Patent Publication Number: US-11646877-B2

Title: Apparatus and method for generating secret key, apparatus and method for generating evaluation key

Description:
CROSS-REFERENCE TO RELATED APPLICATION(S) 
     This application claims the benefit of U.S. Provisional Patent Application No. 62/847,455 filed on May 14, 2019 and the benefit under 35 USC § 119(a) of Korean Patent Application No. 10-2019-0056601 filed on May 14, 2019, in the Korean Intellectual Property Office, the entire disclosures of which are incorporated herein by reference for all purposes. 
    
    
     BACKGROUND 
     1. Field 
     The following description relates to technology for encryption and decryption. 
     2. Description of Related Art 
     In prior arts including U.S. Pat. No. 9,252,942, one trusted user (a secret key manager) generates a public key and a secret key and distributes the public key to all users in order to provide a secure data fusion service among multiple users by using homomorphic encryption. In this case, the users encrypt their own data using the distributed public key and then perform a homomorphic evaluation of the encrypted data. Also, when the general users request the secret key manager for decryption, the secret key manager transmits a decrypted plaintext evaluation result to the general users. 
     In these prior arts, since the secret key is managed by the secret key manager, a problem arises in that the safety of the entire system depends entirely on the safety of the secret key manager. In other words, if the secret key is leaked through the secret key manager, data of all users can be recovered and the safety of the entire system is compromised. In addition, in the case of users who find it difficult to trust each other, it is impossible to set up a single secret key manager that all users can trust. 
     SUMMARY 
     This summary is provided to introduce a selection of concepts in a simplified form that are further described below in the Detailed Description. This summary is not intended to identify key features or essential features of the claimed subject matter, nor is it intended to be used as an aid in determining the scope of the claimed subject matter. 
     In one general aspect, there is provided a method of generating a secret key, which is performed by a computing device comprising one or more processors and a memory in which one or more programs to be executed by the one or more processors are stored, the method including generating a share of each of a user and a plurality of other users for a secret key of the user; providing the share of each of the plurality of other users to a user terminal of each of the plurality of other users; receiving a share of the user for a secret key of each of the plurality of other users from the user terminal of each of the plurality of other users; and generating a new secret key of the user using the share of the user for the secret key of the user and the shares of the user for the secret key of each of the plurality of other users. 
     The generating of the share may include generating the share of each of the user and the plurality of other users for the secret key of the user such that the secret key of the user is allowed to be generated using a predetermined number or more of shares among the shares of the user and the plurality of other users for the secret key of the user. 
     The method may further include generating a partial decryption result using the new secret key of the user with respect to a ciphertext encrypted using a common public key; receiving the partial decryption result with respect to the ciphertext generated using an updated secret key share of each of the predetermined number or more of other users from the user terminal of each of the predetermined number or more of other users among the plurality of other users; and generating a plaintext for the ciphertext using the generated partial decryption result and the received partial decryption result. 
     The common public key may be generated using a public key of the user which corresponds to the secret key of the user and a public key of each of the plurality of other users which corresponds to the secret key of each of the plurality of other users. 
     The generating of the plaintext may include generating the plaintext through linear combination between the generated partial decryption result and the received partial decryption result. 
     In another general aspect, there is provided an apparatus for generating a secret key including one or more processors; a memory; and one or more programs, wherein the one or more programs are stored in the memory and configured to be executed by the one or more processors and the one or more programs include commands for generating a share of each of a user and a plurality of other users for a secret key of the user, providing the share of each of the plurality of other users to a user terminal of each of the plurality of other users, receiving a share of the user for a secret key of each of the plurality of other users from the user terminal of each of the plurality of other users, and generating a new secret key of the user using the share of the user for the secret key of the user and the shares of the user for the secret key of each of the plurality of other users. 
     The generating of the share may include generating the share of each of the user and the plurality of other users for the secret key of the user such that the secret key of the user is allowed to be generated using a predetermined number or more of shares among the shares of the user and the plurality of other users for the secret key of the user. 
     The one or more programs may further include commands for generating a partial decryption result using the new secret key of the user with respect to a ciphertext encrypted using a common public key, receiving the partial decryption result with respect to the ciphertext generated using an updated secret key share of each of the predetermined number or more of other users from the user terminal of each of the predetermined number or more of other users among the plurality of other users, and generating a plaintext for the ciphertext using the generated partial decryption result and the received partial decryption result. 
     The common public key may be generated using a public key of the user which corresponds to the secret key of the user and a public key of each of the plurality of other users which corresponds to the secret key of each of the plurality of other users. 
     The generating of the plaintext may include generating the plaintext through linear combination between the generated partial decryption result and the received partial decryption result. 
     In still another general aspect, there is provided a method of generating an evaluation key, which is performed by a computing device comprising one or more processors and a memory in which one or more programs to be executed by the one or more processors are stored, the method including generating a ciphertext for a secret key of a user using a common public key; providing the ciphertext for the secret key of the user to each of user terminals of a plurality of other users; receiving a ciphertext for a secret key of each of the plurality of other users, which is encrypted using the common public key, from each of the user terminals of the plurality of other users; generating an evaluation key share of the user from the ciphertext for the secret key of the user and the ciphertext for the secret key of each of the plurality of other users using a homomorphic addition operation on the basis of the secret key of the user; receiving, from each of the plurality of other users, an evaluation key share of each of the plurality of other users, which is generated from the ciphertext for the secret key of the user and the ciphertext for the secret key of each of the plurality of other users, using a homomorphic addition operation on the basis of the secret key of each of the plurality of other users; and generating an evaluation key for a homomorphic multiplication operation for a homomorphic multiplication operation for the ciphertext, which is encrypted using the common public key, by using the evaluation key share of the user and the evaluation key share of each of the plurality of other users. 
     The common public key may be generated using a public key of the user which corresponds to the secret key of the user and a public key of each of the plurality of other users which corresponds to the secret key of each of the plurality of other users. 
     The secret key of the user and the secret key of each of the plurality of other users may satisfy Equation 1 below:
 
