Patent Abstract:
Provided are a signature apparatus, a verifying apparatus, a proving apparatus, an encrypting apparatus, and a decrypting apparatus capable of efficiently reducing a signature text counterfeit problem to a discrete logarithm problem. The commitment is a hash value of a set of a value to be committed. Data including a pair of elements of a cyclic group associated with a discrete logarithm problem is used as a public key, and a discrete logarithm of an order of the pair is used as a secret key. Accordingly, it is possible to summarize secret information of an attacker from the commitment without rewinding the attacker and to ensure a higher safety than that of a Schnorr signature scheme. In addition, one-time power residue calculation is performed in each of the signature and verification calculations, so that it is possible to lower an amount of calculation in the signature and verification calculations.

Full Description:
This application is the National Phase of PCT/JP2005/022875, filed Dec. 13, 2005, which claims priority to Japanese Application No. 2005-014891, filed Jan. 12, 2005, the disclosures of which are hereby incorporated by reference in their entirety. 
     1. Technical Field 
     The present invention relates to a signature apparatus, a verifying apparatus, a proving apparatus, an encrypting apparatus, and a decrypting apparatus and, more particularly, to a signature apparatus, a verifying apparatus, a proving apparatus, an encrypting apparatus, and a decrypting apparatus capable of efficiently reducing a signature text counterfeit problem to a discrete logarithm problem. 
     2. Background Art 
     A public key is a cipher which uses different keys for encrypting and decrypting. The key used for the decrypting is maintained in a secret state, while the key used for the encrypting is publicized. The public key needs a system for ensuring an authenticity of a key to be publicized. However, there is no need to distribute the key to a counter party in advance. In addition, due to the public key, it is possible to implement a digital signature capable of authenticating the counter party for communication and verifying the authenticity of received data. For this reason, the public key is widely used as an information security technique in a network such as the Internet. 
     Recently, a crypto scheme having a provable safety and a practicability has been widely researched with respect to the public key. Among the current used crypto schemes, an efficient decryption method has not yet been implemented, and safety of most crypto schemes has not been proven. A probability of presence of the efficient decryption methods associated with the crypto schemes can not completely be denied.
     Non-Patent Document 1: Mihir. Bellare and Phillip. Rogaway. Random Oracles are Practical: A Paradigm for Designing Efficient Protocols. ACM-CCS. 1993. pp. 62-73   Non-Patent Document 2: Mihir Bellare, Phillip Rogaway. The Exact Security of Digital Signatures: How to Sign with RSA and Rabin. In Advances in Cryptology—EUROCRYPT&#39; 96, vol. 1070 of LNCS, pp. 399-416, Springer-Verlag, 1996.   Non-Patent Document 3: Jean-Sebastien Coron. On the Exact Security of Full Domain Hash. In Advances in Cryptology—CRYPTO 2000, vol. 1880 of LNCS, pp. 229-235, Springer Verlag, 2000.   Non-Patent Document 4: Amos. Fiat and Adi. Shamir. How to prove yourself: Practical Solution to Identification and Signature Problems. In Advances in Cryptology—CRYPTO&#39; 86, vol. 263 of LNCS pp. 186-194, Springer-Verlag, 1987.   Non-Patent Document 5: Eu-Jin Goh, Stanislaw Jarecki. A Signature Scheme as Secure as the Diffie-Hellman Problem. In Advances in Cryptology—EUROCRYPT 2003, vol. 2656 of LNCS, pp. 401-415, Springer-Verlag, 2003.   Non-Patent Document 6: Kazuo Ohta, Tatsuaki Okamoto. On Concrete Security Treatment of Signatures Derived from Identification. In Advances in Cryptology—CRYPTO&#39; 98, vol. 1462 of LNCS, pp. 354-369, Springer-Verlag, 1998.   Non-Patent Document 7: Rafael Pass. On Deniability in the Common Reference String and Random Oracle Model. In Advances in Cryptology—CRYPTO 2003, vol. 2729 of LNCS pp. 316-337, Springer Verlag, 2003.   Non-Patent Document 8: David Pointcheval, Jacques Stern: Security Arguments for Digital Signatures and Blind Signatures. J. Cryptology 13(3): 361-396 (2000)   Non-Patent Document 9: R. Rivest, A. Shamir, L. Adleman. A Method for Obtaining Digital Signatures and Public-Key Cryptosystems. Communications of the ACM. Vol. 21, No. 2, pp. 120-126, 1978.   Non-Patent Document 10: C. Schnorr. Efficient Signature Generation by Smart Cards. Journal of Cryptology, 4(3), pp. 161-174, 1991.   Non-Patent Document 11: Handbook of Applied Cryptography. A. Menezes P. Oorshot, and S. Vanstone, CRC Press.   Non-Patent Document 12: Rafael Pass. On Deniability in the Common Reference String and Random Oracle Model. In Advances in Cryptology—CRYPTO 2003, vol. 2729 of LNCS, pp. 316-337, Springer-Verlag, 2003.   Non-Patent Document 13: Markus Jakobsson, Kazue Sako, and Russell Impagliazzo. Designated Verifier Proofs and Their Applications. In Advances in Cryptology—EUROCRYPT&#39; 96, vol. 1070 of LNCS, pp. 143-154, Springer-Verlag, 1996.   

     DISCLOSURE OF THE INVENTION 
     Problems to be Solved by the Invention 
     However, the aforementioned schemes have the following problems. 
     Since an electronic signature scheme was disclosed in Non-Patent Document 9, implementation of a safe and efficient signature scheme has been one of the objects of a cryptology. As approaches for achieving the object, there are signature schemes using Fiat-Shamir heuristic (Non-Patent Document 4) or a hash-then-sign method (Non-Patent Document 1). 
     However, in terms of safety proof (Non-Patent Documents 3, 6, and 8) of the signature schemes, it is disclosed that a signature text cannot be counterfeited during a polynomial time interval while it is not disclosed in detail how much amount of calculation is needed to counterfeit the signature text. For this reason, there is a probability of presence of an attacker who can succeed counterfeiting the signature text with a much smaller amount of calculation than the amount of calculation required for decrypting a base problem (Non-Patent Document 2). 
     The Schnorr signature scheme (Non-Patent Document 10) is one of the signature schemes having such a probability. Although the base problem, that is, a discrete logarithm problem has a safety of about λ bits, the safety proof (Non-Patent Documents 6 and 8) of the Schnorr signature scheme ensures that the Schnorr signature scheme has at most a safety of about λ/2 bits. Therefore, a signature scheme capable of efficiently reducing the signature text counterfeit problem to the discrete logarithm problem is required. 
     It is known that a scheme having such a property can be theoretically implemented. As a scheme satisfying the property, there is a scheme of converting to a cut-and-choose type proving scheme (Non-Patent Document 7) of the discrete logarithm problem. However, the scheme needs a large amount of calculation for signature and verification. It is disclosed that a signature scheme capable of efficiently reducing the signature text counterfeit problem to the discrete logarithm problem and using a small amount of calculation cannot be implemented (Non-Patent Document 5). 
     In Non-Patent Document 2, since there is a need to rewind an attacker during the safety proof, an efficiency of reduction to the base problem is deteriorated, so that the safety is deteriorated compared with the base problem. In the safety proof ¥cite[PS00] of the Schnorr signature scheme, since there is a need to rewind the attacker, the safety is deteriorated compared with the base problem, that is, the discrete logarithm problem. 
     Therefore, an object of the present invention is to provide a signature apparatus, a verifying apparatus, a proving apparatus, an encrypting apparatus, and a decrypting apparatus capable of summarizing secret information of an attacker from a commitment without rewinding the attacker by using a hash value as the commitment. 
     Means for Solving the Problems 
     Various embodiments provide a signature apparatus for generating a signature text by using a commitment, wherein the commitment is a hash value of a set including a committed value, data including a pair of elements of a cyclic group associated with a discrete logarithm problem is used as a public key, and a discrete logarithm of an order of the pair is used as a secret key. 
