Patent Publication Number: US-11032273-B2

Title: Method for authenticating secret information which protects secret information

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application is a continuation-in-part application of International Application No. PCT/KR2018/005012 filed on Apr. 30, 2018, which claims priority to Korean Application No. 10-2017-0055803 filed on Apr. 29, 2017. The present application also claims priority to Korean Application No. 10-2018-0087340 filed on Jul. 26, 2018. These applications are expressly incorporated herein by reference. 
    
    
     TECHNICAL FIELD 
     The present disclosure relates to a method for authenticating secret information such as biometric information and the like, specifically to a method for authenticating the information without exposing the information. 
     BACKGROUND 
     Recently, a variety of methods has been developed for authenticating biometric information such as fingerprint, iris and the like. A person has unique biometric information. Authentication of biometric information is advantageous over the password authentication because there is no need to remember it and there is no risk to be leaked. 
     However, the authentication of biometric information has different feature from the other authentication methods. The recognized biometric information can be varied whenever it is inputted due to the noise generated in the process of recognizing the biometric information. However, the authentication should not be impeded by the noise. Further, the biometric information cannot be altered unlike password when the information is leaked. Therefore, leakage of the biometric information should be completely blocked. 
     The encryption methods such as Fuzzy extractor and Inner product encryption have been developed for encrypting biometric information. The authentication using the conventional encryption method has low speed. Further, the length of the ciphertext is long. 
     SUMMARY 
     The object of the present disclosure is to provide an authentication method based on a secret information such as biometric information, which is fast and uses short ciphertext. 
     The computer-implemented method for authenticating secret information according to an aspect of the present disclosure, comprises receiving, by an authentication server, Q{right arrow over (X)} from a terminal for registering secret information; storing, by the authentication server, the received Q{right arrow over (X)}; receiving, by the authentication server, a vector {right arrow over (Z)} from a terminal for requesting authentication of secret information; calculating, by the authentication server, the inner product of Q{right arrow over (X)} and {right arrow over (Z)}; calculating, by the authentication server, ½(n−the inner product); and determining, by the authentication server, that the authentication is successful if ½(n−the inner product) is within a predetermined value and that the authentication fails otherwise 
     {right arrow over (X)} is a vector of secret information to be registered, the elements of which consist of {−1, 1} n . “Q” is a matrix, the elements of which are randomly selected from    q   m×n . {right arrow over (Y)} is a vector of secret information to be requested for authentication, the elements of which consist of {−1, 1} n . {right arrow over (Z)} is a vector which satisfies Q T {right arrow over (Z)}={right arrow over (Y)}. 
     {right arrow over (Z)} can be a vector which satisfies Q T {right arrow over (Z)}={right arrow over (Y)}+{right arrow over (e)}. {right arrow over (e)} is an error vector, the elements of which are randomly selected from    q   n . 
     The matrix “Q” can be deleted after Q{right arrow over (X)} is received by the authentication server. 
     At least one of Q{right arrow over (X)} and {right arrow over (Z)} can be encrypted and then is transmitted to the authentication server. 
     The computer-implemented method for authenticating secret information according to another aspect of the present disclosure comprises receiving, by an authentication server, a first ciphertext “SK X ” of a vector “{right arrow over (X)}” of secret information to be registered from a terminal for registering secret information; storing, by the authentication server, the received first ciphertext; receiving, by the authentication server, a second ciphertext “C Y ” of a vector “{right arrow over (Y)}” of secret information to be requested for authentication from a terminal for requesting authentication of secret information; calculating, by the authentication server, the inner product of “{right arrow over (X)}” and “{right arrow over (Y)}”; calculating, by the authentication server, a distance for determining similarity based on the inner product; and determining, by the authentication server, that the authentication is successful if the distance is within a predetermined value and that the authentication fails otherwise. 
     The elements of “{right arrow over (X)}” consist of the elements of Z p   k . The first ciphertext “SK X ” is defined by 
                   [           I   k             S         ]     ⁢     X   -&gt;       +     u   -&gt;       ,         
where the elements of the matrix “S” are randomly selected from    q   n×k  and the elements of the vector “{right arrow over (u)}” are randomly selected from    q   m . The elements of “{right arrow over (Y)}” consist of the elements of    p   k . The second ciphertext “C Y ” is defined by
 
