# Patent

## Title
Encryption apparatus, decryption apparatus, encryption method, decryption method, security method, program, and recording medium

## Abstract

Security against CCA is improved without providing space for just improving the security against CCA in a ciphertext space. In encryption processing, a first ciphertext C1 is calculated from a plaintext M and a symmetric key K; a value containing a value corresponding to the symmetric key K and a value corresponding to the first ciphertext C1 is put into a collision-resistant function H to calculate a function value of the function H; and r corresponding to the function value is used to calculate second ciphertext C2=r·(Σμ=1nvμ·bμ)+ρ·bn+1∈G1n+1. In decryption processing, the second ciphertext C2∈G1n+1 and key information D1*=α·(Σμ=1nwμ·bμ*)+bn+1*∈G2n+1 are put into a bilinear function e to calculate a function value S ̃=e(C2, D1*)∈GT; a value corresponding to the function value S ̃ and the first ciphertext C1 are put into the collision-resistant function H to calculate a function value of the function H; and it is judged whether r ̃ corresponding to the function value satisfies C2=r ̃·(Σμ=1nvμ·bμ)+ρ ̃·bn+1∈G1n+1.

## Background
TECHNICAL FIELD
The present invention relates to security techniques, and more specifically, to a cryptographic technique.
BACKGROUND ART
The topics that have been studied in cryptography field include a Chosen Ciphertext Attacks (CCA)-secure cryptographic scheme. Recently, the studies of the CCA-secure cryptographic scheme based on an identity-based encryption (IBE) scheme (refer to non-patent literature 1, for example), which is usually Chosen Plaintext Attacks (CPA) secure, have become active. In non-patent literature 2, for example, a CHK transformation method is proposed. In the CHK transformation method, a one-time signature scheme is used to build the CCA-secure cryptographic scheme based on the CPA-secure IBE scheme. In non-patent literature 3, for example, a BK transformation method is proposed. In the BK transformation method, a message authentication code (MAC) and a bit commitment scheme are used to build the CCA-secure cryptographic scheme based on the CPA-secure IBE scheme.
PRIOR ART LITERATURE
Non-Patent Literature


    Non-patent literature 1: D. Boneh, M. Franklin, “Identity-Based encryption from the Weil pairing,” Crypto 2001, Lecture Notes in Computer Science, Vol. 2139, Springer-Verlag, pp. 213-229, 2001.
    Non-patent literature 2: R. Canetti, S. Halevi, J. Katz, “Chosen-Ciphertext Security from Identity-Based Encryption,” Proc. of EUROCRYPT '04, LNCS 3027, pp. 207-222, 2004.
    Non-patent literature 3: D. Boneh, J. Katz, “Improved Efficiency for CCA-Secure Cryptosystems Built Using Identity-Based Encryption,” Proc. of CT-RSA '05, LNCS 3376, pp. 87-103, 2005.


DISCLOSURE OF THE INVENTION
Problems to be Solved by the Invention
The ciphertext on the CHK transformation method includes the encrypted text, a one-time signature of the encrypted text, and a verifying key of the one-time signature. Accordingly, the ciphertext space thereof includes not only the encrypted text space but also the one-time signature space and the verifying key space. The ciphertext generated on the BK transformation method includes the encrypted text, the MAC, and the bit commitment string. Accordingly, the ciphertext space thereof includes not only the encrypted text space but also the MAC space and the bit commitment string space. This means that the ciphertext spaces on the CHK transformation method and the BK transformation method include extra two-dimensional spaces for the CCA security. However, as the ciphertext space increases, the computation amount and the data amount increase. Therefore, the ciphertext space should be small.
In the IBE scheme, the encrypting party must specify the decrypting party's ID before encryption. It would be convenient, however, if the encrypting party can generate ciphertext without specifying the decrypting party, and a party satisfying given conditions can decrypt the ciphertext.
In view of above, the cryptographic scheme of the present invention provides high convenience and improvement of CCA security level without additional space for CCA security.
Means to Solve the Problems
Encryption processing according to the present invention includes at least the following steps: first a ciphertext C1 is calculated by putting a plaintext M and a symmetric key K into a symmetric key encryption function; a function value of a collision-resistant function H is calculated by putting a value containing a value corresponding to the symmetric key K and a value corresponding to the first ciphertext C1 into the function H; and a second ciphertext C2=r·(Σμ=1nvμ·bμ)+ρ·bn+1∈G1n+1 is calculated by using a value r corresponding to the function value of the function H, a value ρ corresponding to the symmetric key K, the elements vμ (μ=1, . . . , n, n≧1) of an n-dimensional vector v→=(v1, . . . , vn), and n+1 dimensional basis vectors bi∈G1n+1 (i=1, . . . , n+1) composed of n+1 elements of a cyclic group G1.
Decryption processing corresponding to the encryption processing includes at least the following steps: a function value S ̃=e(C2, D1*)∈GT is calculated by putting the second ciphertext C2∈G1n+1 and key information D1*=α·(Σμ=1nwμ·bμ*)+bn+1*∈G2n+1 into a bilinear function e; and a function value of the collision-resistant function H is calculated by putting a value corresponding to the function value S ̃ and the ciphertext C1 into the function H. If the key information D1* is correct, the value corresponding to the function value S ̃ equals the value corresponding to the symmetric key K, which was put into the collision-resistant function H in the encryption processing. It is then judged whether a value ρ ̃, a value r ̃ corresponding to the function value of the function H, the elements vμ of the n-dimensional vector v→, and the basis vectors bi∈G1n+1 (i=1, . . . , n+1) satisfy C2=r ̃·(Σμ=1nvμ·bμ)+ρ ̃·bn+1∈G1n+1.
The value r used in the encryption processing corresponds to the function value obtained by putting the value containing the value corresponding to the symmetric key K and the value corresponding to the first ciphertext C1 into the collision-resistant function H. Accordingly, the value r depends on the first ciphertext C1 and the symmetric key K. Even if an attacker who does not know the symmetric key K can generate first ciphertext C1 from correctly obtained first ciphertext C1′, the ciphertext C1 being different from the ciphertext C1′, the attacker cannot generate a correct value r corresponding to the generated first ciphertext C1. Therefore, security against CCA can be improved by restoring a value r ̃ by putting the first ciphertext C1 and a value corresponding to the function value S ̃ into the collision-resistant function H and judging whether C2=r ̃·(Σμ=1nvμ·bμ)+ρ ̃·bn+1∈G1n+1 is satisfied, in the decryption processing.
A value 2 corresponding to a value 1 means that the value 2 equals the value 1 or that the value 2 depends on at least the value 1.
The second ciphertext C2=r·(Σμ=1nvμ·bμ)+ρ·bn+1∈G1n+1 is formed just of an n dimensional space for an n-dimensional vector v→=(v1, . . . , vn), which is the basic component of the second ciphertext C2, and a one-dimensional space for the value ρ corresponding to a value containing the symmetric key K. This means that the second ciphertext C2 does not require a space used just for improving security against CCA.
In an example of the encryption processing according to the present invention, the first ciphertext C1 is calculated by putting the plaintext M and the symmetric key K into the symmetric key encryption function; the values r and ρ are calculated by putting the symmetric key K and the first ciphertext C1 into the collision-resistant function H; the function value S=gTτ·ρ∈GT, where τ is a constant other than an additive identity element, is calculated; an exclusive OR value of the symmetric key K and a function value R(S) which is a binary sequence obtained by putting the function value S into an injective function R is calculated as a third ciphertext C3; and second ciphertext C2=r·(Σμ=1nvμ·bμ)+ρ·bn+1∈G1n+1 is calculated.
In decryption processing corresponding to the encryption processing, for example, the function value S ̃=e(C2, D1*)∈GT is calculated by putting the second ciphertext C2∈G1n+1 and the key information D1* into the bilinear function e; an exclusive OR value K ̃ of the third ciphertext C3 and the function value R(S ̃), which is a binary sequence obtained by putting the function value S ̃ into the injective function R, is calculated; the values r ̃ and ρ ̃ are calculated by putting the exclusive OR value K ̃ and the first ciphertext C1 into the collision-resistant function H; a function value C2 ̃=r ̃·(Σμ=1nvμ·bμ)+ρ ̃·bn+1∈G1n+1 is calculated; it is judged whether the function value C2 ̃ equals the second ciphertext C2; when the judgment indicates that the function value C2 ̃ equals the second ciphertext C2, a decrypted value M ̃ is calculated by using the exclusive OR value K ̃ as a symmetric key and putting the exclusive OR value K ̃ and the ciphertext C1 into a symmetric key decryption function.
In another example of the encryption processing, the value ρ is selected; the function value S=gTτ·ρ∈GT is calculated, where τ is a constant other than an additive identity element; the first ciphertext C1 is calculated by using the function value R(S) obtained by putting the function value S into the injective function R, as the symmetric key K, and putting the symmetric key K and the plaintext M into the symmetric key encryption function; the value r is calculated by putting the function value S and the first ciphertext C1 into the collision-resistant function H; and the second ciphertext C2=r·(Σμ=1nvμ·bμ)+ρ·bn+1∈G1n+1 is calculated.
In decryption processing corresponding to the encryption processing, for example, the function value S ̃=e(C2, D1*)∈GT is calculated by putting the second ciphertext C2∈G1n+1 and the key information D1* into the bilinear function e; the value r ̃ is calculated by putting the function value S ̃ and the ciphertext C1 into the collision-resistant function H; a function value λ=C2−r ̃·(Σμ=1nvμ·bμ)∈G1n+1 is calculated by using the second ciphertext C2, the value r ̃, and the elements vμ (μ=1, . . . , n) of the n-dimensional vector v→; a function value e(λ, D2*) is calculated by putting the function value λ and auxiliary key information D2*=Σμ=1nβμ·bμ* into the bilinear function e; it is judged whether the function value e(λ, D2*) satisfies e(λ, D2*)=1∈GT; and if the judgment indicates that e(λ, D2*)=1∈GT is satisfied, the decrypted value M ̃ is calculated by using the function value R(S ̃) obtained by putting the function value S ̃ into the injective function R, as the symmetric key K ̃, and by putting the symmetric key K ̃ and the first ciphertext C1 into the symmetric key decryption function.
Preferred examples of the injective function R and the collision-resistant function H are quasi random functions. This will improve the security level.
Effects of the Invention
As described above, the cryptographic scheme of the present invention provide high convenience and improvement of CCA security level without additional space for CCA security.



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a view illustrating the structure of a cryptographic system according to a first embodiment;
FIG. 2 is a view illustrating the structure of an encryption apparatus in FIG. 1;
FIG. 3 is a view illustrating the structure of a decryption apparatus in FIG. 1;
FIG. 4 is a view illustrating the structure of a key generation apparatus in FIG. 1;
FIG. 5 is a view illustrating encryption processing according to the first embodiment;
FIG. 6 is a view illustrating key information generation processing in the first embodiment;
FIG. 7 is a view illustrating decryption processing according to the first embodiment;
FIG. 8 is a view illustrating the structure of a cryptographic system according to a second embodiment;
FIG. 9 is a view illustrating the structure of an encryption apparatus in FIG. 8;
FIG. 10 is a view illustrating the structure of a decryption apparatus in FIG. 8;
FIG. 11 is a view illustrating the structure of a key generation apparatus in FIG. 8;
FIG. 12 is a view illustrating encryption processing according to the second embodiment;
FIG. 13 is a view illustrating key information generation processing in the second embodiment; and
FIG. 14 is a view illustrating decryption processing according to the second embodiment.



DETAILED DESCRIPTION OF THE EMBODIMENTS
Embodiments of the present invention will be described.
DEFINITIONS
Terms and symbols to be used in the embodiments will be defined first.
Matrix: A matrix represents a rectangular arrangement of elements of a set in which an operation is defined. Not only elements of a ring but also elements of a group can form the matrix.
(•)T: (•)T represents a transposed matrix of “•”.
(•)−1: (•)−1 represents an inverse matrix of “•”.
:  represents logical AND.
:  represents logical OR.
Z: Z represents a set of integers.
k: k represents a security parameter (k∈Z, k>0).
{0, 1}*: {0, 1}* represents a binary sequence having a desired bit length. An example thereof is a sequence formed of integers 0 and 1. However, {0, 1}* is not limited to sequences formed of integers 0 and 1. {0, 1}* may be a finite field of order 2 or its extension field.
{0, 1}ζ: {0, 1}ζ represents a binary sequence having a bit length ζ(ζ∈Z, ζ>0). An example thereof is a sequence formed of integers 0 and 1. However, {0, 1}ζ is not limited to sequences formed of integers 0 and 1. {0, 1}ζ may be a finite field of order 2 (when ζ=1) or its extension field (when ζ>1).
(+): (+) represents an exclusive OR operator between binary sequences. For example, the following is satisfied: 10110011(+) 11100001=01010010.
Fq: Fq represents a finite field of order q, where q is an integer equal to or larger than 1. For example, the order q is a prime number or a power of a prime number. In other words, the finite field Fq is a prime field or an extension field over the prime field, for example. Operations in the prime finite field Fq can be easily defined as modulo operations with the order q as modulus, for example. Operations in the extension filed Fq can be easily defined as modulo operations with an irreducible polynomial as modulus, for example. Specific finite field Fq structures are disclosed, for example, in reference literature 1, “ISO/IEC 18033-2: Information technology—Security techniques—Encryption algorithms—Part 2: Asymmetric ciphers”.
0F: 0F represents an additive identity element of the finite field Fq.
1F: 1F represents a multiplicative identity element of the finite field Fq.
δ(i, j): δ(i, j) represents a Kronecker's delta function. When i=j, δ(i, j)=1F. When i≠j, δ(i, j)=0F.
E: E represents an elliptic curve over the finite field Fq. E is defined as a set having a specific point O called a point at infinity and other points (x,y) of x,y∈Fq that satisfy the following Weierstrass equation on affine coordinates:

y2+a1·x·y+a3·y=x3+a2·x2+a4·x+a6  (1)

where a1, a2, a3, a4, a6∈Fq. A binary operation “+” called an elliptic curve addition can be defined for any two points on the elliptic curve E, and a unary operation “−” called an additive inverse can be defined for any one point on the elliptic curve E. It is well known that a finite set of rational points on the elliptic curve E forms a group with respect to the elliptic curve addition. It is also well known that an operation called an elliptic curve scalar multiplication can be defined with the elliptic curve addition. A specific operation method of elliptic operations such as the elliptic curve addition on a computer is also well known. (For example, see the reference literature 1, reference literature 2, “RFC 5091: Identity-Based Cryptography Standard (IBCS) #1: Supersingular Curve Implementations of the BF and BB1 Cryptosystems”, and reference literature 3, Ian F. Blake, Gadiel Seroussi, and Nigel P. Smart, “Elliptic Curves in Cryptography”, Pearson Education, ISBN 4-89471-431-0.)

A finite set of rational points on the elliptic curve E has a subgroup of order p (p≧1). For example, a finite set E[p] of p-division points on the elliptic curve E forms a subgroup of the rational points on the elliptic curve, where #E represents the element count of the finite set of the p-division points on the elliptic curve E and #E is divisible by the large prime p. The p-division points on the elliptic curve E are points A on the elliptic curve E which satisfy the elliptic curve scalar multiplications p·A=O.
G1, G2, GT: G1, G2, GT represent cyclic groups of order q, respectively. Examples of the cyclic groups G1 and G2 include the finite set E[p] of p-division points on the elliptic curve E and subgroups thereof. G1 may equal G2, or G1 may not equal G2. Examples of the cyclic group GT include a finite set forming an extension field over the finite field Fq. A specific example thereof is a finite set of the p-th root of 1 on the algebraic closure of the finite field Fq.
In the embodiments, operations defined on the cyclic groups G1 and G2 are expressed additively, and an operation defined on the cyclic group GT is expressed multiplicatively. More specifically, χ·Ω∈G1 for χ∈Fq and Ω∈G1 means that the operation defined in the cyclic group G1 is applied to Ω∈G1, χ times, and Ω1+Ω2∈G1 for Ω1, Ω2∈G1 means that the operation defined in the cyclic group G1 is applied to Ω1∈G1 and Ω2∈G1. In the same way, χ·Ω∈G2 for χ∈Fq and Ω∈G2 means that the operation defined in the cyclic group G2 is applied to Ω∈G2, χ times, and Ω1+Ω2∈G2 for Ω1, Ω2∈G2 means that the operation defined in the cyclic group G2 is applied to Ω1∈G2 and Ω2∈G2. In contrast, Ωχ∈GT for χ∈Fq and Ω∈GT means that the operation defined in the cyclic group GT is applied to Ω∈GT, χ times, and Ω1·Ω2∈GT for Ω1, Ω2∈GT means that the operation defined in the cyclic group GT is applied to Ω1∈GT and Ω2∈GT.
G1n+1: G1n+1 represents a direct product of (n+1) cyclic groups G1 (n≧1).
G2n+1: G2n+1 represents a direct product of (n+1) cyclic groups G2.
g1, g2, gT: g1, g2, gT represent generators of the cyclic groups G1, G2, GT, respectively.
V: V represents an (n+1)-dimensional vector space formed of the direct product of the (n+1) cyclic groups G1.
V*: V* represents an (n+1)-dimensional vector space formed of the direct product of the (n+1) cyclic groups G2.
e: e represents a function (hereinafter referred to as “bilinear function”) for calculating a non-degenerate bilinear map that maps the direct product G1n+1×G2n+1 of the direct product G1n+1 and the direct product G2n+1 to the cyclic group GT. The bilinear function e outputs an element of the cyclic group GT in response to input (n+1) elements γL (L=1, . . . , n+1) (n≧1) of the cyclic group G1 and (n+1) elements γL*(L=1, . . . , n+1) of the cyclic group G2.

e:G1n+1×G2n+1→GT  (2)

The bilinear function e satisfies the following characteristics:
Bilinearity: The following relationship is satisfied for all Γ1∈G1n+1, Γ2∈G2n+1, and ν, κ∈Fq 

e(ν·Γ1,κ·Γ2)=e(Γ1,Γ2)ν·κ  (3)

Non-degeneracy: This function does not map all

Γ1∈G1n+1,Γ2∈G2n+1  (4)

onto the identity element of the cyclic group GT.