 sk   i =(1, s   i ), i= 1,2, . . . , N   [Equation 1]
 
     where sk i  denotes a secret key of user i among N users including the user and the plurality of other users and s i  denotes an element of a polynomial ring, and each of the ciphertext for the secret key of the user and the ciphertext for each of the plurality of other users may be a ciphertext obtained by encrypting s i  using the common public key. 
     Each of the evaluation key share of the user and the evaluation key share of each of the plurality of other users is the same as a ciphertext obtained by encrypting Σ i   2 +Σ j≠i s i s j  using the common public key. 
     The generating of the evaluation key may include generating the evaluation key by performing the homomorphic addition operation on the evaluation key share of the user and the evaluation key share of each of the plurality of other users and the evaluation key may be the same as a ciphertext obtained by encrypting Σ i s i   2 +Σ i Σ j≠i s i s j  using the common public key. 
     In still another general aspect, there is provided an apparatus for generating an evaluation key including one or more processors; a memory; and one or more programs, 
     wherein the one or more programs are stored in the memory and configured to be executed by the one or more processors and the one or more programs include commands for generating a ciphertext for a secret key of a user using a common public key, providing the ciphertext for the secret key of the user to each of user terminals of a plurality of other users, receiving a ciphertext for a secret key of each of the plurality of other users, which is encrypted using the common public key, from each of the user terminals of the plurality of other users, generating an evaluation key share of the user from the ciphertext for the secret key of the user and the ciphertext for the secret key of each of the plurality of other users using a homomorphic addition operation on the basis of the secret key of the user, receiving, from each of the plurality of other users, an evaluation key share of each of the plurality of other users, which is generated from the ciphertext for the secret key of the user and the ciphertext for the secret key of each of the plurality of other users, using a homomorphic addition operation on the basis of the secret key of each of the plurality of other users, and generating an evaluation key for a homomorphic multiplication operation for the ciphertext, which is encrypted using the common public key, by using the evaluation key share of the user and the evaluation key share of each of the plurality of other users. 
     The common public key may be generated using a public key of the user which corresponds to the secret key of the user and a public key of each of the plurality of other users which corresponds to the secret key of each of the plurality of other users. 
     The secret key of the user and the secret key of each of the plurality of other users may satisfy Equation 1 below:
 
 sk   i =(1, s   i ), i= 1,2, . . . , N   [Equation 1]
 
     where sk i  denotes a secret key of user i among N users including the user and the plurality of other users and s i  denotes an element of a polynomial ring, and each of the ciphertext for the secret key of the user and the ciphertext for each of the plurality of other users is a ciphertext obtained by encrypting s i  using the common public key. 
     Each of the evaluation key share of the user and the evaluation key share of each of the plurality of other users may be the same as a ciphertext obtained by encrypting s i   2 +Σ j≠i s i s j  using the common public key. 
     The generating of the evaluation key may include generating the evaluation key by performing the homomorphic addition operation on the evaluation key share of the user and the evaluation key share of each of the plurality of other users and the evaluation key may be the same as a ciphertext obtained by encrypting Σ i s i   2 +Σ i Σ j≠i s i s j  using the common public key. 
     In another general aspect, there is provided a method of generating an evaluation key, which is performed by a computing device comprising one or more processors and a memory in which one or more programs to be executed by the one or more processors are stored, the method including generating a ciphertext for a secret key of a user using a public key of the user; providing the ciphertext for the secret key of the user to a user terminal of each of a plurality of other users; receiving, from the user terminal of each of the plurality of other users, a ciphertext for a secret key of each of the plurality of other users, which is encrypted using a public key of each of the plurality of other users; generating an intermediate evaluation key using the ciphertext for the secret key of the user and the ciphertext for the secret key of each of the plurality of other users; generating an evaluation key share of the user using the secret key of the user and the intermediate evaluation key; receiving, from each of the plurality of other users, an evaluation key share of each of the plurality of other users, which is generated using the secret key of each of the plurality of other users and the intermediate evaluation key; and generating an evaluation key for a homomorphic multiplication operation for a ciphertext, which is encrypted using a common public key, by using the evaluation key share of the user and the evaluation key share of each of the plurality of other users. 
     the common public key may be generated using the public key of the user and the public key of each of the plurality of other users. 
     The secret key of the user and the secret key of each of the plurality of other users satisfy Equation 1 below:
 