                 C   Y     :=         c   1     -&gt;     =     (           -     S   T       ⁢     a   -&gt;       +       (     q   p     )     ⁢     Y   -&gt;       +     e   -&gt;       ,       a   -&gt;     ⁢     ←   $     ⁢     ℤ   q   n         )         ,         
c 0 =− {right arrow over (u)},{right arrow over (c)} 1   +e*. {right arrow over (e)} and e* are vectors having elements of real numbers which are small. The elements of {right arrow over (e)} are randomly selected from    q   k  and the elements of e* are randomly selected from    q .    q  is defined by {0, 1, 2, . . . q−1} and    q  is defined by [0,q); p, q, k, n and m are positive integers; q&gt;p; and m=k+n.
 
     The elements of the error vectors {right arrow over (e)} and e* can be selected with less distribution such that | {right arrow over (X)},{right arrow over (e)} +e*| is less than 
     
       
         
           
             
               q 
               
                 2 
                 ⁢ 
                 p 
               
             
             . 
           
         
       
     
     If the elements of “{right arrow over (X)}” and “{right arrow over (Y)}” consist of −1 or +1, the distance for determining similarity can be the hamming distance between “{right arrow over (X)}” and “{right arrow over (Y)}” which is ½(k−({right arrow over (X)},{right arrow over (Y)})). 
     The elements of the matrix “S” can be randomly selected from the subset of    q . 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         FIG. 1  is one embodiment of an example diagrammatic view of a device architecture which carries out the present disclosure. 
         FIG. 2  is a flow chart of authenticating process using secret information according to the present disclosure. 
         FIG. 3  is a flow chart of authenticating process using secret information according to another aspect of the present disclosure. 
     
    
    
