Abstract:
The present invention makes it possible, in encrypted data verification, to avoid the leaking of information related to the original plaintext, thereby ensuring safety. The system of the present invention is provided with: means ( 103  in FIG.  1 ) for generating first and second auxiliary data for verifying whether or not the Hamming distance of a plaintext between a first encrypted data in which input data is encrypted and is recorded in a storage device, and a second encrypted data obtained by encrypting input data of a target to be checked is equal to or less than a predetermined value; and means ( 402  and  403  in FIG.  1 ) for taking the difference between the first encrypted data recorded in the storage device, and the second encrypted data, and determining, using the first and second auxiliary data, whether or not the Hamming distance of the plaintext corresponding to the difference between the first encrypted data and the second encrypted data is equal to or less than the predetermined value.

Description:
DESCRIPTION OF RELATED ART 
       [0001]    The present invention is based upon the benefit of priority from Japanese Patent Application No. 2012-157265 (filed on Jul. 13, 2012), the entire contents of which are incorporated herein by reference. 
         [0002]    The present invention is related to an encrypted data (ciphertext) checking (verifying) system, method, and program. 
       BACKGROUND ART 
       [0003]    Recently, along with the popularization of cloud computing, data of a user is stored in a calculation resource that is connected to a network, and service based on the data has been spreading rapidly. In such service, an opportunity to deal with sensitive data of the user has been increased. Therefore, it is important to guarantee the safe management of the data on the user. Under such an environment, research and development of a technology, that enables to manage that data in encrypted state in an open network environment, and execute a search, a statistics processing and the like by using the data without decryption, has been performed actively. 
         [0004]    In addition, recently, a crime, which exploits the vulnerability of personal authentication such like using a password or a magnetic card, occurs frequently. Therefore, a biometric identification technology having further high safety based on a biometric feature, such as a fingerprint and vein, attracts considerable attention. In the biometric identification, in order to verify authentication information, it is necessary to store a template related to biological information in a database (DB). The biological information such the fingerprint and vein is data that is basically not changed through the lifetime. If the biological information is leaked, serious damage occurs by the leakage of the information. Therefore, the biological information is information for which the confidentiality is required the most. Thus, it is necessary to prevent impersonation even if the template is leaked. 
         [0005]    Thus, a biometric identification technology which protects templates (a template protection type biometric identification technology), in which the authentication is performed while template information remains concealed, has become important. 
         [0006]    For example, in Patent literature 1, a method is disclosed in which biometric identification is performed using, as a template, data that is obtained by representing fingerprint data as points on a polynomial expression, adding random points to the points, and concealing the fingerprint data. 
         [0007]    However, in the above-described method disclosed in Patent literature 1, it is known that there is a problem whether or not the biological information is protected with sufficient strength when the biometric identification is repeated plural times. 
         [0008]    In Non-Patent literature 1, a method is disclosed in which biological information is protected by masking a template that is stored in a DB through a random Bose-Chaudhuri-Hocquenghem (BCH) code word. In the technology disclosed in Non-Patent literature 1, a biometric identification template is generated using biological information Z and confidential information S.  FIG. 5  is a diagram based on FIG. 2 of Non-Patent literature 1, and the feature extraction, statistical analysis, quantization, and the like in FIG. 2 of Non-Patent literature 1 are omitted. The enrollment of a template is performed as follows. 
         [0009]    (1) The confidential information S is input to an encoder (ENC). The ENC performs error correcting coding (ECC) on the confidential information S, and generates a code word C. A binary BCH code of parameters (K, s, and d) is used as the ECC. “K” indicates the length of the code word, and “s” indicates the number of information symbols, and “d” indicates the number of correctable errors. 
         [0010]    (2) An XOR (exclusive OR) between “C” and “Z”, that is, “W 2 =C(+)Z” is calculated (hereinafter, the symbol “(+)” indicates bitwise XOR). 
         [0011]    (3) “S” is input to a cryptographic (one-way) hash function H, such as a secure hash algorithm (SHA)-1 or the like, and the hash value H (S) is calculated. 
         [0012]    (4) “W 2 ” and “H(S)” are stored in a DB as template information. 
         [0013]    The verification of whether or not the template, that has been generated as described in (1) to (4), and the other biological information Z′, are obtained from an identical person, is performed as follows. 
         [0014]    (1) The XOR between “Z′” and “W 2 ”, that is, “C′=W 2 (+)Z′=C(+)(Z(+)Z′)” is calculated. 
         [0015]    (2) “C” is input to a decoder (DEC), and error-correcting decoding of the BCH code is performed to calculate “S”. 
         [0016]    (3) “5” is input to the cryptographic (one-way) hash function H, such as the SHA-1 or the like, to calculate a hash value H(S′). 
         [0017]    (4) “H(S)” is read from the DB, and it is verified whether or not “H(S)=H(S′)” is satisfied. When “H(S)=H(S′)” is satisfied, it is determined that the template and the biological information Z′ are obtained from an identical person. When “H(S)=H(S′)” is not satisfied, it is determined that the template and the biological information Z′ are respectively obtained from different persons. 
         [0018]    The method illustrated in  FIG. 5  does not depend on the obtaining method of the biological information Z. Therefore, generally, the method illustrated in  FIG. 5  may be regarded as a method that verifies whether or not the encrypted data is generated by encrypting a plaintext of which distance to presented data is in certain distance. 
       CITATION LIST 
     Patent Literature 
       [0000]    
       
         [PTL 1] Japanese Laid-open Patent Publication No. 2006-158851 
       
     
       Non Patent Literature 
       [0000]    
       
         [NPL 1]: Pim Tuyls, Anton H. M. Akkermans, Tom A. M. Kevenaar, Geert-Jan Schrijen, Asker M. Bazen and Raymond N. J. Veldhuis, “Practical Biometric Authentication with Template Protection”, Proceedings of AVBPA 2005, Lecture Notes in Computer Science, Vol. 3546, Springer Verlag, pp. 436-446, (2005) 
       
     
       SUMMARY OF INVENTION 
     Technical Problem 
       [0021]    The analysis of the related arts is described below. 
         [0022]    As a problem of the above-described related arts, in the verification of the encrypted data in the DB, it is probable that information about the plaintext (decrypted data) is leaked to an administrator or the like who performs the verification. The reason is as follows. 
         [0023]    For example, in the above-described Patent literature 1, the degree of confidentiality of the encrypted data is not sufficient. 
         [0024]    In addition, in the above-described Non-Patent literature 1, in order to enable verification of whether or not the encrypted data is obtained by data that is within a certain Hamming distance from the presented data, it is necessary that the plaintext information is transmitted at the time of the verification. As described above, when the verification processing is performed multiple times, it is probable that the information on the original plaintext is leaked, and for example, the sufficient safety may not be ensured when the data base administrator or the like who performs the verification processing has a malicious intention. 
         [0025]    Thus, the present invention is made in view of the above problems. An object of the present invention is to provide a system, a method, and a program in which in verification of an encrypted data, leakage of information on the original plaintext is able to be avoided, and the safety is able to be ensured. 
       Solution to Problem 
       [0026]    In the present invention, there is provided an encrypted data verification system that includes
       means for generating first and second auxiliary data that are used to verify, that a Hamming distance between a plaintext of a first encrypted data which is encrypted from input data and registered to a storage apparatus, and a plaintext of a second encrypted data which is encrypted from input data of a target to be verified, is a predetermined certain value or less, for the first encrypted data and the second encrypted data respectively; and   means for obtaining a difference between the first encrypted data that is registered to the storage apparatus and the second encrypted data that is obtained by encrypting the input data of the target to be verified, and determining whether or not the Hamming distance of the plaintexts, which corresponds to the difference between the first encrypted data and the second encrypted data, is the predetermined certain value or less, using the first and second auxiliary data.       
 
         [0029]    In the present invention, there is provided a biometric identification system that includes the encrypted data verification system. 
         [0030]    In the present invention, there is provided an encrypted data verification method that includes
       generating first and second auxiliary data that are used to verify, that a Hamming distance between a plaintext of a first encrypted data which is encrypted from input data and registered to a storage apparatus, and a plaintext of a second encrypted data which is encrypted from input data of a target to be verified, is a predetermined certain value or less, for the first encrypted data and the second encrypted data respectively; and   obtaining a difference between the first encrypted data that is registered to the storage apparatus and the second encrypted data that is obtained by encrypting the input data of the target to be verified, and determining whether or not the Hamming distance of the plaintexts, which corresponds to the difference between the first encrypted data and the second encrypted data, is the predetermined certain value or less, using the first and second auxiliary data.       
 