Computability: There exists an algorithm for efficiently calculating e(Γ1, Γ2) for all Γ1∈G1n+1, Γ2∈G2n+1.
In the embodiments, the bilinear function e is formed with following a non-degenerate bilinear function which maps the direct product G1×G2 of the cyclic groups G1 and G2 to the cyclic group GT.

Pair: G1×G2→GT  (5)

The bilinear function e outputs an element of the cyclic group GT in response to an input (n+1)-dimensional vector (γ1, . . . , γn+1) formed of (n+1) elements γL (L=1, . . . , n+1) of the cyclic group G1 and an input (n+1)-dimensional vector (γ1*, . . . , γn+1*) formed of (n+1) elements γL*(L=1, . . . , n+1) of the cyclic group G2.

e=ΠL=1n+1Pair(γL,γL*)  (6)

The bilinear function Pair outputs an element of the cyclic group GT in response to an input element of the cyclic group G1 and an input element of the cyclic group G2, and satisfies the following characteristics:
Bilinearity: The following relationship is satisfied for all Ω1∈G1, Ω2∈G2, and ν, κ∈Fq 

Pair(ν·Ω1,κ·Ω2)=Pair(Ω1,Ω2)ν·κ  (7)

Non-degeneracy: This function does not map all

Ω1∈G1,Ω2∈G2  (8)

onto the identity element of the cyclic group GT.

Computability: There exists an algorithm for efficiently calculating Pair(Ω1, Ω2) for all Ω1∈G1, Ω2∈G2.
A specific example of the bilinear function Pair is a function for performing a pairing computation such as Weil pairing or Tate pairing. (See reference literature 4, Alfred. J. Menezes, “Elliptic Curve Public Key Cryptosystems”, Kluwer Academic Publishers, ISBN 0-7923-9368-6, pp. 61-81, for example.) Depending on the kind of the elliptic curve E, a modified pairing function e(Ω1,phi(Ω2))(Ω1∈G1,Ω2∈G2) which is a combination of a predetermined function phi and the function for pairing computation such as the Tate paring may be used as the bilinear function Pair (see reference literature 2, for example). As the algorithm for performing a pairing computation on a computer, the Miller algorithm (see reference literature 5, V. S. Miller, “Short Programs for Functions on Curves”, 1986, http://crypto.stanford.edu/miller/miller.pdf) or some other known algorithm can be used. Forming methods of a cyclic group and an elliptic curve for effective pairing computation have been well known. (For example, see reference literature 2, described earlier, reference literature 6, A. Miyaji, M. Nakabayashi, and S. Takano, “New Explicit Conditions of Elliptic Curve Traces for FR-Reduction”, IEICE Trans. Fundamentals, Vol. E84-A, No. 5, pp. 1234-1243, May 2001, reference literature 7, P. S. L. M. Barreto, B. Lynn, M. Scott, “Constructing Elliptic Curves with Prescribed Embedding Degrees”, Proc. SCN '2002, LNCS 2576, pp. 259-267, Springer-Verlag. 2003, and reference literature 8, R. Dupont, A. Enge, F. Morain, “Building Curves with Arbitrary Small MOV Degree over Finite Prime Fields”, http://eprint.iacr.org/2002/094/.)
ai (i=1, . . . , n+1): ai (i=1, . . . , n+1) represent (n+1)-dimensional basis vectors having (n+1) elements of the cyclic group G1 as elements. For example, each of the basis vectors ai is the (n+1)-dimensional vector in which i-th element is κ1·g1∈G1 and remain elements are identity elements (each of which is expressed additively as “0”) of the cyclic group G1. In that case, the elements of the (n+1)-dimensional basis vectors ai (i=1, . . . , n+1) can be listed as follows:

a1=(κ1·g1,0,0, . . . ,0)

a2=(0,κ1·g1,0, . . . ,0)

. . .

an+1=(0,0,0, . . . ,κ1·g1)  (9)

Here, κ1 is a constant that is an element of the finite field Fq other than the additive identity element 0F. An example of κ1∈Fq is κ1=1F. The basis vectors ai are orthogonal bases. Each (n+1)-dimensional vector having (n+1) elements of the cyclic group G1 as elements is expressed by a linear combination of the (n+1)-dimensional basis vectors ai (i=1, . . . , n+1). That is the (n+1)-dimensional basis vectors ai span the vector space V, described earlier.
ai* (i=1, . . . , n+1): ai* (i=1, . . . , n+1) represent (n+1)-dimensional basis vectors having (n+1) elements of the cyclic group G2 as elements. For example, each of the basis vectors ai* is the (n+1)-dimensional vector in which i-th element is κ2·g2∈G2 and remain elements are identity elements (each of which is expressed additively as “0”) of the cyclic group G2. In that case, the elements of the (n+1)-dimensional basis vectors ai* (i=1, . . . , n+1) can be listed as follows:

a1*=(κ2·g2,0,0, . . . ,0)

a2*=(0,κ2·g2,0, . . . ,0)

. . .

an+1*=(0,0,0, . . . ,κ2·g2)  (10)

Here, κ2 is a constant that is an element of the finite field Fq other than the additive identity element 0F. An example of κ2∈Fq is κ2=1F. The basis vectors ai* are orthogonal bases. Each (n+1)-dimensional vector having (n+1) elements of the cyclic group G2 as elements is expressed by a linear combination of (n+1)-dimensional basis vectors ai* (i=1, . . . , n+1). That is, the (n+1)-dimensional basis vectors ai* span the vector space V*, described earlier.
The basis vectors ai and the basis vectors ai* satisfy the following expression for an element τ=κ1·κ2 of the finite field Fq other than 0F:

e(ai,aj*)=gTτ·δ(i,j)  (11)

When i=j, the following expression is satisfied from Expressions (6) and (7).





   
  
    
      
        
          
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When i≠j, e(ai, aj*) does not include Pair(κ1·g1, κ2·g2) and is the product of Pair (κ1·g1, 0), Pair (0, κ2·g2), and Pair(0, 0). In addition, the following expression is satisfied from Expression (7).

Pair(g1,0)=Pair(0,g2)=Pair(g1,g2)0 

Therefore, when i≠j, the following expression is satisfied.

e(ai,aj*)=e(g1,g2)0=gT0 

Especially when τ=κ1·κ2=1F (for example, κ1=κ2=1F), the following expression is satisfied.

e(ai,aj*)=gTδ(i,j)  (12)

Here, gT0=1 is the identity element of the cyclic group GT, and gT1=gT is a generator of the cyclic group GT. In that case, the basis vectors ai and the basis vectors ai* are dual normal orthogonal bases, and the vector space V and the vector space V* are a dual vector space in which the bilinear mapping can be defined (dual pairing vector space (DPVS)).

A: “A” represents an (n+1) row by (n+1) column matrix having the basis vectors ai (i=1, . . . , n+1) as elements.
When the basis vectors ai (i=1, . . . , n+1) are expressed by Expression (9), for example, the matrix A is as follows:




  
    
      
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                  ⋮
                
              
              
                
                  
                    a
                    
                      n
                      +
                      1
                    
                  
                
              
            
            )
          
          =
          
            (
            
              
                
                  
                    
                      κ
                      1
                    
                    ·
                    
                      g
                      1
                    
                  
                
                
                  0
                
                
                  ...
                
                
                  0
                
              
              
                
                  0
                
                
                  
                    
                      κ
                      1
                    
                    ·
                    
                      g
                      1
                    
                  
                
                
                  
                    
                  
                
                
                  ⋮
                
              
              
                
                  ⋮
                
                
                  
                    
                  
                
                
                  ⋱
                
                
                  0
                
              
              
                
                  0
                
                
                  ...
                
                
                  0
                
                
                  
                    
                      κ
                      1
                    
                    ·
                    
                      g
                      1
                    
                  
                
              
            
            )
          
        
      
    
    
      
        (
        13
        )
      
    
  




A*: “A*” represents an (n+1) row by (n+1) column matrix having the basis vectors ai* (i=1, . . . , n+1) as elements.
When the basis vectors ai* (i=1, . . . , n+1) are expressed by Expression (10), for example, the matrix A* is as follows:




  
    
      
        
          A
          *
        
        =
        
          
            (
            
              
                
                  
                    a
                    1
                    *
                  
                
              
              
                
                  
                    a
                    2
                    *
                  
                
              
              
                
                  ⋮
                
              
              
                
                  
                    a
                    
                      n
                      +
                      1
                    
                    *
                  
                
              
            
            )
          
          =
          
            (
            
              
                
                  
                    
                      κ
                      2
                    
                    ·
                    
                      g
                      2
                    
                  
                
                
                  0
                
                
                  ...
                
                
                  0
                
              
              
                
                  0
                
                
                  
                    
                      κ
                      2
                    
                    ·
                    
                      g
                      2
                    
                  
                
                
                  
                    
                  
                
                
                  ⋮
                
              
              
                
                  ⋮
                
                
                  
                    
                  
                
                
                  ⋱
                
                
                  0
                
              
              
                
                  0
                
                
                  ...
                
                
                  0
                
                
                  
                    
                      κ
                      2
                    
                    ·
                    
                      g
                      2
                    
                  
                
              
            
            )
          
        
      
    
    
      
        (
        14
        )
      
    
  




X: X represents an (n+1) row by (n+1) column matrix having elements of the finite field Fq as entries. The matrix X is used for coordinate transformation of the basis vectors ai. The matrix X is as follows:




  
    
      
        X
        =
        
          (
          
            
              
                
                  χ
                  
                    1
                    ,
                    1
                  
                
              
              
                
                  χ
                  
                    1
                    ,
                    2
                  
                
              
              
                ...
              
              
                
                  χ
                  
                    1
                    ,
                    
                      n
                      +
                      1
                    
                  
                
              
            
            
              
                
                  χ
                  
                    2
                    ,
                    1
                  
                
              
              
                
                  χ
                  
                    2
                    ,
                    2
                  
                
              
              
                
                  
                
              
              
                ⋮
              
            
            
              
                ⋮
              
              
                
                  
                
              
              
                ⋱
              
              
                ⋮
              
            
            
              
                
                  χ
                  
                    
                      n
                      +
                      1
                    
                    ,
                    1
                  
                
              
              
                
                  χ
                  
                    
                      n
                      +
                      1
                    
                    ,
                    2
                  
                
              
              
                ...
              
              
                
                  χ
                  
                    
                      n
                      +
                      1
                    
                    ,
                    
                      n
                      +
                      1
                    
                  
                
              
            
          
          )
        
      
    
    
      
        (
        15
        )
      
    
  




where each χi,j∈Fq is the entry in the i-th row and the j-th column (i=1, . . . , n+1, j=1, . . . , n+1) of the matrix X.

Here, each entry χi,j of the matrix X is called as a transformation coefficient.
X*: X* represents the transposed matrix of the inverse matrix of the matrix X. X*=(X−1)T. The matrix X* is used for coordinate transformation of the basis vectors ai*. The matrix X* is expressed as follows:




  
    
      
        
          X
          *
        
        =
        
          (
          
            
              
                
                  χ
                  
                    1
                    ,
                    1
                  
                  *
                
              
              
                
                  χ
                  
                    1
                    ,
                    2
                  
                  *
                
              
              
                ...
              
              
                
                  χ
                  
                    1
                    ,
                    
                      n
                      +
                      1
                    
                  
                  *
                
              
            
            
              
                
                  χ
                  
                    2
                    ,
                    1
                  
                  *
                
              
              
                
                  χ
                  
                    2
                    ,
                    2
                  
                  *
                
              
              
                
                  
                
              
              
                ⋮
              
            
            
              
                ⋮
              
              
                
                  
                
              
              
                ⋱
              
              
                ⋮
              
            
            
              
                
                  χ
                  
                    
                      n
                      +
                      1
                    
                    ,
                    1
                  
                  *
                
              
              
                
                  χ
                  
                    
                      n
                      +
                      1
                    
                    ,
                    2
                  
                  *
                
              
              
                ...
              
              
                
                  χ
                  
                    
                      n
                      +
                      1
                    
                    ,
                    
                      n
                      +
                      1
                    
                  
                  *
                
              
            
          
          )
        
      
    
    
      
        (
        16
        )
      
    
  




where the entry in the i-th row and j-th column of the matrix X* is χi,j*∈Fq.

Here, each entry χi,j* of the matrix X* is called as a transformation coefficient.
In that case, X·(X*)T=I is satisfied, where “I” represents an (n+1) row by (n+1) column unit matrix.
In other words, the unit matrix is expressed as follows.




  
    
      
        I
        =
        
          (
          
            
              
                
                  1
                  F
                
              
              
                
                  0
                  F
                
              
              
                ...
              
              
                
                  0
                  F
                
              
            
            
              
                
                  0
                  F
                
              
              
                
                  1
                  F
                
              
              
                
                  
                
              
              
                ⋮
              
            
            
              
                ⋮
              
              
                
                  
                
              
              
                ⋱
              
              
                
                  0
                  F
                
              
            
            
              
                
                  0
                  F
                
              
              
                
                  0
                  F
                
              
              
                ...
              
              
                
                  1
                  ⁢
                  F
                
              
            
          
          )
        
      
    
    
      
        (
        17
        )
      
    
  




The following expression is satisfied.




  
    
      
        
          
            (
            
              
                
                  
                    χ
                    
                      1
                      ,
                      1
                    
                  
                
                
                  
                    χ
                    
                      1
                      ,
                      2
                    
                  
                
                
                  ...
                
                
                  
                    χ
                    
                      1
                      ,
                      
                        n
                        +
                        1
                      
                    
                  
                
              
              
                
                  
                    χ
                    
                      2
                      ,
                      1
                    
                  
                
                
                  
                    χ
                    
                      2
                      ,
                      2
                    
                  
                
                
                  
                    
                  
                
                
                  ⋮
                
              
              
                
                  ⋮
                
                
                  
                    
                  
                
                
                  ⋱
                
                
                  ⋮
                
              
              
                
                  
                    χ
                    
                      
                        n
                        +
                        1
                      
                      ,
                      1
                    
                  
                
                
                  
                    χ
                    
                      
                        n
                        +
                        1
                      
                      ,
                      2
                    
                  
                
                
                  ...
                
                
                  
                    χ
                    
                      
                        n
                        +
                        1
                      
                      ,
                      
                        n
                        +
                        1
                      
                    
                  
                
              
            
            )
          
          ·
          
            (
            
              
                
                  
                    χ
                    
                      1
                      ,
                      1
                    
                    *
                  
                
                
                  
                    χ
                    
                      1
                      ,
                      2
                    
                    *
                  
                
                
                  ...
                
                
                  
                    χ
                    
                      
                        n
                        +
                        1
                      
                      ,
                      1
                    
                    *
                  
                
              
              
                
                  
                    χ
                    
                      1
                      ,
                      2
                    
                    *
                  
                
                
                  
                    χ
                    
                      2
                      ,
                      2
                    
                    *
                  
                
                
                  
                    
                  
                
                
                  ⋮
                
              
              
                
                  ⋮
                
                
                  
                    
                  
                
                
                  ⋱
                
                
                  ⋮
                
              
              
                
                  
                    χ
                    
                      1
                      ,
                      
                        n
                        +
                        1
                      
                    
                    *
                  
                
                
                  
                    χ
                    
                      2
                      ,
                      
                        n
                        +
                        1
                      
                    
                    *
                  
                
                
                  ...
                
                
                  
                    χ
                    
                      
                        n
                        +
                        1
                      
                      ,
                      
                        n
                        +
                        1
                      
                    
                    *
                  
                
              
            
            )
          
        
        =
        
          (
          
            
              
                
                  1
                  F
                
              
              
                
                  0
                  F
                
              
              
                ...
              
              
                
                  0
                  F
                
              
            
            
              
                
                  0
                  F
                
              
              
                
                  1
                  F
                
              
              
                
                  
                
              
              
                ⋮
              
            
            
              
                ⋮
              
              
                
                  
                
              
              
                ⋱
              
              
                
                  0
                  F
                
              
            
            
              
                
                  0
                  F
                
              
              
                
                  0
                  F
                
              
              
                ...
              
              
                
                  1
                  ⁢
                  F
                
              
            
          
          )
        
      
    
    
      
        (
        18
        )
      
    
  




Here, (n+1)-dimensional vectors will be defined below.

χi→=(χi,1, . . . ,χi,n+1)  (19)

χj→=(χj,1*, . . . ,χj,n+1*)  (20)

The Inner Product of the (n+1)-Dimensional Vectors χi→ and χj→* satisfies the following expression from Expression (18).

