, LB n  and RB n  separately contain a subgroup isomorphic to the direct product of F 2 ×F 2 , that is, two free groups with ranks being 2: 
         [0000]      IA=         σ m−5   2 , σ m−4   2 , σ m−2   2 , σ m−1   2           ≦LB n  
 
         [0000]      and 
         [0000]      RA=         σ m+1   2 , σ m+2   2 , σ m+4   2 , σ m+5   2 ,          ≦RB n ,
 
         [0035]    and then a finite presentation group H whose word problem is unsolvable and that is generated by two elements constructs a Mihailova subgroup M LA (H) of LA and a Mihailova subgroup M RA (H) of RA again; the following is 56 generators of M LA (H), where i=m−5; and when i=m+1, 56 generators of M RA (H) can be obtained: 
         [0000]      σ i   2 σ i+3   2 , σ i+1   2 σ i+4   2 , S ij , T ij , j=1, 2, . . . , 27
 
         [0000]    and  27  S u s are (all ( 7 ,s in the following each S u  are replaced with ( 7 , 3 s, and all  6 , +1 s are replaced with  6+4 s to obtain corresponding  27  To, where j= 1 ,  2 , . . . ,  27 ): 
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                         14 
                       
                     
                     ) 
                   
                 
                 9 
               
             
           
         
       
       
         
           
             
                 
             
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                 σ 
                 i 
                 
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                   4 
                 
               
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                 σ 
                 
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                 6 
               
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                     σ 
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                     2 
                   
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                     σ 
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                     8 
                   
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                     σ 
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                     σ 
                     
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                       1 
                     
                     
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                       8 
                     
                   
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                     σ 
                     i 
                     
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                     σ 
                     
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                     σ 
                     
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                       1 
                     
                     8 
                   
                    
                   
                     σ 
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                       4 
                     
                   
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                     σ 
                     
                       i 
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                       14 
                     
                   
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                     σ 
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                    
                   
                     σ 
                     
                       i 
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                       1 
                     
                     2 
                   
                    
                   
                     σ 
                     i 
                     
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                       2 
                     
                   
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                     14 
                   
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                     σ 
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                       1 
                     
                     
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                       14 
                     
                   
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                     σ 
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                     2 
                   
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                     σ 
                     
                       i 
                       + 
                       1 
                     
                     6 
                   
                 
                 ) 
               
               
                 - 
                 1 
               
             
           
         
       
       
         
           
             
                 
             
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                 14 
               
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                     14 
                   
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                 σ 
                 i 
                 
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                 σ 
                 
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                 8 
               
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                     σ 
                     i 
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                     σ 
                     
                       i 
                       + 
                       1 
                     
                     2 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
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                     σ 
                     
                       i 
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                       1 
                     
                     10 
                   
                    
                   
                     σ 
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                     4 
                   
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                     σ 
                     
                       i 
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                       1 
                     
                     
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                       10 
                     
                   
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                     σ 
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                     σ 
                     
                       i 
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                       1 
                     
                     10 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       4 
                     
                   
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                     σ 
                     
                       i 
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                       - 
                       14 
                     
                   
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                     σ 
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                       1 
                     
                     2 
                   
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                     14 
                   
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                       1 
                     
                     
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                       14 
                     
                   
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                     σ 
                     
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                       1 
                     
                     4 
                   
                 
                 ) 
               
               
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                 1 
               
             
           
         
       
       
         
           
             
                 
             
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                     14 
                   
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                 10 
               
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                     2 
                   
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                               - 
                               4 
                             
                           
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                             2 
                           
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                             16 
                           
                         
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                     9 
                   
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                     14 
                   
                 
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                     σ 
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                     4 
                   
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                     4 
                   
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                             16 
                           
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                             16 
                           
                         
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                     9 
                   
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                     16 
                   
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                       1 
                     
                     12 
                   
                 
                 ) 
               
               
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                 1 
               
             
           
         
       
       
         
           
             
                 
             
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                   1 
                 
                 
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                   12 
                 
               
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                 16 
               
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                 σ 
                 
                   i 
                   + 
                   1 
                 
                 16 
               
                
               
                 σ 
                 i 
                 
                   - 
                   4 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 i 
                 2 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 2 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 2 
               
                
               
                 σ 
                 i 
                 4 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 i 
                 2 
               
             
           
         
       
       
         
           
             
               S 
               
                 i 
                  
                 
                     
                 
                  
                 8 
               
             
              
             
               : 
             
              
             
                 
             
              
             
               
                 ( 
                 
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     6 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     6 
                   
                    
                   
                     σ 
                     i 
                     4 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       6 
                     
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     
                       
                         σ 
                         
                           i 
                           + 
                           1 
                         
                         6 
                       
                        
                       
                         ( 
                         
                           
                             σ 
                             i 
                             
                               - 
                               4 
                             
                           
                            
                           
                             σ 
                             
                               i 
                               + 
                               1 
                             
                             
                               - 
                               16 
                             
                           
                            
                           
                             σ 
                             i 
                             2 
                           
                            
                           
                             σ 
                             
                               i 
                               + 
                               1 
                             
                             2 
                           
                            
                           
                             σ 
                             i 
                             
                               - 
                               2 
                             
                           
                            
                           
                             σ 
                             
                               i 
                               + 
                               1 
                             
                             16 
                           
                            
                           
                             σ 
                             i 
                             4 
                           
                            
                           
                             σ 
                             
                               i 
                               + 
                               1 
                             
                             
                               - 
                               16 
                             
                           
                            
                           
                             σ 
                             i 
                             
                               - 
                               2 
                             
                           
                            
                           
                             σ 
                             
                               i 
                               + 
                               1 
                             
                             
                               - 
                               2 
                             
                           
                            
                           
                             σ 
                             i 
                             2 
                           
                            
                           
                             σ 
                             
                               i 
                               + 
                               1 
                             
                             16 
                           
                         
                         ) 
                       
                     
                     9 
                   
                    
                   
                     
 
                   
                    
                   
                       
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       4 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       16 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     2 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     16 
                   
                    
                   
                     σ 
                     i 
                     4 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       16 
                     
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     10 
                   
                 
                 ) 
               
               
                 - 
                 1 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   10 
                 
               
                
               
                 σ 
                 i 
                 2 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 2 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 16 
               
                
               
                 σ 
                 i 
                 4 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   16 
                 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 i 
                 2 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 16 
               
                
               
                 σ 
                 i 
                 
                   - 
                   4 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 i 
                 2 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 2 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 2 
               
                
               
                 σ 
                 i 
                 4 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 i 
                 2 
               
             
           
         
       
       
         
           
             
               S 
               
                 i 
                  
                 
                     
                 
                  
                 9 
               
             
              
             
               : 
             
              
             
                 
             
              
             
               
                 ( 
                 
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     8 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     8 
                   
                    
                   
                     σ 
                     i 
                     4 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       8 
                     
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     
                       
                         σ 
                         
                           i 
                           + 
                           1 
                         
                         2 
                       
                        
                       
                         ( 
                         
                           
                             σ 
                             i 
                             
                               - 
                               4 
                             
                           
                            
                           
                             σ 
                             
                               i 
                               + 
                               1 
                             
                             
                               - 
                               16 
                             
                           
                            
                           
                             σ 
                             i 
                             2 
                           
                            
                           
                             σ 
                             
                               i 
                               + 
                               1 
                             
                             2 
                           
                            
                           
                             σ 
                             i 
                             
                               - 
                               2 
                             
                           
                            
                           
                             σ 
                             
                               i 
                               + 
                               1 
                             
                             16 
                           
                            
                           
                             σ 
                             i 
                             4 
                           
                            
                           
                             σ 
                             
                               i 
                               + 
                               1 
                             
                             
                               - 
                               16 
                             
                           
                            
                           
                             σ 
                             i 
                             
                               - 
                               2 
                             
                           
                            
                           
                             σ 
                             
                               i 
                               + 
                               1 
                             
                             
                               - 
                               2 
                             
                           
                            
                           
                             σ 
                             i 
                             2 
                           
                            
                           
                             σ 
                             
                               i 
                               + 
                               1 
                             
                             16 
                           
                         
                         ) 
                       
                     
                     9 
                   
                    
                   
                     
 
                   
                    
                   
                       
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       4 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     16 
                   
                    
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     2 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     16 
                   
                    
                   
                     σ 
                     i 
                     4 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       16 
                     
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     8 
                   
                 
                 ) 
               
               
                 - 
                 1 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   8 
                 
               
                
               
                 σ 
                 i 
                 2 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 2 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 16 
               
                
               
                 σ 
                 i 
                 4 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   16 
                 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 i 
                 2 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 16 
               
                
               
                 σ 
                 i 
                 
                   - 
                   4 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 i 
                 2 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 2 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 2 
               
                
               
                 σ 
                 i 
                 4 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 i 
                 2 
               
             
           
         
       
       
         
           
             
               S 
               
                 i 
                 , 
                 10 
               
             
              
             
               : 
             
              
             
                 
             
              
             
               
                 ( 
                 
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     10 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     10 
                   
                    
                   
                     σ 
                     i 
                     4 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       10 
                     
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     
                       
                         σ 
                         
                           i 
                           + 
                           1 
                         
                         2 
                       
                        
                       
                         ( 
                         
                           
                             σ 
                             i 
                             
                               - 
                               4 
                             
                           
                            
                           
                             σ 
                             
                               i 
                               + 
                               1 
                             
                             
                               - 
                               16 
                             
                           
                            
                           
                             σ 
                             i 
                             2 
                           
                            
                           
                             σ 
                             
                               i 
                               + 
                               1 
                             
                             2 
                           
                            
                           
                             σ 
                             i 
                             
                               - 
                               2 
                             
                           
                            
                           
                             σ 
                             
                               i 
                               + 
                               1 
                             
                             16 
                           
                            
                           
                             σ 
                             i 
                             4 
                           
                            
                           
                             σ 
                             
                               i 
                               + 
                               1 
                             
                             
                               - 
                               16 
                             
                           
                            
                           
                             σ 
                             i 
                             
                               - 
                               2 
                             
                           
                            
                           
                             σ 
                             
                               i 
                               + 
                               1 
                             
                             
                               - 
                               2 
                             
                           
                            
                           
                             σ 
                             i 
                             2 
                           
                            
                           
                             σ 
                             
                               i 
                               + 
                               1 
                             
                             16 
                           
                         
                         ) 
                       
                     
                     9 
                   
                    
                   
                     
 
                   
                    
                   
                       
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       4 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       16 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     2 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     16 
                   
                    
                   
                     σ 
                     i 
                     4 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       16 
                     
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     6 
                   
                 
                 ) 
               
               
                 - 
                 1 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   6 
                 
               
                
               
                 σ 
                 i 
                 2 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 2 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
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                 σ 
                 
                   i 
                   + 
                   1 
                 
                 16 
               
                
               
                 σ 
                 i 
                 4 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   16 
                 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 i 
                 2 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 16 
               
                
               
                 σ 
                 i 
                 
                   - 
                   4 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 i 
                 2 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 2 
               
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                 σ 
                 i 
                 
                   - 
                   2 
                 
               
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                 σ 
                 
                   i 
                   + 
                   1 
                 
                 2 
               
                
               