         [0033]    In the present invention, there is provided a program that causes a computer to execute
       a processing of generating first and second auxiliary data that are used to verify, that a Hamming distance between a plaintext of a first encrypted data which is encrypted from input data and registered to a storage apparatus, and a plaintext of a second encrypted data which is encrypted from input data of a target to be verified, is a predetermined certain value or less, for the first encrypted data and the second encrypted data respectively; and,   a processing of obtaining a difference between the first encrypted data that is registered to the storage apparatus and the second encrypted data that is obtained by encrypting the input data of the target to be verified, and determining whether or not the Hamming distance of the plaintexts, which corresponds to the difference between the first encrypted data and the second encrypted data, is the predetermined certain value or less, using the first and second auxiliary data. In the present invention, there is provided a computer readable recording medium (magnetic/optical recording medium or semiconductor recording medium) to which the program is recorded.       
 
       Advantageous Effects of Invention 
       [0036]    In the present invention, in verification of an encrypted data, information leakage of the original plaintext can be avoided, and the safety can be ensured. 
     
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         [0037]      FIG. 1  is a diagram illustrating a configuration according to a first exemplary embodiment of the present invention. 
           [0038]      FIG. 2  is a diagram illustrating a configuration according to a second exemplary embodiment of the present invention. 
           [0039]      FIGS. 3(A) and 3(B)  are diagrams respectively illustrating a data registration phase and an encrypted data verification phase according to the first exemplary embodiment of the present invention. 
           [0040]      FIGS. 4(A) and 4(B)  are diagrams respectively illustrating a data registration phase and an encrypted data verification phase according to the second exemplary embodiment of the present invention. 
           [0041]      FIG. 5  is a diagram illustrating a scheme in Non-Patent Literature 1. 
       
    
    