χi→·χj→*=δ(i,j)  (21)

bi: bi represent (n+1)-dimensional basis vectors having (n+1) elements of the cyclic group G1 as elements. The basis vectors bi are obtained by coordinate transformation of the basis vectors ai (i=1, . . . , n+1) with the matrix X. That is, the basis vectors bi are obtained by the following calculation.

bi=Σj=1n+1χi,j·aj  (22)

When the basis vectors aj (j=1, . . . , n+1) are expressed by Expression (9), each element of the basis vectors bi is shown below.

bi=(χi,1·κ1·g1,χi,2·κ1·g1, . . . ,χi,n+1·κ1·g1)  (23)

Each (n+1)-dimensional vector having (n+1) elements of the cyclic group G1 as elements is expressed by a linear combination of (n+1)-dimensional basis vectors bi (i=1, . . . , n+1). That is, the (n+1)-dimensional basis vectors bi span the vector space V, described earlier.
bi*: bi* represent (n+1)-dimensional basis vectors having (n+1) elements of the cyclic group G2 as elements. The basis vectors bi* are obtained by coordinate transformation of the basis vectors ai*(i=1, . . . , n+1) with the matrix X*. That is, the basis vectors bi* are obtained by the following calculation.

bi*=Σj=1n+1χi,j*·aj*  (24)

When the basis vectors aj (j=1, . . . , n+1) are expressed by Expression (10), each element of the basis vectors bi* are shown below.

bi*=(χi,1*·κ2·g2,χi,2·κ2·g2, . . . ,χi,n+1*·κ2·g2)  (25)

Each (n+1)-dimensional vector having (n+1) elements of the cyclic group G2 as elements is expressed by a linear combination of (n+1)-dimensional basis vectors bi*(i=1, . . . , n+1). That is, the (n+1)-dimensional basis vectors bi* span the vector space V*, described earlier.

The basis vectors bi and the basis vectors bi* satisfy the following expression for the elements τ=κ1·κ2 of the finite field Fq other than 0F:

e(bi,bj*)=gTτ·δ(i,j)  26)

The following expression is satisfied from Expressions (6), (21), (23), and (25).





   
  
    
      
        
          
            e
            ⁡
            
              (
              
                
                  b
                  i
                
                ,
                
                  b
                  j
                  *
                
              
              )
            
          
          =
          
          ⁢
          
            
              ∏
              
                L
                =
                1
              
              
                n
                +
                1
              
            
            ⁢
            
              
            
            ⁢
            
              Pair
              ⁡
              
                (
                
                  
                    
                      χ
                      
                        i
                        ⁢
                        
                          
                        
                        ,
                        L
                      
                    
                    ·
                    
                      κ
                      1
                    
                    ·
                    
                      g
                      1
                    
                  
                  ,
                  
                    
                      χ
                      
                        j
                        ,
                        L
                      
                      *
                    
                    ·
                    
                      κ
                      2
                    
                    ·
                    
                      g
                      2
                    
                  
                
                )
              
            
          
        
      
    
    
      
        
          =
          
          ⁢
          
            
              Pair
              ⁡
              
                (
                
                  
                    
                      χ
                      
                        i
                        ,
                        1
                      
                    
                    ·
                    
                      κ
                      1
                    
                    ·
                    
                      g
                      1
                    
                  
                  ,
                  
                    
                      χ
                      
                        j
                        ,
                        1
                      
                      *
                    
                    ·
                    
                      κ
                      2
                    
                    ·
                    
                      g
                      2
                    
                  
                
                )
              
            
            ·
            ...
            ·
          
        
      
    
    
      
        
          
          ⁢
          
            
              (
              
                
                  
                    χ
                    
                      i
                      ,
                      n
                    
                  
                  ·
                  
                    κ
                    1
                  
                  ·
                  
                    g
                    1
                  
                
                ,
                
                  
                    χ
                    
                      j
                      ,
                      n
                    
                    *
                  
                  ·
                  
                    κ
                    2
                  
                  ·
                  
                    g
                    2
                  
                
              
              )
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            Pair
            ⁡
            
              (
              
                
                  
                    χ
                    
                      j
                      ,
                      
                        n
                        +
                        1
                      
                    
                  
                  ·
                  
                    κ
                    1
                  
                  ·
                  
                    g
                    1
                  
                
                ,
                
                  
                    χ
                    
                      j
                      ,
                      
                        n
                        +
                        1
                      
                    
                    *
                  
                  ·
                  
                    κ
                    2
                  
                  ·
                  
                    g
                    2
                  
                
              
              )
            
          
        
      
    
    
      
        
          =
          
          ⁢
          
            
              
                
                  Pair
                  ⁡
                  
                    (
                    
                      
                        g
                        1
                      
                      ,
                      
                        g
                        2
                      
                    
                    )
                  
                
                
                  
                    κ
                    1
                  
                  ·
                  
                    κ
                    2
                  
                  ·
                  
                    χ
                    
                      i
                      ,
                      1
                    
                  
                  ·
                  
                    χ
                    
                      j
                      ,
                      1
                    
                    *
                  
                
              
              ·
              ...
              ·
              
                
                  Pair
                  ⁡
                  
                    (
                    
                      
                        g
                        1
                      
                      ,
                      
                        g
                        2
                      
                    
                    )
                  
                
                
                  
                    κ
                    1
                  
                  ·
                  
                    κ
                    2
                  
                  ·
                  
                    χ
                    
                      i
                      ,
                      2
                    
                  
                  ·
                  
                    χ
                    
                      j
                      ,
                      2
                    
                    *
                  
                
              
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            
              Pair
              ⁡
              
                (
                
                  
                    g
                    1
                  
                  ,
                  
                    g
                    2
                  
                
                )
              
            
            
              
                κ
                1
              
              ·
              
                κ
                2
              
              ·
              
                χ
                
                  i
                  ,
                  
                    n
                    +
                    1
                  
                
              
              ·
              
                χ
                
                  j
                  ,
                  
                    n
                    +
                    1
                  
                
                *
              
            
          
        
      
    
    
      
        
          =
          
          ⁢
          
            
              Pair
              ⁡
              
                (
                
                  
                    g
                    1
                  
                  ,
                  
                    g
                    2
                  
                
                )
              
            
            
              
                κ
                1
              
              ·
              
                
                  κ
                  2
                
                ⁡
                
                  (
                  
                    
                      
                        χ
                        
                          i
                          ,
                          1
                        
                      
                      ·
                      
                        χ
                        
                          j
                          ,
                          1
                        
                        *
                      
                    
                    +
                    
                      
                        χ
                        
                          i
                          ,
                          2
                        
                      
                      ·
                      
                        χ
                        
                          j
                          ,
                          2
                        
                        *
                      
                    
                    +
                    ...
                    +
                    
                      
                        χ
                        
                          i
                          ,
                          
                            n
                            +
                            1
                          
                        
                      
                      ·
                      
                        χ
                        
                          j
                          ,
                          
                            n
                            +
                            1
                          
                        
                        *
                      
                    
                  
                  )
                
              
            
          
        
      
    
    
      
        
          =
          
          ⁢
          
            
              Pair
              ⁡
              
                (
                
                  
                    g
                    1
                  
                  ,
                  
                    g
                    2
                  
                
                )
              
            
            
              
                κ
                1
              
              ·
              
                κ
                2
              
              ·
              
                χ
                i
                →
              
              ·
              
                χ
                j
                
                  →
                  *
                
              
            
          
        
      
    
    
      
        
          =
          
          ⁢
          
            
              Pair
              ⁡
              
                (
                
                  
                    g
                    1
                  
                  ,
                  
                    g
                    2
                  
                
                )
              
            
            
              τ
              ·
              
                δ
                ⁡
                
                  (
                  
                    i
                    ,
                    j
                  
                  )
                
              
            
          
        
      
    
    
      
        
          =
          
          ⁢
          
            g
            T
            
              τ
              ·
              
                δ
                ⁡
                
                  (
                  
                    i
                    ,
                    j
                  
                  )
                
              
            
          
        
      
    
  




Especially when τ=κ1·κ2=1F (for example, κ1=κ2=1F), the following expression is satisfied.

e(bi,bj*)=gTδ(i,j)  (27)

In that case, the basis vectors bi and the basis vectors bi* are the dual normal orthogonal basis of a dual pairing vector space (the vector space V and the vector space V*).

As long as Expression (26) is satisfied, the basis vectors ai and ai* other than those shown in Expressions (9) and (10) as examples, and the basis vectors bi and bi* other than those shown in Expressions (22) and (24) as examples may be used.
B: B represents an (n+1) row by (n+1) column matrix having the basis vectors bi (i=1, . . . , n+1) as elements.
B=X·A is satisfied. When the basis vectors bi are expressed by Expression (23), for example, the matrix B is as follows:




  
    
      
        
          
            
              B
              =
              
                (
                
                  
                    
                      
                        b
                        1
                      
                    
                  
                  
                    
                      
                        b
                        2
                      
                    
                  
                  
                    
                      ⋮
                    
                  
                  
                    
                      
                        b
                        
                          n
                          +
                          1
                        
                      
                    
                  
                
                )
              
            
          
        
        
          
            
              =
              
                (
                
                  
                    
                      
                        
                          χ
                          
                            1
                            ,
                            1
                          
                        
                        ·
                        
                          κ
                          1
                        
                        ·
                        
                          g
                          1
                        
                      
                    
                    
                      
                        
                          χ
                          
                            1
                            ,
                            2
                          
                        
                        ·
                        
                          κ
                          1
                        
                        ·
                        
                          g
                          1
                        
                      
                    
                    
                      ...
                    
                    
                      
                        
                          χ
                          
                            1
                            ,
                            
                              n
                              +
                              1
                            
                          
                        
                        ·
                        
                          κ
                          1
                        
                        ·
                        
                          g
                          1
                        
                      
                    
                  
                  
                    
                      
                        
                          χ
                          
                            2
                            ,
                            1
                          
                        
                        ·
                        
                          κ
                          1
                        
                        ·
                        
                          g
                          1
                        
                      
                    
                    
                      
                        
                          χ
                          
                            2
                            ,
                            2
                          
                        
                        ·
                        
                          κ
                          1
                        
                        ·
                        
                          g
                          1
                        
                      
                    
                    
                      
                        
                      
                    
                    
                      ⋮
                    
                  
                  
                    
                      ⋮
                    
                    
                      
                        
                      
                    
                    
                      ⋱
                    
                    
                      
                        
                          χ
                          
                            n
                            ,
                            
                              n
                              +
                              1
                            
                          
                        
                        ·
                        
                          κ
                          1
                        
                        ·
                        
                          g
                          1
                        
                      
                    
                  
                  
                    
                      
                        
                          χ
                          
                            
                              n
                              +
                              1
                            
                            ,
                            1
                          
                        
                        ·
                        
                          κ
                          1
                        
                        ·
                        
                          g
                          1
                        
                      
                    
                    
                      ...
                    
                    
                      
                        
                          χ
                          
                            
                              n
                              +
                              1
                            
                            ,
                            n
                          
                        
                        ·
                        
                          κ
                          1
                        
                        ·
                        
                          g
                          1
                        
                      
                    
                    
                      
                        
                          χ
                          
                            
                              n
                              +
                              1
                            
                            ,
                            
                              n
                              +
                              1
                            
                          
                        
                        ·
                        
                          κ
                          1
                        
                        ·
                        
                          g
                          1
                        
                      
                    
                  
                
                )
              
            
          
        
      
    
    
      
        (
        28
        )
      
    
  




B*: B* represents an (n+1) row by (n+1) column matrix having the basis vectors bi* (i=1, . . . , n+1) as elements.
B*=X*·A* is satisfied. When the basis vectors bi* (i=1, . . . , n+1) are expressed by Expression (25), for example, the matrix B* is as follows:




  
    
      
        
          
            
              B
              =
              
                (
                
                  
                    
                      
                        b
                        1
                        *
                      
                    
                  
                  
                    
                      
                        b
                        2
                        *
                      
                    
                  
                  
                    
                      ⋮
                    
                  
                  
                    
                      
                        b
                        
                          n
                          +
                          1
                        
                        *
                      
                    
                  
                
                )
              
            
          
        
        
          
            
              =
              
                (
                
                  
                    
                      
                        
                          χ
                          
                            1
                            ,
                            1
                          
                          *
                        
                        ·
                        
                          κ
                          2
                        
                        ·
                        
                          g
                          2
                        
                      
                    
                    
                      
                        
                          χ
                          
                            1
                            ,
                            2
                          
                          *
                        
                        ·
                        
                          κ
                          2
                        
                        ·
                        
                          g
                          2
                        
                      
                    
                    
                      ...
                    
                    
                      
                        
                          χ
                          
                            1
                            ,
                            
                              n
                              +
                              1
                            
                          
                          *
                        
                        ·
                        
                          κ
                          2
                        
                        ·
                        
                          g
                          2
                        
                      
                    
                  
                  
                    
                      
                        
                          χ
                          
                            2
                            ,
                            1
                          
                          *
                        
                        ·
                        
                          κ
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w→: w→ represents an n-dimensional vector having elements of the finite field Fq as elements.

w→=(w1, . . . ,wn)∈Fqn  (30)

wμ: wμ represents the μ-th (μ=1, . . . , n) element of the n-dimensional vector.
v→: v→ represents an n-dimensional vector having elements of the finite field Fq as elements.

v→=(v1, . . . ,vn)∈Fqn  (31)

vμ: vμ represents the μ-th (μ=1, . . . , n) element of the n-dimensional vector.
Collision-resistant function: The collision-resistant function represents a function h that satisfies the following condition with respect to a sufficiently larger security parameter k, or a function regarded as the function h.

Pr[A(h)=(x,y)|h(x)=h(y)x≠y]<ε(k)  (32)

Here, Pr[•] is the probability of the event [•]; A(h) is a probability polynomial time algorithm for calculating x and y (x≠y) that satisfy h(x)=h(y) for the function h; and ε(k) is a polynomial equation of the security parameter k. An example collision-resistant function is a hash function, such as the cryptographic hash function disclosed in the reference literature 1.
Injective function: The injective function represents a function that each element of its codomain is mapped from each one element of its domain, or a function regarded as such. An example of the injection function is a hash function, such as the key derivation function (KDF) disclosed in the reference literature 1.
Quasi random function: The quasi random function represents a function belonging to a subset φζ of a set Φζ, or a function regarded as such, where any probabilistic polynomial time algorithm cannot distinguish the subset φζ from the set Φζ; and the set Φζ is a set of all functions that map an element of a set {0, 1}* to an element of the set {0, 1}ζ. An example of the quasi random function is a hash function such as that described above.
H1: H1 represents a collision-resistant function that outputs two elements (ψ1, ψ2)∈Fq×Fq of the finite field Fq in response to input two binary sequences (ω1, ω2)∈{0, 1}k×{0, 1}*.

H1:{0,1}k×{0,1}*→Fq×Fq  (33)

An example of the function H1 is a function that outputs two elements (ψ1, ψ2)∈Fq×Fq of the finite field Fq in response to the connected bits ω1∥ω2 of input ω1 and ω2. This function may include a hash function such as the cryptographic hash function, a binary-sequence-to-integer conversion function (octet string/integer conversion), and a binary-sequence-to-finite-field-element conversion function (octet string and integer/finite field conversion), which are disclosed in the reference literature 1. Preferably, the function H1 is a quasi random function.
H2: H2 represents a collision-resistant function that outputs an element ψ∈Fq of the finite field Fq in response to an input element of the cyclic group GT and an input binary sequence (ξ, ω2)∈GT×{0, 1}*.

H2:GT×{0,1}*→Fq  (34)

An example of the function H2 is a function that accepts an element ξ∈GT of the cyclic group GT and a binary sequence ω2∈{0, 1}*, inputs the element ξ∈GT of the cyclic group GT to a finite-field-element-to-binary-sequence conversion function (octet string and integer/finite field conversion) disclosed in the reference literature 1 to obtain a binary sequence, applies a hash function such the cryptographic hash function disclosed in the reference literature 1 to the connected bits of the obtained binary sequence and the binary sequence ω2∈{0, 1}*, performs the binary-sequence-to-finite-field-element conversion function (octet string and integer/finite field conversion), and outputs one element ψ∈Fq of the finite field Fq. For ensuring the higher safety, it is preferable that the function H2 is the quasi-random function.
R: R represents an injective function that outputs one binary sequence ω∈{0, 1}k in response to an input element ξ∈GT of the cyclic group GT.

R:GT→{0,1}*  (35)

An example of the injective function R is a function that receives an element ξ∈GT of the cyclic group GT, performs calculations with the finite-field-element-to-binary-sequence conversion function (octet string and integer/finite field conversion) and then with a hash function such as the KDF (key derivation function) disclosed in the reference literature 1, and outputs one binary sequence ω∈{0, 1}k. For ensuring the higher safety, it is preferable that the function R is the collision-resistant function, and it is more preferable that the function R is the quasi random function.
Enc: Enc represents a symmetric key encryption function that indicates encryption processing of a symmetric key cryptosystem. Examples of symmetric key cryptosystems are Camellia and AES.
Enck(M): Enck(M) represents a ciphertext obtained by encrypting a plaintext M by the symmetric key encryption function Enc with a symmetric key K.
Dec: Dec represents a symmetric key decryption function that indicates decryption processing of the symmetric key cryptosystem.
Deck(C): Deck(C) represents a decryption result obtained by decrypting a ciphertext C by the symmetric key decryption function Dec with the symmetric key K.
Inner Product Predicate Encryption
The basic scheme of inner product predicate encryption will be described below.
Predicate Encryption
In the predicate encryption (sometimes called as function encryption), a ciphertext can be decrypted when a combination of attribute information and predicate information makes a predetermined logical formula true. One of the attribute information and predicate information is embedded in the ciphertext and the other is embedded in key information. The conventional predicate encryption is, for example, disclosed in reference literature 9, Jonathan Katz, Amit Sahai and Brent Waters, “Predicate Encryption Supporting Disjunctions, Polynomial Equations, and Inner Products”, one of four papers from Eurocrypt 2008 invited by the Journal of Cryptology.
Inner Product Predicate Encryption
In the inner product predicate encryption, a ciphertext can be decrypted when the inner product of the attribute information and the predicate information which are vectors is zero. In inner product predicate encryption, an inner product of zero is equivalent to the logical formula of true.
Relationship Between Logical Formula and Polynomial
In the inner product predicate encryption, the logical formula formed of a logical OR(s) and/or a logical AND(s) is expressed by a polynomial.
The logical OR (x=η1)(x=η2) of a proposition 1 indicating that x is η1 and a proposition 2 indicating that x is η2 is expressed by the following polynomial.