     Various embodiments also provide a signature apparatus for generating a signature text by using a commitment, wherein the commitment is a hash value of a set including a value to be committed, data including a pair of elements of a cyclic group associated with a discrete logarithm problem is used as a public key, and a discrete logarithm of an order of the pair is used as a secret key, the signature apparatus comprising: committed vector selecting means which selects a committed vector associated with a first commitment; first commitment calculating means which calculates the first commitment; basis vector calculating means which calculates a basis vector; second commitment calculating means which calculates the power residue and calculates a second commitment; vector challenge calculating means which calculates a vector challenge; vector response calculating means which calculates a vector response by using the first commitment, a set used for calculating the power residue, the vector challenge, and the basis vector; and a storage unit which stores the committed vector, the first commitment, the basis vector, the second commitment, the vector challenge, and the vector response, wherein the basis vector and the vector challenge are hash values. 
     In a representative embodiment, the committed vector selecting means selects a plurality of the committed vectors, each component of the plurality of committed vectors and a secret key satisfy a relation equation with a group order as a modulus, and the set is data calculated by using a portion of data selected by the committed vector selecting means, the basis vector, and the vector challenge. 
     In some embodiments each component of the committed vectors and the secret key satisfy a linear equation with the group order as a modulus, an input of the first commitment is data including a random number, a portion of the data is determined by the vector challenge, and the set is represented by a linear equation of the portion of the data and the basis vector. 
     In various embodiments the committed vector includes two components, the one component being a value obtained by adding a secret key to the other component and obtaining a residue with a group order as a modulus, an input of the first commitment includes data for specifying each component of the committed vector, and the set includes a inner product of the portion of the data and the basis vector. 
     In one embodiment, assuming that security parameters are κ, N, and ν, and an order of the cyclic group is q, the committed vector selecting means selects a residue group X —{ 01}, . . . , X —{ 0N}ε(Z/qZ) at random and sets values obtained by adding x to the residue group X —{ 0j} for j=1, . . . N and obtaining a residue with the order q as a modulus to X —{ 1j}, the committed vector for i=0, 1 is Y_i =(X_{il}, . . . , X_{iN}), the first commitment calculating means selects at random a bit column r of ν bits, a hash value of data including the public key, X_{ij}, i, j, r for i=0, 1 is set to the first commitment C_{ij}, the basis vector calculating means sets a hash value of data including the public key and the first commitment C_{ij} to the basis vector V=(u_1, . . . , u_N), the second commitment calculating means calculates an inner product of the basis vector V and the Y — 0 and calculates a second commitment G=g^{X}, the vector challenge calculating means calculates a hash value K=(c — 1, . . . , c_N) of data including the public key, C_{ij}}, G, r, and a message received by the signature apparatus, the vector response calculating means calculates the vector response ξ{j }=X_{c_jj }} for all j=1, . . . , N and Ξ=(ξ_1, . . . , ξ_κ) and a signature text (r, {C_{ij}, G, Ξ) is output. 
     In other embodiments, committed vector selecting means selects the plurality of committed vectors, and each component of the plurality of committed vectors and a secret key satisfy a relational equation. 
     In a suitable embodiment, the relational equation satisfies a linear equation of each component of the plurality of vectors and the secret key, and an input of the first commitment is data including a random number. 
     In some embodiments, the plurality of committed vectors include a plurality of components, the one component is obtained by adding a secret key to another component, and an input of the first commitment includes data for specifying each of the components and data for specifying which is the ordinal number of the component. 
     In an exemplary embodiment, assuming that security parameters are κ, N, and ν, and an integer set R{κ+ξ} satisfies 0≧R{κ+ξ}&lt;2^{κ+ξ}, the committed vector selecting means selects a residue group X_{01}, . . . , X_{0N} ε(Z/qZ) at random and sets values obtained by adding x to the residue group X_{0j} for j=1, . . . N to X_{1j}, the committed vector for i=0, 1 is Y_i=(X_{i1}, . . . , X_{iN}), the first commitment calculating means selects at random a bit column r of ν bits, a hash value of data including the public key, X_{1j}, i, j, r for i=0, 1 is set to the first commitment C_{ij}, the basis vector calculating means sets a hash value of data including the public key and the first commitment C_{ij} to the basis vector V=(u_1, . . . , u_N), the second commitment calculating means calculates an inner product of the basis vector V and the Y_0 and calculates a second commitment G=g^{X}, the vector challenge calculating means calculates a hash value K=(c_1, c_N) of data including the public key, {C_{ij}}, G, r, and a message received by the signature apparatus, the vector response calculating means calculates the vector response ξ_{j}=X_{c_jj} for all j=1, . . . , N and Ξ=(ξ_1, . . . , ξ_κ), and a signature text (r, {C_{ij}}, G, Ξ) is output. 
     In another embodiment, the signature apparatus comprises the committed vector selecting means which selects a committed vector associated with a first commitment; first commitment calculating means which calculates the first commitment; basis vector calculating means which calculates a basis vector; second commitment calculating means which calculates the power residue and generates a second commitment; vector challenge calculating means which calculates a vector challenge; vector response calculating means which calculates a vector response by using the first commitment, a set used for calculating the power residue, the vector challenge, and the basis vector; and a storage unit which stores the committed vector, the first commitment, the basis vector, the second commitment, the vector challenge, and the vector response, wherein the basis vector and the vector challenge are hash values. 
     In various embodiments, the committed vector selecting means selects a plurality of the committed vectors, each having the same configuration as the committed vector, each component of the plurality of committed vectors and a secret key satisfy a relation equation with a group order as a modulus, and the set is data calculated by using a portion of data selected by the committed vector selecting means, the basis vector, and the vector challenge. 
     In a representative embodiment each component of the committed vectors and the secret key satisfy a linear equation with the group order as a modulus, the first commitment is data including a random number, a portion of the data is determined by the vector challenge, and the set is represented by a linear equation of the portion of the data and the basis vector. 
     In some embodiments the one component of the committed vector is a value obtained by adding a secret key to another component and obtaining a residue with a group order as a modulus, the set includes an inner product of the portion of the data and the basis vector, and the basis vector is a value obtained by multiplying a predetermined number t with (1, t^1, t^2, . . . , t^N). 
     In exemplary embodiments assuming that the message is M, the committed vector selecting means selects at random Y — 0=(X — {01}, . . . , X — {0N})εZq^{N}, calculates X — {1j}=x+X — {0j} modq for j=1, . . . , N, and generates the committed vector Y — 1=(X — {11}, . . . X — {1N}), the second commitment calculating means calculates X=&lt;{Y — 0, V}&gt;=Σ_jX — {0j}2^{j−1} modq and calculates the commitment G=g^{X}, the first commitment calculating means selects at random r_{ij}ε{0, 1}^{ν} for each i and j and calculates the first commitment C_{ij}=H — {{0, 1}^ν}(X_{ij}, r_{ij}) of the X_{ij}, the vector challenge calculating means calculates K=(c — {1}, . . . , c_{N})=H — {{0, 1}^{N}(g, h, {C_{ij}}, G, M), the vector response calculating means calculates the vector response ξ_j=X_{c_jj} modq for each j and calculates Ξ=ξ — {1}, . . . , ξ_{N}), and a signature text ({C_{ij}}, {r_{c_jj}}, G, Ξ) is output. 
     Various embodiments provide a verifying apparatus for determining a validity of input data, wherein the input data includes a message and a signature text associated with the message; only if the data is valid, the data is accepted, first commitments are used for verification calculation, and power residue is performed the number of times less than the number of the first commitments; if the validity of the data is authenticated, the first commitments are hash values of data including components of a vector response which is a portion of the data; a public key is data including a pair of elements of a cyclic group associated with a discrete logarithm problem; and a secret key is a discrete logarithm of an order of the pair. 
     Some embodiments provide a basis vector calculating means which calculates a basis vector; vector challenge calculating means which calculates a vector challenge; first commitment validity verifying means which determines a validity of the first commitment; second validity verifying means which calculates power residue and determines a validity of the vector response; and storage means which stores data input to and output from each of the means, wherein the vector challenge and the basis vector are hash values, and, only if the first commitment validity verifying means and the second validity verifying means determine that the signature text is valid, the input of the data is accepted. 