                 σ 
                 i 
                 4 
               
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                 σ 
                 
                   i 
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                   1 
                 
                 
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                   2 
                 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 i 
                 2 
               
             
           
         
       
       
         
           
             
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                 i 
                 , 
                 11 
               
             
              
             
               : 
             
              
             
                 
             
              
             
               
                 ( 
                 
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     2 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     2 
                   
                    
                   
                     σ 
                     i 
                     4 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     
                       i 
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                       1 
                     
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     2 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       4 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       18 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     2 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     18 
                   
                    
                   
                     σ 
                     i 
                     4 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       18 
                     
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     16 
                   
                 
                 ) 
               
               
                 - 
                 1 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   16 
                 
               
                
               
                 σ 
                 i 
                 2 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 2 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 18 
               
                
               
                 σ 
                 i 
                 4 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   18 
                 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 
                   i 
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                   1 
                 
                 
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                   2 
                 
               
                
               
                 σ 
                 i 
                 2 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 18 
               
                
               
                 σ 
                 i 
                 
                   - 
                   4 
                 
               
                
               
                 σ 
                 
                   i 
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                   1 
                 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 i 
                 2 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 2 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
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                 σ 
                 
                   i 
                   + 
                   1 
                 
                 2 
               
                
               
                 σ 
                 i 
                 4 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 i 
                 2 
               
             
           
         
       
       
         
           
             
               S 
               
                 i 
                 , 
                 12 
               
             
              
             
               : 
             
              
             
                 
             
              
             
               
                 ( 
                 
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     σ 
                     
                       i 
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                       1 
                     
                     2 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
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                     σ 
                     
                       i 
                       + 
                       1 
                     
                     4 
                   
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                     σ 
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                     4 
                   
                    
                   
                     σ 
                     
                       i 
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                       1 
                     
                     
                       - 
                       4 
                     
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       2 
                     
                   
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                     σ 
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                     2 
                   
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                     σ 
                     
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                       1 
                     
                     4 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       4 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       18 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
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                     σ 
                     
                       i 
                       + 
                       1 
                     
                     2 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     18 
                   
                    
                   
                     σ 
                     i 
                     4 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       18 
                     
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     14 
                   
                 
                 ) 
               
               
                 - 
                 1 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
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                   14 
                 
               
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                 σ 
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                 σ 
                 
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                   1 
                 
                 2 
               
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                 σ 
                 i 
                 
                   - 
                   2 
                 
               
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                 σ 
                 
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                   1 
                 
                 18 
               
                
               
                 σ 
                 i 
                 4 
               
                
               
                 σ 
                 
                   i 
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                   1 
                 
                 
                   - 
                   18 
                 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 
                   i 
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                   1 
                 
                 
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                   2 
                 
               
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                 σ 
                 i 
                 2 
               
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                 σ 
                 
                   i 
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                   1 
                 
                 18 
               
                
               
                 σ 
                 i 
                 
                   - 
                   4 
                 
               
                
               
                 σ 
                 
                   i 
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                   1 
                 
                 
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                   4 
                 
               
                
               
                 σ 
                 i 
                 2 
               
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                 σ 
                 
                   i 
                   + 
                   1 
                 
                 2 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 
                   i 
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                   1 
                 
                 4 
               
                
               
                 σ 
                 i 
                 4 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   4 
                 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 i 
                 2 
               
             
           
         
       
       
         
           
             
               S 
               
                 i 
                 , 
                 13 
               
             
              
             
               : 
             
              
             
                 
             
              
             
               
                 ( 
                 
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     2 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
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                     σ 
                     
                       i 
                       + 
                       1 
                     
                     6 
                   
                    
                   
                     σ 
                     i 
                     4 
                   
                    
                   
                     σ 
                     
                       i 
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                       1 
                     
                     
                       - 
                       6 
                     
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     6 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       4 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       18 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     σ 
                     
                       i 
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                       1 
                     
                     2 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     18 
                   
                    
                   
                     σ 
                     i 
                     4 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       18 
                     
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
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                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     12 
                   
                 
                 ) 
               
               
                 - 
                 1 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
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                   12 
                 
               
                
               
                 σ 
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                
               
                 σ 
                 
                   i 
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                   1 
                 
                 2 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
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                 σ 
                 
                   i 
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                   1 
                 
                 18 
               
                
               
                 σ 
                 i 
                 4 
               
                
               
                 σ 
                 
                   i 
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                   1 
                 
                 
                   - 
                   18 
                 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 
                   i 
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                   1 
                 
                 
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                   2 
                 
               
                
               
                 σ 
                 i 
                 2 
               
                
               
                 σ 
                 
                   i 
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                   1 
                 
                 18 
               
                
               
                 σ 
                 i 
                 
                   - 
                   4 
                 
               
                
               
                 σ 
                 
                   i 
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                   1 
                 
                 
                   - 
                   6 
                 
               
                
               
                 σ 
                 i 
                 2 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 2 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 6 
               
                
               
                 σ 
                 i 
                 4 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   6 
                 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 i 
                 2 
               
             
           
         
       
       
         
           
             
               S 
               
                 i 
                 , 
                 14 
               
             
              
             
               : 
             
              
             
                 
             
              
             
               
                 ( 
                 
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     2 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     8 
                   
                    
                   
                     σ 
                     i 
                     4 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       8 
                     
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     8 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       4 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       18 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     2 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     18 
                   
                    
                   
                     σ 
                     i 
                     4 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       18 
                     
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     10 
                   
                 
                 ) 
               
               
                 - 
                 1 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   10 
                 
               
                
               
                 σ 
                 i 
                 2 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 2 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 18 
               
                
               
                 σ 
                 i 
                 4 
               
                
               
                 σ 
                 
                   i 
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                   1 
                 
                 
                   - 
                   18 
                 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 i 
                 2 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 18 
               
                
               
                 σ 
                 i 
                 
                   - 
                   4 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   8 
                 
               
                
               
                 σ 
                 i 
                 2 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 2 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 8 
               
                
               
                 σ 
                 i 
                 4 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   8 
                 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 i 
                 2 
               
             
           
         
       
       
         
           
             
               S 
               
                 i 
                 , 
                 15 
               
             
              
             
               : 
             
              
             
                 
             
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                 ( 
                 
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     2 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     10 
                   
                    
                   
                     σ 
                     i 
                     4 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       10 
                     
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     10 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       4 
                     
                   
                    
                   
                     σ 
                     
                       i 
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                       1 
                     
                     
                       - 
                       18 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     2 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
                    
                   
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                       i 
                       + 
                       1 
                     
                     18 
                   
                    
                   
                     σ 
                     i 
                     4 
                   
                    
                   
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                       + 
                       1 
                     
                     
                       - 
                       18 
                     
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
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                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     8 
                   
                 
                 ) 
               
               
                 - 
                 1 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
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                   8 
                 
               
                
               
                 σ 
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                 2 
               
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                 σ 
                 
                   i 
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                   1 
                 
                 2 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
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                 σ 
                 
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                   1 
                 
                 18 
               
                
               
                 σ 
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                 4 
               
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                   i 
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                   1 
                 
                 
                   - 
                   18 
                 
               
                
               
                 σ 
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                   - 
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                 σ 
                 
                   i 
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                   1 
                 
                 
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                   2 
                 
               
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                 σ 
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                 σ 
                 
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                   1 
                 
                 18 
               
                
               
                 σ 
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                   - 
                   4 
                 
               
                
               
                 σ 
                 
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                   1 
                 
                 
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                   10 
                 
               
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                 σ 
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                 2 
               
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                 σ 
                 
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                   1 
                 
                 2 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
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                 10 
               
                
               
                 σ 
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                 4 
               
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                   1 
                 
                 
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                   - 
                   2 
                 
               
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                   1 
                 
                 
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                   2 
                 
               
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                 σ 
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                 2 
               
             
           
         
       
       
         
           
             
               S 
               
                 i 
                 , 
                 16 
               
             
              
             
               : 
             
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                 ( 
                 
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       6 
                     
                   
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                     σ 
                     i 
                     2 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     2 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
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                     σ 
                     
                       i 
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                       1 
                     
                     20 
                   
                    
                   
                     σ 
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                     4 
                   
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                     σ 
                     
                       i 
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                       1 
                     
                     
                       - 
                       20 
                     
                   
                    
                   
                     σ 
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                       - 
                       2 
                     
                   
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                     σ 
                     
                       i 
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                       1 
                     
                     
                       - 
                       2 
                     
                   
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                     σ 
                     i 
                     2 
                   
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                     σ 
                     
                       i 
                       + 
                       1 
                     
                     20 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       4 
                     
                   
                    
                   
                     σ 
                     
                       i 
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                       1 
                     
                     
                       - 
                       14 
                     
                   
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                     σ 
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                     2 
                   
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                     σ 
                     
                       i 
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                       1 
                     
                     2 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
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                       i 
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                       1 
                     
                     14 
                   
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                     σ 
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                     4 
                   
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                       1 
                     
                     
                       - 
                       14 
                     
                   
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                       - 
                       2 
                     
                   
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                       i 
                       + 
                       1 
                     
                     
                       - 
                       2 
                     
                   
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                     2 
                   
                 
                 ) 
               
               
                 - 
                 1 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 σ 
                 i 
                 2 
               
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                 σ 
                 
                   i 
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                   1 
                 
                 2 
               
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                   - 
                   2 
                 
               
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                   1 
                 
                 14 
               
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                 σ 
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                 4 
               
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                   1 
                 
                 
                   - 
                   14 
                 
               
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                   - 
                   2 
                 
               
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                   2 
                 
               
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                 σ 
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                   1 
                 
                 14 
               
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                 σ 
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                   - 
                   4 
                 
               
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                 σ 
                 
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                   1 
                 
                 
                   - 
                   20 
                 
               
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                 σ 
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                 2 
               
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                 σ 
                 
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                   1 
                 
                 2 
               
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                 σ 
                 i 
                 
                   - 
                   2 
                 
               
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                 σ 
                 
                   i 
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                   1 
                 
                 20 
               
                
               
                 σ 
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                 4 
               
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                   1 
                 
                 
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                   20 
                 
               
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                   - 
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                   1 
                 
                 
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                   2 
                 
               
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                   1 
                 
                 6 
               
             
           
         
       
       
         
           
             
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                 , 
                 17 
               
             
              
             
               : 
             
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                 ( 
                 
                   
                     σ 
                     
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                       1 
                     
                     
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                       4 
                     
                   
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                     σ 
                     i 
                     2 
                   
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                     σ 
                     
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                       1 
                     
                     2 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
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                       1 
                     
                     20 
                   
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                     σ 
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                     4 
                   
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                       - 
                       20 
                     
                   
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                       - 
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                       i 
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                       1 
                     
                     
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                       2 
                     
                   
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                     σ 
                     i 
                     2 
                   
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                     σ 
                     
                       i 
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                       1 
                     
                     20 
                   
                    
                   
                     σ 
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                       - 
                       4 
                     
                   
                    
                   
                     σ 
                     
                       i 
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                       1 
                     
                     
                       - 
                       16 
                     
                   
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                     σ 
                     i 
                     2 
                   
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                     σ 
                     
                       i 
                       + 
                       1 
                     
                     2 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     16 
                   
                    
                   
                     σ 
                     i 
                     4 
                   
                    
                   
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                 σ 
                 i 
                 2 
               