     DESCRIPTION OF EMBODIMENTS 
       [0042]    Exemplary embodiments of the present invention are described below. In the exemplary embodiments of the present invention, input data of a target to be checked is encrypted, and registration data (registered data) that is used to perform verification for the input data is encrypted, and a Hamming distance of plaintext is used as an indicator of ambiguity of determination of the verification (matching). Not only the encrypted registration data, the input data for the verification is also encrypted through an encrypting method having a high concealment strength. Even when the verification is performed multiple times using identical input data, key information that is used to perform concealment of the data is changed each time the verification is performed. Therefore, even when the verification is performed multiple times, a possibility about leakage of information about the plaintext may be reduced, and the attack resistance is enhanced to contribute to the improvement of the security. 
         [0043]    In an exemplary embodiment, a system includes means ( 103  in  FIGS. 1 and 303  in  FIG. 2 ) that generates first and second auxiliary data, that are described later. In addition, the system further includes determination means ( 402  and  403  in  FIG. 1 , and  502  and  503  in  FIG. 2 ) determines whether or not a Hamming distance of plaintexts, which corresponds to a difference between a first encrypted data that is described later and a second encrypted data that is also described later, is a predetermined certain value or less. 
         [0044]    Such a first encrypted data is obtained by encrypting input data and is registered to a storage apparatus. Such a second encrypted data is is obtained by encrypting input data of a target to be checked (verified). Such first and second auxiliary data are used to verify that the Hamming distance of the plaintexts between the first encrypted data and the second encrypted data is the predetermined certain value or less. 
         [0045]    In addition, the above-described determination means obtains the difference between the above-described first encrypted data and the above-described second encrypted data. The above-described determination means determines whether or not the Hamming distance of the plaintexts, which corresponds to the difference between the above-described first encrypted data and the above-described second encrypted data, is a predetermined certain value or less, by using the above-described first and second auxiliary data. 
         [0000]    In the exemplary embodiment, the system generates the above-described encrypted data, from the calculation result of an XOR between a code word that is obtained by encoding a key, that is used to perform encoding on the plaintext of the above-described input data, through an error-correcting code having linearity, and the above-described plaintext. Then the system calculates the above-described encrypted data, that is registered to the above-described storage apparatus, and above-described first and second auxiliary data, that are respectively related to the above-described encrypted data of the above-described input data of a target to be verified, based on an XOR between an inner product of the above-described key and a constant, and a cryptographic (one-way) hash function that is executed on the above-described encrypted data and a random number. In addition, in the exemplary embodiment, in a hash function that is used to determine the verification result, the system guarantees that a hash value of the sum of two pieces of data can be calculated from respective hash values of the two pieces of data. As a result, the system enables verification processing between encrypted data, which is not able to be achieved in the above-described Non-Patent literature 1. 
         [0046]    As described above, in the verification processing between encrypted data, data that is transmitted by a user who performs the verification is also encrypted with an encrypting key that is unknown to a database administrator or the like. Therefore, even when the verification processing is performed multiple times, or when the database administrator or the like, who executes the verification processing, has a malicious intention, leakage of information that is related to the original plaintext is able to be avoided. Some exemplary embodiments are described below. 
       First Exemplary Embodiment 
       [0047]    Referring to  FIG. 1 , a system according to a first exemplary embodiment of the present invention includes a registration data generation apparatus  100 , a storage apparatus  200 , a data concealment apparatus  300 , and a specification data verification apparatus  400 . It is noted that these apparatuses may be configured to form a single apparatus in a single site, by integrating themselves, or may be configured so as to form distributed arrangement and to be connected to each other through a communication means. 
         [0048]    The registration data generation apparatus  100  includes an encrypting unit  101 , a key generation unit  102 , and a registration auxiliary data generation unit  103 . 
         [0049]    The encrypting unit  101  accepts following data as inputs. That is, the encrypting unit  101  accepts input data that is to be concealed (concealment target data), and a key that is used to conceal the input data. The encrypting unit  101  outputs the encrypted data that is obtained by executing concealment processing on the input data, by using the key. 
         [0050]    The key generation unit  102  generates the key that is used by the encrypting unit  101  to conceal the input data, and outputs the key to the encrypting unit  101  and to the registration auxiliary data generation unit  103 . 
         [0051]    The registration auxiliary data generation unit  103  accepts the following data as inputs. That is, the registration auxiliary data generation unit  103  accepts the input data, the encrypted data that is output from the encrypting unit  101 , and the key that is output from the key generation unit  102 . The registration auxiliary data generation unit  103  generates and outputs the following data. That is, the registration auxiliary data generation unit  103  generates, input data that corresponds to encrypted data that is output from an encrypting unit  301  of the data concealment apparatus  300 , and data (auxiliary data) that is used to determine that a Hamming distance with the input data that has been input to the encrypting unit  101  is a predetermined certain value or less (within a certain numeric value). 
         [0052]    The encrypted data that is output from the encrypting unit  101  of the registration data generation apparatus  100  satisfies the following relationship. That is, when an encrypted data that is obtained by encrypting input data m 1  by using a key k 1  by the encrypting unit  101  is treated as “c 1 ”, and an encrypted data that is obtained by encrypting input data m 2  by using a key k 2  by the encrypting unit  101  is treated as “c 2 ”, the sum of “c 1 ” and “c 2 ”, that is, “c 1 +c 2 ” becomes an encrypted data that is obtained by encrypting input data m 1 +m 2  by using a key k 1 +k 2 . 
         [0053]    The storage apparatus  200  includes an identifier management unit  201 , an encrypted data storage unit  202 , and an auxiliary data storage unit  203 . The encrypted data storage unit  202  and the auxiliary data storage unit  203  respectively store the encrypted data and the registration auxiliary data that has been output from the registration data generation apparatus  100 . The encrypted data storage unit  202  and the auxiliary data storage unit  203  may be configured as DBs (or, may have file configurations). 
         [0054]    The encrypted data storage unit  202  and the auxiliary data storage unit  203  respectively output encrypted data and auxiliary data that correspond to an identifier that is input from the specification data verification apparatus  400 , under the control of the identifier management unit  201 , when encrypted data are verified. 
         [0055]    The identifier management unit  201  of the storage apparatus  200  manages an identifier that is used to uniquely identify encrypted data and auxiliary data that are input from the registration data generation apparatus  100 . 
         [0056]    When an identifier is input from the specification data verification apparatus  400 , the identifier management unit  201  issues an instruction of output of encrypted data that corresponds to the input identifier, to the encrypted data storage unit  202 . In addition, when the identifier is input from the specification data verification apparatus  400 , the identifier management unit  201  issues an instruction of output of auxiliary data that corresponds to the input identifier, to the auxiliary data storage unit  203 . 
         [0057]    The encrypted data storage unit  202  stores encrypted data that has been output from the encrypting unit  101  of the registration data generation apparatus  100 . When an instruction of output of encrypted data is input from the identifier management unit  201 , the encrypted data storage unit  202  outputs the corresponding encrypted data. 
         [0058]    The auxiliary data storage unit  203  stores auxiliary data that has been output from the registration auxiliary data generation unit  103  of the registration data generation apparatus  100 . When an instruction of output of encrypted data is input from the identifier management unit  201 , the auxiliary data storage unit  203  outputs the corresponding auxiliary data. 
         [0059]    The data concealment apparatus  300  includes an encrypting unit  301 , a key generation unit  302 , and an auxiliary data generation unit  303 . 
         [0060]    The encrypting unit  301  accepts following data as inputs. That is, the encrypting unit  301  accepts input data that is a concealment target (input data of a target to be verified), and a key that is used to perform concealment of the input data. The encrypting unit  301  outputs encrypted data that is obtained by executing the encrypting processing for the input data using the key. 
         [0061]    The key generation unit  302  generates the key that is used to perform concealment of the input data by the encrypting unit  301 . In addition, the key generation unit  302  outputs the generated key to the encrypting unit  301  and the auxiliary data generation unit  303 . 
         [0062]    The auxiliary data generation unit  303  accepts following data as inputs. That is the auxiliary data generation unit  303  accepts the input data, the encrypted data that has been output from the encrypting unit  301 , and the key that has been output from the key generation unit  302 . The auxiliary data generation unit  303  outputs auxiliary data based on such inputs. Such auxiliary data is used to determine whether or not a Hamming distance between the input data (plaintext) that corresponds to the encrypted data that is output from the encrypting unit  101  of the registration data generation apparatus  100  (encrypted registration data), and the data (plaintext) that has been input to the encrypting unit  301 , is a predetermined certain value or less (within a certain numeric value). In other words, such auxiliary data is auxiliary information that is used to determine that the input data (plaintext) that corresponds to the encrypted registration data is matched with the data of the target to be verified (plaintext), which has been input to the encrypting unit  301 , when the Hamming distance between these data is the certain value or less (or, less than the certain value). And also such auxiliary data is auxiliary information that is used to determine that the input data is not matched with the data of the target to be verified when the Hamming distance between the data exceeds the above-described certain value (or, the certain value or more). 
         [0063]    The encrypted data that is output from the encrypting unit  301  of the data concealment apparatus  300  is calculated through the same method (manner) as the encrypting unit  101 . That is, when the encrypted data that is obtained by encrypting the input data m 1  using the key k 1  is treated as “c 1 ”, and the encrypted data that is obtained by encrypting the input data m 2  using the key k 2  is treated as “c 2 ”, the sum of “c 1 ” and “c 2 ”, that is, “c 1 +c 2 ” becomes the encrypted data that is obtained by encrypting the input data m 1 +m 2  by using the key k 1 +k 2 . 
         [0064]    The specification data verification apparatus  400  includes an identifier holding unit  401 , an encrypted data subtraction unit  402 , a match determination unit  403 , and a control unit  404 . 
         [0065]    The identifier holding unit  401  accepts an identifier as an input. The identifier holding unit  401  issues an instruction to output of encrypted data data and auxiliary data that correspond to the identifier that has been input to the storage apparatus  200 , for the identifier management unit  201  of the storage apparatus  200 . 
         [0066]    The encrypted data subtraction unit  402  accepts following data as inputs. That is the encrypted data subtraction unit  402  accepts one piece of data among encrypted data (encrypted registration data) that are stored in the encrypted data storage unit  202  of the storage apparatus  200 , and the encrypted data that is output from the encrypting unit  301  of the data concealment apparatus  300 . The encrypted data subtraction unit  402  outputs a difference between the two pieces of input encrypted data c 1  and c 2 , that is c 1 -c 2 . 
         [0067]    When the encrypted data that is obtained by encrypting the input data m 1  using the key k 1  is treated as “c 1 ”, and the encrypted data that is obtained by encrypting the input data m 2  using the key k 2  is treated as “c 2 ”, due to the feature of the encrypting unit  101  and the encrypting unit  301 , the difference c 1 -c 2  between the two encrypted data c 1  and c 2  becomes an encrypted data that is obtained by encrypting input data m 1 -m 2  using a key k 1 -k 2 . 
         [0068]    The match determination unit  403  accepts the following data as inputs. That is, match determination unit  403  accept one piece of data among auxiliary data that are stored in the auxiliary data storage unit  203  of the storage apparatus  200 , the auxiliary data that is output from the auxiliary data generation unit  303  of the data concealment apparatus  300 , and the difference between the two pieces of encrypted data, which is output from the encrypted data subtraction unit  402 . 
         [0069]    The match determination unit  403  outputs whether or not a Hamming distance between the plaintexts m 1  and m 2  that respectively correspond to the two pieces of encrypted data c 1  and c 2  that has been input to the encrypted data subtraction unit  402 , is the predetermined certain value or less. 
         [0070]    The control unit  404  controls the communication and the like when data is transmitted and received between the data concealment apparatus  300  and the specification data verification apparatus  400 . 
         [0071]    Next, an operation in the first exemplary embodiment is described with reference to the flow diagram illustrated in  FIG. 3 . The operation of the encrypted data verification system according to the first exemplary embodiment is mainly divided into two phases of a data registration phase, and an encrypted data verification phase. 
         [0072]    In the data registration phase, input data is input to the registration data generation apparatus  100 , and such input data is encrypted, and registered to the storage apparatus  200  with auxiliary data. 
         [0073]    In the encrypted data verification phase, data that has been input to the data concealment apparatus  300  is encrypted. In the encrypted data verification phase, it is determined whether or not the encrypted data and auxiliary data, that have been generated through the encrypting process, correspond to plaintexts that are close (the Hamming distance of which is the predetermined certain value or less) to the plaintext that corresponds to the encrypted data and auxiliary data in the storage apparatus, which are specified by an identifier that is input separately. 
         [0074]    In the data registration phase, first, input data that is a target of concealment (concealment target data) is input to the encrypting unit  101  of the registration data generation apparatus  100  (step A 1  in  FIG. 3(A) ). 
         [0075]    Next, the key generation unit  102  of the registration data generation apparatus  100  generates a key that is used to perform concealment of the input data. After that, the key generation unit  102  outputs the generated key to the encrypting unit  101  and the registration auxiliary data generation unit  103  (step A 2  in  FIG. 3(A) ). 
         [0076]    Next, the encrypting unit  101  of the registration data generation apparatus  100  calculates encrypted data that is obtained by encrypting the input data, based on the input data and the key. After that, the encrypting unit  101  stores the calculated encrypted data in the encrypted data storage unit  202  (step A 3  in  FIG. 3(A) ). 
         [0077]    Next, following data are input to the registration auxiliary data generation unit  103 . That is, the input data that has been input in the step A 1 , the key that has been generated in the step A 2 , the encrypted data that has been generated in the step A 3  are input to the registration auxiliary data generation unit  103 . After that, auxiliary data that has been output from the registration auxiliary data generation unit  103  is stored in the auxiliary data storage unit  203  of the storage apparatus  200  (step A 4  in  FIG. 3(A) ). 
         [0078]    The identifier management unit  201  assigns a unique identifier to the data that has been input to the storage apparatus  200 , by the above-described processing. By assigned identifier, the data that has been input to the storage apparatus  200  can be referred (read). 
         [0079]    In the encrypted data verification phase, first, an identifier is input to the identifier holding unit  401  of the specification data verification apparatus  400 . Encrypted data (encrypted registration data) that corresponds to the input identifier is input from the encrypted data storage unit  202  of the storage apparatus  200  to the encrypted data subtraction unit  402 . In addition, auxiliary data that corresponds to the input identifier is input from the auxiliary data storage unit  203  to the match determination unit  403  (step B 1  in  FIG. 3(B) ). 
         [0080]    Next, input data (data of a target to be checked) is input to the encrypting unit  301  of the data concealment apparatus  300  (step B 2  in  FIG. 3(B) ). 
         [0081]    Next, the key generation unit  302  of the data concealment apparatus  300  generates a key that is used to perform concealment of the input data that has been input in the step B 2 . After that, the key generation unit  302  outputs the generated key to the encrypting unit  301  and the auxiliary data generation unit  303  (step B 3  in  FIG. 3(B) ). 
         [0082]    Next, the encrypting unit  301  calculates encrypted data that is obtained by encrypting the input data, based on the input data that has been input in the step B 2  and the key that has been input in the step B 3 . After that, the encrypting unit  301  inputs the calculated encrypted data to the encrypted data subtraction unit  402  of the specification data verification apparatus  400  (step B 4  in  FIG. 3(B) ). 
         [0083]    The encrypted data subtraction unit  402 , to which the encrypted data has been respectively input from the encrypted data storage unit  202  of the storage apparatus  200  and the encrypting unit  301  of the data concealment apparatus  300 , outputs a difference between the two pieces of the encrypted data input, to the match determination unit  403  (step B 5  in  FIG. 3(B) ). 
         [0084]    Next, auxiliary data are input to the match determination unit  403  from the auxiliary data storage unit  203  of the storage apparatus  200 , and from the auxiliary data generation unit  303  of the data concealment apparatus  300 , that are controlled by the control unit  404 , respectively (step B 6  in  FIG. 3(B) ). In this case, the auxiliary data storage unit  203  and the auxiliary data generation unit  303  may respectively input the pieces of auxiliary data to the match determination unit  403  by performing the communication in cooperation. 