     In various embodiments the first commitment validity verifying means inputs a portion of the input data including the vector response to a hash function in a predetermined method and determines that the first commitment is valid only if the calculated hash value is equal to the first commitment, when the to-be-determined data include data called a second commitment and the vector response and the public key includes two elements of a cyclic group associated with a discrete logarithm problem, the second validity verifying means calculates first and second power residues which are power residues of the elements and determines whether or not the first power residue is equal to a value obtained by multiplying the second power residue with the second commitment, the first power residue is obtained by designating elements which are a portion of the public key to an order and designating a Schnorr challenge to a set, the second power residue is obtained by designating elements which are a portion of the public key to an order and designating a Schnorr response to a set, the Schnorr challenge is data calculated by using the vector challenge and the basis vector, and the Schnorr response is data calculated by using the vector response and the basis vector. 
     In exemplary embodiments if a valid signature text is not included in data validly selected at random by the signature apparatus, the data is not accepted, the data selected at random is input to each hash function calculated by the first commitment validity verifying means, one component of the vector response is input to each of the hash functions, the Schnorr challenge is a linear equation of the vector challenge and a linear equation of the basis vector, and the Schnorr response is a linear equation of the vector response and a linear equation of the basis vector. 
     In representative embodiments the Schnorr challenge is an inner product of the vector challenge and the basis vector, and the Schnorr response is an inner product of the vector response and the basis vector. 
     In an exemplary embodiment, assuming that the message is M, and the to-be-determined data is (r, {C_{ij}}, G, Ξ), the basis vector calculating means calculates a hash value of data including the public key and {C_{ij}}, the hash value being the basis vector V=(u — 1, . . . , u_N), the vector challenge calculating means calculates a hash value of data including the public key, {C_{ij}}, G, r, and M, the hash value being the vector challenge K=(c — 1, . . . , c_N), the first commitment validity verifying means determines that C_{c_jj} is valid only if C_{c_jj} for j=1, . . . , N are equal to the ash function determines that {C_{jj}} is valid only if all the C_{c_jj} are valid, the hash function is a hash value of data including the public key, ξ_{j}, c_j, j, and r, and the second validity verifying means determines whether or not g^{&lt;V, Ξ&gt;}=h^{&lt;V, K&gt;}G is satisfied and determines that the signature text is valid if g^{&lt;V, Ξ&gt;}=h^{&lt;V, K&gt;}G is satisfied. 
     A suitable embodiment vector challenge calculating means which calculates the vector challenge; first commitment validity verifying means which determines a validity of the first commitment; and second validity verifying means which calculates the power residue and determines a validity of the vector response, wherein the signature text is accepted only if the validity is authenticated in the first commitment validity verifying means and the second validity verifying means. 
     In some embodiments, if a hash value obtained by inputting a portion of the input data to a hash function is equal to the first commitment, the first commitment validity verifying means determines that the first commitment is valid, the data input to the hash function is used to calculate two power residues, and it is determined whether or not the one of the two power residues is equal to a value obtained by multiplying the other power residue with the second commitment, the to-be-determined data includes the second commitment and the vector response, the public key includes elements of a cyclic group associated with a discrete logarithm problem, each of the power residues is obtained by designating the other element which is a portion of the public key to an order and designating a Schnorr challenge to a set, the Schnorr challenge is calculated by using the vector challenge and the basis vector, and the Schnorr response is calculated by using the vector response and the basis vector. 
     In various embodiments, if data which should be selected at random is not included in a case where the signature apparatus generates a signature text validly, the data is rejected, the data which should be selected at random is input to each of the hash functions calculated by the first commitment validity verifying means, one component of the vector response is input to each of the hash functions, the Schnorr challenge is a linear equation of the vector challenge and a linear equation of the basis vector, and the Schnorr response is a linear equation of the vector response and a linear equation of the basis vector. 
     In an exemplary embodiment, the Schnorr challenge is an inner product of the vector challenge and the basis vector, and the Schnorr response is an inner product of the vector response and the basis vector. 
     In a representative embodiment, assuming that the message is M, and the to-be-verified signature text is ({C_{jj}}, {r_{cjj}}, G, Ξ), the vector challenge calculating means calculates K=(c — {1}, . . . , c_{N})=H — {{0, 1}^{N}}(g, h, {C_{ij}, G, M}; the first commitment validity verifying means determines whether or not C_{c_jj}=H — {{0, 1}^ν}(ξ_j, r_{c_jj}) for all j=1, . . . , N is satisfied, if the relation for all j is satisfied, it is determined to be b=1, and if not, it is determined to be b=0; when b=0, the second validity verifying means determines whether or not g^{&lt;V, Ξ&gt;}=h^{&lt;V, K&gt;}G is satisfied; if g^{&lt;V, Ξ&gt;}=h^{&lt;V, K&gt;}G is not satisfied, b=0 is designated, and data indicating that the signature text is rejected is output; and if g^{&lt;V, Ξ&gt;}=h^{&lt;V, K&gt;}G is satisfied, b=1 is designated, and data indicating that the signature text is accepted is output. 
     Some embodiments are characterized by using a method of determining a validity of a proof text by the methods discussed-above. 
     Other embodiments are characterized by using a method of determining a validity of a signature text by the methods discussed-above. 
     Various embodiments are characterized by using a method of determining a validity of a signature text generated by the method s discussed-above. 
     Some embodiments provide a proving apparatus for determining a validity of a public key in a verifying apparatus using a verifier-designated proving scheme, wherein the public key of the verifying apparatus includes two data, the first data and the second data belong to the same cyclic group, and a secret key of a verifier or a portion thereof is obtained by designating the first data to an order and designating the second data to a discrete logarithm. 
     Several embodiments provide a proving apparatus for determining a validity of a public key in a verifying apparatus using a verifier-designated proving scheme, wherein the public key of the verifying apparatus includes two data, the first data and the second data belong to the same cyclic group, a secret key of a verifier or a portion thereof is obtained by designating the first data to an order and designating the second data to a discrete logarithm, a signature text generated by the signature apparatus according to claim  5  is used as a proof text or a portion thereof for determining validity. 
     A representative embodiment provides a proving apparatus for determining a validity of a public key in a verifying apparatus using a verifier-designated proving scheme, wherein the public key of the verifying apparatus includes two data, the first data and the second data belong to the same cyclic group, a secret key of a verifier or a portion thereof is obtained by designating the first data to an order and designating the second data to a discrete logarithm, and a signature text generated by the signature apparatus according to claim  9  is used as a proof text or a portion thereof for determining validity. 
     Various embodiments provide a proving apparatus for determining a validity of a public key in a verifying apparatus using a verifier-designated proving scheme, the public key of the verifying apparatus includes two data, the first data and the second data belong to the same cyclic group, a secret key of a verifier or a portion thereof is obtained by designating the first data to an order and designating the second data to a discrete logarithm, and a signature text generated by the signature apparatus according to claim  14  is used as a proof text or a portion thereof for determining validity. 
     A plurality of embodiments provide a verifying apparatus for verifying a validity of a proof text for a public key of a verifier-designated proving scheme verifying apparatus, wherein the verifying is performed by using the method according to claim  20 . 
     Some embodiments provide a verifying apparatus for verifying a validity of a proof text for a public key of a verifier-designated proving scheme verifying apparatus, wherein the verifying is performed by using the method according to claim  25 . 
     A number of embodiments are characterized by using a signature text generated by using the methods discussed-above as a proof text or a portion thereof for proving a knowledge of a random number used to generate a cipher text. 
     Various embodiments are characterized by using a signature text generated by using the methods discussed-above as a proof text or a portion thereof for proving a knowledge of a random number used to generate a cipher text. 
     Representative embodiments are characterized by using a signature text generated by using the methods discussed-above as a proof text or a portion thereof for proving a knowledge of a random number used to generate a cipher text. 
     Some embodiments are characterized by including a proof text as a portion of a cipher text and verifying the proof text by using the methods discussed-above. 
     Various embodiments are characterized by including a proof text as a portion of a cipher text and verifying the proof text by using the methods discussed-above. 