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                 16 
               
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                 σ 
                 i 
                 4 
               
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                   1 
                 
                 
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                 i 
                 
                   - 
                   2 
                 
               
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                 σ 
                 
                   i 
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                   1 
                 
                 
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                 σ 
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                 σ 
                 
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                 σ 
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                   20 
                 
               
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                   1 
                 
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                 i 
                 
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                   2 
                 
               
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                 σ 
                 
                   i 
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                   1 
                 
                 20 
               
                
               
                 σ 
                 i 
                 4 
               
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                   i 
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                   1 
                 
                 
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                   20 
                 
               
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                   i 
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                   1 
                 
                 
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                 4 
               
             
           
         
       
       
         
           
             
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                 i 
                 , 
                 18 
               
             
              
             
               : 
             
              
             
                 
             
              
             
               
                 ( 
                 
                   
                     σ 
                     i 
                     
                       - 
                       4 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
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                       12 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
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                     σ 
                     
                       i 
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                       1 
                     
                     2 
                   
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                     σ 
                     i 
                     
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                       2 
                     
                   
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                       i 
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                       1 
                     
                     12 
                   
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                     σ 
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                     4 
                   
                    
                   
                     σ 
                     
                       i 
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                       1 
                     
                     
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                       12 
                     
                   
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                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
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                     σ 
                     
                       i 
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                       1 
                     
                     
                       - 
                       2 
                     
                   
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                     σ 
                     
                       i 
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                       1 
                     
                     10 
                   
                    
                   
                     σ 
                     i 
                     
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                       2 
                     
                   
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                     σ 
                     
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                       1 
                     
                     2 
                   
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                     σ 
                     
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                       1 
                     
                     2 
                   
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                     σ 
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                       4 
                     
                   
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                     σ 
                     
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                       1 
                     
                     
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                       2 
                     
                   
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                     2 
                   
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                       1 
                     
                     2 
                   
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                     σ 
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                     4 
                   
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                       1 
                     
                     
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                       1 
                     
                     2 
                   
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                     σ 
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                     2 
                   
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                       1 
                     
                     2 
                   
                    
                   
                     σ 
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                       - 
                       4 
                     
                   
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                     σ 
                     
                       i 
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                       1 
                     
                     
                       - 
                       2 
                     
                   
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                     σ 
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                     σ 
                     
                       i 
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                       1 
                     
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
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                     σ 
                     
                       i 
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                       1 
                     
                     
                       - 
                       18 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     
 
                   
                    
                   
                       
                   
                    
                   
                     σ 
                     
                       i 
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                       1 
                     
                     2 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
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                     σ 
                     
                       i 
                       + 
                       1 
                     
                     20 
                   
                    
                   
                     σ 
                     i 
                     4 
                   
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                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       20 
                     
                   
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                     σ 
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                       - 
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                     σ 
                     
                       i 
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                       1 
                     
                     
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                       2 
                     
                   
                    
                   
                     
                       σ 
                       i 
                       2 
                     
                     
                       - 
                       1 
                     
                   
                    
                   
                     
                       
                         σ 
                         
                           i 
                           + 
                           1 
                         
                         20 
                       
                        
                       
                         ( 
                         
                           
                             σ 
                             i 
                             
                               - 
                               4 
                             
                           
                            
                           
                             σ 
                             
                               i 
                               + 
                               1 
                             
                             
                               - 
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                             σ 
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                             σ 
                             
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                             2 
                           
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                             σ 
                             i 
                             
                               - 
                               2 
                             
                           
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                             σ 
                             
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                             2 
                           
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                             σ 
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                               1 
                             
                             2 
                           
                         
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                     2 
                   
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                    
                   
                     σ 
                     i 
                     
                       - 
                       4 
                     
                   
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                       1 
                     
                     
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                 1 
               
             
           
         
       
       
         
           
             
                 
             
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                       σ 
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                         2 
                       
                     
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                         i 
                         + 
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                       2 
                     
                   
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                   i 
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                 20 
               
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                 12 
               
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                 i 
                 , 
                 19 
               
             
              
             
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                     σ 
                     
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                     18 
                   
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                     18 
                   
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                       14 
                     
                   
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                       1 
                     
                     2 
                   
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                       - 
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                     14 
                   
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                     4 
                   
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                     14 
                   
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                     16 
                   
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                 14 
               
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                     8 
                   
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                     16 
                   
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                     16 
                   
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                 23 
               
             
              
             
               : 
             
              
             
                 
             
              
             
               
                 ( 
                 
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       4 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     2 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
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                     σ 
                     
                       i 
                       + 
                       1 
                     
                     18 
                   
                    
                   
                     σ 
                     i 
                     4 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       18 
                     
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     
                       
                         σ 
                         
                           i 
                           + 
                           1 
                         
                         18 
                       
                        
                       
                         ( 
                         
                           
                             σ 
                             i 
                             
                               - 
                               4 
                             
                           
                            
                           
                             σ 
                             
                               i 
                               + 
                               1 
                             
                             
                               - 
                               14 
                             
                           
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                             σ 
                             i 
                             2 
                           
                            
                           
                             σ 
                             
                               i 
                               + 
                               1 
                             
                             2 
                           
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                             σ 
                             i 
                             
                               - 
                               2 
                             
                           
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                             σ 
                             
                               i 
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                               1 
                             
                             14 
                           
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                             σ 
                             i 
                             4 
                           
                            
                           
                             σ 
                             
                               i 
                               + 
                               1 
                             
                             
                               - 
                               14 
                             
                           
                            
                           
                             σ 
                             i 
                             
                               - 
                               2 
                             
                           
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                             σ 
                             
                               i 
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                               1 
                             
                             
                               - 
                               2 
                             
                           
                            
                           
                             σ 
                             i 
                             2 
                           
                            
                           
                             σ 
                             
                               i 
                               + 
                               1 
                             
                             14 
                           
                         
                         ) 
                       
                     
                     5 
                   
                    
                   
                     
 
                   
                    
                   
                       
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       4 
                     
                   
                    
                   
                     σ 
                     
                       i 
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                       1 
                     
                     
                       - 
                       10 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     σ 
                     
                       i 
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                       1 
                     
                     2 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
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                     σ 
                     
                       i 
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                       1 
                     
                     10 
                   
                    
                   
                     σ 
                     i 
                     4 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       10 
                     
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
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                     σ 
                     
                       i 
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                       1 
                     
                     
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                       2 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     σ 
                     
                       i 
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                       1 
                     
                     10 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       4 
                     
                   
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                     σ 
                     
                       i 
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                       1 
                     
                     
                       - 
                       6 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     2 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     6 
                   
                    
                   
                     σ 
                     i 
                     4 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       6 
                     
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     
                       i 
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                       1 
                     
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     6 
                   
                    
                   
                     
 
                   
                    
                   
                       
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       4 
                     
                   
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                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     σ 
                     
                       i 
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                       1 
                     
                     2 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
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                     σ 
                     
                       i 
                       + 
                       1 
                     
                     2 
                   
                    
                   
                     σ 
                     i 
                     4 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     
                       
                         σ 
                         
                           i 
                           + 
                           1 
                         
                         2 
                       
                        
                       
                         ( 
                         
                           
                             σ 
                             i 
                             
                               - 
                               4 
                             
                           
                            
                           
                             σ 
                             
                               i 
                               + 
                               1 
                             
                             
                               - 
                               16 
                             
                           
                            
                           
                             σ 
                             i 
                             2 
                           
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                             σ 
                             
                               i 
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                               1 
                             
                             2 
                           
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                             σ 
                             i 
                             
                               - 
                               2 
                             
                           
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                               i 
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                               1 
                             
                             16 
                           
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                             σ 
                             i 
                             4 
                           
                            
                           
                             σ 
                             
                               i 
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                               1 
                             
                             
                               - 
                               16 
                             
                           
                            
                           
                             σ 
                             i 
                             
                               - 
                               2 
                             
                           
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                             σ 
                             
                               i 
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                               1 
                             
                             
                               - 
                               2 
                             
                           
                            
                           
                             σ 
                             i 
                             2 
                           
                            
                           
                             σ 
                             
                               i 
                               + 
                               1 
                             
                             16 
                           
                         
                         ) 
                       
                     
                     4 
                   
                    
                   
                     
 
                   
                    
                   
                       
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       4 
                     
                   
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                     σ 
                     
                       i 
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                       1 
                     
                     
                       - 
                       16 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     2 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
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                     σ 
                     
                       i 
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                       1 
                     
                     16 
                   
                    
                   
                     σ 
                     i 
                     4 
                   
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                     σ 
                     
                       i 
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                       1 
                     
                     
                       - 
                       16 
                     
                   
                    
                   
                     σ 
                     i 
                     
                       - 
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                     σ 
                     
                       i 
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                       1 
                     
                     
                       - 
                       2 
                     
                   
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                     σ 
                     i 
                     2 
                   
                 
                 ) 
               
               
                 - 
                 1 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 σ 
                 i 
                 2 
               
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                 σ 
                 
                   i 
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                   1 
                 
                 2 
               
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                 σ 
                 i 
                 
                   - 
                   2 
                 
               
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                 14 
               
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                     σ 
                     
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                     14 
                   
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                     ( 
                     
                       
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                         i 
                         
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                           4 
                         
                       
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                         14 
                       
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                         σ 
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                         4 
                       
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                           i 
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                           1 
                         
                         
                           - 
                           14 
                         
                       
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                         σ 
                         i 
                         
                           - 
                           2 
                         
                       
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                           i 
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                           1 
                         
                         
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                           2 
                         
                       
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                         σ 
                         i 
                         2 
                       
                        
                       
                         σ 
                         
                           i 
                           + 
                           1 
                         
                         14 
                       
                     
                     ) 
                   
                 
                 4 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 σ 
                 i 
                 
                   - 
                   4 
                 
               
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                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   6 
                 
               
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                 σ 
                 i 
                 2 
               
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                 σ 
                 
                   i 
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                   1 
                 
                 2 
               
                
               
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                 i 
                 
                   - 
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                 6 
               
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                 6 
               
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                   - 
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                 10 
               
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                 10 
               
             
           
         
       
       
         
           
             
                 
             
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                   ( 
                   
                     
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                       i 
                       
                         - 
                         4 
                       
                     
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                         i 
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                         1 
                       
                       
                         - 
                         16 
                       
                     
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                       2 
                     
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                       16 
                     
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                         i 
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                         1 
                       
                       
                         - 
                         2 
                       
                     
                      
                     
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                       i 
                       2 
                     
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                         i 
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                         1 
                       
                       16 
                     
                   
                   ) 
                 
                 3 
               
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                   - 
                   4 
                 
               
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                   i 
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                   1 
                 
                 2 
               
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                 18 
               
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                 σ 
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                 4 
               
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                 σ 
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                 2 
               
             
           
         
       
       
         
           
             
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                 i 
                 , 
                 24 
               
             
              
             
               : 
             
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              
             
               
                 ( 
                 
                   
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                       1 
                     
                     
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                       4 
                     
                   
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                     2 
                   
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                       1 
                     
                     2 
                   
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                     18 
                   
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                       18 
                     
                   
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                       - 
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                           i 
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                         18 
                       
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                             i 
                             
                               - 
                               4 
                             
                           
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                               1 
                             