         [0085]    As described above, to the match determination unit  403 , the difference between the two pieces of encrypted data is input from the encrypted data subtraction unit  402  in the step B 5 , and the auxiliary data are respectively input from the auxiliary data storage unit  203  and the auxiliary data generation unit  303  in the step B 6 . The match determination unit  403  determines whether or not a Hamming distance between the plaintext of the encrypted data that has been input to the encrypted data subtraction unit  402  in the step B 1  and, the plaintext of the encrypted data that has been input to the encrypted data subtraction unit  402  in the step B 4 , is a predetermined certain value or less, from these input data. The match determination unit  403  outputs the determination result (step B 7  in  FIG. 3(B) ). 
         [0086]    It is noted that the apparatuses  100 ,  200 ,  300 , and  400  of  FIG. 1  may be integrated into a single computer system, or may be configured as respective apparatuses. Alternatively, the units in each of the apparatuses  100 ,  200 ,  300 , and  400  may be configured as the respective apparatuses. Processing of each of the units in each of the apparatuses may be achieved by a program that is executed in a computer. In this case, in the present invention, a recording medium (semiconductor memory or magnetic/optical disk) to which the program is recorded may be provided. 
       Second Exemplary Embodiment 
       [0087]    A second exemplary embodiment of the present invention is described below. In the above-described encrypted data verification system according to the first exemplary embodiment, input data and an identifier are input to the system, and verification is performed between a plaintext of encrypted data that corresponds to the identifier, and the input data. On the contrary, in the second exemplary embodiment, merely input data is input to the system, and the system outputs an identifier of encrypted data that is matched with the input data. 
         [0088]    In the above-described first exemplary embodiment, the verification called as “one-to-one verification” can be achieved, and in the second exemplary embodiment, “1-to-many verification” can be achieved. 
         [0089]    Referring to  FIG. 2 , the system according to the second exemplary embodiment includes a registration data generation apparatus  100 , a storage apparatus  200 , a data concealment apparatus  300 , and a data verification apparatus  500 . A configuration of the registration data generation apparatus  100 , the storage apparatus  200 , and the data concealment apparatus  300  is similar to that of the above-described first exemplary embodiment. In the system according to second exemplary embodiment, the configuration of the data verification apparatus  500  is different from the above-described system according to first exemplary embodiment. 
         [0090]    The registration data generation apparatus  100  includes an encrypting unit  101 , a key generation unit  102 , and a registration auxiliary data generation unit  103 . The encrypting unit  101  accepts input data that is a concealment target (concealment target data) and a key that is used to perform concealment of the input data, as inputs. In addition, the encrypting unit  101  outputs encrypted data that is obtained by executing the encrypting processing for the input data using the key. 
         [0091]    The key generation unit  102  generates the key that is used to perform concealment of the input data by the encrypting unit  101 . After that (key generation), the key generation unit  102  outputs the generated key to the encrypting unit  101  and the registration auxiliary data generation unit  103 . 
         [0092]    The registration auxiliary data generation unit  103  accepts the input data, the encrypted data that has been output from the encrypting unit, and the key that has been output from the key generation unit  102 , as inputs. After that, the registration auxiliary data generation unit  103  outputs data (auxiliary data) that is used to determine that a Hamming distance, between input data that corresponds to encrypted data that is output from the encrypting unit  301  of the data concealment apparatus  300  and the input data that is input to the encrypting unit  101 , is within a certain numeric value, from the accepted inputs. 
         [0093]    The encrypted data that is output from the encrypting unit  101  of the registration data generation apparatus  100  satisfies the following relationship. That is, the relationship includes that when the encrypting unit  101  treats an encrypted data that is obtained by encrypting input data m 1  using a key k 1 , as “c 1 ”, and treats an encrypted data, that is obtained by encrypting input data m 2  using a key k 2 , as “c 2 ”, the sum of “c 1 ” and “c 2 ”, that is, “c 1 +c 2 ” becomes an encrypted data that is obtained by encrypting input data m 1 +m 2  by using a key k 1 +k 2 . 
         [0094]    The storage apparatus  200  includes an identifier management unit  201 , an encrypted data storage unit  202 , and an auxiliary data storage unit  203 . The storage apparatus  200  stores encrypted data and registration auxiliary data that is output from the registration data generation apparatus. In addition, the storage apparatus  200  outputs encrypted data and auxiliary data that correspond to an identifier that is input from the data verification apparatus  500 , when the encrypted data is verified. 
         [0095]    The identifier management unit  201  manages an identifier that is used to uniquely identify encrypted data and auxiliary data that are input from the registration data generation apparatus  100 . After that, when the identifier has been input from the data verification apparatus  500 , the identifier management unit  201  issues an instruction of output of encrypted data and auxiliary data that correspond to the input identifier, to the encrypted data storage unit  202  and the auxiliary data storage unit  203 . The encrypted data storage unit  202  stores the encrypted data that has been output from the encrypting unit  101  of the registration data generation apparatus  100 . After that, when the instruction of output of encrypted data has been input from the identifier management unit  201 , the encrypted data storage unit  202  outputs the corresponding encrypted data. The auxiliary data storage unit  203  stores the auxiliary data that has been output from the registration auxiliary data generation unit  103  of the registration data generation apparatus  100 . After that, when the instruction of output of encrypted data has been input from the identifier management unit  201 , the auxiliary data storage unit  203  performs outputs the corresponding auxiliary data. 
         [0096]    The data concealment apparatus  300  includes an encrypting unit  301 , a key generation unit  302 , and an auxiliary data generation unit  303 . 
         [0097]    The encrypting unit  301  accepts input data that is a concealment target (concealment target data), and a key that is used to perform concealment of the input data, as inputs. After that, the encrypting unit  301  performs output of encrypted data that is obtained by executing the encrypting processing for the input data by using the key. The key generation unit  302  generates a key that is used to perform concealment of the input data by the encrypting unit  301 , and outputs the generated key to the encrypting unit  301  and the auxiliary data generation unit  303 . 
         [0098]    The auxiliary data generation unit  303  accepts the input data, the encrypted data that has been output from the encrypting unit, and the key that has been output from the key generation unit  302 , as inputs. After that, based on the input data, the auxiliary data generation unit  303  outputs auxiliary data that is used to determine that a Hamming distance, between input data that corresponds to the encrypted data that is output from the encrypting unit  101  of the registration data generation apparatus  100  and the input data that has been input to the encrypting unit  301 , is less than a predetermined certain value. 
         [0099]    The encrypted data that is output from the encrypting unit  301  of the data concealment apparatus  300  is calculated by the same method as that of the encrypting unit  101 . That is, when the encrypting unit  301  treats the encrypted data that is obtained by encrypting the input data m 1  using the key k 1 , as “c 1 ”, and treats the encrypted data that is obtained by encrypting the input data m 2  using the key k 2 , as “c 2 ”, the sum of “c 1 ” and “c 2 ”, that is, “c 1 +c 2 ” becomes the encrypted data that is obtained by encrypting the input data m 1 +m 2  using the key k 1 +k 2 . 
         [0100]    The data verification apparatus  500  includes an entire-data request unit  501 , an encrypted data subtraction unit  502 , a match determination unit  503 , a control unit  504 , and an identifier output unit  505 . 
         [0101]    The entire-data request unit  501  inputs an instruction of sequential read of all data that are stored in the storage apparatus, to the identifier management unit  201 , in response to an instruction from the identifier output unit  505 . 
         [0102]    The encrypted data subtraction unit  502  accepts one piece of data among encrypted data that are stored in the encrypted data storage unit  202  of the storage apparatus  200 , and the encrypted data that is output from the encrypting unit of the data concealment apparatus  300 , as inputs. After that, the encrypted data subtraction unit  502  performs output of a difference c 1 -c 2  between the input encrypted data c 1  and c 2 . 
         [0103]    When the encrypted data that is obtained by encrypting the input data m 1  using the key k 1 , is treated as “c 1 ”, and the encrypted data that is obtained by encrypting the input data m 2  using the key k 2 , is treated as “c 2 ” due to the feature of the encrypting unit  101  and the encrypting unit  301 , a difference c 1 -c 2  between the two encrypted data c 1  and c 2  becomes the encrypted data that is obtained by encrypting the input data m 1 -m 2  using the key k 1 -k 2 . 
         [0104]    The match determination unit  503  accepts the following data, as inputs. That is, the match determination unit  503  accepts one piece of data among auxiliary data that are stored in the auxiliary data storage unit  203  of the storage apparatus  200 , the auxiliary data that is output from the auxiliary data generation unit  303  of the data concealment apparatus  300 , and the encrypted data that is output from the encrypted data subtraction unit  402 . 
         [0105]    The encrypted data subtraction unit  502  outputs whether or not a Hamming distance, between the plaintexts m 1  and m 2  that respectively correspond to the two pieces of encrypted data c 1  and c 2  that has been input to the encrypted data subtraction unit  502 , is a predetermined certain value or less (or less than the certain value), based on the accepted pieces of input data. 
         [0106]    The control unit  504  controls the communication when data is transmitted and received between the data concealment apparatus  300  and the data verification apparatus  500 . 
         [0107]    The identifier output unit  505  accepts, an identifier that has been used by the identifier management unit  201  to issue an instruction of output of data to the encrypted data storage unit  202  and the auxiliary data storage unit  203 , and a determination result that has been output from the match determination unit  503 , as inputs. When the match determination unit  503  determines that the matching is performed successfully (that is, the determination result indicates that input plaintext matches plaintext of registered data), the identifier output unit  505  outputs the identifier that has been input from the identifier management unit  201 . 
         [0108]    An operation in the second exemplary embodiment is described below with reference to the flow diagram illustrated in  FIG. 4 . The operation of the encrypted data verification system according to the second exemplary embodiment is divided into two phases of a data registration phase, and an encrypted data verification phase. 
         [0109]    In the data registration phase, input data is input to the registration data generation apparatus  100 , and such input data is encrypted, and registered to the storage apparatus  200  with auxiliary data. In the encrypted data verification phase, data that has been input to the data concealment apparatus  300  is encrypted. In addition, in the encrypted data verification phase, the identifier, corresponds to the encrypted data, stored in the storage apparatus  200  is output. That is, the identifier, corresponds to the encrypted data, to which the plaintext, that is close to the encrypted data and the auxiliary data (the Hamming distance between the plaintext and plaintexts of those data is small) that are generated through encrypting of input data, is encrypted. 
         [0110]    In the data registration phase, first, input data that is a conceal target is input to the encrypting unit  101  of the registration data generation apparatus  100  (step C 1  in  FIG. 4(A) ). 
         [0111]    Next, the key generation unit  102  of the registration data generation apparatus  100  generates a key that is used to perform concealment of the input data. After that, the key generation unit  102  outputs the generated key to the encrypting unit  101  and the registration auxiliary data generation unit  103  (step C 2  in  FIG. 4(A) ). 
         [0112]    Next, the encrypting unit  101  calculates encrypted data that is obtained by encrypting the input data, from the input data and the key. After that, the encrypting unit  101  stores the calculated encrypted data in the encrypted data storage unit  202  (step C 3  in  FIG. 4(A) ). 
         [0113]    Next, the input data that has been input in the step C 1 , the key that has been generated in the step C 2 , and the encrypted data that has been generated in the step C 3  are input to the registration auxiliary data generation unit  103 . After that, an output (auxiliary data) of the registration auxiliary data generation unit  103  is stored in the auxiliary data storage unit  203  of the storage apparatus  200  (step C 4  in  FIG. 4(A) ). 
         [0114]    Through the above-described processing, a unique identifier is assigned to data (encrypted data and auxiliary data) that is stored in the storage apparatus  200 , in the identifier management unit  201 . The pieces of data that are stored in the storage apparatus  200  can be referred (read) by the assigned identifier. 
         [0115]    In the encrypted data verification phase, first, input data is input to the encrypting unit of the data concealment apparatus  300  (step D 1  in FIG.  4 (B)). 
         [0116]    Next, the key generation unit  302  of the data concealment apparatus  300  generates a key that is used to perform concealment of the input data. After that, the key generation unit  302  outputs the generated key to the encrypting unit  301  and the auxiliary data generation unit  303  (step D 2  in  FIG. 4(B) ). 
         [0117]    Next, the encrypting unit  301  calculates encrypted data that is obtained by encrypting the input data, from the input data that has been input in the step D 1 , and the key that has been input in the step D 2 . After that, the encrypting unit  301  inputs the calculated encrypted data to the encrypted data subtraction unit  502  of the data verification apparatus  500  (step D 3  in  FIG. 4(B) ). 
         [0118]    Next, an identifier is input from the entire-data request unit  501  to the identifier management unit  201 . After that, the encrypted data that corresponds to the input identifier is input from the encrypted data storage unit  202  of the storage apparatus  200  to the encrypted data subtraction unit  502 . In addition, auxiliary data that corresponds to the input identifier is input from the auxiliary data storage unit  203  to the match determination unit  503  (step D 4  in  FIG. 4(B) ). 
         [0119]    The encrypted data subtraction unit  502  to which the encrypted data has been respectively input from the encrypted data storage unit  202  of the storage apparatus  200  and the encrypting unit  301  of the data concealment apparatus  300 , outputs a difference between the two pieces of input encrypted data, to the match determination unit  503  (step D 5  in  FIG. 4(B) ). 
         [0120]    Next, the pieces of auxiliary data are respectively input from the auxiliary data storage unit  203  of the storage apparatus  200  and the auxiliary data generation unit  303  of the data concealment apparatus  300 , to the match determination unit  503  (step D 6  in  FIG. 4(B) ). In this case, the auxiliary data storage unit  203  and the auxiliary data generation unit  303 , that are controlled by the control unit  504 , respectively input those data to the match determination unit  503  by, communicating in coordination. 
         [0121]    As described above, to the match determination unit  503 , the difference between the two pieces of encrypted data is input in the step D 5 , and the auxiliary data is input in the step D 6 . 
         [0122]    The match determination unit  503  determines whether or not a Hamming distance between the plaintext of the encrypted data that has been input to the encrypted data subtraction unit  502  in the step D 3 , and the plaintext of the encrypted data that has been input to the encrypted data subtraction unit  502  in the step D 4 , is a predetermined certain value or less, from these input data. In addition, the match determination unit  503  outputs the determination result (step D 7  in  FIG. 4(B) ). 
         [0123]    When it is determined that the matching is successfully performed in the result of the step D 7 , that is, the determination result indicates that input plaintext matches plaintext of registered data, the identifier output unit  505  performs output of the identifier that has been input to the identifier management unit  201  in step D 4  (step D 8  in  FIG. 4(B) ). 
         [0124]    The processing from the steps D 4  to D 8  is repeated for all identifiers (corresponding to all of encrypted data and auxiliary data) that are stored in the storage apparatus  200 , which are managed by the identifier management unit  201  of the storage apparatus  200 . 
         [0125]    It is noted that the apparatuses  100 ,  200 ,  300 , and  500  in  FIG. 2  may be integrated into a single computer system. In addition, the apparatuses may be configured as respective apparatuses. Alternatively, the units in each of the apparatuses  100 ,  200 ,  300 , and  400  may be configured as respective apparatuses. The processing of each of the units in each of the apparatuses in  FIG. 1  may be achieved by a program that is executed by a computer. In this case, in the present invention, a recording medium (semiconductor memory or magnetic/optical disk) to the program has been recorded is provided. The above exemplary embodiments are described below with reference to a further specific example. 
       First Example 
       [0126]    Next, a first example of the present invention is described in detail with reference to  FIG. 1 . The first example is a specific example of the above-described first exemplary embodiment. 
         [0127]    In the data registration phase, first, as input data, a binary sequence (string) “Z” of “N” bits is input to the encrypting unit  101  of the registration data generation apparatus  100 . 
         [0128]    Next, the key generation unit  102  of the registration data generation apparatus  100  generates a key (random number of “K” bits) “S”, and outputs the generated key to the encrypting unit  101  and the registration auxiliary data generation unit  103 . 
         [0129]    Next, the encrypting unit  101  calculates encrypted data “W 1 ” of “N” bits, which is obtained by calculating an XOR between a code word “C” of “N” bits, which has been obtained by encoding the input key “S” of “K” bits using a binary BCH code, and the input data “Z” of “N” bits (see, the following Eqn. (1)). In addition, the encrypting unit  101  stores the calculated encrypted data “W 1 ” in the encrypted data storage unit  202  of the storage apparatus  200 . 
         [0000]        W 1 =C (+) Z   (1)
 