     Effect of the Invention 
     According to the present invention, the hash value is used as a commitment, so that it is possible to summarize secret information of an attacker from the commitment without rewinding the attacker and to ensure a higher safety than that of a Schnorr signature scheme. In addition, one-time power residue calculation is performed in each of the signature and verification calculations, thus it is possible to lower an amount of calculation in the signature and verification calculations. 
     DESCRIPTION OF EXEMPLARY EMBODIMENTS 
     Hereinafter, configurations and operations of a signature apparatus and a verifying apparatus according to exemplary embodiments will be described. 
     First Exemplary Embodiment 
       FIG. 1  is a block diagram illustrating configurations of a signature apparatus SBN 0  and a verifying apparatus VBN 0  according to a first exemplary embodiment. The signature apparatus SBN 0  receives data by using a receiving apparatus RBN 0  and transmits data through a transmitting apparatus SeBN 0 . The verifying apparatus VBN 0  receives data by using a receiving apparatus RBN 1 . For example, LAN or Internet can be used as a channel used for data communication, but the present invention is not limited thereto. 
     Now, symbols used in the embodiment are described. 
     Symbol A denotes a cyclic group of which order is q. The number of bits of the order q is κ. Symbol g denotes a base point of the cyclic group A. It is assumed that, although the order q of the cyclic group A is publicized, the discrete logarithm problem associated with the cyclic group A is hard to falsify. 
     Symbol Z denotes a ring of all integers. Symbol N denotes a set of all natural numbers. An i-th component of a vector “a” is denoted by a_i. An inner product is denoted by &lt;•, •&gt;. An inner product of a vector “a” and a vector “b” is represented by &lt;a, b&gt;=a — 1b — 1+ . . . a_Nb_N. An X-value hash function of a set X is denoted by H_X. 
     Now, a key generating method is described. An xε(Z/qZ)\{0} is taken at random, and h=g^x is obtained. A public key and a secret key are (g, h, q) and x, respectively. The signature apparatus SBN 0  reserves the public key and the secret key in a storage unit SB 0 . It is assumed that the public key is reserved in a location, from which the verifying apparatus VBN 0  can acquire the public key in any type of an acquisition method. The acquisition method is, for example, means for reserving the public key in a public key table publicized on the Internet or means for directly acquiring the public key from the signature apparatus SBN 0 . The verifying apparatus VBN 0  acquires the public key and reserves the public key in the storage unit SB 0  if needed. Details of the key generating method are disclosed in Non-Patent Document 11. Hereinafter, the description is made under the state that the verifying apparatus VBN 0  has already acquired the public key. 
     The operations of the signature apparatus SBN 0  are described with reference to  FIGS. 1 to 3 . 
     When the receiving apparatus RBN 0  receives a message, the signature apparatus SBN 0  inputs the message to an input unit SB 1 . A signature text is generated and output in a committed vector selecting unit SB 2 , a first commitment calculating unit SB 3 , a basis vector calculating unit SB 4 , a second commitment calculating unit SB 5 , a vector challenge calculating unit SB 6 , a vector response calculating unit SB 7 , and a signature text output unit SB 8 . 
     Each of the units (SB 1  to SB 7 ) reads the data from the storage unit SB 0 , processes the data, and store the data in the storage unit SB 0  if needed. 
     Now, detailed processes of each of the units (SB 1  to SB 7 ) are described. 
     The input unit SB 1  receives a message M from the receiving apparatus RBN 0  and stores the message M in the storage unit SB 0 . 
     Processes of the committed vector selecting unit SB 2  are described. 
     When the message M is stored in the storage unit SB 0 , the committed vector selecting unit SB 2  reads the order q from the storage unit SB 0  (SF 2 ). When the order q is read, the committed vector selecting unit SB 2  selects at random a residue group of order q, that is, X — {01}, . . . , X — {0N}ε(Z/qZ) (SF 3 ). The X — {1j}=x+X — {0j} modq for all the j=1, . . . , N is calculated (SF 4 ). The X — {0j} for i=0 and j=1, . . . , N is set to Y — 0, and the X — {1j} for i=0 and j=1, . . . , N is set to Y — 1 (SF 5 ). The Y — 0 and the Y — 1 are referred to as i-th committed vectors. The Y — 0 and the Y — 1 are stored in the storage unit SB 0  (SF 6 ). 
     Processes of the first commitment calculating unit SB 3  are described. 
     The first commitment calculating unit SB 3  reads (ν, g, h, {X_{ij}}), i, j, r) from the storage unit SB 0  (SF 7 ). The first commitment calculating unit SB 3  selects at random a bit column r of ν bits (SF 8 ). A hash value C_{ij}=H — {{0, 1}^ν}(g, h, X_{ij}, i, j, r) of data including the bit column r and the public key (g, h, q) is calculated (SF 9 ). Here, i=0 and 1. In the embodiment, the hash value C_{ij} calculated by the first commitment calculating unit SB 3  is set to a first commitment, and {C_{ij}}_{i=0, 1, j=1, . . . , N} is set to a first commitment vector. 
     The first commitment vector (r, {C_{ij}}) calculated by the first commitment calculating unit SB 3  is stored in the storage unit SB 0  (SF 10 ). 
     Processes of the basis vector calculating unit SB 4  are described. 
     The basis vector calculating unit SB 4  reads (q, N, g, h, {C_{ij}}) from the storage unit SB 0  (SF 11 ). 
     The basis vector calculating unit SB 4  calculates a hash value V=(u — 1, . . . , u_N)=H_{((Z/qZ)\{0})^{N}}(g, h, {C_{ij}}) of data including the public key (q, g, h) and the first commitment {C_{ij}} (SF 12 ) and stores the V as a basis vector in the storage unit SB 0  (SF 13 ). 
     The second commitment calculating unit SB 5  is described. 
     The second commitment calculating unit SB 5  reads (q, g, V, Y — 0) from the storage unit SB 0  (SF 14 ) and calculates an inner product of the basis vector V and the Y — 0 (SF 15 ). The second commitment calculating unit SB 5  calculates a second commitment G=g^{X} (SF 16 ) and stores the second commitment G in the storage unit SB 0  (SF 17 ). 
     Operations of the vector challenge calculating unit SB 6  are described. 
     The vector challenge calculating unit SB 6  reads (g, h, {C_{ij}}, G, r, M) from the storage unit SB 0  (SF 18 ), calculates a vector challenge K=(c — 1, . . . , c_N)=H — {{0, 1}^N}(g, h, {C_{ij}, G, r, M} (SF 19 ), and stores the vector challenge in the storage unit SB 0  (SF 20 ). 
     Operations of the vector response calculating unit SB 7  are described. 
     The vector response calculating unit SB 7  reads ({X_{ij}}, {c_j}) from the storage unit SB 0  (SF 21 ), calculates ξ_{j}=c_jj} for j=1, . . . , N (SF 23 ), and stores a vector response Ξ=(ξ — 1, . . . , ξ_κ) in the storage unit SB 0  (SF 24 ). 
     Operations of the signature text output unit SB 8  are described. 
     The signature text output unit SB 8  reads a signature text (r, {C_{ij}}, G, Ξ) (SF 25 ) and outputs the signature text (r, {C_{ij}, G, Ξ} to the verifying apparatus VBN 0  (SF 26 ). 
     Now, a configuration and operations of the verifying apparatus VBN 0  are described with reference to  FIGS. 1 and 4 . 
     When the receiving apparatus RBN 1  receives a message M, an input unit VB 1  stores the message M and its signature text (r, {C_{ij}}, G, Ξ) in the storage unit VB 0  (VF 1 ). When the message M and the signature text (r, {C_{ij}}, G, Ξ) are stored in the storage unit VB 0 , a validity of the signature text is verified through the later-described verifying processes of a basis vector calculating unit VB 2 , a vector challenge VB 3 , a first validity verifying unit VB 4 , a second validity verifying unit VB 5 , and an output unit VB 6 . 
     Operations of the basis vector calculating unit VB 2  are described. 