                             
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                               14 
                             
                           
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                               - 
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                             14 
                           
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                             i 
                             
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                               2 
                             
                           
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                             2 
                           
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                               i 
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                               1 
                             
                             14 
                           
                         
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                     6 
                   
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                       4 
                     
                   
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                     2 
                   
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                       1 
                     
                     2 
                   
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                       - 
                       2 
                     
                   
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                       1 
                     
                     10 
                   
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                     4 
                   
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                     2 
                   
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                       1 
                     
                     10 
                   
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                       - 
                       4 
                     
                   
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                       1 
                     
                     
                       - 
                       8 
                     
                   
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                     2 
                   
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                       1 
                     
                     2 
                   
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                       - 
                       2 
                     
                   
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                       i 
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                       1 
                     
                     8 
                   
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                     4 
                   
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                       1 
                     
                     
                       - 
                       8 
                     
                   
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                       - 
                       2 
                     
                   
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                       1 
                     
                     
                       - 
                       2 
                     
                   
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                     2 
                   
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                       i 
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                       1 
                     
                     8 
                   
                    
                   
                     
 
                   
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                       - 
                       4 
                     
                   
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                       1 
                     
                     
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                       4 
                     
                   
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                       1 
                     
                     2 
                   
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                       - 
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                       1 
                     
                     4 
                   
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                     4 
                   
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                       1 
                     
                     
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                       - 
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                     2 
                   
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                           i 
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                           1 
                         
                         4 
                       
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                         ( 
                         
                           
                             σ 
                             i 
                             
                               - 
                               4 
                             
                           
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                               i 
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                               1 
                             
                             
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                               16 
                             
                           
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                             2 
                           
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                             2 
                           
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                               - 
                               2 
                             
                           
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                             16 
                           
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                             i 
                             4 
                           
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                               1 
                             
                             
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                               2 
                             
                           
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                             16 
                           
                         
                         ) 
                       
                     
                     5 
                   
                    
                   
                     
 
                   
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                       4 
                     
                   
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                       i 
                       + 
                       1 
                     
                     
                       - 
                       16 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     2 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     16 
                   
                    
                   
                     σ 
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                     4 
                   
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                     σ 
                     
                       i 
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                       1 
                     
                     
                       - 
                       16 
                     
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
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                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       2 
                     
                   
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                     σ 
                     i 
                     2 
                   
                 
                 ) 
               
               
                 - 
                 1 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 σ 
                 i 
                 2 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 2 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 14 
               
                
               
                 σ 
                 i 
                 4 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   14 
                 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 i 
                 2 
               
                
               
                 
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     14 
                   
                    
                   
                     ( 
                     
                       
                         σ 
                         i 
                         
                           - 
                           4 
                         
                       
                        
                       
                         σ 
                         
                           i 
                           + 
                           1 
                         
                         
                           - 
                           14 
                         
                       
                        
                       
                         σ 
                         i 
                         2 
                       
                        
                       
                         σ 
                         
                           i 
                           + 
                           1 
                         
                         2 
                       
                        
                       
                         σ 
                         i 
                         
                           - 
                           2 
                         
                       
                        
                       
                         σ 
                         
                           i 
                           + 
                           1 
                         
                         14 
                       
                        
                       
                         σ 
                         i 
                         4 
                       
                        
                       
                         σ 
                         
                           i 
                           + 
                           1 
                         
                         
                           - 
                           14 
                         
                       
                        
                       
                         σ 
                         i 
                         
                           - 
                           2 
                         
                       
                        
                       
                         σ 
                         
                           i 
                           + 
                           1 
                         
                         
                           - 
                           2 
                         
                       
                        
                       
                         σ 
                         i 
                         2 
                       
                        
                       
                         σ 
                         
                           i 
                           + 
                           1 
                         
                         14 
                       
                     
                     ) 
                   
                 
                 5 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 σ 
                 i 
                 
                   - 
                   4 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   8 
                 
               
                
               
                 σ 
                 i 
                 2 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 2 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 8 
               
                
               
                 σ 
                 i 
                 4 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   8 
                 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 i 
                 2 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 8 
               
                
               
                 σ 
                 i 
                 
                   - 
                   4 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   10 
                 
               
                
               
                 σ 
                 i 
                 2 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 2 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 10 
               
                
               
                 σ 
                 i 
                 4 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   10 
                 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
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                   1 
                 
                 
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                   2 
                 
               
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                 σ 
                 i 
                 2 
               
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                 σ 
                 
                   i 
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                   1 
                 
                 10 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   ( 
                   
                     
                       σ 
                       i 
                       
                         - 
                         4 
                       
                     
                      
                     
                       σ 
                       
                         i 
                         + 
                         1 
                       
                       
                         - 
                         16 
                       
                     
                      
                     
                       σ 
                       i 
                       2 
                     
                      
                     
                       σ 
                       
                         i 
                         + 
                         1 
                       
                       2 
                     
                      
                     
                       σ 
                       i 
                       
                         - 
                         2 
                       
                     
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                         i 
                         + 
                         1 
                       
                       16 
                     
                      
                     
                       σ 
                       i 
                       4 
                     
                      
                     
                       σ 
                       
                         i 
                         + 
                         1 
                       
                       
                         - 
                         16 
                       
                     
                      
                     
                       σ 
                       i 
                       
                         - 
                         2 
                       
                     
                      
                     
                       σ 
                       
                         i 
                         + 
                         1 
                       
                       
                         - 
                         2 
                       
                     
                      
                     
                       σ 
                       i 
                       2 
                     
                      
                     
                       σ 
                       
                         i 
                         + 
                         1 
                       
                       16 
                     
                   
                   ) 
                 
                 6 
               
                
               
                 σ 
                 i 
                 
                   - 
                   4 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   18 
                 
               
                
               
                 σ 
                 i 
                 2 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 2 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 18 
               
                
               
                 σ 
                 i 
                 4 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   18 
                 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 i 
                 2 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 2 
               
             
           
         
       
       
         
           
             
               S 
               
                 i 
                 , 
                 25 
               
             
              
             
               : 
             
              
             
                 
             
              
             
               
                 ( 
                 
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       4 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     2 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
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                     σ 
                     
                       i 
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                       1 
                     
                     18 
                   
                    
                   
                     σ 
                     i 
                     4 
                   
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                     σ 
                     
                       i 
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                       1 
                     
                     
                       - 
                       18 
                     
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
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                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
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                         σ 
                         
                           i 
                           + 
                           1 
                         
                         18 
                       
                        
                       
                         ( 
                         
                           
                             σ 
                             i 
                             
                               - 
                               4 
                             
                           
                            
                           
                             σ 
                             
                               i 
                               + 
                               1 
                             
                             
                               - 
                               14 
                             
                           
                            
                           
                             σ 
                             i 
                             2 
                           
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                             σ 
                             
                               i 
                               + 
                               1 
                             
                             2 
                           
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                             σ 
                             i 
                             
                               - 
                               2 
                             
                           
                            
                           
                             σ 
                             
                               i 
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                             14 
                           
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                             σ 
                             i 
                             4 
                           
                            
                           
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                               i 
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                               1 
                             
                             
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                               14 
                             
                           
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                             σ 
                             i 
                             
                               - 
                               2 
                             
                           
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                             σ 
                             
                               i 
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                               1 
                             
                             
                               - 
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                             σ 
                             i 
                             2 
                           
                            
                           
                             σ 
                             
                               i 
                               + 
                               1 
                             
                             14 
                           
                         
                         ) 
                       
                     
                     7 
                   
                    
                   
                     
 
                   
                    
                   
                       
                   
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                     i 
                     
                       - 
                       4 
                     
                   
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                       i 
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                       1 
                     
                     
                       - 
                       6 
                     
                   
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                     i 
                     2 
                   
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                       1 
                     
                     2 
                   
                    
                   
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                     i 
                     
                       - 
                       2 
                     
                   
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                       i 
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                       1 
                     
                     6 
                   
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                     i 
                     4 
                   
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                       i 
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                       1 
                     
                     
                       - 
                       6 
                     
                   
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                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
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                       i 
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                       1 
                     
                     
                       - 
                       2 
                     
                   
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                     i 
                     2 
                   
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                       i 
                       + 
                       1 
                     
                     6 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       4 
                     
                   
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                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       8 
                     
                   
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                     σ 
                     i 
                     2 
                   
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                       i 
                       + 
                       1 
                     
                     2 
                   
                    
                   
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                     i 
                     
                       - 
                       2 
                     
                   
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                       i 
                       + 
                       1 
                     
                     8 
                   
                    
                   
                     σ 
                     i 
                     4 
                   
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                       i 
                       + 
                       1 
                     
                     
                       - 
                       8 
                     
                   
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                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
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                       i 
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                       1 
                     
                     
                       - 
                       2 
                     
                   
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                     σ 
                     i 
                     2 
                   
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                     σ 
                     
                       i 
                       + 
                       1 
                     
                     8 
                   
                    
                   
                     
 
                   
                    
                   
                       
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       4 
                     
                   
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                       i 
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                       1 
                     
                     
                       - 
                       6 
                     
                   
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                     σ 
                     i 
                     2 
                   
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                       i 
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                       1 
                     
                     2 
                   
                    
                   
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                     i 
                     
                       - 
                       2 
                     
                   
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                     σ 
                     
                       i 
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                       1 
                     
                     6 
                   
                    
                   
                     σ 
                     i 
                     4 
                   
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                     σ 
                     
                       i 
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                       1 
                     
                     
                       - 
                       6 
                     
                   
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                       - 
                       2 
                     
                   
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                       i 
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                       1 
                     
                     
                       - 
                       2 
                     
                   
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                     σ 
                     i 
                     2 
                   
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                     σ 
                     
                       i 
                       + 
                       1 
                     
                     6 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       4 
                     
                   
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                     σ 
                     
                       i 
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                       1 
                     
                     
                       - 
                       10 
                     
                   
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                     σ 
                     i 
                     2 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     2 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
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                       i 
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                       1 
                     
                     10 
                   
                    
                   
                     σ 
                     i 
                     4 
                   
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                       i 
                       + 
                       1 
                     
                     
                       - 
                       10 
                     
                   
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                     i 
                     
                       - 
                       2 
                     
                   
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                       i 
                       + 
                       1 
                     
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     10 
                   
                    
                   
                     
 
                   
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                       ( 
                       
                         
                           σ 
                           i 
                           
                             - 
                             4 
                           
                         
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                             i 
                             + 
                             1 
                           
                           
                             - 
                             16 
                           
                         
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                           2 
                         
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                             i 
                             + 
                             1 
                           
                           2 
                         
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                           i 
                           
                             - 
                             2 
                           
                         
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                             i 
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                           16 
                         
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                           i 
                           4 
                         
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                             i 
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                             1 
                           
                           
                             - 
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                           i 
                           
                             - 
                             2 
                           
                         
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                             i 
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                             1 
                           
                           
                             - 
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                           σ 
                           i 
                           2 
                         
                          
                         
                           σ 
                           
                             i 
                             + 
                             1 
                           
                           16 
                         
                       
                       ) 
                     
                     6 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       4 
                     
                   
                    
                   
                     σ 
                     
                       i 
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                       1 
                     
                     
                       - 
                       16 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
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                       i 
                       + 
                       1 
                     