(x−η1)·(x−η2)  (36)

Then, the relationships between truth values and the function values of Expression (36) are shown in the following table.











TABLE 1





Proposition 1
Proposition 2
Logical OR
Function value


(x = η1)
(x = η2)
(x = η1)   (x = η2)
(x = η1) · (x = η2)







True
True
True
0


True
False
True
0


False
True
True
0


False
False
False
Other than 0









As understood from Table 1, when the logical OR (x=η1)(x=η2) is true, the function value of Expression (36) is zero; and when the logical OR (x=η1)(x=η2) is false, the function value of Expression (36) is a value other than zero. In other words, the logical OR (x=η1)(x=η2) of true is equivalent to the function value of zero in Expression (36). Therefore, the logical OR can be expressed by Expression (36).
The logical AND (x=η1)(x=η2) of the proposition 1 indicating that x is η1 and the proposition 2 indicating that x is η2 is expressed by the following polynomial

ι1·(x−η1)+ι2·(x−η2)  (37)

where ι1 and ι2 are random numbers. Then, the relationships between truth values and the function values of Expression (37) are shown in the following table.











TABLE 2








Function value


Proposition 1
Proposition 2
Logical AND
l1 · (x − η1) + l2 · (x −


(x = η1)
(x = η2)
(x = η1)   (x = η2)
η2)







True
True
True
0


True
False
False
Other than 0


False
True
False
Other than 0


False
False
False
Other than 0









As understood from Table 2, when the logical AND (x=η1)(x=η2) is true, the function value of Expression (37) is zero; and when the logical AND (x=η1)(x=η2) is false, the function value of Expression (37) is a value other than zero. In other words, a logical AND (x=η1)(x=η2) of true is equivalent to a function value of zero in Expression (37). Therefore, the logical AND can be expressed by Expression (37).
As described above, by using Expressions (36) and (37), a logical formula formed of a logical OR(s) and/or a logical AND(s) can be expressed by a polynomial f(x). An example will be shown below.

Logical formula: {(x=η1)(x=η2)(x=η3)}(x=η4)(x=η5)

Polynomial: f(x)=ι1·{(x−η1)·(x−η2)·(x−η3)}+ι2·(x−=η4)+ι3·(x−=η5)

In Expression (36), one indeterminate element x is used to express the logical OR. A plurality of indeterminate elements can also be used to express a logical OR. For example, when two indeterminate elements x0 and x1 are used, the logical OR (x0=η0)(x1=η1) of the proposition 1 indicating that x0 is η0 and the proposition 2 indicating that x1 is η1 can be expressed by the following polynomial.

(x0−η0)·(x1−η1)

Three or more indeterminate elements can also be used to express a logical OR by a polynomial.

In Expression (37), one indeterminate element x is used to express the logical AND. A plurality of indeterminate elements can also be used to express a logical AND. For example, the logical AND (x0=η0)(x1=η1) of the proposition 1 indicating that x0 is η0 and the proposition 2 indicating that x1 is η1 can be expressed by the following polynomial.

ι0·(x0−η0)+ι1·(x1−η1)

Three or more indeterminate elements can also be used to express a logical AND by a polynomial.

A logical formula including a logical OR(s) and/or a logical AND(s) is expressed with H(H≧1) types of indeterminate elements x0, . . . , xH−1 as the polynomial f(x0, . . . , xH−1). It is assumed that a proposition for each of the indeterminate elements x0, . . . , xH−1 is “xh is ηh”, where ηh (h=0, . . . , H−1) is a constant determined for each proposition. Then, in the polynomial f(x0, . . . , xH−1) indicating the logical formula, the proposition indicating that an indeterminate element xh is a constant ηh is expressed by the polynomial indicating the difference between the indeterminate element xh and the constant ηh; the logical OR of propositions is expressed by the product of the polynomials indicating the propositions; and the logical AND of propositions or the logical ORs of propositions is expressed by a linear combination of the polynomials indicating the propositions or the logical ORs of propositions. For example, five indeterminate elements x0, . . . , x4 are used to express a logical formula

{(x0=η0)(x1=η1)(x2=η2)}(x3=η3)(x4=η4)

by the following polynomial

f(x0, . . . , x4)=ι0·{(x0−η0)·(x1−η1)·(x2−η2)}+ι1·(x3−=η3)+ι2·(x4−=η4)

Relationship Between Polynomial and Inner Product
The polynomial f(x0, . . . , xH−1) indicating a logical formula can be expressed by the inner product of two n-dimensional vectors. More specifically, the polynomial f(x0, . . . , xH−1) is equal to the inner product of a vector

v→=(v1, . . . ,vn)

which has the indeterminate elements of the terms of the polynomial f(x0, . . . , xH−1) as elements, and a vector

w→=(w1, . . . ,wn)

which has the coefficients of the terms of the polynomial f(x0, . . . , xH−1) as elements.

f(x0, . . . ,xH−l)=w→·v→

In other words, whether the polynomial f(x0, . . . , xH−1) indicating a logical formula is zero is equivalent to whether the inner product of the vector v→ having the indeterminate elements of the terms of the polynomial f(x0, . . . , xH−1) as elements and the vector w→ having the coefficients of the terms of the polynomial f(x0, . . . , xH−1) as elements is zero.

f(x0, . . . ,xH−1)=0←→w→·v→=0

For example, a polynomial f(x)=θ0·x0+θ1·x+ . . . +θn−1·xn−1 expressed with one indeterminate element x can be expressed by the inner product of two n-dimensional vectors as follows.

w→=(w1, . . . ,wn)=(θ0, . . . ,θn−1)  (39)

v→=(v1, . . . ,vn)=(x0, . . . ,xn−1)  (40)

f(x)=w→·v→  (41)

In other words, whether the polynomial f(x) indicating a logical formula is zero is equivalent to whether the inner product in Expression (41) is zero.

f(x)=0←→w→·v→0  (42)

When a vector having the indeterminate elements of the terms of the polynomial f(x0, . . . , xH−1) as elements is expressed by

w→=(w1, . . . ,wn)

and a vector having the coefficients of the terms of the polynomial f(x0, . . . , xH−1) as elements is expressed by

v→=(v1, . . . ,vn)

whether the polynomial f(x0, . . . , xH−1) indicating a logical formula is zero is equivalent to whether the inner product of the vector w→ and the vector v→ is zero.

For example, when the following expressions are used instead of Expressions (39) and (40),

w→=(w1, . . . ,wn=(x0, . . . ,xn−1)  (43)

v→=(v1, . . . ,vn=(θ0, . . . ,θn−1)  (44)

whether the polynomial f(x) indicating a logical formula is zero is equivalent to whether the inner product in Expression (41) is zero.

In the inner product predicate encryption, one of the vectors v→=(v0, . . . , vn−1) and w→=(w0, . . . , wn−1) is used as the attribute information and the other is used as the predicate information. One of the attribute information and predicate information is embedded in ciphertext and the other is embedded in key information. For example, an n-dimensional vector (θ0, . . . , θn−1) is used as the predicate information, another n-dimensional vector (x0, . . . , xn−1) is used as the attribute information, one of the attribute information and predicate information is embedded in ciphertext, and the other is embedded in key information. It is assumed in the following description that an n-dimensional vector embedded in key information is w→=(w1, . . . , wn) and another n-dimensional vector embedded in ciphertext is v→=(v1, . . . , vn). For example,

Predicate information: w→=(w1, . . . ,wn)=(θ0, . . . ,θn−1)

Attribute information: v→=(v1, . . . , vn)=(x0, . . . ,xn−1)

Alternatively,

Predicate information: v→=(v1, . . . ,vn)=(θ0, . . . ,θn−1)

Attribute information: w→=(w1, . . . ,wn)=(x0, . . . ,xn−1)

Basic Scheme of Inner Product Predicate Encryption
An example of basic scheme of a key encapsulation mechanism (KEM) using the inner product predicate encryption will be described below. This scheme includes Setup(1k), GenKey(MSK, w→), Enc(PA, v→), and Dec(SKw, C2).
Setting up Setup(1k)
Input: Security parameter k
Output: Master key information MSK, public parameter PK
In an example of Setup(1k), the security parameter k is used as n, the (n+1) row by (n+1) column matrix A having the (n+1)-dimensional basis vectors ai (i=1, . . . , n+1) as elements, the (n+1) row by (n+1) column matrix A* having the basis vectors ai* (i=1, . . . , n+1) as elements, and the (n+1) row by (n+1) column matrixes X and X* used for coordinate transformation are selected. Then, the (n+1)-dimensional basis vectors bi (i=1, . . . , n+1) are calculated through coordinate transformation by Expression (22), and the (n+1)-dimensional basis vectors bi* (i=1, . . . , n+1) are calculated through coordinate transformation by Expression (24). Then, the (n+1) row by (n+1) column matrix B* having the basis vectors bi* (i=1, . . . , n+1) as elements is output as the master key information MSK; and the vector spaces V and V*, the (n+1) row by (n+1) column matrix B having the basis vectors bi (i=1, . . . , n+1) as elements, the security parameter k, the finite field Fq, the elliptic curve E, the cyclic groups G1, G2, and GT, the generators g1, g2, and gT, the bilinear function e, and others are output as the public parameter PK.
Key information generation GenKey(MSK, w→)
Input: Master key information MSK, vector w→
Output: Key information D* corresponding to vector w→
In an example of GenKey(MSK, w→), an element α∈Fq is selected from the finite field Fq. Then, the matrix B*, which is the master key information MSK, is used to generate and output the key information D* corresponding to the vector w→ in the following way.

D*=α·(Σμ=1nwμ·bμ*)+bn+1*∈G2n+1  (45)

If it is difficult to solve a discrete logarithmic problem on the cyclic group G2, it is difficult to separate and extract the components of wμ·bμ* and bn+1* from the key information D*.

Encryption Enc(PA, v→)
Input: Public parameter PK, vector v→
Output: Ciphertext C2, symmetric key K
In an example of Enc(PA, v→), the symmetric key K and a random number υ1 which is an element of the finite field Fq, are generated. Then, the public parameter PK, such as the matrix B, an element υ2 corresponding to a value that includes the symmetric key K, in the finite field Fq, the vector v→, and the random number υ1 are used to generate ciphertext C2 in the following way.

C2=υ1·(Σμ=1nvμ·bμ)+υ2·bn+1∈G1n+1  (46)

The ciphertext C2 and the symmetric key K are output. An example of the symmetric key K is gTτ·υ2∈GT, where υ2 means υ2. An example of τ is 1F, as described above. If it is difficult to solve a discrete logarithmic problem on the cyclic group G1, it is difficult to separate and extract the components of vμ·bμ and υ2·bn+1 from the ciphertext C2.

Decryption and key sharing Dec(SKw, C2)
Input: Key information D1* corresponding to vector w→, ciphertext C2 
Output: Symmetric key K
In an example of Dec(SKw, C2), the ciphertext C2 and the key information D1* are input to the bilinear function e of Expression (2).
Then, from the characteristics of Expressions (3) and (26), the following is satisfied.




   
  
    
      
        
          
            
              
                
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When the inner product w→·v→ is zero, Expression (47) can be deformed to the following form.




   
  
    
      
        
          
            
              
                
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From this result, the symmetric key K is generated and output. An example of the symmetric key K is gTτ·υ2∈GT.
Security Against CCA
The scheme is CCA-secure if the following relation is satisfied on the supposition given below:

Pr[bit=bit′]<(1⁄2)−FNK(k)  (49)

where FNK(k) is a function of k. If [3] follows [2], the scheme is secure against CCA2. If [2] follows [3], the scheme is secure against CCA1. CCA2 is stronger than CCA1.

[1] The attacker is given a public parameter PK.
[2] The attacker passes two pieces of plaintext M0 and M1, which are two bit strings, to an encryption oracle having the public parameter PK. The encryption oracle selects bit∈{0, 1} at random, encrypts either piece of plaintext Mbit, and gives the ciphertext C to the attacker.
[3] The attacker can pass ciphertext C(σ) (C(σ)≠C) to a decryption oracle having key information D* and receive a decrypted result of the ciphertext C(σ) from the decryption oracle.
[4] The attacker outputs bit′∈{0, 1}.
Security against CCA in basic scheme of inner product predicate encryption
The basic scheme of the inner product predicate encryption described above is not CCA-secure. This will be described by showing a simple example. In the simple example, τ=1F; the symmetric key K is K=MAP (gTυ2); and the ciphertext obtained by the symmetric key encryption of the plaintext M with the symmetric key K is C1=K (+) M. MAP (gTυ2) represents mapping of gTυ2∈GT to a bit string. In this example, the attacker can take the following strategy (hereafter called as assumed strategy).
[1] The attacker is given a public parameter PK.
[2] The attacker passes a vector v→ and two pieces of plaintext M0 and M1 to an encryption oracle having the public parameter PK. The encryption oracle selects bit∈{0, 1} at random, generates the ciphertext

C1=K(+)Mbit  (50)

obtained by encrypting either piece of plaintext Mbit with the symmetric key K=gTυ2, generates ciphertext

C2=υ1·(Σμ=1nvμ·bμ)+υ2·bn+1∈G1n+1  (51)

of the symmetric key K=gTυ2, and gives the ciphertext (C1, C2) to the attacker.

[3] The attacker passes the ciphertext

(C1(+)ΔM,C2)  (52)

to a decryption oracle having the key information D* and receives the decrypted results of the ciphertext C1(1) and C1(2) from the decryption oracle.

Here, if bit=0, then C1=K (+) M0, and the decrypted result of C1 (+) ΔM is M0 (+) ΔM. If bit=1, then C1=K (+) M1, and the decrypted result of C1 (+) ΔM is M1 (+) ΔM.
[4] When the decrypted result of C1 (+) ΔM is M0 (+) ΔM, the attacker outputs bit′=0. When the decrypted result is M1 (+) ΔM, the attacker outputs bit′=1.
Then, Pr[bit=bit′]=1, and Expression (49) is not satisfied.
Inner Product Predicate Encryption in Embodiments
As described above, the basic scheme of the inner product predicate encryption is not CCA-secure. If the CHK transformation method or BK transformation method is used to make the basic scheme of inner product predicate encryption CCA-secure, a two-dimensional space used just for improving the level of CCA-security becomes necessary in the ciphertext space, and the number of dimensions of the vector spaces V and V* must be n+3. In the embodiments, the CCA security is improved without providing a space used just for improving the level of CCA security.
Improved Scheme of Inner Product Predicate Encryption
An improved scheme of the inner product predicate encryption in the embodiments will be overviewed below.
Encryption Processing
An encryption apparatus for performing encryption processing executes the following processing.
(ENC-1) A symmetric key encryption unit puts plaintext M and a symmetric key K both of which are read from the storage into a symmetric key encryption function to calculate ciphertext C1 and outputs the ciphertext C1.
(ENC-2) A function operation unit puts a value containing a value corresponding to the symmetric key K and a value corresponding to the ciphertext C1 into the collision-resistant function H to calculate the function value of the function H and outputs the function value of the function H.
(ENC-3) A vector operation unit calculates the ciphertext

C2=r·(Σμ=1nvμ·bμ)+ρ·bn+1∈G1n+1  (53)

by using the element r of the finite field Fq, corresponding to the function value of the function H, the element ρ of the finite field Fq, corresponding to the symmetric key K, the elements vμ (μ=1, . . . , n, n≧1) of the n-dimensional vector v→ (v1, . . . , vn) composed of the elements of the finite field Fq, and the n+1 dimensional basis vectors b1∈G1n+1, and outputs the ciphertext C2.

Decryption Processing
A decryption apparatus for performing decryption processing executes the following processing.
(DEC-1) A value containing the ciphertext C1 and the ciphertext C2∈G1n+1 is input to an input unit.
(DEC-2) A bilinear function operation unit puts the ciphertext C2∈G1n+1 and key information

D1*=α·(Σμ=1nwμ·bμ*)+bn+1*  (54)

into the bilinear function e to calculate the function value

S ̃=e(C2,D1*)∈GT  (55)

and outputs the function value S ̃.

(DEC-3) A function operation unit puts the value corresponding to the function value S ̃ and the ciphertext C1 into the collision-resistant function H to calculate the function value of the function H and outputs the function value of the function H. If the key information D1* is correct, the value corresponding to the function value S ̃ is equal to the value corresponding to the symmetric key K, which is input to the collision-resistant function H in (ENC-2) of the encryption processing.
(DEC-4) A decision unit decides whether the ciphertext C2, the element r ̃ of the finite field Fq, corresponding to the function value of the function H, the element ρ ̃ of the finite field Fq, the elements vμ of the n-dimensional vector v→, and the basis vectors bi∈G1n+1 (i=1, . . . , n+1) satisfy the following relation, and outputs the decision.

C2=r ̃·(Σμ=1nvμ·bμ)+ρ ̃·bn+1∈G1n+1  (56)

(DEC-5) If Expression (56) is satisfied, the decrypted result of the ciphertext C1 is output. If Expression (56) is not satisfied, an error message (⊥) is output.
Security Against CCA in Improved Method of Inner Product Predicate Encryption
A case in which the assumed strategy described above is applied to the improved scheme will be described.
[1] The attacker is given the public parameter PK.
[2] The attacker passes the vector v→ and the two pieces of plaintext M0 and M1 to the encryption oracle having the public parameter PK. The encryption oracle selects bit∈{0, 1} at random, encrypts either piece of plaintext Mbit with a symmetric key K=gTυ2, and generates the ciphertext

C1=K(+)Mbit  (57)

(ENC-1). The encryption oracle further puts the value containing the value corresponding to the symmetric key K and the value corresponding to the ciphertext C1 into the collision-resistant function H to calculate the function value of the function H (ENC-2). Then, the encryption oracle generates the ciphertext

C2=r·(Σμ=1nvμ·bμ)+ρbn+1∈G1n+1  (58)

of the symmetric key K=gTυ2 by using the element r of the finite field Fq, corresponding to the function value of the function H, the element ρ of the finite field Fq, corresponding to the symmetric key K, the elements vμ (μ=1, . . . , n, n≧1) of the n-dimensional vector v→=(v1, . . . , vn) composed of elements of the finite field Fq, and the n+1 dimensional basis vectors bi∈G1n+1 (ENC-3), and gives the ciphertext (C1, C2) to the attacker.