     It should be understood that the above-referenced drawings are not necessarily to scale, presenting a somewhat simplified representation of various preferred features illustrative of the basic principles of the disclosure. The specific design features of the present disclosure will be determined in part by the particular intended application and use environment 
     DETAILED DESCRIPTION 
     Hereinafter, the present disclosure will be described in detail with reference to the accompanying drawings. As those skilled in the art would realize, the described embodiments may be modified in various different ways, all without departing from the spirit or scope of the present disclosure. Further, throughout the specification, like reference numerals refer to like elements. 
     In this specification, the order of each step should be understood in a non-limited manner unless a preceding step must be performed logically and temporally before a following step. That is, except for the exceptional cases as described above, although a process described as a following step is preceded by a process described as a preceding step, it does not affect the nature of the present disclosure, and the scope of rights should be defined regardless of the order of the steps. In addition, in this specification, “A or B” is defined not only as selectively referring to either A or B, but also as including both A and B. In addition, in this specification, the term “comprise” has a meaning of further including other components in addition to the components listed. 
     The terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of the disclosure. As used herein, the singular forms “a,” “an,” and “the” are intended to include the plural forms as well, unless the context clearly indicates otherwise. It will be further understood that the terms “comprise” and/or “comprising,” when used in this specification, specify the presence of stated features, integers, steps, operations, elements, and/or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and/or groups thereof. As used herein, the term “and/or” includes any and all combinations of one or more of the associated listed items. The term “coupled” denotes a physical relationship between two components whereby the components are either directly connected to one another or indirectly connected via one or more intermediary components. Unless specifically stated or obvious from context, as used herein, the term “about” is understood as within a range of normal tolerance in the art, for example within 2 standard deviations of the mean. “About” can be understood as within 10%, 9%, 8%, 7%, 6%, 5%, 4%, 3%, 2%, 1%, 0.5%, 0.1%, 0.05%, or 0.01% of the stated value. Unless otherwise clear from the context, all numerical values provided herein are modified by the term “about.” 
     The method according to the present disclosure can be carried out by an electronic arithmetic device (referred to as “terminal” or “server” in this specification) such as a computer, tablet, mobile phone, portable computing device, stationary computing device, server computer etc. Additionally, it is understood that one or more various methods, or aspects thereof, may be executed by at least one processor. The processor may be implemented on a computer, tablet, mobile device, portable computing device, etc. A memory configured to store program instructions may also be implemented in the device(s), in which case the processor is specifically programmed to execute the stored program instructions to perform one or more processes, which are described further below. Moreover, it is understood that the below information, methods, etc. may be executed by a computer, tablet, mobile device, portable computing device, etc. including the processor, in conjunction with one or more additional components, as described in detail below. Furthermore, control logic may be embodied as non-transitory computer readable media on a computer readable medium containing executable program instructions executed by a processor, controller/control unit or the like. Examples of the computer readable mediums include, but are not limited to, ROM, RAM, compact disc (CD)-ROMs, magnetic tapes, floppy disks, flash drives, smart cards and optical data storage devices. The computer readable recording medium can also be distributed in network coupled computer systems so that the computer readable media is stored and executed in a distributed fashion, e.g., by a telematics server or a Controller Area Network (CAN). 
     A variety of devices can be used herein.  FIG. 1  illustrates an example diagrammatic view of an exemplary device architecture according to embodiments of the present disclosure. As shown in  FIG. 1 , a device  109  may contain multiple components, including, but not limited to, a processor (e.g., central processing unit (CPU)  110 , a memory  120 , a wired or wireless communication unit  130 , one or more input units  140 , and one or more output units  150 . It should be noted that the architecture depicted in  FIG. 1  is simplified and provided merely for demonstration purposes. The architecture of the device  109  can be modified in any suitable manner as would be understood by a person having ordinary skill in the art, in accordance with the present claims. Moreover, the components of the device  109  themselves may be modified in any suitable manner as would be understood by a person having ordinary skill in the art, in accordance with the present claims. Therefore, the device architecture depicted in  FIG. 1  should be treated as exemplary only and should not be treated as limiting the scope of the present disclosure. 
     The processor  110  is capable of controlling operation of the device  109 . More specifically, the processor  110  may be operable to control and interact with multiple components installed in the device  109 , as shown in  FIG. 1 . For instance, the memory  120  can store program instructions that are executable by the processor  110  and data. The process described herein may be stored in the form of program instructions in the memory  120  for execution by the processor  110 . The communication unit  130  can allow the device  109  to transmit data to and receive data from one or more external devices via a communication network. The input unit  140  can enable the device  109  to receive input of various types, such as audio/visual input, user input, data input, and the like. To this end, the input unit  140  may be composed of multiple input devices for accepting input of various types, including, for instance, one or more cameras  142  (i.e., an “image acquisition unit”), touch panel  144 , microphone (not shown), sensors  146 , keyboards, mice, one or more buttons or switches (not shown), and so forth. The term “image acquisition unit,” as used herein, may refer to the camera  142 , but is not limited thereto. The input devices included in the input  140  may be manipulated by a user. The output unit  150  can display information on the display screen  152  for a user to view. The display screen  152  can also be configured to accept one or more inputs, such as a user tapping or pressing the screen  152 , through a variety of mechanisms known in the art. The output unit  150  may further include a light source  154 . The device  109  is illustrated as a single component, but the device may also be composed of multiple, separate components that are connected together and interact with each other during use. 
     The device  109  can thus be programmed in a manner allowing it to generate various exemplary files having column-oriented layouts, and more specifically, to generate various exemplary files having column-oriented layouts that improve data security and query speed by selective encryption to the columns. 
     In this specification, biometric information is described as an exemplary one of secret information. However, the present disclosure can be applied to the other type of secret information such as password, secret key, privately sensitive information and the like. 
       FIG. 2  shows a flow chart of method of authenticating secret information according to the embodiments of the present disclosure. The authentication method can be carried out in the environment comprising a terminal for registering secret information  10 , a terminal for requesting authentication of secret information  20 , authentication server  30 . The terminal for registering secret information  10  and the terminal for requesting authentication of secret information  20  can be the same terminal or separate terminal. In this specification, the present disclosure will be described in the assumption that the two terminals are separate each other. The secret information can be biometric information such as fingerprint. 
     The terminal for registering secret information  10  transforms the inputted secret information into a vector {right arrow over (X)}∈{−1, 1} n  having length of “n,” the elements of which consist of −1 and 1 and, then calculates Q{right arrow over (X)}, where Q is a m×n matrix (Q∈   q   m×n ) (step  200 ). “m,” “n,” and “q” are positive integers which are determined according to the security parameters. The numbers are shared with the terminals  10 ,  20  and the server  30 . The matrix Q can be deleted from the terminal  10  after the Q{right arrow over (X)} is received by the authentication server  30  or registered in the authentication server  30 . The deletion of Q can remove the risk of leakage thereof. 
     The calculated Q{right arrow over (X)} is transmitted to the authentication server  30  (step  210 ). The transmitted Q{right arrow over (X)} can be a kind of ciphertext of the secret information because Q is known only to the terminal for registering secret information  10 . The authentication server  30  registers Q{right arrow over (X)} as a ciphertext of the secret information (step  220 ). 
     Afterwards, a user who wants authentication of secret information provides the terminal for requesting authentication of secret information  20  with his/her secret information through a recognition device of the secret information. The terminal for requesting authentication of secret information  20  transforms the inputted secret information into a vector {right arrow over (Y)}∈{−1, 1} n  having length of “n,” the elements of which consist of −1 and 1. 
     The terminal for requesting authentication of secret information  20  calculates a vector {right arrow over (Z)} which satisfies the following relationship (step  230 ):
 