                     2 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
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                     σ 
                     
                       i 
                       + 
                       1 
                     
                     16 
                   
                    
                   
                     σ 
                     i 
                     4 
                   
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                       i 
                       + 
                       1 
                     
                     
                       - 
                       16 
                     
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
                 
                 ) 
               
               
                 - 
                 1 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 σ 
                 i 
                 2 
               
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                   i 
                   + 
                   1 
                 
                 2 
               
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                 i 
                 
                   - 
                   2 
                 
               
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                   1 
                 
                 14 
               
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                 σ 
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                 4 
               
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                   i 
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                   1 
                 
                 
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                   14 
                 
               
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                 2 
               
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                     σ 
                     
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                       + 
                       1 
                     
                     14 
                   
                    
                   
                     ( 
                     
                       
                         σ 
                         i 
                         
                           - 
                           4 
                         
                       
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                           i 
                           + 
                           1 
                         
                         
                           - 
                           14 
                         
                       
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                         2 
                       
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                         2 
                       
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                           2 
                         
                       
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                         14 
                       
                        
                       
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                         i 
                         4 
                       
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                           i 
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                           1 
                         
                         
                           - 
                           14 
                         
                       
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                           - 
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                           1 
                         
                         
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                         i 
                         2 
                       
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                           i 
                           + 
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                         14 
                       
                     
                     ) 
                   
                 
                 6 
               
             
           
         
       
       
         
           
             
                 
             
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                 σ 
                 i 
                 
                   - 
                   4 
                 
               
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                   i 
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                   6 
                 
               
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                 2 
               
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                 6 
               
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                 4 
               
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                 6 
               
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                   4 
                 
               
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                 8 
               
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                 8 
               
             
           
         
       
       
         
           
             
                 
             
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                 i 
                 
                   - 
                   4 
                 
               
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                 6 
               
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                 6 
               
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                   1 
                 
                 
                   - 
                   10 
                 
               
                
               
                 σ 
                 i 
                 2 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 2 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 10 
               
                
               
                 σ 
                 i 
                 4 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   10 
                 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 i 
                 2 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 10 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   ( 
                   
                     
                       σ 
                       i 
                       
                         - 
                         4 
                       
                     
                      
                     
                       σ 
                       
                         i 
                         + 
                         1 
                       
                       
                         - 
                         16 
                       
                     
                      
                     
                       σ 
                       i 
                       2 
                     
                      
                     
                       σ 
                       
                         i 
                         + 
                         1 
                       
                       2 
                     
                      
                     
                       σ 
                       i 
                       
                         - 
                         2 
                       
                     
                      
                     
                       σ 
                       
                         i 
                         + 
                         1 
                       
                       16 
                     
                      
                     
                       σ 
                       i 
                       4 
                     
                      
                     
                       σ 
                       
                         i 
                         + 
                         1 
                       
                       
                         - 
                         16 
                       
                     
                      
                     
                       σ 
                       i 
                       
                         - 
                         2 
                       
                     
                      
                     
                       σ 
                       
                         i 
                         + 
                         1 
                       
                       
                         - 
                         2 
                       
                     
                      
                     
                       σ 
                       i 
                       2 
                     
                      
                     
                       σ 
                       
                         i 
                         + 
                         1 
                       
                       16 
                     
                   
                   ) 
                 
                 7 
               
                
               
                 σ 
                 i 
                 
                   - 
                   4 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   18 
                 
               
                
               
                 σ 
                 i 
                 2 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 2 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 18 
               
                
               
                 σ 
                 i 
                 4 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   18 
                 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 i 
                 2 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 2 
               
             
           
         
       
       
         
           
             
               S 
               
                 i 
                 , 
                 26 
               
             
              
             
               : 
             
              
             
                 
             
              
             
               
                 ( 
                 
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       4 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     2 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     18 
                   
                    
                   
                     σ 
                     i 
                     4 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       18 
                     
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     
                       
                         σ 
                         
                           i 
                           + 
                           1 
                         
                         18 
                       
                        
                       
                         ( 
                         
                           
                             σ 
                             i 
                             
                               - 
                               4 
                             
                           
                            
                           
                             σ 
                             
                               i 
                               + 
                               1 
                             
                             
                               - 
                               14 
                             
                           
                            
                           
                             σ 
                             i 
                             2 
                           
                            
                           
                             σ 
                             
                               i 
                               + 
                               1 
                             
                             2 
                           
                            
                           
                             σ 
                             i 
                             
                               - 
                               2 
                             
                           
                            
                           
                             σ 
                             
                               i 
                               + 
                               1 
                             
                             14 
                           
                            
                           
                             σ 
                             i 
                             4 
                           
                            
                           
                             σ 
                             
                               i 
                               + 
                               1 
                             
                             
                               - 
                               14 
                             
                           
                            
                           
                             σ 
                             i 
                             
                               - 
                               2 
                             
                           
                            
                           
                             σ 
                             
                               i 
                               + 
                               1 
                             
                             
                               - 
                               2 
                             
                           
                            
                           
                             σ 
                             i 
                             2 
                           
                            
                           
                             σ 
                             
                               i 
                               + 
                               1 
                             
                             14 
                           
                         
                         ) 
                       
                     
                     8 
                   
                    
                   
                     
 
                   
                    
                   
                       
                   
                    
                   
                     
                       ( 
                       
                         
                           σ 
                           i 
                           
                             - 
                             4 
                           
                         
                          
                         
                           σ 
                           
                             i 
                             + 
                             1 
                           
                           
                             - 
                             2 
                           
                         
                          
                         
                           σ 
                           i 
                           2 
                         
                          
                         
                           σ 
                           
                             i 
                             + 
                             1 
                           
                           2 
                         
                          
                         
                           σ 
                           i 
                           
                             - 
                             2 
                           
                         
                          
                         
                           σ 
                           
                             i 
                             + 
                             1 
                           
                           2 
                         
                          
                         
                           σ 
                           i 
                           4 
                         
                          
                         
                           σ 
                           
                             i 
                             + 
                             1 
                           
                           
                             - 
                             2 
                           
                         
                          
                         
                           σ 
                           i 
                           
                             - 
                             2 
                           
                         
                          
                         
                           σ 
                           
                             i 
                             + 
                             1 
                           
                           
                             - 
                             2 
                           
                         
                          
                         
                           σ 
                           i 
                           2 
                         
                          
                         
                           σ 
                           
                             i 
                             + 
                             1 
                           
                           2 
                         
                       
                       ) 
                     
                     3 
                   
                    
                   
                     
                       ( 
                       
                         
                           σ 
                           i 
                           
                             - 
                             4 
                           
                         
                          
                         
                           σ 
                           
                             i 
                             + 
                             1 
                           
                           
                             - 
                             16 
                           
                         
                          
                         
                           σ 
                           i 
                           2 
                         
                          
                         
                           σ 
                           
                             i 
                             + 
                             1 
                           
                           2 
                         
                          
                         
                           σ 
                           i 
                           
                             - 
                             2 
                           
                         
                          
                         
                           σ 
                           
                             i 
                             + 
                             1 
                           
                           16 
                         
                          
                         
                           σ 
                           i 
                           4 
                         
                          
                         
                           σ 
                           
                             i 
                             + 
                             1 
                           
                           
                             - 
                             16 
                           
                         
                          
                         
                           σ 
                           i 
                           
                             - 
                             2 
                           
                         
                          
                         
                           σ 
                           
                             i 
                             + 
                             1 
                           
                           
                             - 
                             2 
                           
                         
                          
                         
                           σ 
                           i 
                           2 
                         
                          
                         
                           σ 
                           
                             i 
                             + 
                             1 
                           
                           16 
                         
                       
                       ) 
                     
                     7 
                   
                    
                   
                     
 
                   
                    
                   
                       
                   
                    
                   
                     
                       σ 
                       i 
                       
                         - 
                         4 
                       
                     
                      
                     
                       σ 
                       
                         i 
                         + 
                         1 
                       
                       
                         - 
                         16 
                       
                     
                      
                     
                       σ 
                       i 
                       2 
                     
                      
                     
                       σ 
                       
                         i 
                         + 
                         1 
                       
                       2 
                     
                      
                     
                       σ 
                       i 
                       
                         - 
                         2 
                       
                     
                      
                     
                       σ 
                       
                         i 
                         + 
                         1 
                       
                       16 
                     
                      
                     
                       σ 
                       i 
                       4 
                     
                      
                     
                       σ 
                       
                         i 
                         + 
                         1 
                       
                       
                         - 
                         16 
                       
                     
                      
                     
                       σ 
                       i 
                       
                         - 
                         2 
                       
                     
                      
                     
                       σ 
                       
                         i 
                         + 
                         1 
                       
                       
                         - 
                         2 
                       
                     
                      
                     
                       σ 
                       i 
                       2 
                     
                   
                 
                 ) 
               
               
                 - 
                 1 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 σ 
                 i 
                 2 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 2 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 14 
               
                
               
                 σ 
                 i 
                 4 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   14 
                 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 i 
                 2 
               
                
               
                 
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     14 
                   
                    
                   
                     ( 
                     
                       
                         σ 
                         i 
                         
                           - 
                           4 
                         
                       
                        
                       
                         σ 
                         
                           i 
                           + 
                           1 
                         
                         
                           - 
                           14 
                         
                       
                        
                       
                         σ 
                         i 
                         2 
                       
                        
                       
                         σ 
                         
                           i 
                           + 
                           1 
                         
                         2 
                       
                        
                       
                         σ 
                         i 
                         
                           - 
                           2 
                         
                       
                        
                       
                         σ 
                         
                           i 
                           + 
                           1 
                         
                         14 
                       
                        
                       
                         σ 
                         i 
                         4 
                       
                        
                       
                         σ 
                         
                           i 
                           + 
                           1 
                         
                         
                           - 
                           14 
                         
                       
                        
                       
                         σ 
                         i 
                         
                           - 
                           2 
                         
                       
                        
                       
                         σ 
                         
                           i 
                           + 
                           1 
                         
                         
                           - 
                           2 
                         
                       
                        
                       
                         σ 
                         i 
                         2 
                       
                        
                       
                         σ 
                         
                           i 
                           + 
                           1 
                         
                         14 
                       
                     
                     ) 
                   
                 
                 7 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 σ 
                 i 
                 
                   - 
                   4 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   6 
                 
               
                
               
                 σ 
                 i 
                 2 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 2 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 6 
               
                
               
                 σ 
                 i 
                 4 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   6 
                 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 i 
                 2 
               
                
               
                 
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     6 
                   
                    
                   
                     ( 
                     
                       
                         σ 
                         i 
                         
                           - 
                           4 
                         
                       
                        
                       
                         σ 
                         
                           i 
                           + 
                           1 
                         
                         
                           - 
                           2 
                         
                       
                        
                       
                         σ 
                         i 
                         2 
                       
                        
                       
                         σ 
                         
                           i 
                           + 
                           1 
                         
                         2 
                       
                        
                       
                         σ 
                         i 
                         
                           - 
                           2 
                         
                       
                        
                       
                         σ 
                         
                           i 
                           + 
                           1 
                         
                         2 
                       
                        
                       
                         σ 
                         i 
                         4 
                       
                        
                       
                         σ 
                         
                           i 
                           + 
                           1 
                         
                         
                           - 
                           2 
                         
                       
                        