     The basis vector calculating unit VB 2  reads (N, g, h, {C_{ij}}) from the storage unit VB 0  (VF 2 ), calculates a basis vector V=(u — 1, . . . , u_N)=H_{((Z/qZ)\{0})^{N}}(g, h, {C_{ij}} (VF 3 ), and stores the basis vector in the storage unit VB 0  (VF 4 ). 
     Operations of the vector challenge calculating unit VB 3  are described. 
     The vector challenge calculating unit VB 3  reads (ν, g, h, {C_{ij}, G, r, M} from the storage unit VB 0  (VF 5 ), calculates a vector challenge K=(c — 1, . . . , c_N)=H — {{0, 1}^ν}(g, h, {C_{ij}}, G, r, M) (VF 6 ), and stores the vector challenge in the storage unit VB 0  (VF 7 ). 
     Operations of the first commitment validity verifying unit VB 4  are described. 
     The first commitment validity verifying unit VB 4  reads (ν, g, h, ξ_{j}, {c_j}, {C_{c_jj}}) from the storage unit VB 0  (VF 8 ). It is verified whether or not H — {{0, 1}^{ν}(g, h, ξ_{j}, c_j, j, r)=C_{c_jj} for j=1, . . . , N is satisfied. For the j in which H — {{0, 1}^{ν}}(g, h, ξ_{j}, c_j, j, r)=C_{c_jj} is satisfied, b=1 is designated, and for the j in which H — {{0, 1^{ν}}(g, h, ξ_{j}, c_j, j, r)=C_{c_jj} is not satisfied, b=0 is designated, and the data are stored in the storage unit VB 0  (VF 9 ). In addition, the first commitment validity verifying unit VB 4  determines whether or not b corresponding to j=1, . . . , N is 0 (VF 10 ). When b=0 (VF 10 /YES), the verification is ended. When b=1 (VF 10 /NO), processes of the second validity verifying unit VB 5  are performed. 
     The second validity verifying unit VB 5  reads (b, g, h, V, Ξ, K, G) from the storage unit VB 0  (VF 12 ). Next, it is checked whether or not g^{&lt;V, Ξ&gt;}=h^{&lt;V, K&gt;}G is satisfied, and when the equation is satisfied, the b=1 stored in the step VF 9  is replaced with b=0 (VF 13 , VF 14 ). Here, the &lt;V, Ξ&gt; is set to a Schnorr response, and the &lt;V, K&gt; is set to a Schnorr challenge. 
     Finally, the output unit VB 6  outputs data indicating that the signature text is accepted if b=1 and data indicating that the signature text is rejected if b=0. 
     Second Exemplary Embodiment 
       FIG. 1  is a block diagram illustrating configurations of a signature apparatus SBN 0  and a verifying apparatus VBN 0  according to a second exemplary embodiment. The signature apparatus SBN 0  receives data by using a receiving apparatus RBN 0  and transmits data through a transmitting apparatus SeBN 0 . The verifying apparatus VBN 0  receives data by using a receiving apparatus RBN 1 . For example, LAN or Internet can be used for transmission/reception of data, but the present invention is not limited thereto. 
     Now, symbols used in the embodiment are described. 
     Symbol A denotes a cyclic group of which order is q. The number of bits of the order q is κ. Symbol g denotes a base point of the cyclic group A. In addition, it is assumed that, although the order q of the cyclic group A is publicized, the discrete logarithm problem associated with the cyclic group A is hard to falsify. 
     Symbol Z denotes a ring of all integers. Symbol N denotes a set of all natural numbers. An i-th component of a vector “a” is denoted by a_i. Inner product is denoted by &lt;•, •&gt;. An inner product of a vector “a” and a vector “b” is represented by &lt;a, b&gt;=a — 1b — 1+ . . . a_Nb_N. An X-value hash function of a set X is denoted by H_X. In addition, R_{κ+ζ}=Z∩[0, 2^{κ+ζ}] is defined. A hash value function of the set X is denoted by H_X. 
     Now, a key generating method is described. An xε(Z/qZ)\{0 } is taken at random, and h=g^x is obtained. A public key and a secret key are (g, h, q) and x, respectively. The signature apparatus SBN 0  stores the public key and the secret key in a storage unit SB 0 . It is assumed that the public key is reserved in a location, from which the verifying apparatus VBN 0  can acquire the public key in any type of an acquisition method. The acquisition method is, for example, means for using a means for reserving the public key in a public key table publicized on the Internet or means for directly acquiring the public key from the signature apparatus SBN 0 . The verifying apparatus VBN 0  acquires the public key and stores the public key in the storage unit SB 0  if needed. Details of the key generating method are disclosed in Non-Patent Document 11. Hereinafter, the description is made under the state that the verifying apparatus VBN 0  has already acquired the public key. 
     Specific operations of the signature apparatus SBN 0  according to the embodiment are described with reference to  FIGS. 1 ,  5 , and  6 . 
     When the receiving apparatus RBN 0  receives a message, the signature apparatus SBN 0  inputs the message to an input unit SB 1 . A signature text is generated and output in a committed vector selecting unit SB 2 , a first commitment calculating unit SB 3 , a basis vector calculating unit SB 4 , a second commitment calculating unit SB 5 , a vector challenge calculating unit SB 6 , a vector response calculating unit SB 7 , and a signature text output unit SB 8 . 
     Specific operations of the committed vector selecting unit SB 2  are described. 
     The committed vector selecting unit SB 2  reads (M, κ, ζ) from the storage unit SB 0  (SF 22 ). The committed vector selecting unit SB 2  selects a residue group X — {01}, . . . , X — {0N} from R_{κ+ζ} (SF 23 ). The X — {1j}=x+X — {0j} for all the j=1, . . . , N is calculated (SF 24 ). The X — {0j} for i=0 and j=1, . . . , N is set to Y — 0, and the X — {1j} for i=0 and j=1, . . . , N is set to Y — 1 (SF 25 ). The Y — 0 and the Y — 1 are referred to as i-th committed vectors. The Y — 0 and the Y — 1 are stored in the storage unit SB 0  (SF 26 ). 
     Processes of the first commitment calculating unit SB 3  are described. 
     The first commitment calculating unit SB 3  reads (N, {c_{j}}, {X_{ij}}) from the storage unit SB 0  (SF 27 ). The first commitment calculating unit SB 3  selects at random a bit column r of ν bits (SF 28 ). A hash value C_{ij}=H — {{0, 1}^ν}(g, h, X_{ij}, i, j, r) of data including the bit column r and the public key (g, h, q) is calculated (SF 29 ). Here, i=0 and 1. In the embodiment, the hash value C_{ij} calculated by the first commitment calculating unit SB 3  is set to a first commitment, and the {C_{ij}})_{i=0, 1, j=1, . . . , N} is set to a first commitment vector. 
     The first commitment vector (r, {C_{ij}}) calculated by the first commitment calculating unit SB 3  is stored in the storage unit SB 0  (SF 210 ). 
     Processes of the basis vector calculating unit SB 4  are described. 
     The basis vector calculating unit SB 4  reads (κ, ζ, N, g, h, {C_{ij}}) from the storage unit SB 0  (SF 211 ). 
     The basis vector calculating unit SB 4  calculates a hash value V=(u — 1, . . . , u_N)=H_{(R — {κ+ζ}\{0})^{N}}(g, h, {C_{ij}}) of data including the public key (q, g, h) and the first commitment {C_{ij}} (SF 212 ) and stores the V as a basis vector in the storage unit SB 0  (SF 213 ). 
     The second commitment calculating unit SB 5  is described. 
     The second commitment calculating unit SB 5  reads (V, Y — 0, G) from the storage unit SB 0  (SF 214 ) and calculates an inner product of the basis vector V and the Y — 0 (SF 215 ). The second commitment calculating unit SB 5  calculates a second commitment G=g^{X} (SF 216 ) and stores the second commitment G in the storage unit SB 0  (SF 217 ). 
     Operations of the vector challenge calculating unit SB 6  are described. 