 Q   T   {right arrow over (Z)}={right arrow over (Y)} 
 
     The vector {right arrow over (Z)} can be calculated by randomly selecting the elements of “m−n” and then calculating the remaining “n” elements such that the vector satisfies the above relationship. 
     It is desirable that “m” is greater than “n.” “n” is a dimension of the secret information. The security level is determined by “m.” The safety level of LWE problem is influenced by “k (=m−n).” It is general that “k” is determined between 500 and 1,000 according to the desired security level and the other factors. For example, “n” can be 2,000 and “m” can be 2,500 when the secret information is iris information. 
     The calculated {right arrow over (Z)} is transmitted to the authentication server  30  (step  240 ). 
     The hamming distance d({right arrow over (X)},{right arrow over (Y)}) between {right arrow over (X)} and {right arrow over (Y)} can be calculated by the following equation:
 
 d ( {right arrow over (X)},{right arrow over (Y)} )=½( n−     {right arrow over (X)},{right arrow over (Y)}   )  [Mathematical Equation 1]
 
where  {right arrow over (X)},{right arrow over (Y)}  is an inner product of {right arrow over (X)} and {right arrow over (Y)}.
 
     The hamming distance of two vectors means the number of different elements when the elements having the same position in two vectors are compared. For example, the hamming distance of the vector “1100” and the vector “1000” is “1,” and the hamming distance of the vector “1111” and the vector “1100” is “2.” That is, the less hamming distance of two vectors which are transformed from two inputted secret information respectively is, the closer the two inputted secret information are. 
     The authentication server  30  calculates the inner product of Q{right arrow over (X)} that is the registered ciphertext of the secret information earlier and {right arrow over (Z)} that is received from the terminal for requesting authentication of secret information  20  (step  250 ). The inner product satisfies the following relationship:
 
   Q{right arrow over (X)},{right arrow over (Z)}     =     {right arrow over (X)},Q   T   {right arrow over (Z)}     =     {right arrow over (X)},{right arrow over (Y)}         [Mathematical Equation 2]
 
