Patent ID: 11909873
Assignee: MITSUBISHI ELECTRIC CORPORATION
Field: Digital communication (Electrical engineering)
Classification: CPC H  G | IPC G  H

Claim 5:
6. The decryption device according to claim 4,
wherein the processing circuitry acquires a ciphertext CT indicated in Formula 5,
acquires a decryption key SK(f) indicated in Formula 6,
computes the operation values Λ′0, Λ′j, and Λj by performing a pairing operation as indicated in Formula 7,
computes a value Λ as indicated in Formula 8, and computes a value ΛΛ′0 as the output value, CT
         =
         
          (
          
           
            x
            →
           
           ,
           
            
             {
             
              c
              
               ′
               ⁡
               (
               
                ι
                ′
               
               )
              
             
             }
            
            
             
              ι
              ′
             
             ∈
             
              [
              
               0
               ,
               
                n
                ′
               
              
              ]
             
            
           
           ,
            
           
            
             {
             
              c
              
               (
               ι
               )
              
             
             }
            
            
             ι
             ∈
             
              [
              n
              ]
             
            
           
          
          )
         
        
        ,
       
      
     
     
      
       
        ω
        ,
        ξ
        ,
        
         
          ϕ
          0
          ′
         
         
          ⟵
          U
         
         
          𝔽
          q
         
        
        ,
       
      
     
     
      
       
        
         
          c
          
           ′
           ⁡
           (
           0
           )
          
         
         =
         
          
           (
           
            ω
            ,
            
             
              0
              →
             
             2
            
            ,
            ξ
            ,
            0
            ,
            
             ϕ
             0
             ′
            
           
           )
          
          
           𝔹
           0
          
         
        
        ,
       
      
     
     
      
       
        
         for
         ⁢
            
         
          ι
          ′
         
        
        ∈
        
         [
         
          n
          '
         
         ]
        
       
      
     
     
      
       
        
         
          ϕ
          
           ι
           ′
          
          ′
         
         
          ⟵
          U
         
         
          𝔽
          q
         
        
        ,
        
         
          c
          
           ′
           ⁡
           (
           
            ι
            ′
           
           )
          
         
         =
         
          
           (
           
            
             ω
             ⁡
             (
             
              1
              ,
              
               x
               
                ι
                ′
               
              
             
             )
            
            ,
            
             
              0
              →
             
             4
            
            ,
             
            
             
              0
              →
             
             2
            
            ,
             
            
             ϕ
             
              ι
              ′
             
             ′
            
           
           )
          
          
           𝔹
           
            ι
            ′
           
          
         
        
        ,
       
      
     
     
      
       
        
         for
         ⁢
            
         ι
        
        ∈
        
         [
         n
         ]
        
       
      
     
     
      
       
        
         
          ϕ
          ι
         
         
          ⟵
          U
         
         
          𝔽
          q
         
        
        ,
        
         
          c
          
           (
           ι
           )
          
         
         =
         
          
           (
           
            
             ω
             ⁡
             (
             
              1
              ,
              
               z
               ι
              
             
             )
            
            ,
            
             
              0
              →
             
             4
            
            ,
            
             
              0
              →
             
             2
            
            ,
            
             ϕ
             ι
            
           
           )
          
          
           𝔹
           
            
             n
             ′
            
            +
            ι
           
          
         
        
        ,
       
      
     
    
   
   
    
     [
     
      Formula
      ⁢
         
      5
     
     ]
    
   
  
 

where
n′,n∈,
({right arrow over (x)},{right arrow over (z)})∈qn′×qn are attribute vectors,, SK
          ⁢
          
           (
           f
           )
          
         
         =
         
          (
          
           f
           ,
            
           
            
             {
             
              β
              
               j
               ′
              
              ′
             
             }
            
            
             
              j
              ′
             
             ∈
             
              [
              m
              ]
             
            
           
           ,
           
            
             {
             
              β
              j
             
             }
            
            
             j
             ∈
             
              [
              n
              ]
             
            
           
           ,
            
           
            
             {
             
              
               k
               ′
              
              
               (
               
                j
                ′
               
               )
              
             
             }
            
            
             
              j
              ′
             
             ∈
             
              [
              
               0
               ,
               m
              
              ]
             
            
           
           ,
            
           
            
             {
             
              k
              
               (
               j
               )
              
             
             }
            
            
             j
             ∈
             
              [
              n
              ]
             
            
           
          
          )
         
        
        ,
       
      
     
     
      
       
        
         
          (
          
           
            (
            
             
              
               {
               
                σ
                j
               
               }
              
              
               j
               ∈
               
                [
                n
                ]
               
              
             
             ,
             
              
               {
               
                
                 α
                 
                  j
                  ′
                 
                
                ,
                
                 γ
                 
                  j
                  ′
                 
                
               
               }
              
              
               
                j
                ′
               
               ∈
               
                [
                m
                ]
               
              
             
            
            )
           
           ,
            
           
            
             ρ
             ⁢
             
              :
              [
              m
              ]
             
            
            →
            
             [
             
              n
              ′
             
             ]
            
           
          
          )
         
         
          ⟵
          R
         
         
          PGB
          ⁡
          (
          f
          )
         
        
        ,
       
      
     
     
      
       
        
         r
         0
        
        ,
        
         κ
         0
         ′
        
        ,
        
         ς
         
          ⟵
          U
         
         
          𝔽
          q
         
        
        ,
       
      
     