     The vector challenge calculating unit SB 6  reads (g, h, {C_{ij}}, G, r, M) from the storage unit SB 0  (SF 18 ), calculates a vector challenge K=(c — 1, . . . , c_N)=H — {{0, 1}^N}(g, h, {C_{ij}, G, r, M} (SF 19 ), and stores the vector challenge in the storage unit SB 0  (SF 220 ). 
     Operations of the vector response calculating unit SB 7  are described. 
     The vector response calculating unit SB 7  reads (N, {c_{j}}, {X_{ij}}) from the storage unit SB 0  (SF 221 ), calculates ξ_{j}=X_{c_jj} for j=1, . . . , N (SF 223 ), and stores the ξ_{j}=X_{c_jj} as a vector response Ξ=(ξ — 1, . . . , ξ_κ) in the storage unit SB 0  (SF 24 ). 
     Operations of the signature text output unit SB 8  are described. 
     The signature text output unit SB 8  reads a signature text (r, {C_{ij}}, G, Ξ) (SF 225 ) and outputs the signature text (r, {C_{ij}, G, Ξ}) to the verifying apparatus VBN 0  (SF 226 ). 
     A configuration and operations of the verifying apparatus VBN 0  are described with reference to  FIGS. 1 and 7 . 
     When the receiving apparatus RBN 1  receives a message M, an input unit VB 1  stores the message M and its signature text (r, {C_{ij}}, G, Ξ) in the storage unit VB 0  (VF 1 ). When the message M and the signature text (r, {C_{ij}}, G, Ξ) are stored in the storage unit VB 0 , a validity of the signature text is verified through the later-described verifying processes in a basis vector calculating unit VB 2 , a vector challenge VB 3 , a first validity verifying unit VB 4 , a second validity verifying unit VB 5 , and an output unit VB 6 . 
     Operations of the basis vector calculating unit VB 2  are described. 
     The basis vector calculating unit VB 2  reads (N, κ, ζ, g, h, {C_{ij}}) from the storage unit VB 0  (VF 22 ), calculates a basis vector V=(u — 1, . . . , u_N)=H_{(R — {κ+ζ}\{0})^{N}}(g, h, {C_{ij}} (VF 23 ), and stores the basis vector in the storage unit VB 0  (VF 24 ). 
     Operations of the vector challenge calculating unit VB 3  are described. 
     The vector challenge calculating unit VB 3  reads (N, g, h, {C_{ij}}, G, r, M) from the storage unit VB 0  (VF 25 ), calculates a vector challenge K=(c — 1, . . . , c_N)=H — {{0, 1}^N}(g, h, {C_{ij}}, G, r, M) (VF 26 ), and stores the vector challenge in the storage unit VB 0  (VF 27 ). 
     Operations of the first commitment validity verifying unit VB 4  are described. 
     The first commitment validity verifying unit VB 4  reads (ν, g, h, ξ_{j}, {c_j}, {C_{c_jj}}) from the storage unit VB 0  (VF 28 ). It is verified whether or not H — {{0, 1}^{ν}}(g, h, ξ_{j}, c_j, j, r)=C_{c_jj} for j=1, . . . , N is satisfied. For the j in which H — {{0, 1}^{ν}}(g, h, ξ_{j}, c_j, j, r)=C_{c_jj} is satisfied, b=1 is designated, and for the j in which H — {{0, 1}^{ν}}(g, h, ξ_{j}, c_j, j, r)=C_{c_jj} is not satisfied, b=0 is designated (VF 210 ). In addition, the first commitment validity verifying unit VB 4  determines whether or not b corresponding to j=1, . . . , N is 0 (VF 210 ). When b=0 (VF 210 /NO), the verification is ended. When b=1 (VF 210 /YES), b=1 is stored in the storage unit VB 0  (VF 211 ). 
     The second validity verifying unit reads (b, g, h, V, Ξ, K, G) from the storage unit VB 0  (VF 212 ). It is checked whether or not g^{&lt;V, Ξ&gt;}=h^{&lt;V, K&gt;}G is satisfied, and when the equation is satisfied, the b=1 stored in the step VF 9  is replaced with b=0 (VF 213 , VF 214 )). Here, the &lt;V, Ξ&gt; is set to a Schnorr response, and the &lt;V, K&gt; is set to a Schnorr challenge. 
     Finally, the output unit VB 6  reads b from the storage unit VB 0  (VF 215 ), and outputs data indicating that the signature text is accepted if b=1 and data indicating that the signature text is rejected if b=0 (VF 216 ). 
     Third Exemplary Embodiment 
       FIG. 8  is a block diagram illustrating configurations of a signature apparatus SB 30  and a verifying apparatus VBN 30  according to a third exemplary embodiment. The signature apparatus SB 30  receives data by using a receiving apparatus RBN 30  and transmits data through a transmitting apparatus SeBN 30 . The verifying apparatus VBN 30  receives data by using a receiving apparatus RBN 31 . For example, LAN or Internet can be used for transmission/reception of data, but the present invention is not limited thereto. 
     Now, symbols used in the embodiment are described. 
     Symbol A denotes a cyclic group of which order is q. The number of bits of the order q is κ. Symbol g denotes a base point of the cyclic group A. It is assumed that, although the order q of the cyclic group A is publicized, the discrete logarithm problem associated with the cyclic group A is hard to falsify. 
     Symbol Z denotes a ring of all integers. Symbol N denotes a set of all natural numbers. An i-th component of a vector “a” is denoted by a_i. An inner product is denoted by &lt;•, •&gt;. An inner product of a vector “a” and a vector “b” is represented by &lt;a, b&gt;=a — 1b — 1+ . . . a_Nb_N modq. An X-value hash function of a set X is denoted by H_X. A basis vector is defined as V=(u — 1, . . . , u_{N})=(2^{0}, . . . , 2^{N−1}). 
     Now, a key generating method is described. An xε(Z/qZ)\{0} is taken at random, and h=g^x is obtained. A public key and a secret key are (g, h, q) and x, respectively. The signature apparatus SBN 30  stores the public key and the secret key in a storage unit SB 30 . It is assumed that the public key is reserved in a location, from which the verifying apparatus VBN 30  can acquire the public key in any type of an acquisition method. The acquisition method is, for example, a method of means for reserving the public key in a public key table publicized on the Internet or means for directly acquiring the public key from the signature apparatus SBN 30 . The verifying apparatus VBN 30  acquires the public key and reserves the public key in the storage unit SB 30  if needed. Details of the key generating method are disclosed in Non-Patent Document 11. Hereinafter, the description is made under the state that the verifying apparatus VBN 0  has already acquired the public key. 
     Specific operations of the signature apparatus SBN 30  according to the embodiment are described with reference to  FIGS. 8 ,  9 , and  10 . 
     When the receiving apparatus RBN 30  receives a message, the signature apparatus SBN 30  inputs the message to an input unit SB 31 . A signature text is generated and output in a committed vector selecting unit SB 32 , a second commitment calculating unit SB 33 , a first commitment calculating unit SB 34 , a vector challenge calculating unit SB 35 , a vector response calculating unit SB 36 , and a signature text output unit SB 37 . 
     The input unit SB 31  receives a message M from the receiving apparatus RBN 30  and stores the message M in a storage unit SB 30 . 
     Specific operations of the committed vector selecting unit SB 32  are described. 
     The committed vector selecting unit SB 2  reads (q, x) from the storage unit SB 30  (SF 32 ). The committed vector selecting unit SB 2  selects a residue group X — {01}, X — {0N} from (Z/qZ) (SF 33 ). The X — {1j}=x+X — {0j} for all the j=1, . . . , N is calculated (SF 34 ). The X — {0j} for i=0 and j=1, . . . , N is set to Y — 0, and the X — {1j} for i=0 and j=1, . . . , N is set to Y — 1 (SF 35 ). The Y — 0 and the Y — 1 are referred to as i-th committed vectors. The Y — 0 and the Y — 1 are stored in the storage unit SB 0  (SF 36 ). 
     Processes of the second commitment calculating unit SB 33  are described. 