[3] Since the attacker does not know the symmetric key K, the attacker cannot generate C2′=r′·(Σμ=1nvμ·bμ)+ρ·bn+1∈G1n+1 corresponding to an element r′ of the finite field Fq, corresponding to the function value obtained by putting the value containing ciphertext C1 (+) ΔM and the value corresponding to the symmetric key K into the collision-resistant function H. Therefore, the attacker passes the following combination of C1 (+) ΔM and the ciphertext C2 corresponding to the ciphertext C1 to a decryption oracle.

(C1(+)ΔM,C2)  (59)

The decryption oracle to which the ciphertext of Expression (59) has been input (DEC-1) generates the function value S ̃ of Expression (55) (DEC-2), calculates the function value of the collision-resistant function H by putting the ciphertext C1 (+) ΔM and the value corresponding to the function value S ̃ into the function H, and outputs the function value of the function H (DEC-3). The decryption oracle judges whether the second ciphertext C2, an element r ̃ΔM of the finite field Fq, corresponding to the function value of the function H, the element ρ ̃ of the finite field Fq, the elements vμ of the n-dimensional vector v ̃, and the basis vectors bi∈G1n+1 (i=1, . . . , n+1) satisfy the following relation.

C2=r ̃ΔM·(Σμ=1nvμ·bμ)+ρ ̃bn+1∈G1n+1  (60)

(DEC-4). Here, since r ̃ΔM≠r ̃, Expression (60) is not satisfied, and the decryption oracle outputs the error message (⊥).

[4] Since the attacker cannot obtain the decrypted result of C1 (+) ΔM, the attacker is not able to output bit′=0 when the decrypted result is M0 (+) ΔM or to output bit′=1 when the decrypted result is M1 (+) ΔM. Accordingly, the assumed strategy of the attacker will end in failure.
First Embodiment
A first embodiment of the improved scheme will be described below.
Overview
An overview of the first embodiment will be given next.
Encryption Processing
An encryption apparatus for performing encryption processing executes the following processing.
(ENC-11) A symmetric key encryption unit puts the plaintext M and the symmetric key K both of which are read from a storage into the symmetric key encryption function to calculate the ciphertext C1 and outputs the ciphertext C1.
(ENC-21) A function operation unit puts the symmetric key K and the ciphertext C1 into the collision-resistant function H to calculate the elements r and ρ of the finite field Fq and outputs the elements r and ρ. The symmetric key K in (ENC-21) corresponds to the value corresponding to the symmetric key K mentioned earlier in (ENC-2).
(ENC-31) A group operation unit calculates the function value S=gTτ·ρ∈GT by using the element ρ and outputs the function value S.
(ENC-32) An exclusive-OR operation unit calculates the ciphertext C3 that is the exclusive OR value of the symmetric key K and the function value R(S) which is the binary sequence obtained by putting the function value S into the injective function R, and outputs the ciphertext C3.
(ENC-33) A vector operation unit calculates C2=r·(Σμ=1nvμ·bμ)+ρ·bn+1∈G1n+1 by using the elements vμ (μ=1, . . . , n) of the n-dimensional vector v→ composed of the elements of the finite field Fq and the elements r and ρ, and outputs the ciphertext C2.
Decryption Processing
A decryption apparatus for performing decryption processing executes the following processing.
(DEC-11) A value containing the ciphertext C1, the ciphertext C2, and the ciphertext C3 is input to an input unit.
(DEC-21) A bilinear function operation unit puts the ciphertext C2∈G1n+1 and the key information D1*=α·(Σμ=1nwμ·bμ*)+bn+1* into the bilinear function e to calculate the function value S ̃=e(C2, D1*)∈GT and outputs the function value S ̃.
If the inner product satisfies w→·v→=0, the characteristics of Expressions (3) and (26) lead to the following expression.




   
  
    
      
        
          
            
              
                
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(DEC-31) An exclusive-OR operation unit calculates the exclusive OR value K ̃ of the ciphertext C3 and the function value R(S ̃) which is the binary sequence obtained by putting the function value S ̃ into the injective function R, and outputs the exclusive OR value K ̃. If Expression (61) is satisfied, the following relation is satisfied.




   
  
    
      
        
          
            
              
                
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(DEC-32) A function operation unit puts the exclusive OR value K ̃ and the ciphertext C1 into the collision-resistant function H to calculate the elements r ̃ and ρ ̃ of the finite field Fq and outputs the elements r ̃ and ρ ̃. The elements r ̃ and ρ ̃ in (DEC-31) correspond to the value containing the element r ̃ mentioned earlier in (DEC-3).
(DEC-41) A vector operation unit calculates the function value C2 ̃=r ̃·(Σμ=1nvμ·bμ)+ρ ̃·bn+1∈G1n+1 by using the elements vμ (μ=1, . . . , n) of the n-dimensional vector v→ and the elements r ̃ and ρ ̃, and outputs the function value C2 ̃.
(DEC-42) A comparison unit judges whether the function value C2 ̃ equals the ciphertext C2 and outputs the judgment. This judgment corresponds to the judgment in (DEC-4).
(DEC-51) If the judgment output from the comparison unit indicates that the function value C2 ̃ equals the ciphertext C2, a symmetric key decryption unit uses the exclusive OR value K ̃ as the symmetric key, puts the exclusive OR value K ̃ and the ciphertext C1 into the symmetric key decryption function to calculate the decrypted value M ̃, and outputs the decrypted value M ̃. If the judgment output from the comparison unit indicates that the function value C2 ̃ does not equal the ciphertext C2, an error message (⊥) is output.
Security Against CCA in First Embodiment
A case in which the assumed strategy described above is applied to the scheme of the first embodiment will be described. In this example, τ=1F, and the ciphertext obtained by symmetric key encryption of the plaintext M with the symmetric key K is C1=K (+) M.
[1] The attacker is given the public parameter PK.
[2] The attacker passes the vector v→ and two pieces of plaintext M0 and M1 to the encryption oracle having the public parameter PK. The encryption oracle selects bit∈{0, 1} at random, encrypts either piece of plaintext Mbit with the symmetric key K=gTυ2, and generates ciphertext

C1=K(+)Mbit  (63)

(ENC-11). The encryption oracle further puts the symmetric key K and the ciphertext C1 into the collision-resistant function H to calculate the elements r and ρ of the finite field Fq (ENC-21), calculates the function value S=gTτ·ρ∈GT (ENC-31), calculates the ciphertext C3 by obtaining an exclusive OR of the function value R(S) and the symmetric key K (ENC-32), calculates C2=r·(Σμ=1nvμ·bμ)+ρ·bn+1∈G1n+1 (ENC-33), and gives the ciphertext (C1, C2, C3) to the attacker.

[3] Since the attacker does not know the symmetric key K, the attacker cannot generate C2′=r′·(Σμ=1nvμ·bμ)+ρ′·bn+1∈G1n+1 corresponding to the elements r′ and ρ′ of the finite field Fq, obtained by putting the symmetric key K and ciphertext C1 (+) ΔM into the collision-resistant function H. Therefore, the attacker passes the following combination of C1 (+) ΔM, the ciphertext C2 corresponding to the ciphertext C1, and the ciphertext C3 to a decryption oracle.

(C1(+)ΔM,C2,C3)  (64)

The decryption oracle to which the ciphertext of Expression (64) has been input (DEC-11) generates the function value S ̃=e(C2, D1*)∈GT (DEC-21), calculates the exclusive OR value K ̃ of the ciphertext C3 and the function value R(S ̃) which is the binary sequence obtained by putting the function value S ̃ into the injective function R, and calculates the elements r ̃ΔM and ρ ̃ΔM of the finite field Fq by putting the exclusive OR value K ̃ and ciphertext C1 (+) ΔM into the collision-resistant function H (DEC-32). The decryption oracle calculates the following function value C2 ̃ by using the elements vμ of v→, the element r ̃ΔM, the basis vectors bi∈G1n+1 (i=1, . . . , n+1), and the element ρ ̃ΔM (DEC-41).

C2 ̃=r ̃ΔM·(Σμ=1nvμ·bμ)+ρ ̃ΔM·bn+1∈G1n+1  (65)

Then, the decryption oracle judges whether the function value C2 ̃ equals the ciphertext C2 ̃. Here, since r ̃ΔM≠r ̃ and ρ ̃ΔM≠ρ ̃, Expression (65) does not satisfy this relation, and the decryption oracle outputs an error message (⊥).

[4] Since the attacker cannot obtain the decrypted result of C1 (+) ΔM, the attacker is not able to output bit′=0 when the decrypted result is M0 (+) ΔM or to output bit′=1 when the result is M1 (+) ΔM. Accordingly, the strategy of the attacker will end in failure.
Details
The first embodiment will be described next in detail. While τ=1F is satisfied in this example, this will not limit the present invention, and τ may be an element of the finite field Fq other than 0F or 1F.
Overall Structure
FIG. 1 is a view illustrating the structure of a cryptographic system 1 of the first embodiment.
As shown in FIG. 1, the cryptographic system 1 of this embodiment includes an encryption apparatus 110, a decryption apparatus 120, and a key generation apparatus 130, which are connected through a network 140 so that they can communicate with one another.
Encryption Apparatus
FIG. 2 is a view illustrating the structure of the encryption apparatus 110 in FIG. 1. Arrows in FIG. 2 indicate the flow of data, but the arrows, indicating data input to and output from a control unit 111 and a temporary storage 113b, are omitted.
As shown in FIG. 2, the encryption apparatus 110 of this embodiment includes the control unit 111, a random number generator 112a, a symmetric key encryption unit 112b, a function operation unit 112c, a group operation unit 112d, an exclusive-OR operation unit 112e, a vector operation unit 112f, a storage 113a, the temporary storage 113b, and a transmitter 114.
The encryption apparatus 110 is, for example, a special apparatus that includes a special program and a known or specialized computer provided with a central processing unit (CPU), a random access memory (RAM), a read-only memory (ROM), or the like. The control unit 111, the random number generator 112a, the symmetric key encryption unit 112b, the function operation unit 112c, the group operation unit 112d, the exclusive-OR operation unit 112e, and the vector operation unit 112f are processing units implemented by the CPU executing the special program. At least some of the processing units may be a specialized integrated circuit: For example, the random number generator 112a may be a known random number generation integrated circuit. The storage 113a and the temporary storage 113b are, for example, an auxiliary storage such as a RAM, a register, a cache memory, an element in a chip, or a hard disk, or a storage area formed by combining at least some of these devices. The transmitter 114 is a communication device such as a modem or a local area network (LAN) card, for example.
The encryption apparatus 110 executes processing under the control of the control unit 111. Each piece of data output from each processing unit is stored in the temporary storage 113b or the storage 113a, the data stored in the temporary storage 113b or the storage 113a is read, input to a processing unit, and used for processing thereof, when necessary, while these data flow descriptions are simplified below.
Decryption Apparatus
FIG. 3 is a view illustrating the structure of the decryption apparatus 120 in FIG. 1. Arrows in FIG. 3 indicate the flow of data, but the arrows, indicating data input to and output from a control unit 121 and a temporary storage 123b, are omitted.
As shown in FIG. 3, the decryption apparatus 120 of this embodiment includes the control unit 121, a bilinear function operation unit 122a, an exclusive-OR operation unit 122b, a function operation unit 122c, a decision unit 122d, a symmetric key decryption unit 122e, a storage 123a, the temporary storage 123b, a transmitter 124, and a receiver 125 (corresponding to the input unit). The decision unit 122d includes a vector operation unit 122da and a comparison unit 122db. 
The decryption apparatus 120 is, for example, a special apparatus that includes a special program and a known or specialized computer provided with a CPU, a RAM, a ROM, or the like. The control unit 121, the bilinear function operation unit 122a, the exclusive-OR operation unit 122b, the function operation unit 122c, the decision unit 122d, and the symmetric key decryption unit 122e are processing units implemented by the CPU executing the special program, for example. At least some of the processing units may be a specialized integrated circuit: For example, the bilinear function operation unit 122a may be a known pairing computation integrated circuit. The storage 123a and the temporary storage 123b are, for example, an auxiliary storage such as a RAM, a register, a cache memory, an element in a chip, or a hard disk, or a storage area formed by combining at least some of these devices. The transmitter 124 and the receiver 125 are a communication device such as a modem or a LAN card, for example.
The decryption apparatus 120 executes processing under the control of the control unit 121. Each piece of data output from each processing unit is stored in the temporary storage 123b or the storage 123a, the data stored in the temporary storage 123b or the storage 123a is read, input to a processing unit, and used for processing thereof, when necessary, while these data flow descriptions are simplified below.
Key Generation Apparatus
FIG. 4 is a view illustrating the structure of the key generation apparatus 130 in FIG. 1. Arrows in FIG. 4 indicate the flow of data, but the arrows, indicating data input to and output from a control unit 131 and a temporary storage 133b, are omitted.
As shown in FIG. 4, the key generation apparatus 130 of this embodiment includes the control unit 131, a random number generator 132a, a vector operation unit 132b, a storage 133a, a temporary storage 133b, a receiver 134, and a transmitter 135.
The key generation apparatus 130 is, for example, a special apparatus that includes a special program and a known or specialized computer provided with a CPU, a RAM, a ROM, or the like. The control unit 131, the random number generator 132a, and the vector operation unit 132b are processing units implemented by the CPU executing the special program, for example. At least some of the processing units may be a specialized integrated circuit: The storage 133a and the temporary storage 133b are, for example, an auxiliary storage such as a RAM, a register, a cache memory, an element in an integrated circuit, or a hard disk, or a storage area formed by combining at least some of these devices. The transmitter 134 and the receiver 135 are a communication device such as a modem or a LAN card, for example.
The key generation apparatus 130 executes processing under the control of the control unit 131. The flow of data will be described just briefly below. Each piece of data output from each processing unit is stored in the temporary storage 133b or the storage 133a, the data stored in the temporary storage 133b or the storage 133a is read, input to a processing unit, and used for processing thereof, when necessary, while these data flow descriptions are simplified below.
Preparatory Settings
Preparatory settings for executing the processing in this embodiment will be described next.
A public parameter PK is specified in the encryption apparatus 110, the decryption apparatus 120, and the key generation apparatus 130. The public parameter PK includes the vector spaces V and V*, the (n+1) row by (n+1) column matrix B having the basis vectors bi (i=1, . . . , n+1) as its elements, the security parameter k, the finite field Fq, the elliptic curve E, the cyclic groups G1, G2, and GT, the generators g1, g2, and gT, the bilinear function e, and the integer n greater than or equal to 1. The collision-resistant function H1 shown in Expression (33) and the injective function R shown in Expression (35) are specified in the encryption apparatus 110 and the decryption apparatus 120.
The (n+1) row by (n+1) column matrix B*, which is the master key information MSK, is stored securely in the storage 133a of the key generation apparatus 130. The matrix B* is secret information and is kept secret.
The n-dimensional vector v→=(v1, . . . , vn) composed of the elements of the finite field Fq, shown in Expression (31), is stored in the storage 113a of the encryption apparatus 110, and the n-dimensional vector v→=(v1, . . . , vn) and the n-dimensional vector w→=(w1, . . . , wn) composed of the elements of the finite field Fq, shown in Expression (30), are stored in the storage 123a of the decryption apparatus 120. These vectors satisfy following relation.

w→·v→=0  (66)

The predicate information and the attribute information may be w→=(w1, . . . , wn) and v→=(v1, . . . , vn), respectively. Alternatively, the predicate information and the attribute information may be v→=(v1, . . . , vn) and w→=(w1, . . . , wn), respectively.

The storage 113a of the encryption apparatus 110 stores the plaintext M∈{0, 1}* to be encrypted.
Encryption Processing
FIG. 5 is a view illustrating encryption processing in the first embodiment. The encryption processing in this embodiment will be described next.
The random number generator 112a in the encryption apparatus 110 (FIG. 2) generates the k-bit binary-sequence random number, and that is stored in the storage 113a as the following symmetric key (step S101).

K∈{0,1}k  (67)

The plaintext M and symmetric key K read from the storage are input to the symmetric key encryption unit 112b. The symmetric key encryption unit 112b puts them into the symmetric key encryption function Enc to obtain the following ciphertext C1, and outputs the ciphertext C1 (step S102).

C1=Enck(M)∈{0,1}k  (68)

The ciphertext C1 and the symmetric key K are input to the function operation unit 112c. The function operation unit 112c puts the symmetric key K and ciphertext C1 into the collision-resistant function H1 to obtain the following elements r and ρ of the finite field Fq, and outputs the elements r and ρ (step S103).

(r,ρ)=H1(K,C1)∈Fq×Fq  (69)

The element ρ of the finite field Fq is input to the group operation unit 112d. The group operation unit 112d calculates the following function value S by using the element ρ, and outputs the function value S (step S104).

S=gTρ∈GT  (70)

The function value S and the symmetric key K are input to the exclusive-OR operation unit 112e. The exclusive-OR operation unit 112e uses the symmetric key K and the function value

R(S)∈{0,1}k  (71)

which is the k-bit binary sequence obtained by putting the function value S into the injective function R, to obtain an exclusive-OR value of them as the ciphertext

C3=R(S)(+)K∈{0,1}k  (72)

and outputs the ciphertext C3 (step S105).