         [0130]    Here, calculation symbol “(+)” indicates a bitwise XOR. It is assumed that the binary BCH code used herein is a code that outputs data of “N”-bits from input data of “K”-bits (“N”&gt;“K”). And, It is also assumed that such BCH code is a code that guarantees that Hamming distance between different code words is at least “d” or more. 
         [0131]    Next, the input data “Z”, the key “S”, and the encrypted data “W 1 ” are input to the registration auxiliary data generation unit  103 . The registration auxiliary data generation unit  103  calculates auxiliary data “W 2 ”, based on the inputs, in accordance with the following Eqn. (2). 
         [0000]        W 2=( c,S )(+) h ( W 1, n )  (2)
 
         [0132]    Here, in the above-described Eqn. (2), the “c” is a constant of “K” bits. The “n” is a random number of “k” bits (“k” is a security parameter). The security parameter is a parameter that indicates the strength of the safety, and is a predetermined value that has been defined by the system. The “(c,S)” indicates an inner product. That is, “(A,B)” indicates an inner product of “A” and “B”, when regarding the pieces of data of “A” and “B” (of which size respectively is “K=(m*k) bits”) as arranged vectors into which divided “A” and “B” is divided for each k bits (it is assumed that the calculation is performed on the Galois field GF (2 k)). In addition, “(+)” indicates a bitwise XOR. In addition, “h” is a cryptographic (one-way) hash function which generates the output data of “k” bits (for example, SHA-256 or the like). 
         [0133]    H (x,y,z) is defined as a function that is represented by the following Eqn. (3). 
         [0000]        H ( x,y,z )=( c,z )(+) h ( y,z )  (3)
 