                       
                         σ 
                         i 
                         
                           - 
                           2 
                         
                       
                        
                       
                         σ 
                         
                           i 
                           + 
                           1 
                         
                         
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                           2 
                         
                       
                        
                       
                         σ 
                         i 
                         2 
                       
                        
                       
                         σ 
                         
                           i 
                           + 
                           1 
                         
                         2 
                       
                     
                     ) 
                   
                 
                 3 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   ( 
                   
                     
                       σ 
                       i 
                       
                         - 
                         4 
                       
                     
                      
                     
                       σ 
                       
                         i 
                         + 
                         1 
                       
                       
                         - 
                         16 
                       
                     
                      
                     
                       σ 
                       i 
                       2 
                     
                      
                     
                       σ 
                       
                         i 
                         + 
                         1 
                       
                       2 
                     
                      
                     
                       σ 
                       i 
                       
                         - 
                         2 
                       
                     
                      
                     
                       σ 
                       
                         i 
                         + 
                         1 
                       
                       16 
                     
                      
                     
                       σ 
                       i 
                       4 
                     
                      
                     
                       σ 
                       
                         i 
                         + 
                         1 
                       
                       
                         - 
                         16 
                       
                     
                      
                     
                       σ 
                       i 
                       
                         - 
                         2 
                       
                     
                      
                     
                       σ 
                       
                         i 
                         + 
                         1 
                       
                       
                         - 
                         2 
                       
                     
                      
                     
                       σ 
                       i 
                       2 
                     
                      
                     
                       σ 
                       
                         i 
                         + 
                         1 
                       
                       16 
                     
                   
                   ) 
                 
                 8 
               
                
               
                 σ 
                 i 
                 
                   - 
                   4 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   18 
                 
               
                
               
                 σ 
                 i 
                 2 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 2 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 18 
               
                
               
                 σ 
                 i 
                 4 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   18 
                 
               
                
               
                 σ 
                 i 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 
                   - 
                   2 
                 
               
                
               
                 σ 
                 i 
                 2 
               
                
               
                 σ 
                 
                   i 
                   + 
                   1 
                 
                 2 
               
             
           
         
       
       
         
           
             
               S 
               
                 i 
                 , 
                 27 
               
             
              
             
               : 
             
              
             
                 
             
              
             
               
                 ( 
                 
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       4 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     2 
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     18 
                   
                    
                   
                     σ 
                     i 
                     4 
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       18 
                     
                   
                    
                   
                     σ 
                     i 
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     
                       i 
                       + 
                       1 
                     
                     
                       - 
                       2 
                     
                   
                    
                   
                     σ 
                     i 
                     2 
                   
                    
                   
                     
                       
                         σ 
                         
                           i 
                           + 
                           1 
                         
                         18 
                       
                        
                       
                         ( 
                         
                           
                             σ 
                             i 
                             
                               - 
                               4 
                             
                           
                            
                           
                             σ 
                             
                               i 
                               + 
                               1 
                             
                             
                               - 
                               14 
                             
                           
                            
                           
                             σ 
                             i 
                             2 
                           
                            
                           
                             σ 
                             
                               i 
                               + 
                               1 
                             
                             2 
                           
                            
                           
                             σ 
                             i 
                             
                               - 
                               2 
                             
                           
                            
                           
                             σ 
                             
                               i 
                               + 
                               1 
                             
                             14 
                           
                            
                           
                             σ 
                             i 
                             4 
                           
                            
                           
                             σ 
                             
                               i 
                               + 
                               1 
                             
                             
                               - 
                               14 
                             
                           
                            
                           
                             σ 
                             i 
                             
                               - 
                               2 
                             
                           
                            
                           
                             σ 
                             
                               i 
                               + 
                               1 
                             
                             
                               - 
                               2 
                             
                           
                            
                           
                             σ 
                             i 
                             2 
                           
                            
                           
                             σ 
                             
                               i 
                               + 
                               1 
                             
                             14 
                           
                         
                         ) 
                       
                     
                     9 
                   
                    
                   
                     
 
                   
                    
                   
                       
                   
                    
                   
                     
                       ( 
                       
                         
                           σ 
                           i 
                           
                             - 
                             4 
                           
                         
                          
                         
                           σ 
                           
                             i 
                             + 
                             1 
                           
                           
                             - 
                             2 
                           
                         
                          
                         
                           σ 
                           i 
                           2 
                         
                          
                         
                           σ 
                           
                             i 
                             + 
                             1 
                           
                           2 
                         
                          
                         
                           σ 
                           i 
                           
                             - 
                             2 
                           
                         
                          
                         
                           σ 
                           
                             i 
                             + 
                             1 
                           
                           2 
                         
                          
                         
                           σ 
                           i 
                           4 
                         
                          
                         
                           σ 
                           
                             i 
                             + 
                             1 
                           
                           
                             - 
                             2 
                           
                         
                          
                         
                           σ 
                           i 
                           
                             - 
                             2 
                           
                         
                          
                         
                           σ 
                           
                             i 
                             + 
                             1 
                           
                           
                             - 
                             2 
                           
                         
                          
                         
                           σ 
                           i 
                           2 
                         
                          
                         
                           σ 
                           
                             i 
                             + 
                             1 
                           
                           2 
                         
                       
                       ) 
                     
                     3 
                   
                    
                   
                     
                       ( 
                       
                         
                           σ 
                           i 
                           
                             - 
                             4 
                           
                         
                          
                         
                           σ 
                           
                             i 
                             + 
                             1 
                           
                           
                             - 
                             16 
                           
                         
                          
                         
                           σ 
                           i 
                           2 
                         
                          
                         
                           σ 
                           
                             i 
                             + 
                             1 
                           
                           2 
                         
                          
                         
                           σ 
                           i 
                           
                             - 
                             2 
                           
                         
                          
                         
                           σ 
                           
                             i 
                             + 
                             1 
                           
                           16 
                         
                          
                         
                           σ 
                           i 
                           4 
                         
                          
                         
                           σ 
                           
                             i 
                             + 
                             1 
                           
                           
                             - 
                             16 
                           
                         
                          
                         
                           σ 
                           i 
                           
                             - 
                             2 
                           
                         
                          
                         
                           σ 
                           
                             i 
                             + 
                             1 
                           
                           
                             - 
                             2 
                           
                         
                          
                         
                           σ 
                           i 
                           2 
                         
                          
                         
                           σ 
                           
                             i 
                             + 
                             1 
                           
                           16 
                         
                       
                       ) 
                     
                     8 
                   
                    
                   
                     
 
                   
                    
                   
                       
                   
                    
                   
                     
                       σ 
                       i 
                       
                         - 
                         4 
                       
                     
                      
                     
                       σ 
                       
                         i 
                         + 
                         1 
                       
                       
                         - 
                         16 
                       
                     
                      
                     
                       σ 
                       i 
                       2 
                     
                      
                     
                       σ 
                       
                         i 
                         + 
                         1 
                       
                       2 
                     
                      
                     
                       σ 
                       i 
                       
                         - 
                         2 
                       
                     
                      
                     
                       σ 
                       
                         i 
                         + 
                         1 
                       
                       16 
                     
                      
                     
                       σ 
                       i 
                       4 
                     
                      
                     
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         [0036]    2. An Embodiment for Establishing Core Protocol 1 of Public Key Cryptographic Protocols System: 
         [0037]    In this embodiment, two entities of the protocol are separately Alice and Bob, 
         [0038]    1) Alice and Bob jointly choose an element g in B n , Alice chooses two elements b 1 , b 2 ∈LB n  as private keys, and Bob chooses two elements d 1 , d 2 ∈RB n  as private keys; 
         [0039]    2) Bob chooses two elements c 1 , c 2 ∈RB n , computes y=d 1 c 1 gc 2 d 2 , and sends y to Alice; 
         [0040]    3) Alice chooses four elements a 1 , a 2 , b 3 , b 4 ∈LB n , computes 
         [0000]        x=b   1 a 1 ga 2 b 2  and z=b 3 a 1 ya 2 b 4 =b 3 a 1 d 1 c 1 gc 2 d 2 a 2 b 4 , 
         [0041]    and sends (x, z) to Bob; 
         [0042]    4) Bob chooses two elements d 3 , d 4 ∈RB n , computes 
         [0000]      w=d 3 c 1 xc 2 d 4 =d 3 c 1 b 1 a 1 ga 2 b 2 c 2 d 4    
         [0000]      and 
         [0000]        v=d   1   −1   zd   2   1   =d   1   −1   b   3   a   1   d   1   c   1   gc   2   d   2   a   2   b   4   d   2   −1   =b   3   a   1   c   1   gc   2   a   2   b   4 , 
         [0043]    and sends (w, v) to Alice; and 
         [0044]    5) Alice computes 
         [0000]        u=b   1   −1   wb   2   −1   =b   1   −1   =d   3   c   1   b   1   a   1   ga   2   b   2   c   2   d   4   b   2   −1   =d   3   c   1   a   1   ga   2   c   2   d   4 , 
         [0000]      u=b 1   −1   wb   2   −1   =b   1   −1   d   3   c   1   b   1   a   1   ga   2   b   2   c   2   d   4   b   2   −1   =d   3   c   1   a   1   ga   2   c   2   d   4 , 
         [0000]    and sends u to Bob, 
         [0045]    In step 4) of the foregoing protocol, because d 1 , d 2 ∈RB n , and a 1 , a 2 , b 3 , b 4 ∈LB n , d 1   −1  and d 2   −1  separately commute with b 3  and a 1  and with b 4  and a 2  in multiplication, so that a final equation in the step is obtained. Likewise, a final equation in step 5) is obtained. 
         [0046]    On the basis of this embodiment, an exemplary embodiment for establishing a key exchange protocol is: 
         [0047]    The following procedures are performed after the five steps in the core protocol: 
         [0048]    6) Alice computes K A =b 3   −1 vb 4   −1 =a 1 c 1 gc 2 a 2  and Bob computes K B =d 3   −1 ud 4   −1 =c 1 a 1 ga 2 c 2 . 
         [0049]    Because a 1 , a 2 ∈LB n , and c 1 , c 2 ∈RB n , a 1  and c 1  separately commute with a 2  and c 2  in multiplication, so that Alice and Bob reach a shared key K=K A =K B . 
         [0050]    On the Basis of this Embodiment, an Exemplary Embodiment for Establishing a Data Encryption Protocol is: 
         [0051]    It is given that to-be-encrypted plaintext information (encoded) is m∈ {0, 1} k  (that is, a 0-1 string with a length of k), and it is given that Θ: B n Δ{0, 1} k  is a collision-resistant Hash function from the group B n  to a plaintext space {0, 1} k . The private keys of Alice are (B n , LB n , RB n , g, Θ), and a 1 , a 2 , b 1 , b 2 , b 3 , b 4 ∈LB n  are chosen, and the private keys are b 1  and b 2 . Bob chooses c 1 , c 2 , d 1 , d 2 , d 3 , d 4 ∈RB n , and uses d 1  and d 2  as the private keys. The following procedures are performed after the five steps in the core protocol: 
         [0052]    6) Encrypting: Bob first computes K B =d 3   −1 ud 4   −1 =c 1 a 1 ga 2 c 2 , then computes (encrypts) t=Θ(K B )⊕m, uses t as ciphertext, and sends the ciphertext to Alice. ⊕ herein is the exclusive or operation. 
         [0053]    7) Decrypting: Alice first computes K A =b 3   −1 vb 4   −1 =a 1 c 1 gc 2 a 2 , then computes (decrypts) 
         [0000]        m′= Θ( K   A )⊕ t =Θ( K   A )⊕(Θ( K   B )⊕ m )
 
         [0054]    verification of m′=m: K A =K B  is known according to a key exchange protocol, and therefore, 
         [0000]        m ′=Θ( K   A )⊕(Θ( K   B )⊕ m )=Θ( K   B )⊕(Θ( K   B )⊕ m )=(Θ( K   B )⊕Θ( K   B ))⊕ m=m.  
 