     The second commitment calculating unit SB 33  reads (q, V, Y — 0, g) from the storage unit SB 30  (SF 37 ) and calculates a set X=&lt;{Y — 0, V}&gt;=Σ_jX — {0j}2^{j−1} modq (SF 38 ). The second commitment calculating unit SB 33  calculates a commitment G=g^{X} (SF 39 ) and stores the commitment G in the storage unit SB 30  (SF 310 ). 
     Processes of the first commitment calculating unit  34  are described. 
     The first commitment calculating unit  34  reads (ν, {X_{ij}}) (SF 311 ). The first commitment calculating unit  34  selects at random a bit column r of ν bits for each of the i, j (SF 312 ). A hash value C_{ij}=H — {{0, 1}^ν}(X_{ij}, r_{ij}) of data including the bit column r and the public key (g, h, q) is calculated (SF 313 ). Here, i=0 and 1. In the embodiment, the hash value C_{ij} calculated by the first commitment calculating unit SB 34  is set to a first commitment. 
     The first commitment vector {C_{ij}} calculated by the first commitment calculating unit SB 3  is stored in the storage unit SB 0  (SF 314 ). 
     Operations of the vector challenge calculating unit SB 35  are described. 
     The vector challenge calculating unit SB 35  reads (N, g, h, {C_ij}}, G, M) from the storage unit SB 0  (SF 315 ), calculates a vector challenge K=(c — 1, . . . , c_N)=H — {{0, 1}^N}(g, h, {C_{ij}}, G, M) (SF 316 ), and stores the vector challenge in the storage unit SB 0  (SF 317 ). 
     Operations of the vector response calculating unit SB 36  are described. 
     The vector response calculating unit SB 36  reads ({c_{j}}, {X_{ij}}) from the storage unit SB 0  (SF 318 ), calculates ξ_{j}=X_{c_jj} for j=1, . . . , N (SF 319 ), and stores a vector response Ξ=(ξ — 1, . . . , ξ_κ) in the storage unit SB 30  (SF 320 , SF 321 ). 
     Operations of the signature text output unit SB 37  are described. 
     The signature text output unit SB 37  reads a signature text ({C_{ij}}, {r_{c_jj}}, G, Ξ) (SF 322 ) and outputs the signature text ({C_{ij}}, {r_{c_jj}}, G, Ξ) to the verifying apparatus VBN 30  (SF 323 ). 
     A configuration and operations of the verifying apparatus VBN 30  are described with reference to  FIGS. 8 and 11 . 
     When the receiving apparatus RBN 31  receives a message M, input unit VB 31  stores the message M and its signature text ({C_{ij}}, {r_{c_jj}}, G, Ξ) in a storage unit VB 0  (VF 31 ). When the message M and the signature text ({C_{ij}}, {r_{c_{jj}}, G, Ξ} are stored in the storage unit VB 30 , a validity of the signature text is verified through the later-described verifying processes of a vector challenge VB 32 , a first validity verifying unit VB 34 , a second validity verifying unit VB 33 , and an output unit VB 35 . 
     Operations of the vector challenge calculating unit VB 32  are described. 
     The vector challenge calculating unit VB 32  reads (N, g, h, {C_{ij}}, G, M) from the storage unit VB 30  (VF 32 ), calculates a vector challenge K=(c — 1, . . . , c_N)=H — {{0, 1}^N}(g, h, {C_{ij}}, G, M) (VF 33 ), and stores the vector challenge in the storage unit VB 30  (VF 34 ). 
     Operations of the first commitment validity verifying unit VB 33  are described. 
     The first commitment validity verifying unit VB 33  reads (ν, ξ_j, r_{c_jj}, {C_{c_jj}}) from the storage unit VB 30  (VF 35 ). It is verified whether or not H — {{0, 1}^{ν}}(ξ_j, r_{c_jj}, j, r)=C_{c_jj} for each of j=1, . . . , N is satisfied. For the j in which H — {{0, 1}^{ν}}(g, h, ξ_{j}, c_j, j, r)=C_{c_jj} is satisfied, b=1 is designated, and for the j in which H — {{0, 1}^{ν}}(g, h, ξ_{j}, c_j, j, r)=C_{c_jj} is not satisfied, b=0 is designated (VF 36 ). In addition, the first commitment validity verifying unit VB 4  determines whether or not b corresponding to j=1, . . . , N is 0 (VF 37 ). When b=0 (VF 37 /NO), the verification is ended. When b=1 (VF 37 /YES), b=1 is stored in the storage unit VB 0  (VF 38 ). 
     The second validity verifying unit  33  reads (b, g, h, V, Ξ, K, G) from the storage unit VB 30  (VF 39 ). It is checked whether or not g^{&lt;V, Ξ&gt;}=h^{&lt;V, K&gt;}G is satisfied, and when the equation is satisfied, the b=1 stored in the step VF 9  is replaced with the b=0 (VF 310 , VF 311 ). Here, the &lt;V, Ξ&gt; is set to a Schnorr response, and the &lt;V, K&gt; is set to a Schnorr challenge. 
     Finally, the output unit VB 35  reads b from the storage unit VB 30  (VF 312 ), and the output unit VB 35  outputs data indicating that the signature text is accepted if b=1 and data indicating that the signature text is rejected if b=0. 
     Example 1 
     An example of the first exemplary embodiment to which a straight-line extractable proving scheme of a discrete logarithm (Non-Patent Document 12) is applied is described. 
     Configurations of a proving apparatus and a verifying apparatus are illustrated in  FIG. 12 . As illustrated in  FIG. 12 , in the example, there are the proving apparatus PSPBN 0  and the verifying apparatus VSPBN 0 , which correspond to the signature apparatus SBN 0  and the verifying apparatus VBN 0  according to the first exemplary embodiment, respectively. 
     In the example, instead of a message M, a predetermined ID or random number is used. Operations of each units are the same as those of the first exemplary embodiment except that the ID or the random number is used instead of the message M. 
     In addition, in the example, a straight-line extractable proving scheme of the discrete logarithm may be applied to the second or third exemplary embodiment. 
     Example 2 
     Example 2 is an example where a verifier-designated proving scheme is applied to a proving apparatus DVSSBN 0  and a verifying apparatus DVSVBN 0  according to Example 1. 
     The proving apparatus DVSSBN 0  receives data by using a receiving apparatus DVSRBN 0  and transmits data through a communication apparatus DVSCBN 0 . The verifying apparatus DVSVBN 0  receives data by using a receiving apparatus DVSRBN 1 . For example, LAN or Internet can be used as a channel used for data communication. 
     The proving apparatus DVSSBN 0  includes an input unit DVSSB 1 , a key validity proof text verifying unit DVSSB 2 , a proving unit DVSSB 3 , and a storage unit DVSSB 0 . The data input through the input unit DVSSB 0  of the proving apparatus DVSSBN 0  is stored in the storage unit DVSSB 0 . 
     The verifying apparatus DVSVBN 0  includes an output unit DVSVB 1 , a key validity proof text generating unit DVSSB 2 , a verifying unit DVSVB 3 , and a storage unit DVSVB 0 . The data input through the communication unit DVSCBN 0  is stored in the storage unit DVSVB 0 . 
     The key validity proof text generating unit DVSSB 2  reads required data from the storage unit DVSSB 0  and stores a calculation result thereof in the storage unit DVSSB 0 . 
     Similarly, the proving unit DVSSB 3  reads required data from the storage unit DVSSB 0  and stores a calculation result thereof in the storage unit DVSSB 0 . 
     The key validity proof text generating unit DVSVB 2  reads required data from the storage unit DVSVB 0  and stores a calculation result thereof in the storage unit DVSVB 0 . 
     Similarly, the verifying unit DVSVB 3  reads required data from the storage unit DVSVB 0  and stores a calculation result thereof in the storage unit DVSVB 0 . 
     The output unit DVSVB 1  reads a required data from the storage unit DVSSB 0  and outputs the data. 
     An instance which is used for proving the corresponding secret is shared by the proving apparatus DVSSBN 0  and the verifying apparatus DVSVBN 0  in advance. For example, there is a method where, the to-be-proven secret is transmitted and received through a channel by using a transmitting/receiving apparatus, and the secret is hard-coded when the proving apparatus DVSSBN 0  and the verifying apparatus DVSVBN 0  are produced. 