The elements r and ρ of the finite field Fq and the n-dimensional vector v→=(v1, . . . , vn) are input to the vector operation unit 112f. The vector operation unit 112f calculates ciphertext

C2=r·(Σμ=1nvμ·bμ)+ρ·bn+1∈G1n+1  (73)

by using the elements vμ (μ=1, . . . , n) of the n-dimensional vector v→ and the elements r and ρ, and outputs the ciphertext C2 (step S106).

The ciphertext C1, ciphertext C2, and ciphertext C3 are input to the transmitter 114. The transmitter 114 sends a value containing them through the network 140 to the decryption apparatus 120 (step S107).
Key Information Generation Processing
Key information generation processing is performed especially when the key information D1* is not stored in the storage 123a of the decryption apparatus 120. When the key information D1* is stored in the storage 123a of the decryption apparatus 120, this processing may be omitted.
FIG. 6 is a view illustrating the key information generation processing in the first embodiment. The key information generation processing in this embodiment will be described next.
The n-dimensional vector w→=(w1, . . . , wn) stored in the decryption apparatus 120 (FIG. 3) is read and input to the transmitter 124. The transmitter 124 sends the n-dimensional vector w→=(w1, . . . , wn) through the network 140 to the key generation apparatus 130 (step S111).
The sent n-dimensional vector w→=(w1, . . . , wn) is received by the receiver 134 of the key generation apparatus 130 (FIG. 4) (step S112). This causes the random number generator 132a of the key generation apparatus 130 to generate a random number α which is an element of the finite field Fq and to output the random number α (step S113).

α∈Fq  (74)

The n-dimensional vector w→=(w1, . . . , wn), the random number α, and the matrix B* are input to the vector operation unit 132b. By using them, the vector operation unit 132b generates the following key information D1*, and outputs the key information D1* (step S114).

D1*=α·(Σμ=1nwμ·bμ*)+bn+1*∈G1n+1  (75)

The key information D1* is input to the transmitter 135. The transmitter 135 sends the key information D1* through the network 140 to the decryption apparatus 120 (step S115).
The sent key information D1* is received by the receiver 125 of the decryption apparatus 120 (FIG. 3) and stored in the storage 123a (step S116).
Decryption Processing
FIG. 7 is a view illustrating decryption processing in the first embodiment. The decryption processing in this embodiment will be described next.
The receiver 125 of the decryption apparatus 120 (FIG. 3) receives the value containing the ciphertext C1, the ciphertext C2∈G1n+1, and the ciphertext C3 (step S121).
The ciphertext C2∈G1n+1 and the key information D1* are input to the bilinear function operation unit 122a. The bilinear function operation unit 122a puts the ciphertext C2∈G1n+1 and the key information D1* into the bilinear function e to calculate the following function value S ̃, and outputs the function value S ̃ (step S122).

S ̃=e(C2,D1*)∈GT  (76)

The function value S ̃ and the ciphertext C3 are input to the exclusive-OR operation unit 122b. The exclusive-OR operation unit 122b calculates the following exclusive OR value K ̃ of the ciphertext C3 and the function value R(S ̃) which is the binary sequence obtained by putting the function value S ̃ into the injective function R, and outputs the exclusive OR value K ̃ (step S123).

K ̃=R(S ̃)(+)C3  (77)

The exclusive OR value K ̃ and the ciphertext C1 are input to the function operation unit 122c. The function operation unit 122c puts the exclusive OR value K ̃ and the ciphertext C1 into the collision-resistant function H1, calculates the following elements r ̃ and ρ ̃, and outputs the elements r ̃ and ρ ̃ (step S124).

(r ̃,ρ ̃)=H1(K ̃,C1)∈Fq×Fq  (78)

The elements vμ (μ=1, . . . , n) of the n-dimensional vector v→ and the elements r ̃ and ρ ̃ are input to the vector operation unit 122da of the decision unit 122d. By using them, the vector operation unit 122da calculates the following function value C2 ̃, and outputs the function value C2 ̃ (step S125).

C2 ̃=r ̃·(Σμ=1nvμ·bμ)+ρ ̃·bn+1∈G1n+1  (79)

The function value C2 ̃ and the ciphertext C2 are input to the comparison unit 122 db of the decision unit 122d. The comparison unit 122 db judges whether the function value C2 ̃ equals the ciphertext C2 and outputs the judgment (step S126).
If the judgment output from the comparison unit 122 db indicates that the function value C2 ̃ equals the ciphertext C2, the exclusive OR value K ̃ and the ciphertext C1 are input to the symmetric key decryption unit 122e, and the symmetric key decryption unit 122e puts the exclusive OR value K ̃ and the ciphertext C1 into the symmetric key decryption function to obtain the following decrypted value M ̃, and outputs the decrypted value M ̃ (step S127).

M ̃=DecK ̃(C1)  (80)

If the judgment output from the comparison unit 122 db indicates that the function value C2 ̃ differs from the ciphertext C2, the symmetric key decryption unit 122e outputs an error message (⊥) (step S128).

Second Embodiment
A second embodiment of the improved scheme described above will be described next.
Overview
An overview of the second embodiment will be given first.
Encryption Processing
An encryption apparatus for performing encryption processing executes the following processing.
(ENC-111) A selection unit selects an element ρ of the finite field Fq and outputs the element ρ.
(ENC-112) A group operation unit calculates the function value S=gTτ·ρ∈GT by using the element ρ and outputs the function value S.
(ENC-113) A symmetric key encryption unit uses, as a symmetric key K, the function value R(S) obtained by putting the function value S into the injective function R, puts the symmetric key K and plaintext M into the symmetric key encryption function to obtain the ciphertext C1, and outputs the ciphertext C1.
(ENC-211) A function operation unit puts the function value S and the ciphertext C1 into the collision-resistant function H to obtain the element r of the finite field Fq and outputs the element r.
(ENC-311) A vector operation unit calculates the ciphertext C2=r·(Σμ=1nvμ·bμ)+ρ·bn+1∈G1n+1 by using the elements r and ρ and the elements vμ (μ=1, . . . , n) of the n-dimensional vector v→ composed of the elements of the finite field Fq, and outputs the ciphertext C2.
Decryption Processing
In a decryption apparatus for performing decryption processing, the following processing is executed.
(DEC-101) A value containing the ciphertext C1 and the ciphertext C2∈G1n+1 is input to an input unit.
(DEC-201) A bilinear function operation unit puts the ciphertext C2∈G1n+1 and the key information D1*=α·(Σμ=1nwμ·bμ*)+bn+1* into the bilinear function e to obtain the function value S ̃=e(C2, D1*)∈GT and outputs the function value S ̃. If the inner product w→·v→=0, S ̃=S is satisfied because of the characteristics of Expressions (3) and (26).
(DEC-301) A function operation unit puts the function value S ̃ and the ciphertext C1 into the collision-resistant function H to calculate the element r ̃ of the finite field Fq and outputs the element r ̃. If S ̃=S is satisfied and if the ciphertext C1 has been generated authentically, r ̃=r is satisfied.
(DEC-401) A vector operation unit calculates the function value λ=C2−r ̃·(Σμ=1nvμ·bμ)∈G1n+1 by using the ciphertext C2, the element r ̃, and the elements vμ (μ=1, . . . , n) of the n-dimensional vector v→, and outputs the function value λ. If r ̃=r is satisfied, λ=ρ·bn+1∈G1n+1 is satisfied.
(DEC-402) The bilinear function operation unit calculates the function value e(λ, D2*) by putting the function value λ and the auxiliary key information D2*=Σμ=1n·βμ·bμ* into the bilinear function e, and outputs the function value e(λ, D2*). If λ=ρ·bn+1∈G1n+1 is satisfied, e(λ, D2*)=1∈GT is satisfied because of the characteristic of Expression (3).
(DEC-403) A comparison unit judges whether the function value e(λ, D2*) satisfies e(λ, D2*)=1∈GT and outputs the judgment. In this embodiment, whether Expression (56) is satisfied is determined by judging whether the function value e(λ, D2*) satisfies e(λ, D2*)=1∈GT. This judgment corresponds to the judgment in (DEC-4).
(DEC-501) If the judgment output from the comparison unit indicates that e(λ, D2*)=1∈GT is satisfied, a symmetric key decryption unit uses, as the symmetric key K ̃, the function value R(S) obtained by putting the function value S ̃ into the injective function R, puts the symmetric key K ̃ and the first ciphertext C1 into the symmetric key decryption function to obtain the decrypted value M ̃, and outputs the decrypted value M ̃. If the judgment output from the comparison unit indicates that e(λ, D2*)=1∈GT is not satisfied, an error message (⊥) is output.
Security Against CCA in Second Embodiment
A case in which the assumed strategy described above is applied to the scheme of the second embodiment will be described. In this example, τ=1F, and the ciphertext obtained by symmetric key encryption of the plaintext M with the symmetric key K is C1=K (+) M.
[1] An attacker is given the public parameter PK.
[2] The attacker passes the vector v→ and two pieces of plaintext M0 and M1 to the encryption oracle having the public parameter PK. The encryption oracle selects an element ρ of the finite field Fq (ENC-111), calculates the function value S=gTτ·ρ∈GT (ENC-112), uses a function value R(S) as the symmetric key K (ENC-113), and generates the following ciphertext (ENC-113).

C1=K(+)Mbit  (81)

The encryption oracle further puts the function value S and the ciphertext C1 into the collision-resistant function H to obtain the element r of the finite field Fq (ENC-211). Then, the encryption oracle generates the ciphertext C2=r·(Σμ=1nvμ·bμ)+ρ·bn+1∈G1n+1 by using the elements r and ρ and the elements vμ (μ=1, . . . , n) of the n-dimensional vector v→ composed of the elements of the finite field Fq (ENC-311) and gives the two pieces of ciphertext (C1, C2) to the attacker.

[3] Since the attacker does not know the symmetric key K=R(S) and/or the function value S (value corresponding to the symmetric key K), the attacker cannot generate C2′=r′·(Σμ=1nvμ·bμ)+ρ·bn+1∈G1n+1 corresponding to the element r′ of the finite field Fq, obtained by putting the function value S and the ciphertext C1 (+) ΔM into the collision-resistant function H. Therefore, the attacker passes the following combination of C1 (+) ΔM and ciphertext C2 corresponding to the ciphertext C1, to a decryption oracle.

(C1(+)ΔM,C2)  (82)

The decryption oracle to which the ciphertext of Expression (82) has been input (DEC-101) calculates the function value S ̃=e(C2, D1*)∈GT (DEC-201), calculates the element r ̃ΔM of the finite field Fq by putting the function value S ̃ and the ciphertext C1 (+) ΔM into the collision-resistant function H (DEC-301), and obtains the function value λΔM=C2−r ̃ΔM·(Σμ=1nvμ·bμ)∈G1n+1 (DEC-401). Because r ̃ΔM≠r is satisfied, λΔM=(r−r ̃ΔM)·(Σμ=1nvμ·bμ)+ρ·bn+1≠ρ·bn+1 is satisfied. The decryption oracle calculates a function value e(λΔM, D2*) (DEC-402) and judges whether a function value e(λ, D2*) satisfies e(λ, D2*)=1∈GT (DEC-403). Here, since λΔM≠ρ·bn+1 is satisfied, e(λ, D2*)=1∈GT is not satisfied (DEC-403). Therefore, the decryption oracle outputs an error message (⊥).
[4] Since the attacker cannot obtain the decrypted result of C1 (+) ΔM, the attacker is not able to output bit′=0 when the decrypted result is M0 (+) ΔM or to output bit′=1 when the decrypted result is M1 (+) ΔM. Accordingly, the strategy of the attacker will end in failure.
Details
The second embodiment will be described next in detail. While τ=1F is satisfied in this example, this will not limit the present invention, and τ may be an element of the finite field Fq other than 0F or 1F. Each component of the second embodiment identical to that of the first embodiment will be described by using the same reference numeral as used in the first embodiment.
Overall Structure
FIG. 8 is a view illustrating the structure of a cryptographic system of the second embodiment.
As shown in FIG. 8, the cryptographic system 2 of this embodiment includes an encryption apparatus 210, a decryption apparatus 220, and a key generation apparatus 230, which are connected through a network 140 so that they can communicate with one another.
Encryption Apparatus
FIG. 9 is a view illustrating the structure of the encryption apparatus 210 in FIG. 8. Arrows in FIG. 9 indicate the flow of data, but the arrows, indicating data input to and output from a control unit 111 and a temporary storage 113b, are omitted.
As shown in FIG. 9, the encryption apparatus 210 of this embodiment includes the control unit 111, a selection unit 212a, a group operation unit 112d, a function operation unit 212c, a symmetric key generation unit 212e, a symmetric key encryption unit 112b, a vector operation unit 112f, a storage 113a, the temporary storage 113b, and a transmitter 114.
The encryption apparatus 210 is, for example, a special apparatus that includes a special program and a known or specialized computer provided with a CPU, a RAM, a ROM, or the like. The control unit 111, a random number generator 212a (corresponding to the selection unit), the group operation unit 112d, the function operation unit 212c, the symmetric key generation unit 212e, the symmetric key encryption unit 112b, and the vector operation unit 112f are processing units implemented by the CPU executing the special program, for example. At least some of the processing units may be a specialized integrated circuit: For example, the random number generator 212a may be a known random number generation integrated circuit.
The encryption apparatus 210 executes processing under the control of the control unit 111. Each piece of data output from each processing unit is stored in the temporary storage 113b or the storage 113a, the data stored in the temporary storage 113b or the storage 113a is input to a processing unit and used for processing thereof, when necessary, while these data flow descriptions are simplified below.
Decryption Apparatus
FIG. 10 is a view illustrating the structure of the decryption apparatus 220 in FIG. 8. Arrows in FIG. 10 indicate the flow of data, but the arrows, indicating data input to and output from a control unit 121 and a temporary storage 123b, are omitted.
As shown in FIG. 10, the decryption apparatus 220 in this embodiment includes the control unit 121, a bilinear function operation unit 122a, function operation units 222b and 222c, a decision unit 222d, a symmetric key decryption unit 122e, a storage 123a, the temporary storage 123b, a transmitter 124, and a receiver 125 (corresponding to the input unit). The decision unit 222d includes a vector operation unit 222da, a bilinear function operation unit 222 db, and a comparison unit 222dc. 
The decryption apparatus 220 is, for example, a special apparatus that includes a special program and a known or specialized computer provided with a CPU, a RAM, a ROM, or the like. The control unit 121, the bilinear function operation unit 122a, the function operation units 222b and 222c, the decision unit 222d, and the symmetric key decryption unit 122e are processing units implemented by the CPU executing the special program, for example. At least some of the processing units may be a specialized integrated circuit.
The decryption apparatus 220 executes processing under the control of the control unit 121. Each piece of data output from each processing unit is stored in the temporary storage 123b or the storage 123a, the data stored in the temporary storage 123b or the storage 123a is read, input to a processing unit, and used for processing thereof, when necessary, while these data flow descriptions are simplified below.
Key Generation Apparatus
FIG. 11 is a view illustrating the structure of the key generation apparatus 230 in FIG. 8. Arrows in FIG. 11 indicate the flow of data, but the arrows, indicating data input to and output from a control unit 131 and a temporary storage 133b, are omitted.
As shown in FIG. 11, the key generation apparatus 230 of this embodiment includes the control unit 131, a random number generator 232a, vector operation units 132b and 232c, a storage 133a, the temporary storage 133b, a receiver 134, and a transmitter 135.
The key generation apparatus 230 is, for example, a special apparatus that includes a special program and a known or specialized computer provided with a CPU, a RAM, a ROM, or the like. The control unit 131, the random number generator 232a, and the vector operation units 132b and 232c are processing units implemented by the CPU executing the special program, for example. At least some of the processing units may be a specialized integrated circuit.
The key generation apparatus 230 executes processing under the control of the control unit 131. Each piece of data output from each processing unit is stored in the temporary storage 133b or the storage 133a, the data stored in the temporary storage 133b or the storage 133a is read, input to a processing unit, and used for processing thereof, when necessary, while these data flow descriptions are simplified below.
Preparatory Settings
In the preparatory settings of the second embodiment, the public parameter PK, as in the first embodiment, is specified in the encryption apparatus 210, the decryption apparatus 220, and the key generation apparatus 230. The collision-resistant function H2 shown in Expression (34) is specified in the encryption apparatus 210 and the decryption apparatus 220.
The (n+1) row by (n+1) column matrix B*, which is the master key information MSK, is stored securely in the storage 133a of the key generation apparatus 230. The matrix B* is secret information and is kept secret.
The n-dimensional vector v→=(v1, . . . , vn) composed of elements of the finite field Fq, shown in Expression (31), is stored in the storage 113a of the encryption apparatus 110, and the n-dimensional vector v→=(v1, . . . , vn) and the n-dimensional vector w→=(w1, . . . , wn) composed of the elements of the finite field Fq, shown in Expression (30), are stored in the storage 123a of the decryption apparatus 120. These vectors satisfy Expression (66). The predicate information and the attribute information may be w→=(wl, . . . , wn) and v→=(v1, . . . , vn), respectively. Alternatively, the predicate information and the attribute information may be v→=(v1, . . . , vn) and w→=(w1, . . . , wn), respectively.
The storage 113a of the encryption apparatus 210 stores the plaintext M∈{0, 1}* to be encrypted.
Encryption Processing
FIG. 12 is a view illustrating encryption processing in the second embodiment. The encryption processing in this embodiment will be described next.
The random number generator 212a of the encryption apparatus 210 (FIG. 9) first selects an element

ρ∈Fq  (83)

of the finite field Fq at random, and outputs the element ρ (step S201).

The element ρ is input to the group operation unit 112d. By using the element ρ, the group operation unit 112d calculates the following function value S, and outputs the function value S (step S202).