         [0134]    H (x,y,z) satisfies the following Eqn. (4). 
         [0000]        H ( a 1 ,b 1 ,c 1)(+) H ( a 2 ,b 2 ,c 2)= H ( a 1(+) a 2 ,b 1 ,c 1)(+) h ( b 2 ,c 2)  (4)
 
         [0135]    In addition, for a random number “r” of “(K−k)” bits, a code word data that is obtained by executing error-correcting coding to the data represented with following Eqn. (5), by using the BCH code, is treated as “C 3 ” (here, “//” is a calculation symbol that indicates bit concatenation). 
         [0000]      “ h ( W 1 ,N )// r   (5)”
 
         [0000]    And “W 3 ” is calculated from “C 3 ” and “Z” in accordance with the following Eqn. (6). 
         [0000]        W 3 =C 3(+) Z   (6)
 
         [0136]    The registration auxiliary data generation unit  103  registers the set of “(W 2 ,W 3 )” that has been calculated in accordance with the above-described Eqns. (2) and (6), to the auxiliary data storage unit  203 , as auxiliary data. 
         [0137]    In the above-described processing, to the data that has been input to the storage apparatus  200 , a unique identifier is assigned by the identifier management unit  201 . After that, the input data can be referred by using the assigned identifier. 
         [0138]    Hereinafter, encrypted data “W 1 ”, and pieces of auxiliary data “W 2 ” and “W 3 ” that are associated with an identifier “i” are respectively represented as “W 1 [i]”, “W 2 [i]”, and “W 3 [i]”. 
         [0139]    In the encrypted data verification phase, first, the identifier “i” is input to the identifier holding unit  401  of the specification data verification apparatus  400 . The encrypted data “W 1 [i]” that corresponds to the input identifier “i” is read (referred) from the encrypted data storage unit  202  of the storage apparatus  200 , and is input to the encrypted data subtraction unit  402 . In addition, the pieces of auxiliary data “W 2 [i]” and “W 3 [i]” that correspond to the input identifier “i” are read (referred) from the auxiliary data storage unit  203 , and are input to the match determination unit  403 . 
         [0140]    Next, binary sequence input data “Z” of “N” bits (data to be verified) is input to the encrypting unit  301  of the data concealment apparatus  300 . 
         [0141]    Next, the key generation unit  302  of the data concealment apparatus  300  generates a key (random number of “K” bits) “S” that is used to perform concealment of the input data “Z”, and outputs the generated key to the encrypting unit  301  and the auxiliary data generation unit  303 . 
         [0142]    The encrypting unit  301  of the data concealment apparatus  300  calculates the encrypted data “W 1 ′” that is obtained by calculating an XOR between the code word “C” and the input data “Z”. Here, “C” is obtained by performing error-correcting coding by using the binary BCH code on the key “S”, that has been input from the key generation unit  302 . After that, the encrypting unit  301  inputs the calculated encrypted data “W 1 ′”, to the encrypted data subtraction unit  402  of the specification data verification apparatus  400 . 
         [0000]        W 1 ′=C ′(+) Z′   (7)
 
         [0143]    To the encrypted data subtraction unit  402 , the encrypted data “W 1 ′” from the encrypting unit  301  of the data concealment apparatus  300 , and the encrypted data “W 1 [i]” that corresponds to the identifier “i” from the encrypted data storage unit  202  of the storage apparatus  200  are input. The encrypted data subtraction unit  402  calculates a difference (XOR) between the two pieces of input encrypted data “W 1 ′” and “W 1 [i]”, that is, calculated with following Eqn. (8). 
         [0000]      “ W 1′(+) W[i]   (8)”
 
         [0144]    And the encrypted data subtraction unit  402  outputs the calculated difference to the match determination unit  403 . 
         [0145]    Next, for a random number “ns” and an element (generator) “g” of a group “G” that has been defined beforehand (multiplicative group “Zp”), the control unit  404  calculates following Eqn. (9). 
         [0000]      “ g   —   s=g**ns”   (9)”.
 