     Therefore, the hamming distance d({right arrow over (X)},{right arrow over (Y)}) of {right arrow over (X)} and {right arrow over (Y)} can be calculated by the following equation (step  260 ):
 
 d ( {right arrow over (X)},{right arrow over (Y)} )=½( n−     Q{right arrow over (X)},{right arrow over (Z)}   )
 
     If the calculated hamming distance is within a predetermined value, the secret information inputted in the terminal  20 , which is requested for authentication, is authenticated and otherwise, it is determined that the authentication fails (step  270 ). The result of authentication can be transmitted to the terminal  20  (step  280 ). 
     If the secret information is not a biometric information which can inherently have noise when it is inputted, it can be determined that the authentication is successful only when the hamming distance between {right arrow over (X)} and {right arrow over (Y)} is “0.” 
     The security level can be improved if at least one of Q{right arrow over (X)} and {right arrow over (Z)} is encrypted by the public key of the authentication server  30  before it is transmitted to the server  30 . 
     According to another embodiments, at step  130 , the vector {right arrow over (Z)} can be calculated to satisfy the following relationship:
 
 Q   T   {right arrow over (Z)}={right arrow over (Y)}+{right arrow over (e)},  
 
where {right arrow over (e)} is a noise vector of small size.
 
     The noise vector has the elements which are randomly selected from    q   n  in LWE problem. “q” is a positive integer. According to the embodiment, the authentication server  30  calculates the inner product of Q{right arrow over (X)} and {right arrow over (Z)} and then calculates the hamming distance between {right arrow over (X)} and {right arrow over (Y)}+{right arrow over (e)} using the inner product of Q{right arrow over (X)} and {right arrow over (Z)}. 
     The vectors {right arrow over (X)} and {right arrow over (Y)}+{right arrow over (e)} satisfies the following relationship:
 
   {right arrow over (X)},{right arrow over (Y)}+{right arrow over (e)}     =     {right arrow over (X)},{right arrow over (Y)}     +     {right arrow over (X)},{right arrow over (e)}         [Mathematical Equation 3]
 
     This means that  {right arrow over (X)},{right arrow over (Y)}+{right arrow over (e)}  can be the approximate value of  {right arrow over (X)},{right arrow over (Y)} , which has a small error of  {right arrow over (X)},{right arrow over (e)} . Therefore,  {right arrow over (X)},{right arrow over (Y)}+{right arrow over (e)}  can be used for authentication of secret information. According to the embodiment, the inputted secret information to be transmitted to the server  30  for request for authentication can be encrypted after a noise is added in LWE problem, thereby improving security level. 
       FIG. 3  shows a flow chart of method of authenticating secret information according to another aspect of the present disclosure. 
     At step  300 , the terminal for registering secret information  10  generates master key (MSK). If the terminal  10  is separate one from the terminal  20 , the terminals  10  and  20  share the master key (MSK). Alternatively, the master key can be stored in a separate place, for example cloud storage, other than the terminal  10  and/or terminal  20 . If the master key (MSK) is stored in the separate place, another authentication is needed for using the master key. The master key (MSK) can comprise a random vector a and a random matrix “S” which are defined as following:
 
 MSK:={right arrow over (u)}       q   m   ; S       q   n×k   [Mathematical Equation 4]
 