S·gTρ∈GT  (84)

The function value S is input to the symmetric key generation unit 212e. The symmetric key generation unit 212e puts the function value S into the injective function R to calculate the following symmetric key K, and stores the symmetric key K in the storage 113a (step S203).

K=R(S)∈{0,1}k  (85)

The symmetric key K and plaintext M are input to the symmetric key encryption unit 112b. The symmetric key encryption unit 112b puts the symmetric key K and the plaintext M into a symmetric key encryption function Enc to calculate the following ciphertext C1, and outputs the ciphertext C1 (step S204).

C1=Enck(M)∈{0,1}k  (86)

The ciphertext C1 and the function value S are input to the function operation unit 212c. The function operation unit 212c puts the function value S and the first ciphertext C1 into the collision-resistant function H2 to calculate the following element r of the finite field Fq, and outputs the element r (step S205).

r=H2(S,C1)∈Fq  (87)

The elements r and ρ and the elements v1(μ=1, . . . , n) of the n-dimensional vector v→ are input to the vector operation unit 112f. The vector operation unit 112f calculates the following ciphertext C2 by using the elements vμ (μ=1, . . . , n) of the n-dimensional vector v→ and the elements r and ρ, and outputs the ciphertext C2 (step S206).

C2=r·(Σμ=1nvμ·bμ)+ρ·bn+1∈G1n+1  (88)

The ciphertext C1 and the ciphertext C2 are input to the transmitter 114. The transmitter 114 sends a value containing them through the network 140 to the decryption apparatus 120 (step S207).
Key Information Generation Processing
Key information generation processing is performed especially when neither key information D1* nor auxiliary key information D2* is stored in the storage 123a of the decryption apparatus 220. When the key information D1* and the auxiliary key information D2* are stored in the storage 123a of the decryption apparatus 220, this processing may be omitted.
FIG. 13 is a view illustrating the key information generation processing in the second embodiment. The key information generation processing in this embodiment will be described next.
The n-dimensional vector w→=(w1, . . . , wn) stored in the decryption apparatus 220 (FIG. 10) is read and input to the transmitter 124. The transmitter 124 sends the n-dimensional vector w→=(w1, . . . , wn) through the network 140 to the key generation apparatus 230 (step S111).
The sent n-dimensional vector w→=(w1, . . . , wn) is received by the receiver 134 of the key generation apparatus 230 (FIG. 11) (step S112). This causes the random number generator 232a and the vector operation unit 132b of the key generation apparatus 230 to generate the key information D1* shown in Expression (75) by performing processing in steps S113 and S114 of the first embodiment and to output the key information D1*.
The random number generator 232a generates a random number

βμ(μ=1, . . . ,n)∈Fq  (89)

of an element of the finite field Fq and outputs the random number βμ (step S215).

The random number βμ and the matrix B* are input to the vector operation unit 232c. By using them, the vector operation unit 232b generates the following auxiliary key information D2*, and outputs the auxiliary key information D2* (step S216).

D2*=Σμ=1nβμ·bμ*∈G2n  (90)

The key information D1* and the auxiliary key information D2* are input to the transmitter 135. The transmitter 135 sends the key information D1* and the auxiliary key information D2* through the network 140 to the decryption apparatus 220 (step S217).
The key information D1* and the auxiliary key information D2* sent are received by the receiver 125 of the decryption apparatus 220 (FIG. 10) and stored in the storage 123a (step S218).
Decryption Processing
FIG. 14 is a view illustrating decryption processing in the second embodiment. The decryption processing in this embodiment will be described next.
The receiver 125 of the decryption apparatus 220 (FIG. 10) receives the value containing ciphertext C1 and ciphertext C2∈G1n+1 (step S221).
The ciphertext C2∈G1n+1 and the key information D1* are input to the bilinear function operation unit 122a. The bilinear function operation unit 122a puts the ciphertext C2∈G1n+1 and the key information D1* into a bilinear function e to calculate the following function value S ̃, and outputs the function value S ̃ (step S222).

S ̃=e(C2,D1*)∈GT  (91)

The function value S ̃ and the ciphertext C1 are input to the function operation unit 222c. The function operation unit 222c calculates the following element r ̃ of the finite field Fq by putting the function value S ̃ and the first ciphertext C1 into the collision-resistant function H2 and outputs the element r ̃ (step S223).

r ̃=H2(S ̃,C1)∈Fq  (92)

The ciphertext C2, the element r ̃, and the elements vμ (μ=1, . . . , n) of the n-dimensional vector v→ are input to the vector operation unit 222da of the decision unit 222d. By using the ciphertext C2, the element r ̃, and the elements vμ (μ=1, . . . , n) of the n-dimensional vector v→, the vector operation unit 222da calculates the following function value λ, and outputs the function value λ (step S224).

λ=C2−r ̃·(Σμ=1nvμ·bμ)∈G1n+1  (93)

The function value λ and the auxiliary key information D2* is input to the bilinear function operation unit 222db of the decision unit 222d. The bilinear function operation unit 222db puts the function value λ and the auxiliary key information D2* into the bilinear function e to calculate the following function value e(λ, D2*), and outputs the function value e(λ, D2*) (step S225).

e(λ,D2*)∈GT  (94)

The function value e(λ, D2*) is input to the comparison unit 222dc of the decision unit 222d. The comparison unit 222dc judges whether the function value e(λ, D2*) satisfies the following relation, and outputs the judgment (step S226).

e(λ,D2*)=1∈GT  (95)

If the judgment output from the comparison unit 222dc indicates that e(λ, D2*)=1∈GT is satisfied, the function value S ̃ is input to the function operation unit 222b. The function operation unit 222b puts the function value S ̃ into the injective function R to obtain the following symmetric key K ̃, and outputs the symmetric key K ̃ (step S227).

K ̃=R(S ̃)  (96)

The symmetric key K ̃ and the ciphertext C1 are input to the symmetric key decryption unit 222b, and the symmetric key decryption unit 222b puts the symmetric key K ̃ and the ciphertext C1 into the symmetric key decryption function Dec to calculate the following decrypted value M ̃, and outputs the decrypted value M ̃ (step S228).

M ̃=DecK ̃(C1)  (97)

If the judgment output from the comparison unit 222dc indicates that e(λ, D2*)=1∈GT is not satisfied, the symmetric key decryption unit 122e outputs an error message (⊥) (step S229).
Modifications and Others
The present invention is not limited to the embodiments described above. For example, although the symmetric key K in step S101 of the first embodiment is a binary-sequence random number, the symmetric key K may be a value selected from a plurality of predetermined binary sequences in accordance with a rule determined. Although α in step S113 of the first embodiment is a random number which is the element of the finite field Fq, α may be selected from the elements of the finite field Fq in accordance with a rule determined. Values generated from the other random numbers can be modified similarly.
The function values obtained by Expression (69) are used directly as r and ρ in the first embodiment. Function values obtained by putting the function values obtained by Expression (69) into an additional function may be used as r and ρ. In that case, function values obtained by putting the function values obtained by Expression (78) into the additional function are used as r ̃ and ρ ̃. The function value obtained by Expression (87) is used directly as r in the second embodiment. A function value obtained by putting the function value obtained by Expression (87) into an additional function may be used as r. In that case, a function value obtained by putting the function value obtained by Expression (92) into the additional function is used as r ̃.
In the decryption processing in the first and second embodiments, an error message is output when the judgment condition in step S126 or S226 is not satisfied. When the judgment condition is not satisfied, the decryption apparatus may output a random number unrelated to the decrypted result or may output nothing.
The various types of processing described above need not always be executed in the order in which it has been described and may be performed in parallel or independently in accordance with the processing capabilities of the units executing the processing or when needed. For example, the processing in steps S105 and S106 in the first embodiment may be executed in parallel, and the processing in step S105 may be executed after the processing in step S106.
Each operation defined on the finite field Fq may be replaced with an operation defined on a finite ring Zq of order q. An example of replacing each operation defined on the finite field Fq with an operation defined on the finite ring Zq is a method of permitting q other than a prime number or a power thereof. It is of course possible that modifications can be added within the scope of the present invention.
When the above described structure is implemented by a computer, the processing details of the functions that should be provided by each apparatus are described in a program. When the program is executed by a computer, the processing functions described above are implemented on the computer.
The program containing the processing details can be recorded in a computer-readable storage medium. The computer-readable storage medium can be any type of medium, such as a magnetic storage device, an optical disc, a magneto-optical storage medium, and a semiconductor memory.
The program is distributed by selling, transferring, or lending a portable recording medium such as a DVD or a CD-ROM with the program recorded on it, for example. The program may also be distributed by storing the program in a storage unit of a server computer and transferring the program from the server computer to another computer through the network.
A computer that executes this type of program first stores the program recorded on the portable recording medium or the program transferred from the server computer in its storage unit. Then, the computer reads the program stored in its storage unit and executes processing in accordance with the read program. In a different program execution form, the computer may read the program directly from the portable recording medium and execute processing in accordance with the program, or the computer may execute processing in accordance with the program each time the computer receives the program transferred from the server computer. Alternatively, the processing may be executed by a so-called application service provider (ASP) service, in which the processing function is implemented just by giving a program execution instruction and obtaining the results without transferring the program from the server computer to the computer.
DESCRIPTION OF REFERENCE NUMERALS


    1, 2: Cryptographic systems
    110, 210: Encryption apparatuses
    120, 220: Decryption apparatuses

## Claims

1. An encryption apparatus comprising:
a symmetric key encryption unit adapted to calculate a ciphertext C1 by putting a plaintext M and a symmetric key K into a symmetric key encryption function and to output the ciphertext C1;
a function operation unit adapted to calculate a function value of a collision-resistant function H by putting a value containing a value corresponding to the symmetric key K and a value corresponding to the ciphertext C1 into the function H; and
a vector operation unit adapted to calculate a second ciphertext C2=r·(Σμ=1nvμ·bμ)+ρ·bn+1∈G1n+1 by using a value r corresponding to the function value of the function H, a value ρ corresponding to the symmetric key K, elements vμ (μ=1, . . . , n) (n≧1) of an n-dimensional vector v→=(v1, . . . , vn), and n+1 dimensional basis vectors bi∈G1n+1 (i=1, . . . , n+1) composed of n+1 elements of a cyclic group G1 and to output the second ciphertext C2.

2. The encryption apparatus according to claim 1, wherein the symmetric key K is a binary sequence; and
the function operation unit is adapted to calculate the values r and ρ by putting the symmetric key K and the ciphertext C1 into the collision-resistant function H;
the encryption apparatus further comprising:
a group operation unit adapted to calculate a function value S=gTτ·ρ∈GT by using the value ρ, where gT is a generator of a cyclic group GT and τ is a constant other than an additive identity element; and
an exclusive-OR operation unit adapted to calculate a third ciphertext C3 and to output the third ciphertext C3, the third ciphertext C3 being an exclusive OR of a function value R(S) and the symmetric key K, the function value R(S) being a binary sequence obtained by putting the function value S into an injective function R.

3. The encryption apparatus according to claim 1, further comprising:
a selection unit adapted to select the value ρ;
a group operation unit adapted to calculate a function value S=gTτ·ρ∈GT by using the value ρ, where gT is a generator of a cyclic group GT and τ is a constant other than an additive identity element; and
a symmetric key generation unit adapted to calculate a function value R(S) as the symmetric key K by putting the function value S into an injective function R, wherein
the function operation unit is adapted to calculate the value r by putting the function value S and the ciphertext C1 into the collision-resistant function H.

4. The encryption apparatus according to claim 2, wherein the injective function R is a quasi random function.

5. The encryption apparatus according to claim 3, wherein the injective function R is a quasi random function.

6. The encryption apparatus according to claim 1, wherein the collision-resistant function H is a quasi random function.

7. The encryption apparatus according to claim 2, wherein the collision-resistant function H is a quasi random function.

8. The encryption apparatus according to claim 3, wherein the collision-resistant function H is a quasi random function.

9. The encryption apparatus according to claim 2, wherein the cyclic group G1 and the cyclic group GT are sets in which a non-degenerate bilinear function e is defined, the non-degenerate bilinear function e outputting a single element of the cyclic group GT in response to input of n+1 elements γL (L=1, . . . , n+1) of the cyclic group G1 and n+1 elements γL* (L=1, . . . , n+1) of a cyclic group G2.

10. The encryption apparatus according to claim 9, wherein the values r and ρ and the constant τ are elements of a finite field Fq; and
the order of each of the cyclic groups G1 and G2 equals the order q (q≧1) of the finite field Fq.

11. The encryption apparatus according to claim 9 or 10, wherein the bilinear function e outputs an element e=Πi=1n+1Pair(γL, γL*) of the cyclic group GT in response to the input of the n+1 elements γL (L=1, . . . , n+1) of the cyclic group G1 and the n+1 elements γL* (L=1, . . . , n+1) of the cyclic group G2, where Pair is a non-degenerate bilinear function that translates a combination of a single element of the cyclic group G1 and a single element of the cyclic group G2 to a single element of the cyclic group GT.

12. The encryption apparatus according to claim 3, wherein the cyclic group G1 and the cyclic group GT are sets in which a non-degenerate bilinear function e is defined, the non-degenerate bilinear function e outputting a single element of the cyclic group GT in response to input of n+1 elements γL (L=1, . . . , n+1) of the cyclic group G1 and n+1 elements γL* (L=1, . . . , n+1) of a cyclic group G2.

13. The encryption apparatus according to claim 12, wherein the values r and ρ and the constant τ are elements of a finite field Fq; and
the order of each of the cyclic groups G1 and G2 equals the order q (q≧1) of the finite field Fq.

14. The encryption apparatus according to claim 12 or 13, wherein the bilinear function e outputs an element e=Πi=1n+1Pair(γL, γL*) of the cyclic group GT in response to the input of the n+1 elements γL (L=1, . . . , n+1) of the cyclic group G1 and the n+1 elements γL* (L=1, . . . , n+1) of the cyclic group G2, where Pair is a non-degenerate bilinear function that translates a combination of a single element of the cyclic group G1 and a single element of the cyclic group G2 to a single element of the cyclic group GT.

15. A non-transitory computer-readable recording medium having stored thereon a program causing a computer to function as an encryption apparatus according to any one of claims 1 to 3.

16. A decryption apparatus comprising:
an input unit to which a value containing a ciphertext C1 and a second ciphertext C2∈G1n+1 is input;
a bilinear function operation unit adapted to calculate a function value S ̃=e(C2, D1*)∈GT by putting the second ciphertext C2∈G1n+1 and key information D1*∈G2n+1 into a bilinear function e;
a function operation unit adapted to calculate a function value of a collision-resistant function H by putting a value corresponding to the function value S ̃ and the ciphertext C1 into the function H; and
a decision unit adapted to judge whether the second ciphertext C2, a value ρ ̃, a value r ̃ corresponding to the function value of the function H, elements vμ (μ=1, . . . , n) of an n-dimensional vector v→=(v1, . . . , vn), and basis vectors bi∈G1n+1 (i=1, . . . , n+1) satisfy C2=r ̃·(τμ=1nvμ·bμ)+ρ ̃·bn+1∈G1n+1 and to output a result of judgment, wherein
the bilinear function e is a non-degenerate bilinear function which outputs a single element of a cyclic group GT in response to input of n+1 elements γL (L=1, . . . , n+1) (n≧1) of a cyclic group G1 and n+1 elements γL* (L=1, . . . , n+1) of a cyclic group G2; each of the basis vectors bi∈G1n+1 (i=1, . . . , n+1) is an n+1 dimensional basis vector composed of n+1 elements of the cyclic group G1; each bi*∈G2n+1 (i=1, . . . , n+1) is an n+1 dimensional basis vector composed of n+1 elements of the cyclic group G2; a function value obtained by putting the elements of the basis vectors bi∈G1n+1 (i=1, . . . , n+1) and the elements of the basis vectors bj*∈G2n+1 (j=1, . . . , n+1) into the bilinear function e is expressed as gTτ·δ(i, j)∈GT by using Kronecker's delta function δ(i, j); τ is a constant other than an additive identity element; gT is a generator of the cyclic group GT; w→=(w1, . . . , wn) is an n-dimensional vector; wμ (μ=1, . . . , n) are elements of the n-dimensional vector w→=(w1, . . . , wn); α is a selected value; and the key information D1*∈G2n+1 is D1*=α·(Σμ=1nwμ·bμ*)+bn+1*∈G2n+1.

17. The decryption apparatus according to claim 16, wherein the value containing the ciphertext C1 and the second ciphertext C2∈G1n+1, input to the input unit, further contains a third ciphertext C3 which is a binary sequence;
the decryption apparatus further comprising an exclusive-OR operation unit adapted to calculate an exclusive OR value K ̃ of the third ciphertext C3 and a function value R(S ̃) which is a binary sequence obtained by putting the function value S ̃ into an injective function R;
the function operation unit is adapted to calculate the values r ̃ and ρ ̃ by putting the exclusive OR value K ̃ and the ciphertext C1 into the collision-resistant function H; and
the decision unit includes:
a vector operation unit adapted to calculate a function value C2 ̃=r ̃·(Σμ=1nvμ·bμ)+ρ ̃·bn+1∈G1n+1 by using the elements vμ (μ=1, . . . , n) of the n-dimensional vector v′ and the values r ̃ and ρ ̃; and
a comparison unit adapted to judge whether the function value C2 ̃ equals the second ciphertext C2;

the decryption apparatus further comprising a symmetric key decryption unit adapted to use the exclusive OR value K ̃ as a symmetric key, to calculate a decrypted value M ̃ by putting the exclusive OR value K ̃ and the ciphertext C1 into a symmetric key decryption function, and to output the decrypted value M ̃, when judgment by the comparison unit indicates that the function value C2 ̃ equals the second ciphertext C2.