         [0146]    In the Eqn. (9), it is assumed that “g**ns” indicates the ns-th power of “g” on the group “G” (multiplicative group “Zp”) (“**” is an exponentiation operator). The group “G” is a cyclic group for the multiplicative. The group “G” is constituted by a multiplicative group “Zp (=Z/pZ)” of a digit number “p”, using “p” as a prime number. And the value of “g**ns” is given by a mod “p” using the prime number “p” as modulo. 
         [0147]    The control unit  404  outputs “W 3 [i]” and “g_s” to the auxiliary data generation unit  303 . 
         [0148]    Next, the auxiliary data generation unit  303  of the data concealment apparatus  300  applies decoding processing of the binary BCH code to a value that is obtained by calculating an XOR between “W 3 [i]” and the input data “Z”, that is calculated with following Eqn. (10). 
         [0000]        W 3[ i ](+) Z′   (10)
 
         [0149]    As the result of the calculation, the auxiliary data generation unit  303  obtains “h” that is the decryption result. 
         [0150]    The auxiliary data generation unit  303  calculates “W 2 ′” and “g_c”, from the key “S”, the encrypted data “W 1 ′”, “h′”, “g”, and “g_s”, and the random number “nc”, based on the following Eqns. (11a) and (11b), and outputs the calculated “W 2 ” and “g_c” to the match determination unit  403  of the data verification apparatus  400 . 
         [0000]        W 2′= H ( S′,W 1′, g   —   s**nc )(+) h′   (11a)
 
         [0000]        g   —   c=g**nc   (11b)
 
         [0151]    In the above-described Eqn. (11a), when “g” is treated as a generator of the multiplicative group “Zp”, “g” and “p” are published, and two parties “X” and “Y” respectively calculate “A=g**ns (=g**ns mod p)” and “B=g**nc (=g**nc mod p)” using “ns” and “nc” (private keys). “X” transmits “A” to “Y”, and “Y” transmits “B” to “X”. “X” calculates “B**ns=g**(nc) (=g**(ns*nc)mod p)” using “ns” and “B” that has been received from “Y”. And “Y” calculates “A**nc=g** (ns*nc) (=g**(ns*nc)mod p)” using “nc” and “A” that has been received from “X”. The calculated “g**(ns*nc)mod p” is used as a key of common key encryption by “X” and “Y” (Diffie-Hellman key exchange). Even if a third party obtains “A” and “B” by eavesdropping or the like, there is no method of calculating “g** (ns*nc)mod p” from “A” and “B”, so that it is difficult to generate the key. In addition, in the calculation of “g**ns”, a value, on which Diffie-Hellman key exchange has been performed, is set as a random number component (NONCE), so that for example, defense against replay attacks is achieved in the application to biometric identification, that is described later. 
         [0152]    Next, the match determination unit  403  applies the decoding processing of the binary BCH code, to the difference between the two pieces of input encrypted data “W 1 ′” and “W[i]”, that is calculated with following Eqn. (12). 
         [0000]        W 1′(+) W[i]   (12)
 
         [0153]    And the match determination unit  403  calculates “T” that is the decryption result of the difference between the two pieces of encrypted data “W 1 ” and “W[i]”. 
         [0154]    In addition, the match determination unit  403  determines whether or not a calculation result of an XOR between “H(T,W 1 ′,g_c**ns)(=(g_c**ns,T)+h(W 1 ′,g_c**ns))” and “W 2 ′” is equal to “W 2 [i]”. The equation “H(T,W 1 ′,g_c**ns)(=(g_c**ns,T)+h(W 1 ′,g_c**ns))” is calculated using the decryption result “T” of the difference between the two pieces of encrypted data “W 1 ′” and “W[i]”, “W 1 ′”, and “g_c**ns”. “W 2 ′” is calculated in accordance with the Eqn. (11a). In other words, the match determination unit  403  checks (verifies) whether or not the next Eqn. (13) is satisfied. 
         [0000]        W 2[ i]=H ( T,W 1 ′,g   —   c**ns )(+) W 2′  (13)
 
         [0155]    When the above-described Eqn. (13) is satisfied, the match determination unit  403  determines that a Hamming distance between the original data (plaintext) of “W 1 [i]” and the input data (plaintext) “Z′” is “d” or less. When the above-described Eqn. (13) is not satisfied, the match determination unit  403  determines that the Hamming distance between the original data (plaintext) of “W 1 [i]” and the input data (plaintext) “Z′” exceeds “d”. After that, the match determination unit  403  outputs the determination result. It is noted that, in the above-described BCH coding, a Hamming distance between given different code words is assumed to be a value that exceeds “d” at least. 
         [0156]    Here, “g_s**nc(=(g**ns)**nc)” in the auxiliary data “W 2 ′=H(S′,W 1 ′,g_s**nc)(+)h′” that is generated by the auxiliary data generation unit  303  of the data concealment apparatus  300 , and “g_c**ns(=(g**nc)**ns)” in the match determination unit  403  of the specification data verification apparatus  400 , may be generated, for example, in both of the auxiliary data generation unit  303  and the match determination unit  403  through the known Diffie-Hellman key exchange method. 
       Second Example 
       [0157]    Next, a second example is described in detail with reference to  FIG. 2 . The second example is a specific example of the above-described second exemplary embodiment. 
         [0158]    In the data registration phase, first, as input data, a binary sequence “Z” of “N” bits is input to the encrypting unit  101  of the registration data generation apparatus  100 . 
         [0159]    Next, the key generation unit  102  of the registration data generation apparatus  100  generates a random number “S” of “K” bits. In addition, the key generation unit  102  outputs the generated random number “S” (key) to the encrypting unit  101  and the registration auxiliary data generation unit  103 . 
         [0160]    Next, the encrypting unit  101  calculates encrypted data “W 1 ” that is obtained by calculating an XOR between the code word “C” that is obtained by coding the input key “S” through the binary BCH code, and the input data “Z”. In addition, the encrypting unit  101  stores the calculated encrypted data “W 1 ” in the encrypted data storage unit  202 . The binary BCH code used herein is a code outputs data of “N”-bits from input data of “K”-bits (“N”&gt;“K”). And also, such BCH code is a code that guarantees that Hamming distance between different code words is at least “d” or more. 
         [0161]    Next, the input data “Z”, the key “S”, and the encrypted data “W 1 ” are input to the registration auxiliary data generation unit  103 . The registration auxiliary data generation unit  103  calculates “W 2 ”, based on the inputs, in accordance with the following Eqn. (14). 
         [0000]        W 2=( c,S )(+) h ( W 1 ,n )  (14)
 
         [0162]    Here, in the above-described Eqn. (14), “c” is a constant of “K” bits. In addition, “n” is a random number of “k” bits (“k” is a security parameter). In addition, “(A,B)” indicates an inner product of “A” and “B” when each of the two pieces of data “A” and “B” of “K=(m*k)” bits is regarded as a vector into which “A” and “B” is divided for each “k” bits (it is assumed that the calculation is performed on the Galois field GF (2 k)). The calculation symbol “(+)” indicates a bitwise XOR. The symbol “h” is a cryptographic (one-way) hash function in which the output corresponds to k bits (for example, SHA-256 or the like). 
         [0163]    In addition, “H(x,y,z)” is defined as a function that is represented by the Eqn. (15) (same as the above-described Eqn. (3)). 
         [0000]        H ( x,y,z )=( c,x )(+) h ( y,z )  (15)
 
         [0164]    For the random number “r” of “(K−k)” bits, the code word data that is obtained by encoding the data represented with following Eqn. (16), by using the BCH code, is treated as “C 3 ” (here, “//” is a symbol that indicates bit concatenation). 
         [0000]        h ( W 1, N )// r   (16)
 
         [0165]    And “W 3 ” is calculated in accordance with the following Eqn. (17), from “C 3 ” and “Z”. 
         [0000]        W 3 =C 3(+) Z   (17)
 
         [0000]    The registration auxiliary data generation unit  103  registers the set of “(W 2 ,W 3 )” that has been generated as described above, to the auxiliary data storage unit  203 , as auxiliary data. 
         [0166]    In the above-described processing, a unique identifier is assigned to the data that has been input to the storage apparatus  200  by the identifier management unit  201 . And after that, the data that is input to the storage apparatus  200  can be referred by the assigned identifier. Hereinafter, “W 1 ”, “W 2 ”, and “W 3 ” that are associated with the identifier “i” are respectively represented as “W 1 [i]”, “W 2 [i]”, and “W 3 [i]”. 
         [0167]    In the encrypted data verification phase, first, input data “Z” (data to be checked) is input to the encrypting unit  301  of the data concealment apparatus  300 . 
         [0168]    Next, the key generation unit  302  of the data concealment apparatus  300  generates a key “S” (random number of “K” bits) that is used to perform concealment of the input data “Z”. In addition, the key generation unit  302  outputs the generated key “S” to the encrypting unit  301  and the auxiliary data generation unit  303 . 
         [0169]    The encrypting unit  301  calculates encrypted data “W 1 ′” that is obtained by calculating an XOR between the code word “C” that is obtained by coding the input key “S” through the binary BCH code, and the input data “Z”. That is, “W 1 ′” is calculated in accordance with following Eqn. (18). 
         [0000]        W 1′═ C ′(+) Z′   (18)
 