         q :={0, 1, 2, . . . q−1} 
     m=n+k 
     “p,” “q,” “n,” and “m” are positive integers. 
     The symbol “:=” means that the left of the symbol is defined by the right of the symbol. The symbol “$” means that the elements of the left of the symbol are randomly selected from the right of the symbol. 
     The security level is determined by “n.” The greater “n” is, the greater security level is. Generally, “n” can be determined between 900 and 1,300 for security of 128 bits. 
     The elements of the matrix “S” can be randomly selected from a subset of    q , the elements of which consist of “0” and “1.” In that case, the size of the master key (MSK) can be significantly reduced. 
     The terminal for registering secret information  10  transforms the secret information that a user inputs for registering the secret information into a vector “{right arrow over (X)}” having a length of “k.” The elements of the vector “{right arrow over (X)}” can consist of “−1” and “1” for authentication using hamming distance. 
     The elements of the vector “{right arrow over (X)}” can consist of the elements of    p   k . In that case, the authentication can be carried out by use of Euclidean distance. 
     The terminal for registering secret information  10  generates a ciphertext “SK X ” of the inputted secret information for registration by use of the master key (MSK) as follows (step  310 ); 
                       SK   X     :=         [           I   k             S         ]     ⁢     X   -&gt;       +     u   -&gt;         ,           [     Mathematical   ⁢           ⁢   Equation   ⁢           ⁢   5     ]               
where I k  is an identity matrix of k×k.
 
     The ciphertext “SK X ” is transmitted to the authentication server  30  (step  320 ). The authentication server  30  registers the received ciphertext “SK X ” (step  330 ). 
     A user who wants authentication of secret information provides the terminal for requesting authentication of secret information  20  with his/her secret information. 
     The terminal for requesting authentication of secret information  20  transforms the secret information that the user inputted for requesting of authentication into a vector “{right arrow over (Y)}” having a length of “k.” The elements of the vector “{right arrow over (Y)}” can consists of “−1” and “1” for authentication using hamming distance. 
     As in the vector “{right arrow over (X)},” the elements of the vector “{right arrow over (Y)}” can consist of the elements of    p   k . 
     The terminal for requesting authentication of secret information  20  generates the ciphertext “C Y ” of the secret information which is inputted for requesting for authentication of the secret information as follows: 
     
       
         
           
             
               
                 
                   
                     
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     “p” is a positive integer which is less than “q.” 
     {right arrow over (e)} and e* are vectors having elements of real numbers which is small. The elements of {right arrow over (e)} can be randomly selected from    q   k  and the elements of e* can be randomly selected from    q .
 
   q :=[0, q )
 
     In this specification, “[” or “]” which is used for indicating the range of numbers inside therebetween means including the number beside it. “(” or “)” which is used for indicating the range of numbers inside therebetween means excluding the number beside it. 
     In this specification, the aforementioned values of “k,” “q,” “n,” and “p” are determined according to security parameters and are shared with the terminals  10 ,  20  and the server  30 . 
     The ciphertext “C Y ” is transmitted to the authentication server  30  (step  350 ). 
     The authentication server  30  calculates the hamming distance between the pre-registered vector of secret information “{right arrow over (X)}” and the vector of secret information to be requested for authentication “{right arrow over (Y)}” as follows (step  360 ): 
     
       
         
           
             
               
                 
                   
                     
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                     ⁢ 
                     7 
                   
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     └number┐ outputs the rounded number of the number which is included in inside of └ ┐. 
     The authentication server  30  determines that the authentication is successful if the hamming distance is within a predetermined value and determines that the authentication fails otherwise (step  370 ). 
     The following is for proving that the equation of the hamming distance is true. First of all, “2k” should be less than “p.” 
     We establish the following relationship: 
     
       
         
           
             
               
                 
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     Then, the following relationship is established:
 
   T{right arrow over (X)} ,{right arrow over ( c   1 )} =   {right arrow over (X)},T   T {right arrow over ( c   1 )} 
 
     The left side can be transformed as follows:
 
   T{right arrow over (X)} ,{right arrow over ( c   1 )} =   SK   X   −{right arrow over (u)} ,{right arrow over ( c   1 )} =   SK   X ,{right arrow over ( c   1 )} −   {right arrow over (u)} ,{right arrow over ( c   1 )} 
 
     The right side can be transformed as follows: 
     
       
         
           
             
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                         X 
                         -&gt; 
                       
                       , 
                       
                         Y 
                         -&gt; 
                       
                     
                     〉 
                   
                 
                 + 
                 
                   〈 
                   
                     
                       X 
                       -&gt; 
                     