     
      
       
        
         
          k
          
           ′
           ⁡
           (
           0
           )
          
         
         =
         
          
           (
           
            
             -
             
              r
              0
             
            
            ,
            
             
              0
              →
             
             2
            
            ,
            1
            ,
             
            
             κ
             0
             ′
            
            ,
            0
           
           )
          
          
           𝔹
           0
           *
          
         
        
        ,
       
      
     
     
      
       
        
         for
         ⁢
            
         
          j
          ′
         
        
        ∈
        
         [
         m
         ]
        
       
      
     
     
      
       
        
         
          β
          
           j
           ′
          
          ′
         
         
          ⟵
          U
         
         
          𝔽
          q
         
        
        ,
        
         
          
           κ
           →
          
          
           (
           
            j
            ′
           
           )
          
         
         
          ⟵
          U
         
         
          𝔽
          q
          2
         
        
        ,
       
      
     
     
      
       
        
         
          γ
          
           j
           ′
          
          +
         
         =
         
          
           γ
           
            j
            ′
           
          
          +
          
           β
           j
           ′
          
         
        
        ,
        
         r
         0
        
        ,
        
         
          
           k
           ′
          
          (
          
           j
           ′
          
          )
         
         =
         
          
           (
           
            
             (
             
              
               γ
               
                j
                ′
               
               +
              
              ,
              
               α
               
                j
                ′
               
              
             
             )
            
            ,
            
             
              0
              →
             
             4
            
            ,
            
             
              
               κ
               →
              
              ′
             
             (
             
              j
              ′
             
             )
            
            ,
            0
           
           )
          
          
           𝔹
           
            ρ
            ⁡
            (
            
             j
             ′
            
            )
           
           *
          
         
        
        ,
       
      
     
     
      
       
        
         for
         ⁢
            
         j
        
        ∈
        
         [
         n
         ]
        
       
      
     
     
      
       
        
         
          β
          j
         
         
          ⟵
          U
         
         
          𝔽
          q
         
        
        ,
        
         
          
           κ
           →
          
          
           (
           j
           )
          
         
         
          ⟵
          U
         
         
          𝔽
          q
          2
         
        
        ,
       
      
     
     
      
       
        
         
          σ
          j
          +
         
         =
         
          
           
            σ
            j
           
           +
           
            
             β
             j
            
            ⁢
            
             r
             0
            
            ⁢
            
             k
             ⁡
             (
             j
             )
            
           
          
          =
          
           
            (
            
             
              (
              
               
                σ
                j
                +
               
               ,
               ς
              
              )
             
             ,
              
             
              
               0
               →
              
              ⁢
              4
             
             ,
              
             
              
               κ
               →
              
              
               (
               j
               )
              
             
             ,
             0
            
            )
           
           
            𝔹
            
             
              n
              ′
             
             +
             j
            
            *
           
          
         
        
        ,
       
      
     
    
   
   
    
     [
     
      Formula
      ⁢
         
      6
     
     ]
    
   
  
 

where
PGB(f) is an algorithm that takes as input a function f, and outputs a set of
constants ({σj}j∈[n], {αj′,γj′}j′∈[m])∈Fqn×(Fq2)m, which are elements of a predicate vector, along with a function ρ:[m]→[n],

Λ′0=e(c′(0),k′(0)),

Λ′j′=e(c′(ρ(j′)),k′(j′)) for j′∈[m], 

Λj=e(c(j),k(j)) for j∈[n]  [Formula 7], Λ
         =
         
          
           (
           
            
             ∏
             
              
               j
               ′
              
              ∈
              
               [
               m
               ]
              
             
            
            
             Λ
             
              j
              ′
             
             
              ′Ω
              
               j
               ′
              
              ′
             
            
           
           )
          
          ⁢
          
           
            (
            
             
              ∏
              
               j
               ∈
               
                [
                n
                ]
               
              
             
             
              Λ
              j
              
               Ω
               j
              
             
            
            )
           
           
            1
            
             
              
               ∑
               
                j
                ∈
                
                 [
                 m
                 ]
                
               
               ′
              
              
               
                Ω
                
                 j
                 ′
                
                ′
               
               ⁢
               
                β
                
                 j
                 ′
                
                ′
               
              
             
             +
             
              
               ∑
               
                
                 j
                 ∈
                 
                  [
                  n
                  ]
                 
                
               
              
              
               
                Ω
                j
               
               ⁢
               
                β
                j
               
              
             
            
           
          
         
        
        ,
       
      
     
     
      
       
        
         
          (
          
           
            
             {
             
              Ω
              j
             
             }
            
            
             j
             ∈
             
              [
              n
              ]
             
            
           
           ,
           
            
             {
             
              Ω
              
               j
               ′
              
              ′
             
             }
            
            
             
              j
              ′
             
             ∈
             
              [
              m
              ]
             
            
           
          
          )
         
         =
         
          R
          ⁢
          E
          ⁢
          
           C
           ⁡
           (
           
            f
            ,
            
             x
             →
            
           
           )
          
         
        
        ,
       
      
     
    
   
   
    
     [
     
      Formula
      ⁢
         
      8
     
     ]
    
   
  
 

where
REC(f,{right arrow over (x)}) is an algorithm that takes as input a function f and x→∈Fqn′, and outputs
a set of coefficients ({Ωj}j∈[n], {Ω′j′}j′∈[m])∈Fqn+m′.