         [0170]    In addition, the encrypting unit  301  inputs the calculated “W 1 ′” to the encrypted data subtraction unit  502  of the data verification apparatus  500 . 
         [0171]    Next, the identifier “i” is input from the entire-data request unit  501  to the identifier management unit  201 . The encryption data “W 1 [i]” that corresponds to the input identifier “i” is read from the encrypted data storage unit  202  of the storage apparatus  200 , and is input to the encrypted data subtraction unit  502 . In addition, the pieces of auxiliary data “W 2 [i]” and “W 3 [i]” that correspond to the identifier “i” are read from the auxiliary data storage unit, and are input to the match determination unit  503 . The encrypted data subtraction unit  502  accepts the encrypted data “W 1 [i]” from the encrypted data storage unit  202  of the storage apparatus  200 , and the encrypted data “W 1 ′” from the data concealment apparatus  300 , as inputs. After that, the encrypted data subtraction unit  502  outputs a difference (XOR) between the two pieces of input encrypted data “W 1 ′” and “W 1 [i]”, that is calculated in accordance with following Eqn. (19), to the match determination unit  503 . 
         [0000]        W 1′(+) W 1[ i]   (19)
 
         [0172]    Next, the control unit  504  calculates 
         [0000]      “ g   —   s=g**ns   (20)”
 
         [0000]    for a random number ns and an element (generator) “g” of a group (multiplicative group “Zp”) “G” that is defined beforehand, and outputs the calculated value to the auxiliary data generation unit  303 . 
         [0173]    Next, the auxiliary data generation unit  303  of the data concealment apparatus  300  randomly selects “S 1 ′” and “S 2 ′” that satisfy following Eqn. (21). 
         [0000]        S′=S 1′(+) S 2′  (21)
 
         [0174]    The auxiliary data generation unit  303  of the data concealment apparatus  300  calculates “W 2 ” and “g_c”, based on the following Eqns. (22a) and (22b). 
         [0000]        W 2′= H ( S 1′, W 1′, g   —   s**nc )  (22a)
 
         [0000]        g   —   c=g**nc   (22b)
 
         [0175]    Next, the auxiliary data generation unit  303  of the data concealment apparatus  300  calculates “W 3 ” from “C 3 ” and “Z”. “C 3 ” obtained by performing the binary BCH error-correcting coding on data that is obtained by performing bit concatenation on an inner product “(c,S 2 ′)” and a random number “r”. That is, the data is calculated in accordance with following Eqn. (23). 
         [0000]      ( c,S 2′)// r′   (23)
 
         [0176]    “W 3 ” is calculated in accordance with following Eqn. (24). 
         [0000]        W 3′= C 3(+) Z′   (24)
 
         [0177]    In addition, the auxiliary data generation unit  303  outputs “W 1 ′”, “W 2 ”, “W 3 ”, and “g_c”, to the match determination unit  503  of the data verification apparatus  500 . 
         [0178]    Next, the match determination unit  503  applies the decoding processing of the binary BCH code, to the difference between the pieces of input encrypted data, that is calculated in accordance with following Eqn. (25). 
         [0000]        W 1′(+) W 1 [i]   (25)
 
         [0179]    Then the match determination unit  503  calculates “T” that is the decryption result of the difference between the two pieces of encrypted data “W 1 ′” and “W 1 [i]”. 
         [0180]    In addition, the match determination unit  503  applies the decoding processing of the binary BCH code, to an XOR between “W 3 [i]” and “W 3 ′”, that is calculated in accordance with following Eqn. (26) 
         [0000]        W 3[ i ](+) W 3′  (26)
 
         [0181]    Then the match determination unit  503  calculates “w 3 ” that is the decryption result of “W 3 [i](+)W 3 ”. 
         [0182]    The match determination unit  503  checks whether or not the result that has been obtained by calculating a bitwise XOR between “W 2 ′”, “w 3 ”, and “H(T,W 1 ′, g_c**ns)”, that has been calculated using the decryption result “T” of the difference between the two pieces of encrypted data “W 1 ′” and “g_c**ns”, is equal to “W 2 [i]”. In other words, the match determination unit  503  checks (verifies) whether or not the Eqn. (27) is satisfied. 
         [0000]        W 2[ i]=H ( T,W 1′, g   —   c**ns )(+) W 2′(+) w 3  (27)
 
         [0183]    When the above-described Eqn. (27) is satisfied, the match determination unit  503  determines that a Hamming distance between the original data of “WIN” and “Z” is “d” or less. In this case, the identifier output unit  505  outputs the identifier “i”. When the above-described Eqn. (27) is not satisfied, the match determination unit  503  determines that the Hamming distance exceeds “d”. In this case, the identifier output unit  505  does not output the identifier “i”. 
         [0184]    The above-described operation is performed on all identifiers “i” that are managed by the storage apparatus, and the output of all identifiers is performed that include the original data in which a Hamming distance with the input data “Z” becomes “d” or less. 
         [0185]    In the second example, similar to the above-described first example, “g_s**nc(=(g**ns)**nc)” in the auxiliary data “W 2 ′=H(S′,W 1 ′,g_s**nc)(+)h′” that is generated in the auxiliary data generation unit  303  of the data concealment apparatus  300 , and “g_c**ns (=(g**nc)**ns)” in the match determination unit  503  of the data verification apparatus  500  may be generated in both of the auxiliary data generation unit  303  and the match determination unit  503 , for example, by the known Diffie-Hellman key exchange method. 
         [0186]    As an application example of the first and second examples, there is authentication to protect biological information. The outline of the authentication is described below. 
         [0187]    In this case, biological information that is obtained from a fingerprint, vein, or the like is treated as input data in the data registration phase and input data in the encrypted data verification phase. 
         [0188]    In the above-described described system, it can be determined whether or not encrypted biometric data, that is stored in the storage apparatus, and encrypted biometric data, that has been transmitted from the data concealment apparatus, are obtained from an identical person, while the biological information remains concealed (encrypted). In other words, whether or not the pieces of biometric data are obtained from an identical person can be determined depending on whether or not a Hamming distance between these two pieces of input data is a predetermined certain value or less. In addition, in the above-described described system, the authentication can be performed based on such a determination result. Regarding biological information, identical data may not be always obtained stably. However, it can be assumed that pieces of data that are obtained from an identical person are similar to each other (pieces of data of which the Hamming distance is small, may be obtained), so that the authentication is preferably applied to biometric identification. It is noted that in the biometric identification application, for example, each value of parameters (“K”, “s”, “d”) of the BCH may be obtained experimentally. 
         [0189]    It is noted that each disclosure of the above-described Patent literature and Non-Patent literature is assumed to be incorporated by reference herein. Within the entire disclosure of the present invention (including claims), based on the basic technical concept, the exemplary embodiments and the examples can be modified and adjusted. In addition, within the claims of the present invention, a variety of combinations or selections can be made from various disclosure elements (including elements in each of claims, elements in each of the examples, elements in each of the drawings, and the like). In other words, the present invention includes various changes and modifications that would be made by those skilled in the art in accordance with the entire disclosure including claims and the technical idea, of course. In particular, regarding the numerical range described herein, it should be understood that any number or sub-ranges contained within the claims is specifically described even if it is not otherwise stated. 
       REFERENCE SIGNS LIST 
       [0000]    
       
           100  registration data generation apparatus 
           101  encrypting unit 
           102  key generation unit 
           103  registration auxiliary data generation unit 
           200  storage apparatus 
           201  identifier management unit 
           202  encrypted data storage unit 
           203  auxiliary data storage unit 
           300  data concealment apparatus 
           301  encrypting unit 
           302  key generation unit 
           303  auxiliary data generation unit 
           400  specification data verification apparatus 
           401  identifier holding unit 
           402  encrypted data subtraction unit 
           403  match determination unit 
           404  control unit 
           500  data verification apparatus 
           501  entire-data request unit 
           502  encrypted data subtraction unit 
           503  match determination unit 
           504  control unit 
           505  identifier output unit