         [0055]    On the Basis of this Embodiment, an Exemplary Embodiment for Establishing a Digital Signature Protocol is: 
         [0056]    It is given that to-be-encrypted plaintext information (encoded) is m, and it is given that Θ: B n →{0, 1} k  is a collision-resistant Hash function. The public keys of Alice are (B n , LB n , RB n , g, Θ), and a 1 , a 2 , b 1 , b 2 , b 3 , b 4 ∈LB n  are chosen, and the private keys are b 1  and b 2 . Bob chooses c 1 , c 2 , d 1 , d 2 , d 3 , d 4 ∈RB n , and uses d 1  and d 2  as the private keys. The following procedures are performed after the five steps in the core protocol: 
         [0057]    6) Signing: Alice computes K A =b 3   −1 vb 4   −1 =a 1 c 1 gc 2 a 2  and S=Θ(mK A ), and Alice uses S as a signature of Alice for a file m and sends (S, m) to Bob. 
         [0058]    7) Verifying: Bob computes K B =d 3   −1 ud 4   −1 =c 1 a 1 ga 2 c 2  and S′=Θ(mK B ), and if S′=S, Bob acknowledges that S is the signature of Alice for the file m; otherwise, Bob refuses to accept that S is the signature of Alice for the file m. 
         [0059]    On the Basis of this Embodiment, an Exemplary Embodiment for an Identity Authentication Protocol on the Basis of the Core Protocol is: 
         [0060]    Alice chooses an element g in B n , four elements a 1 , a 2 , b 1 , b 2 ∈LB n  and a collision-resistant Hash function Θ: B n →{0, 1} k , and computes x=b 1 a 1 ga 2 b 2 . The public keys of Alice are (B n , LB n , RB n , g, x, Θ), and the private keys are b 1  and b 2 . 
         [0061]    An authentication process is: 
         [0062]    It is given that Alice is a prover and Bob is a verifier. 
         [0063]    1) Bob chooses six elements c 1 , c 2 , d 1 , d 2 , d 3 , d 4 ∈RB n , the private keys are d 1  and d 2 . Bob computes 
         [0000]      y=d 1 c 1 gc 2 d 2  and w=d 3 c 1 xc 2 d 4 , 
         [0000]    uses (y, w) as challenge 1, and sends the challenge 1 to Alice; 
         [0064]    2) Alice chooses two elements b 3 , b 4 ∈LB n , computes 
         [0000]      z=b 3 a 1 ya 2 b 4  and u=b 1   1 =d 3   c   1 a 1 ga 2 c 2 d 4 , 
         [0000]    uses (z, u) as a response, and sends the response to Bob; 
         [0065]    3) Bob computes v=d 1   −1   zd   2   −1   =b   3   a   1   c   1   gc   2   a   2   b   4 , uses v as challenge 2, and sends the challenge 2 to Alice; 
         [0066]    4) Alice computes t=Θ(b 3   −1 vb 4   −1 1 )=Θ(a 1 c 1 gc 2 a 2 ), uses t as a commitment, and sends the commitment to Bob; 
         [0067]    5) Bob computes t′=Θ(d 3   −1 ud 4   −1 )=Θ(c 1 a 1 ga 2 c 2 ), and verifies whether t=t′, 
         [0068]    and if t=t′, Bob acknowledges an identity of Alice; otherwise, Bob refuses to acknowledge the identity. 
         [0069]    3. An Embodiment for Establishing Core Protocol 2 of Public Key Cryptographic Protocols System: 
         [0070]    In this embodiment, two entities of the protocol are separately Alice and Bob, 
         [0071]    1.1) Alice and Bob jointly choose an element g in B n , Alice chooses two elements b 1 ∈LB n  and d 2 ∈RB n  as private keys, and Bob chooses two elements b 2 ∈LB n  and d 1 ∈RB n  as private keys; 
         [0072]    2.1) Bob chooses two elements a 2 ∈LB n  and c 1 ∈RB n , computes y=d 1 c 1 ga 2 b 2 , and sends y to Alice; 
         [0073]    3.1) Alice chooses four elements a 1 , b 4 ∈LB n  and c 2 , d 4 ∈RB n , computes 
         [0000]      x=b 1 a 1 gc 2 d 2  and z=b 4 a 1 yc 2 d 4 =b 4 a 1 d 1 c 1 ga 2 b 2 c 2 d 4 , 
         [0074]    and sends (x, z) to Bob; 
         [0075]    4.1) Bob chooses two elements b 3 ∈LB n  and d 3 ∈RB n , computes 
         [0000]      w=d 3 c 1 xa 2 b 3 =d 3 c 1 b 1 a 1 gc 2 d 2 a 2 b 3    
         [0000]      and 
         [0000]        v=d   1   −1   zb   2   −1   =d   1   −1   b   4   a   1   d   1   c   1   ga   2   b   2   c   2   d   4   b   2   −1   =b   4   a   1   c   1   ga   2   c   2   d   4 , 
         [0076]    and sends (w, v) to Alice; and 
         [0077]    5.1) Alice computes 
         [0000]        u=b   1   −1   wd   2   −1   =b   1   −1   d   3   c   1   b   1   a   1   gc   2   d   2   a   2   b   3   d   2   −1   =d   3   c   1   a   1   gc   2   a   2   b   3 , 
         [0078]    and sends u to Bob; 
         [0079]    In step 4) of the foregoing protocol, because d 1 , d 2 ∈RB n  and a 1 , a 2 , b 3 , b 4 ∈LB n , d 1   −1 , d 2   −1  separately commute with b 3  and a 1 , and with b 4  and a 2  in multiplication, so that a final equation in the step is obtained. Likewise, a final equation in step 5) is obtained. 
         [0080]    3.3 An application protocol 
         [0081]    The following application protocol is established on the basis of the core protocol. 
         [0082]    On the Basis of this Embodiment, an Exemplary Embodiment for Establishing a Key Exchange Protocol is: 
         [0083]    the following procedures are performed after the five steps in the core protocol: 
         [0084]    6.1) Alice computes K A =b 4   1 vd 4   1 =a 1 c 1 ga 2 c 2  and Bob computes K B =d 3   1 =c 1 a 1 gc 2 a 2 . 
         [0085]    Because a 1 , a 2 ∈LB n , and c 1 , c 2 ∈RB n , a 1  and c 1  are separately commute with a 2  and c 2  in multiplication, so that Alice and Bob reach a shared key K=K A =K B . 
         [0086]    On the Basis of this Embodiment, an Exemplary Embodiment for Establishing a Data Encryption Protocol is: 
         [0087]    It is given that to-be-encrypted plaintext information (encoded) is m∈{0, 1} k  (that is, a 0-1 string with a length of k), and it is given that Θ: B n →{0, 1} k  is a collision-resistant Hash function from the group B n  to a plaintext space {0, 1} k . The public keys of Alice are (B n , LB n , RB n , g, Θ), a 1 , b 1 , b 4 ∈LB n  and c 2 , d 2 , d 4 ∈RB n  are chosen, and the private keys are b 1  and d 2 . Bob chooses a 2 , b 2 , b 3 ∈LB n  and c 1 , d 1 , d 3 ∈RB n , and uses d 1  and b 2  as the private keys. The following procedures are performed after the five steps in the core protocol: 
         [0088]    6.1) Encrypting: Bob first computes K B =d 3   −2 ub 3   −1 =c 1 a 1 gc 2 a 2 , then computes (encrypts) t=Θ(K B )⊕m, uses t as ciphertext, and sends the ciphertext to Alice. ⊕ herein is the exclusive or operation. 
         [0089]    7.1) Decrypting: Alice first computes K A =b 4   −1 vd 4   −1 =a 1 c 1 ga 2 c 2 , then computes (decrypts) 
         [0000]        m′ =( K   A )⊕ t =Θ( K   A )⊕(Θ( K   B )⊕ m )
 
         [0090]    verification of m′=m: K A =K B  is known according to a key exchange protocol, and therefore, 
         [0000]        m ′=Θ( K   A )⊕(Θ( K   B )⊕ m )=Θ( K   B )⊕(Θ( K   B )⊕ m )=(Θ( K   B )⊕Θ( K   B ))⊕ m=m.  
 