                     , 
                     
                       e 
                       -&gt; 
                     
                   
                   〉 
                 
               
             
           
         
       
     
     Therefore, the following relationship can be satisfied: 
     
       
         
           
             
               〈 
               
                 
                   SK 
                   X 
                 
                 , 
                 
                   
                     c 
                     1 
                   
                   -&gt; 
                 
               
               〉 
             
             = 
             
               
                 〈 
                 
                   
                     u 
                     -&gt; 
                   
                   , 
                   
                     
                       c 
                       1 
                     
                     -&gt; 
                   
                 
                 〉 
               
               + 
               
                 
                   ( 
                   
                     q 
                     p 
                   
                   ) 
                 
                 ⁢ 
                 
                   〈 
                   
                     
                       X 
                       -&gt; 
                     
                     , 
                     
                       Y 
                       -&gt; 
                     
                   
                   〉 
                 
               
               + 
               
                 〈 
                 
                   
                     X 
                     -&gt; 
                   
                   , 
                   
                     e 
                     -&gt; 
                   
                 
                 〉 
               
             
           
         
       
     
     By applying Mathematical Equation 6, the following relationship is satisfied: 
     
       
         
           
             
               
                 c 
                 0 
               
               + 
               
                 〈 
                 
                   
                     SK 
                     X 
                   
                   , 
                   
                       
                   
                   ⁢ 
                   
                     
                       c 
                       1 
                     
                     -&gt; 
                   
                 
                 〉 
               
             
             = 
             
               
                 
                   ( 
                   
                     q 
                     p 
                   
                   ) 
                 
                 ⁢ 
                 
                   〈 
                   
                     
                       X 
                       -&gt; 
                     
                     , 
                     
                       Y 
                       -&gt; 
                     
                   
                   〉 
                 
               
               + 
               
                 〈 
                 
                   
                     X 
                     -&gt; 
                   
                   , 
                   
                     e 
                     -&gt; 
                   
                 
                 〉 
               
               + 
               
                 e 
                 * 
               
             
           
         
       
     
     This equation means that the following is true when |({right arrow over (X)},{right arrow over (e)})+e*| is bounded by 
               q     2   ⁢   p       :           v=     {right arrow over (X)},{right arrow over (Y)}   (mod  p ) 
     It is desirable that the elements of the error vectors {right arrow over (e)} and e* are selected with low distribution for satisfying the above equation with high probability. 
     Because “2k” is less than “p,” the following relation is satisfied when v= {right arrow over (X)},{right arrow over (Y)}  (mod p):
 
 v=     {right arrow over (X)},{right arrow over (Y)}       
 
     If the elements of the vectors of secret information {right arrow over (X)},{right arrow over (Y)} are selected from    p , the similarity of the two vectors of the secret information is determined by the Euclidean distance which is calculated based on the hamming distance between {right arrow over (X)} and {right arrow over (Y)} by the above method. According to the embodiments of the present disclosure, the inner product of two vectors of secret information {right arrow over (X)} and {right arrow over (Y)} can be calculated from v without disclosing the two vectors. Thus, present disclosure can be applied to any methods which determines the similarity based on the inner product of the vectors. The embodiments of this aspect of the present disclosure is faster than the method related to  FIG. 2  which should calculate {right arrow over (Z)} which satisfies Q T {right arrow over (Z)}={right arrow over (Y)}. 
     According to the present disclosure, the calculation for authenticating secret information is significantly simple, thereby enabling fast authentication. Further, the size of the master key (MSK) is small to reduce calculation load for authentication. 
     Although the present disclosure has been described with reference to accompanying drawings, the scope of the present disclosure is determined by the claims described below and should not be interpreted as being restricted by the embodiments and/or drawings described above. It should be clearly understood that improvements, changes and modifications of the present disclosure disclosed in the claims and apparent to those skilled in the art also fall within the scope of the present disclosure. Accordingly, this description is to be taken only by way of example and not to otherwise limit the scope of the embodiments herein.