     The proving apparatus DVSSBN 0  is assumed to have the to-be-proven secret in advance. For example, there is a method where the secret is transmitted through the channel by using the transmitting/receiving apparatus or a method where the secret is hard-coded when the proving apparatus DVSSBN 0  is produced. 
     A key generating method in the verifying apparatus DVSVB 0  is described. An xε(Z/qZ)^* is taken at random, and h=g^x is obtained. A public key and a secret key are (g, h, q) and x, respectively. The storage unit DVSVB 0  of the verifying apparatus DVSVBN 0  stores the public key (g, h, q) and the secret key x. An acquisition method for the public key (g, h, q) is, for example, an method of reserving the public key in a public key table published on the Internet or a method of directly acquiring the public key from the verifying apparatus DVSVBN 00 . The proving apparatus DVSSBN 0  acquires the public key and stores the public key in the storage unit DVSSB 0  if needed. Details of the key generating method are disclosed in Non-Patent Document 11. Hereinafter, the description is made under the state that the proving apparatus DVSSBN 0  has already acquired the public key. 
     Now, operations of each unit of the apparatuses are described. 
     The verifying apparatus DVSVBN 0  operates key validity proof text verifying unit DVSVB 2  to generate a proof text indicating that the verifying apparatus has a secret key corresponding to the its own public key. The key validity proof text generating unit DVSVB 2  which generates the proof text performs operations which are the same as those of the proving apparatus PSPBN 0  ( FIG. 12 ) according to Example 1. 
     The verifying apparatus DVSVBN 0  transmits the generated proof text through the communication apparatus DVSCBN 1  to the proving apparatus DVSSBN 0 . The proving apparatus DVSSBN 0  receives the proof text through the communication apparatus DVSCBN 0 . The proving apparatus DVSPBN 0  verifies a validity of the proof text in the key validity proof text verifying unit DVSSB 2 . The key validity proof text verifying unit DVSSB 2  performs operations which are the same as those of the verifying apparatus VSPBN 0  ( FIG. 12 ) according to Example 1. 
     The proving apparatus DVSPBN 0  proves whether or not there is a secret corresponding to the instance in the proving unit DVSSB 3  or whether or not there is a secret key corresponding to the public key of a verifier. The verifying apparatus DVSVBN 0  verifies a validity of the proof in the verifying unit DVSVB 3 . 
     Since the proving unit DVSSB 3  and the verifying unit DVSVB 3  are the same as a proving method and a verifying method in Non-Patent Document 13, the description thereof is not repeated. Details thereof are disclosed in Non-Patent Document 13. 
     Example 3 
     In Example 3, a crypto scheme added to the configurations and operations of Example 1 is implemented (see  FIG. 14 ). 
     In the example, there are an encrypting apparatus EVEN 0  and a decrypting apparatus DVEN 1 . The encrypting apparatus EVEN 1  receives data by using a receiving apparatus RBS 0  and transmits data through a transmitting apparatus SeBE 0 . The decrypting apparatus DBEN 0  receives data by using a receiving apparatus RBE 1 . For example, LAN or Internet can be used for data communication, but not limited thereto. 
     A key generating method in the encrypting apparatus EBEN 0  is described. An xε(Z/qZ)^* is taken at random, and h=g^x is obtained. A public key and a secret key are (g, h, q) and x, respectively. A storage unit EEB 0  of the encrypting apparatus EBEN 0  stores the public key and the secret key. The public key is reserved in a location from which the decrypting apparatus DBEN 0  can acquires the public key in any type of an acquisition method. The acquisition method is, for example, means for reserving the public key in a public key table publicized on the Internet or means for directly acquiring the public key from the encrypting apparatus EBEN 0 . The decrypting apparatus DBEN 0  acquires the public key and reserves the public key in the storage unit DBE 0  if needed. Details of the key generating method are disclosed in Non-Patent Document 11. Hereinafter, the description is made under the state that the decrypting apparatus DVEN 0  has already acquired the public key in advance. 
     In the encrypting apparatus EBEN 0 , processes of an input unit EBE 1 , an encrypting unit EBE 2 , and a proving unit EBE 3  are sequentially performed. 
     When the input unit EBE 1  receives a to-be-encrypted message mεG, the message mεG is stored in the storage unit EBE 0 . 
     In the encrypting unit EBE 2 , an ElGamal cipher text of the message mεG is generated. More specifically, the encrypting unit EBE 2  reads required data from the storage unit EBE 0  and selects yεZ/qZ at random. I=g^{y} and J=mh^{y} are calculated, and a cipher text (I, J) is generated. The generated cipher text (I, J) is stored in the storage unit EBE 0 . 
     The proving unit EBE 3  reads required data from the storage unit EBE 0  and generates a proof text P which is associated with a discrete logarithm problem of I with g as a base in the same manner as that of the proving apparatus PSPBN 0  (see  FIG. 12 ) according to Example 1. Finally, a cipher text (I, J, P) is stored in the storage unit EBE. 
     When the receiving apparatus RBE 1  receives cipher text (I, J, P), the receiving apparatus RBE 1  stores the cipher text (I, J, P) in the storage unit DBE 0 . Processes of a verifying unit DBE 1 , a decrypting unit DBE 3 , and an output unit DBE 4  are sequentially performed. 
     The verifying unit DBE 1  reads required data from the storage unit DBE and verifies the cipher text P in the same manner as that of the verifying apparatus VSPBN 0  (see  FIG. 12 ) according to Example 1. If it is determined that the cipher text P is valid, b=1 is designated, and if not, b=0 is designated. The determination result is stored in the storage unit DBE. 
     The decrypting apparatus DBEN 0  reads b from the storage unit DBE 0 . If b=0, data indicating that “cipher text is invalid” is output, and if b=1, processes of the decrypting unit DBE 3  are performed. 
     The decrypting unit DBE 3  decrypts the ElGamal cipher text by using a typical decrypting operation. More specifically, the decrypting unit DBE 3  reads required data from the storage unit DBE 0  and calculates m′=J/I^y. The calculation result is stored in the storage unit DBE 0 . 
     The output unit DBE 4  reads the m′ from the storage unit DBE 0  and outputs the m′. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a block diagram illustrating configurations of a signature apparatus and a verifying apparatus. 
         FIG. 2  is a flowchart of processes of the signature apparatus. 
         FIG. 3  is a flowchart of processes of the signature apparatus. 
         FIG. 4  is a flowchart of processes of the verifying apparatus. 
         FIG. 5  is a flowchart of processes of the signature apparatus. 
         FIG. 6  is a flowchart of processes of the signature apparatus. 
         FIG. 7  is a flowchart of processes of the verifying apparatus. 
         FIG. 8  is a block diagram illustrating configuration of a signature apparatus and a verifying apparatus. 
         FIG. 9  is a flowchart of processes of the signature apparatus. 
         FIG. 10  is a flowchart of processes of the signature apparatus. 
         FIG. 11  is a flowchart of processes of the verifying apparatus. 
         FIG. 12  is a block diagram illustrating configuration of a signature apparatus and a verifying apparatus. 
         FIG. 13  is a block diagram illustrating configuration of a proving apparatus and a verifying apparatus. 
         FIG. 14  is a block diagram illustrating configuration of an encrypting apparatus and a decrypting apparatus. 
     
    
    
     REFERENCE NUMERALS 
     SBN 0 : signature apparatus 
     SB 0 : storage unit 
     SB 1 : input unit 
     SB 2 : committed vector selecting unit 
     SB 3 : first commitment calculating unit 
     SB 4 : basis vector calculating unit 
     SB 5 : second commitment calculating unit 
     SB 6 : vector challenge calculating unit 
     SB 7 : vector response calculating unit 
     SB 8 : signature text output unit 
     VBN 0 : verifying apparatus 
     VB 0 : storage unit 
     VB 1 : input unit 
     VB 2 : basis vector calculating unit 
     VB 3 : vector challenge calculating unit 
     VB 4 : first validity verifying unit 
     VB 5 : second validity verifying unit 
     VB 6 : output unit

Technology Classification (CPC): 7