         [0091]    On the Basis of this Embodiment, an Exemplary Embodiment for Establishing a Digital Signature Protocol is: 
         [0092]    It is given that to-be-encrypted plaintext information (encoded) is m, and it is given that Θ: B n →{0, 1} k  is a collision-resistant Hash function. The public keys of Alice are (B n , LB n , RB n , g, Θ), a 1 , b 1 , b 4 ∈LB n  and c 2 , d 2 , d 4  ∈RB n  are chosen, and the private keys are b 1  and d 2 . Bob chooses a 2 , b 2 , b 3 ∈LB n  and c 1 , d 1 , d 3 ∈RB n , and uses d 1  and b 2  as the private keys. The following procedures are performed after the five steps in the core protocol: 
         [0093]    6.1) Signing: Alice computes K A =b 4   −1 vd 4   −1 =a 1 c 1 ga 2 c 2  and S=Θ(mK A ), and Alice uses S as a signature of Alice for a file m and sends (S, m) to Bob. 
         [0094]    6.2) Verifying: Bob computes K B =d 3   −1 ub 3   −1 =c 1 a 1 gc 2 a 2  and S′=Θ(mK B ), and if S′=S, Bob acknowledges that S is the signature of Alice for the file m; otherwise, Bob refuses to accept that S is the signature of Alice for the file m. 
         [0095]    On the Basis of this Embodiment, an Exemplary Embodiment for an Identity Authentication Protocol on the Basis of the Core Protocol is: 
         [0096]    Alice chooses an element g in B n , four elements a 1 , b 1 ∈LB n  and c 2 , d 2 ∈RB n , and a collision-resistant Hash function Θ: B n →{0, 1} k , and computes x=b 1 a 1 gc 2 d 2 . The public keys of Alice are (B a , LB n , RB n , g, x, Θ), and the private keys are b 1  and d 2 . 
         [0097]    An authentication process is: 
         [0098]    It is given that Alice is a prover and Bob is a verifier. 
         [0099]    6.1) Bob chooses six elements c 1 , d 1 , d 3 ∈RB n  and a 2 , b 2 , b 3 ∈LB n , and the private keys are b 2  and d 1 . Bob computes 
         [0000]      y=d 1 c 1 ga 2 b 2  and w=d 3 c 1 xa 2 b 3 , 
         [0000]    uses (y, w) as challenge 1, and sends the challenge 1 to Alice; 
         [0100]    6.2) Alice chooses two elements b 4 ∈LB n  and d 4 ∈RB n , computes 
         [0000]      z=b 4 a 1 yc 2 d 4  and u=b 1   −1 wd 2   −1 =d 3 c 1 a 1 gc 2 a 2 b 3 , 
         [0101]    uses (z, u) as a response, and sends the response to Bob; 
         [0102]    6.3) Bob computes v=d 1   −1 zb 2   −1 =b 4 a 1 c 1 ga 2 c 2 d 4 , uses v as challenge 2, and sends the challenge 2 to Alice; 
         [0103]    6.4) Alice computes t=Θ(b 4   −1 vd 4   −1 )=Θ(a 1 c 1 ga 2 c 2 ), uses t as a commitment, and sends the commitment to Bob; 
         [0104]    6.5) Bob computes t′=(d 3   −1 ub 3   −1 )=Θ(c 1 a 1 gc 2 a 2 ), and verifies whether t=t′, and if t=t′, Bob acknowledges an identity of Alice; otherwise, Bob refuses to acknowledge the identity. 
         [0105]    4. Security Analysis 
         [0106]    We may only provide the security of a key exchange protocol. 
         [0107]    First, definitions of three determining problems of a group are provided. 
         [0108]    a subgroup membership problem (subgroup membership problem or generalized word problem, GWP for short): given a subgroup H whose generator set is X in group G, whether any element g in G can be represented by a word on Xis determined, that is, whether g is an element in H is determined. 
         [0109]    an element decomposition search problem (decomposition search problem, DSP for short): given that g and h are two elements in group G. It is known that two elements c and d exist in G, so that h=cgd. Decide whether two elements c′ and d′ in G can be obtained, so that h=c′gd′ 
         [0110]    a generalized element decomposition search problem (generalized decomposition search problem, GDSP for short): given that g and h are two elements in group G, and H and K are two subgroups in G. It is known that an element c of H and an element d of K exist, so that h=cgd. Decide whether an element c′ of H and an element d′ of K can be obtained, so that h=c′gd′. 
         [0111]    The DSP can be solved easily by letting c′=g −1  and d′=h. The decidability of the GDSP is not determined. However, for a decomposition equation h=cgd (c and d are unknown) in an infinite non-abelian group, it is impossible to certainly solve c and d. Because people do not know values of c and d, even if they enable h=c′gd′ by using so-called “solutions” c′ and d′ which are obtained through computation by solving the GDSP problem, they also cannot determine whether c′=c and d′=d. Particularly, if c and d are separately taken from subgroups C and D with an unsolvable GWP problem, a solver not only cannot determine whether c′=c and d′=d, but also cannot determine whether c′ and d′ respectively are elements in C and D. 
         [0112]    In core protocol 1, information that can be acquired by an attacker Eve by using disclosed information and an interactive process with Alice and Bob is as follows: 
         [0113]    an infinite non-abelian group G and two subgroups A and B in G, so that for any α∈A and any b∈B, ab=ba is true, an element g in G, and the following elements in G: 
         [0000]      y=d 1 c 1 gc 2 d 2 , x=b 1 a 1 ga 2 b 2 , z=b 3 a 1 d 1 c 1 gc 2 d 2 a 2 b 4 , w=d 3 c 1 b 1 a 1 ga 2 b 2 c 2 d 4 , and 
         [0000]      v=b 3 a 1 c 1 gc 2 a 2 b 4  and u=d 3 c 1 a 1 ga 2 c 2 d 4    
         [0114]    It should be noted that Eve only knows x, y, z, w, u and v, but does not know corresponding decomposition expressions. If Eve can obtain c 1 ′, c 2 ′∈B, and a 1 ′, a 2 ′∈A by solving the GDSP problem, so that a 1 ′ga 2 ′=a 1 ga 2  and c 1 ′gc 2 ′=c 1 gc 2 , according to the multiplication commutativity of elements in A and B, it is obtained that 
         [0000]      c 1   ′a   1   ′ga   2   ′c   2   ′=c   1   ′a   1   ga   2   c   1   ′=a   1   c   1   ′gc   2   ′a   2   =a   1   c   1   gc   2   a   2   =K    
         [0115]    and therefore, Eve needs to first obtain elements a 1 ga 2  and c 1 gc 2 . 
         [0116]    Because Eve does not know a 1 ga 2  and c 1 gc 2 , she cannot strip b 1  and b 2  from x to obtain a 1 ga 2 , or strip d 1  and d 2  from y to obtain c 1 gc 2 . Eve knows w=b 1 ub 2  and z=d 1 vd 2  (but does not know b 1  and b 2 , and d 1  and d 2 ). Now, even if Eve can solve the GDSP problem, to obtain b 1 ′, b 2 ′ ∈A, and d 1 ′, d 2 ′ ∈B, so that b 1 ub 2 ′=b 1 ub 2  and d 1 ′vd 2 ′=d 1 vd 2 , she also cannot determine b 1 ′=b 1 , b 2 ′=b 2 , and d 1 ′=d 1 , d 2 ′=d 2 . Therefore, Eve still cannot strip b 1  and b 2  from x to obtain a 1 ga 2 , or strip d 1  and d 2  from y to obtain c 1 gc 2 . 
         [0117]    Particularly, in a specific implementation solution, a braid group B, with Tz 12  is taken as an infinite non-abelian group G, subgroups LB n  and RB n  of B, are taken as A and B respectively, and private keys b 1  and b 2 , and private keys d 1  and d 2  are respectively chosen from a Mihailova subgroup M LA (H) of LB n  and a Mihailova subgroup M RA ( H ) of RB n . In the foregoing attack of Eve, she obtains b 1 ′, b 2 ′∈LB n  and d 1 ′, d 2 ′∈RB n  by solving the GDSP problem, so that b 1 ′ub 2 ′=b 1 ub 2  and d 1 ′vd 2 ′=d 1 vd 2 . She must determine b 1 ′=b 1 , b 2 ′=b 2  and d 1 ′=d 1 , d 2 ′=d 2 . Because b 1 , b 2 EM M (H) and d 1 , d 2  EM RA (H), she must first determine whether b 1 ′, b 2 ′∈M LA (H), and whether d 1 ′, d 2 ′ ∈M RA (H). However, the GWP problems of M LA (H) and M RA (H) are unsolvable, so that Eve cannot carry out an attack even if she uses a quantum computational system. 
         [0118]    In core protocol 2, information that can be acquired by an attacker Eve by using disclosed information and an interactive process with Alice and Bob is as follows: 
         [0119]    an infinite non-abelian group G and two subgroups A and B in G, so that for any a∈A and any b∈B, ab=ba is true, an element g in G, and the following elements in G: 
         [0000]      y=cl 1 c 1 ga 2 b 2 , x=b 1 a 1 gc 2 d 2 , z=b 4 a 1 d 1 c 1 ga 2 b 2 c 2 d 4 , w=d 3 c 1 b 1 a 1 gc 2 d 2 a 2 b 3 , 
         [0000]      and 
         [0000]      v=b 4 a 1 c 1 ga 2 c 2 d 4  and u=d 3 c 1 a 1 gc 2 a 2 b 3    
         [0120]    It should be noted that, Eve only knows x, y, z, w, u, and v, but does not know corresponding decomposition expressions. If Eve can obtain c 1 ′, c 2 ′∈B, and a 1 ′, a 2 ′∈A by solving the GDSP problem, so that a 1 ′gc 2 ′=a 1 gc 2  and c 1 ′ga 2 ′=c 1 ga 2 , according to the multiplication commutativity of elements in A and B, it is obtained that 
         [0000]        c   1 ′a 1 ′gc 2 ′a 2 ′=c 1 ′a 1 gc 2 c 1 ′=a 1 c 1 ′ga 2 ′c 2 =a 1 c 1 ga 2 c 2 =K
 
         [0121]    and therefore, Eve needs to first obtain elements a 1 gc 2  and c 1 ga 2 . 
         [0122]    Because Eve does not know a 1 gc 2  and c 1 ga 2 , she cannot strip b 1  and d 2  from x to obtain a 1 gc 2 , or strip d 1  and b 2  from y to obtain c 1 ga 2 . Eve knows w=b 1 ud 2  and z=d 1 vb 2  (but does not know b 1  and b 2 , and d 1  and d 2 ). Now, even if Eve can solve the GDSP problem, to obtain b 1 ′, b 2 ′∈A, and d 1 ′, d 2 ′∈B, so that b 1 ′ud 2 ′=b 1 ud 2  and d 1 ′vb 2 ′=d 1 vb 2 , she also cannot determine b 1 ′=b 1 , b 2 ′=b 2  and d 1 ′=d 1 , d 2 ′=d 2 . Therefore, Eve still cannot strip b 1  and d 2  from x to obtain a 1 gc 2 , or strip d 1  and b 2  from y to obtain c 1 ga 2 . 
         [0123]    Particularly, in a specific implementation solution, a braid group B, with i ,   12  is taken as an infinite non-abelian group G, subgroups LB n  and RB n  of B n  are taken as A and B respectively, and private keys b 1  and b 2 , and private keys d 1  and d 2  are respectively chosen from a Mihailova subgroup M LA (H) of LB, and a Mihailova subgroup M RA (H) of RB n . In the foregoing attack of Eve, she obtains b 1 ′, b 2 ′ ∈LB n  and d 1 ′, d 2 ′∈RB n  by solving the GDSP problem, so that b 1 ′ud 2 ′=b 1 ud 2  and d 1 ′vb 2 ′=d 1 vb 2 . She must determine b 1 ′=b 1 , b 2 ′=b 2  and d 1 ′=d 1 , d 2 ′=d 2 . Because b 1 , b 2 ∈ M RA (H) and d 1 , d 2 E M RA (H), she must first determine whether b 1 ′, b 2 ′∈M LA (H), and whether d 1 ′, d 2 ′∈M RA (H). However, the GWP problems of M LA (H) and M RA (H) are unsolvable, so that Eve cannot carry out an attack even if she uses a quantum computational system. 
         [0124]    5. Choosing of a Parameter 
         [0125]    In an exemplary embodiment, a braid group B, has an exponent of n≧12, subgroups in each protocol are A=LB n  and B=RB n , choosing of a 1 , a 2 , c 1 , and c 2  needs to satisfy that their product a 1 a 2 c 1 c 2  is not less than 256 bits, each of private keys b 1 , b 2 , d 1  and d 2  is not less than 256 bits, and each of protection layer elements b 3 , b 4 , d 3 , and d 4  is not less than 128 bits. 
         [0126]    It is particularly pointed out that, to resist the quantum computational attack, it is suggested that private keys b 1  and b 2 , and d 1  and d 2  be respectively chosen from Mihailova subgroups M LA (H) and M RA (H) of the braid group B n . Therefore, because of the unsolvability of the GWP of M LA (H) and M RA (H), as described in the security analysis, even if a quantum computational system is used, b 1  and b 2 , and d 1  and d 2  also cannot be attacked. 
         [0127]    The foregoing describes the method of establishing public key cryptographic protocols against the quantum computational attack according to the present invention, so as to help to understand the present invention. However, the implementation manners of the present invention are not limited by the foregoing embodiments, any variation, modification, replacement, combination, and simplification made without departing from the principle of the present invention shall be an equivalent replacement manner and fall within the protection scope of the present invention.