18. The decryption apparatus according to claim 16, wherein the function operation unit is adapted to calculate the value r ̃ by putting the function value S ̃ and the ciphertext C1 into the collision-resistant function H; and
the decision unit includes:
a vector operation unit adapted to calculate a function value λ=C2−r ̃·Σμ=1nvμ·bμ∈G1n+1 by using the second ciphertext C2, the value r ̃, and the elements vμ (μ=1, . . . , n) of the n-dimensional vector v→;
a second bilinear function operation unit adapted to calculate a function value e(λ, D2*) by putting the function value X and auxiliary key information D2*=Σμ=1nβμ·bμ* concerning a value βμ (μ=1, . . . , n) into the bilinear function e; and
a comparison unit adapted to judge whether the function value e(λ, D2*) satisfies e(λ, D2*)=1∈GT;

the decryption apparatus further comprising a symmetric key decryption unit which is adapted to use a function value R(S ̃) obtained by putting the function value S ̃ into an injective function R as a symmetric key K ̃ if judgment by the comparison unit indicates that e(X, D2*)=1∈GT is satisfied, to calculate a decrypted value M ̃ by putting the symmetric key K ̃ and the ciphertext C1 into a symmetric key decryption function, and to output the decrypted value M ̃.

19. The decryption apparatus according to claim 17 or 18, wherein the injective function R is a quasi random function.

20. The decryption apparatus according any one of claims 16 to 18, wherein the collision-resistant function H is a quasi random function.

21. The decryption apparatus according to any one of claims 16 to 18, wherein the basis vector bi satisfies bi=Σj=1n+1χi,j·aj, where χi,j indicates the element of an (n+1) row by (n+1) column matrix X in the i-th row and the j-th column, and ai∈G1n+1 (i=1, . . . , n+1) indicates an n+1 dimensional basis vector having κ1·g1 as the i-th element and having the identity elements of the cyclic group G1 as the remaining n elements, g1 being a generator of the cyclic group G1 and κ1 being a constant other than an additive identity element; and
the basis vector bi* satisfies bi*=Σj=1n+1χi,j*·aj* where χi,j* indicates the element of an (n+1) row by (n+1) column matrix X* in the i-th row and the j-th column, and ai*∈G2n+1 (i=1, . . . , n+1) indicates an n+1 dimensional basis vector having κ2·g2 as the i-th element and having the identity elements of the cyclic group G2 as the remaining n elements, g2 being a generator of the cyclic group G2 and κ2 being a constant other than the additive identity element.

22. The decryption apparatus according to any one of claims 16 to 18, wherein the values r ̃ and the constant τ, and the value a are elements of a finite field Fq; and
the order of each of the cyclic groups G1 and G2 equals the order q (q≧1) of the finite field Fq.

23. The decryption apparatus according to any one of claims 16 to 18, wherein the bilinear function e outputs an element e=ΠL=1n+1Pair(γL, γL*) of the cyclic group GT in response to the input of the n+1 elements γL (L=1, . . . , n+1) of the cyclic group G1 and the n+1 elements γL* (L=1, . . . , n+1) of the cyclic group G2, where Pair is a non-degenerate bilinear function that translates a combination of a single element of the cyclic group G1 and a single element of the cyclic group G2 to a single element of the cyclic group GT.

24. A non-transitory computer-readable recording medium having stored thereon a program causing a computer to function as a decryption apparatus according to any one of claims 16 to 18.

25. A security method executed by an encryption apparatus and a decryption apparatus, the security method comprising:
(A) a step of calculating, in the encryption apparatus, a ciphertext C1 by putting a plaintext M and a symmetric key K into a symmetric key encryption function and outputting the ciphertext C1;
(B) a step of calculating, in the encryption apparatus, a first function value of a collision-resistant function H by putting a value containing a value corresponding to the symmetric key K and a value corresponding to the ciphertext C1 into the function H;
(C) a step of calculating, in the encryption apparatus, a second ciphertext C2=r·(Σμ=1nvμ·bμ)+ρ·bn+1∈G1n+1 by using a value r corresponding to the first function value of the function H, a value ρ corresponding to the symmetric key K, elements vμ (μ=1, . . . , n, n≧1) of an n-dimensional vector v→=(v1, . . . , vn), and n+1 dimensional basis vectors bi∈G1n+1 (i=1, . . . , n+1) composed of n+1 elements of the cyclic group G1 and outputting the second ciphertext C2;
(D) a step of inputting a value containing the ciphertext C1 and the second ciphertext C2∈G1n+1 to the decryption apparatus;
(E) a step of calculating, in the decryption apparatus, a function value S ̃=e(C2, D1*)∈GT by putting the second ciphertext C2∈G1n+1 and key information D1* into a bilinear function e;
(F) a step of calculating, in the decryption apparatus, a second function value of the collision-resistant function H by putting a value corresponding to the function value S ̃ and the ciphertext C1 into the function H; and
(G) a step of judging, in the decryption apparatus, whether the second ciphertext C2, a value ρ ̃, a value r ̃ corresponding to the second function value of the function H, the elements vμ of the n-dimensional vector v→, and the basis vectors bi∈G1n+1 (i=1, . . . , n+1) satisfy C2=r ̃·(Σμ=1nvμ·bμ)+ρ ̃·bn+1∈G1n+1, wherein
the bilinear function e is a non-degenerate bilinear function which outputs a single element of a cyclic group GT in response to input of n+1 elements γL (L=1, . . . , n+1) (n≧1) of the cyclic group G1 and n+1 elements γL* (L=1, . . . , n+1) of a cyclic group G2; each of the basis vectors bi∈G1n+1 (i=1, . . . , n+1) is an n+1 dimensional basis vector composed of the n+1 elements of the cyclic group G1; each of basis vectors bi*∈G2n+1 (i=1, . . . , n+1) is an n+1 dimensional basis vector composed of n+1 elements of the cyclic group G2; a function value obtained by putting the elements of the basis vectors bi∈G1n+1 (i=1, . . . , n+1) and the elements of the basis vectors bj*∈G2n+1 (j=1, . . . , n+1) into the bilinear function e is expressed as gTτ·δ(i, j)∈GT by using Kronecker's delta function δ(i, j); τ is a constant other than an additive identity element; gT is a generator of the cyclic group GT; α is a selected value; w→=(w1, . . . , wn) is an n-dimensional vector; wμ (μ=1, . . . , n) are elements of the n-dimensional vector w→=(w1, . . . , wn); and the key information D1* is D1*=α·(Σμ=1nwμ·bμ*)+bn+1*∈G2n+1.

26. The security method according to claim 25, wherein the step (B) includes a step of calculating the values r and ρ by putting the symmetric key K and the ciphertext C1 into the collision-resistant function H;
the step (C) includes:
a step of calculating, in the decryption apparatus, a function value S=gTτ·ρ∈GT by using the value ρ;
a step of calculating, in the decryption apparatus, a third ciphertext C3 and outputting the third ciphertext C3, the third ciphertext C3 being an exclusive OR value of a function value R(S) and the symmetric key K, the function value R(S) being a binary sequence obtained by putting the function value S into an injective function R; and
a step of calculating, in the decryption apparatus, the second ciphertext C2 by using the elements vμ (μ=1, . . . , n) of the n-dimensional vector v→ and the values r and ρ and outputting the second ciphertext C2;

the step (D) includes a step of inputting, in the decryption apparatus, a value containing the ciphertext C1, the second ciphertext C2∈G1n+1, and the third ciphertext C3 to the decryption apparatus;
the step (F) includes:
a step of calculating, in the decryption apparatus, an exclusive OR K ̃ value of the third ciphertext C3 and a function value R(S ̃) which is a binary sequence obtained by putting the function value S ̃ into the injective function R; and
a step of calculating, in the decryption apparatus, the values r ̃ and ρ ̃ by putting the exclusive OR value K ̃ and the ciphertext C1 into the collision-resistant function H; and

the step (G) includes:
a step of calculating, in the decryption apparatus, a function value C2 ̃=r ̃·(Σμ=1nvμ·bμ)+ρ ̃·bn+1∈G1n+1 by using the elements vμ (μ=1, . . . , n) of the n-dimensional vector v→ and the values r ̃ and ρ ̃; and
a step of judging, in the decryption apparatus, whether the function value C2 ̃ equals the second ciphertext C2, if the function value C2 ̃ equals the second ciphertext C2, using the exclusive OR value K ̃ as a symmetric key, calculating a decrypted value M ̃ by putting the exclusive OR value K ̃ and the ciphertext C1 into a symmetric key decryption function, and outputting the decrypted value M ̃.

27. The security method according to claim 25, wherein the step (A) includes:
a step of selecting the value ρ in the encryption apparatus;
a step of calculating, in the encryption apparatus, a function value S=gTτ·ρ∈GT by using the value ρ; and
a step of using, in the encryption apparatus, a function value R(S) obtained by putting the function value S into an injective function R, as a symmetric key K, calculating the ciphertext C1 by putting the symmetric key K and the plaintext M into the symmetric key encryption function, and outputting the ciphertext C1;
the step (B) includes a step of calculating, in the encryption apparatus, the value r by putting the function value S and the ciphertext C1 into the collision-resistant function H;
the step (C) includes a step of calculating, in the encryption apparatus, the second ciphertext by using the elements vμ (μ=1, . . . , n) of the n-dimensional vector v→ and the values r and ρ and outputting the second ciphertext C2;
the step (F) includes a step of calculating, in the decryption apparatus, the value r ̃ by putting the function value S ̃ and the ciphertext C1 into the collision-resistant function H; and
the step (G) includes:
a step of calculating, in the decryption apparatus, a function value λ=C2 ̃=r ̃·(Σμ=1nvδ·bμ)∈G1n+1 by using the second ciphertext C2, the value r ̃, and the elements vμ (μ=1, . . . , n) of the n-dimensional vector v→;
a step of calculating, in the decryption apparatus, a function value e(λ, D2*) by putting the function value λ and auxiliary key information D2* into the bilinear function e; and
a step of judging, in the decryption apparatus, whether the function value e(λ, D2*) satisfies e(λ, D2*)=1∈GT, if e(λ, D2*)=1∈GT is satisfied, using a function value R(S ̃) obtained by putting the function value S ̃ into an injective function R, as a symmetric key K ̃, calculating a decrypted value M ̃ by putting the symmetric key K ̃ and the ciphertext C1 into a symmetric key decryption function, and outputting the decrypted value M.

28. An encryption method comprising:
(A) a step of calculating, in a first means of an encryption apparatus, a ciphertext C1 by putting a plaintext M and a symmetric key K into a symmetric key encryption function and outputting the ciphertext C1;
(B) a step of calculating, in a second means of the encryption apparatus, a function value of a collision-resistant function H by putting a value containing a value corresponding to the symmetric key K and a value corresponding to the ciphertext C1 into the function H; and
(C) a step of calculating, in a third means of the encryption apparatus, a second ciphertext C2=r·(Σμ=1nvμ·bμ)+ρ·bn+1∈G1n+1 by using a value r corresponding to the function value of the function H, a value ρ corresponding to the symmetric key K, elements vμ (μ=1, . . . , n) (n≧1) of an n-dimensional vector v→=(v1, . . . , vn), and n+1 dimensional basis vectors bi∈G1n+1 (i=1, . . . , n+1) composed of n+1 elements of a cyclic group G1 and outputting the second ciphertext C2.

29. The encryption method according to claim 28, wherein the symmetric key K is a binary sequence; and
the step (B) includes a step of calculating, in a fourth means of the encryption apparatus, the values r and ρ by putting the symmetric key K and the ciphertext C1 into the collision-resistant function H;
the encryption method further comprising:
a step of calculating, in a fifth means of the encryption apparatus, a function value S=gTτ·ρ∈GT by using the value ρ, where gT is a generator of the cyclic group GT and τ is a constant other than an additive identity element; and
a step of calculating, in a sixth means of the encryption apparatus, a third ciphertext C3 and outputting the third ciphertext C3, the third ciphertext C3 being as an exclusive OR value of a function value R(S) and the symmetric key K, the third ciphertext C3 being a binary sequence obtained by putting the function value S into an injective function R.

30. The encryption method according to claim 28, further comprising:
a step of selecting the value ρ in a seventh means of the encryption apparatus;
a step of calculating, in an eighth means of the encryption apparatus, a function value S=gTτ·ρ∈GT by using the value ρ, where gT is a generator of a cyclic group GT and τ is a constant other than an additive identity element; and
a step of calculating, in a ninth means of the encryption apparatus, a function value R(S) as the symmetric key K by putting the function value S into an injective function R, wherein
the step (B) includes a step of calculating, in a tenth means of the encryption apparatus, the value r by putting the function value S and the ciphertext C1 into the collision-resistant function H.

31. A decryption method comprising:
(A) a step of inputting a value containing a ciphertext C1 and a second ciphertext C2∈G1n+1 to a decryption apparatus;
(B) a step of calculating, in a first means of the decryption apparatus, a function value S ̃=e(C2, D1*)∈GT by putting the second ciphertext C2∈G1n+1 and key information D1*∈G2n+1 into a bilinear function e;
(C) a step of calculating, in a second means of the decryption apparatus, a function value of a collision-resistant function H by putting a value corresponding to the function value S ̃ and the ciphertext Cl into the function H; and
(D) a step of judging, in a third means of the decryption apparatus, whether the second ciphertext C2, a value r ̃ corresponding to the function value of the function H, a value ρ ̃, elements vμ (μ=1, . . . , n) of an n-dimensional vector v→=(v1, . . . , vn), and basis vectors bi∈G1n+1 (i=1, . . . , n+1) satisfy C2=r ̃·(Σμ=nvμ·bμ)+ρ ̃·bn+1∈G1n+1 and outputting a result of judgment, wherein
the bilinear function e is a non-degenerate bilinear function which outputs a single element of a cyclic group GT in response to input of n+1 elements γL (L=1, . . . , n+1) (n≧1) of a cyclic group G1 and n+1 elements γL* (L=1, . . . , n+1) of a cyclic group G2; each of the basis vectors bi∈G1n+1 (i=1, . . . , n+1) is an n+1 dimensional basis vector composed of n+1 elements of the cyclic group G1; each of bi*∈G2n+1 (i=1, . . . , n+1) is an n+1 dimensional basis vector composed of n+1 elements of the cyclic group G2; a function value obtained by putting the elements of the basis vectors bi∈G1n+1 (i=1, . . . , n+1) and the elements of the basis vectors bj*∈G2n+1 (j=1, . . . , n+1) into the bilinear function e is expressed as gTτ·δ(i, j)∈GT by using Kronecker's delta function δ(i, j); τ is a constant other than an additive identity element; gT is a generator of the cyclic group GT; w→=(w1, . . . , wn) is an n-dimensional vector; wμ (μ=1, . . . , n) are elements of the n-dimensional vector w→=(w1, . . . , wn); α is a selected value; and the key information D1*∈G2n+1 is D1*=α·(Σμ=1nwμ·bμ*)+bn+1∈G2n+1.

32. The decryption method according to claim 31, wherein the value containing the ciphertext C1 and the second ciphertext C2∈G1n+1, input in the step (A), further contains a third ciphertext C3 which is a binary sequence;
the decryption method further comprising a step of calculating, in a fourth means of the decryption apparatus, an exclusive OR value K ̃ of the third ciphertext C3 and a function value R(S ̃) which is a binary sequence obtained by putting the function value S ̃ into an injective function R;
the step (C) includes a step of calculating, in a fifth means of the decryption apparatus, the values r ̃ and ρ ̃ by putting the exclusive OR value K ̃ and the ciphertext C1 into the collision-resistant function H;
the step (D) includes:
(D-1) a step of calculating, in a sixth means of the decryption apparatus, a function value C2 ̃=r ̃·(Σμ=1nvμ·bμ)+ρ ̃·bn+1∈G1n+1 by using the elements vμ (μ=1, . . . , n) of the n-dimensional vector v→ and the values r ̃ and ρ ̃; and
(D-2) a step of judging, in a seventh means of the decryption apparatus, whether the function value C2 ̃ equals the second ciphertext C2;

the decryption method further comprising a step of using, in an eighth means of the decryption apparatus, the exclusive OR K ̃ as a symmetric key, calculating a decrypted value M ̃ by putting the exclusive OR value K ̃ and the ciphertext C1 into a symmetric key decryption function, and outputting the decrypted value M ̃, when judgment in the step (D-2) indicates that the function value C2 ̃ equals the second ciphertext C2.

33. The decryption method according to claim 31, wherein the step (C) includes a step of calculating, in a ninth means of the decryption apparatus, the value r′ by putting the function value S ̃ and the ciphertext C1 into the collision-resistant function H; and
the step (D) includes:
(D-1) a step of calculating, in a tenth means of the decryption apparatus, a function value λ=C2−r ̃·Σμ=1nvμ·bμ∈G1n+1 by using the second ciphertext C2, the value r ̃, and the elements vμ (μ=1, . . . , n) of the n-dimensional vector v→;
(D-2) a step of calculating, in an eleventh means of the decryption apparatus, a function value e(λ, D2*) by putting the function value λ and auxiliary key information D2*=Σμ=1nβμ·bμ* concerning a value βμ (μ=1, . . . , n) into the bilinear function e; and
(D-3) a step of judging, in a twelfth means of the decryption apparatus, whether the function value e(λ, D2*) satisfies e(λ, D2*)=1∈GT;

the decryption method further comprising a step of using, in a thirteenth means of the decryption apparatus, a function value R(S ̃) obtained by putting the function value S ̃ into an injective function R, as a symmetric key K ̃, calculating a decrypted value M ̃ by putting the symmetric key K ̃ and the ciphertext C1 into a symmetric key decryption function, and outputting the decrypted value M ̃, when judgment in the step (D-3) indicates that e(λ, D2*)=1∈GT is satisfied.