 sk   i =(1, s   i ), i= 1,2, . . . , N   [Equation 1]
 
     where sk i  denotes a secret key of user i among N users including the user and the plurality of other users and s i  denotes an element of a polynomial ring, and each of the ciphertext for the secret key of the user and the ciphertext for each of the plurality of other users may be a ciphertext obtained by encrypting s 1 . 
     The generating of the intermediate evaluation key may include generating the intermediate evaluation key by performing a homomorphic addition operation between the ciphertext for the secret key of the user and the ciphertext for the secret key of each of the plurality of other users and the intermediate evaluation key may be the same as a ciphertext obtained by encrypting Σ i=1   N s i  using the common public key. 
     Each of the evaluation key share of the user and the evaluation key share of each of the plurality of other users may be the same as a ciphertext obtained by encrypting s i   2 +Σ j≠i s i s j  using the common public key. 
     The generating of the evaluation key may include generating the evaluation key by performing the homomorphic addition operation on the evaluation key share of the user and the evaluation key share of each of the plurality of other users and the evaluation key is the same as a ciphertext obtained by encrypting Σ i s i   2 +Σ i Σ j≠i s i s j  using the common public key. 
     In another general aspect, there is provided an apparatus for generating an evaluation key including one or more processors; a memory; and one or more programs, wherein the one or more programs are stored in the memory and configured to be executed by the one or more processors and the one or more programs include commands for generating a ciphertext for a secret key of a user using a public key of the user, providing the ciphertext for the secret key of the user to a user terminal of each of a plurality of other users, receiving, from the user terminal of each of the plurality of other users, a ciphertext for a secret key of each of the plurality of other users, which is encrypted using a public key of each of the plurality of other users, generating an intermediate evaluation key using the ciphertext for the secret key of the user and the ciphertext for the secret key of each of the plurality of other users, generating an evaluation key share of the user using the secret key of the user and the intermediate evaluation key, receiving, from each of the plurality of other users, an evaluation key share of each of the plurality of other users, which is generated using the secret key of each of the plurality of other users and the intermediate evaluation key, and generating an evaluation key for a homomorphic multiplication operation for a ciphertext, which is encrypted using a common public key, by using the evaluation key share of the user and the evaluation key share of each of the plurality of other users. 
     The common public key may be generated using the public key of the user and the public key of each of the plurality of other users. 
     The secret key of the user and the secret key of each of the plurality of other users may satisfy Equation 1 below:
 
 sk   i =(1, s   i ), i= 1,2, . . . , N   [Equation 1]
 
     where sk i  denotes a secret key of user i among N users including the user and the plurality of other users and s i  denotes an element of a polynomial ring, and each of the ciphertext for the secret key of the user and the ciphertext for each of the plurality of other users may be a ciphertext obtained by encrypting s 1 . 
     The generating of the intermediate evaluation key may include generating the intermediate evaluation key by performing a homomorphic addition operation between the ciphertext for the secret key of the user and the ciphertext for the secret key of each of the plurality of other users and the intermediate evaluation key may be the same as a ciphertext obtained by encrypting Σ i=1   N s i  using the common public key. 
     Each of the evaluation key share of the user and the evaluation key share of each of the plurality of other users may be the same as a ciphertext obtained by encrypting s i   2 +Σ j≠i s i s j  using the common public key. 
     The generating of the evaluation key may include generating the evaluation key by performing the homomorphic addition operation on the evaluation key share of the user and the evaluation key share of each of the plurality of other users and the evaluation key is the same as a ciphertext obtained by encrypting Σ i s i   2 +Σ i Σ j≠i s i s j  using the common public key. 
     Other features and aspects will be apparent from the following detailed description, the drawings, and the claims. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG.  1    is a diagram illustrating a configuration of an encryption system according to one embodiment of the present disclosure. 
         FIG.  2    is a flowchart illustrating a process of generating a common public key according to one embodiment of the present disclosure. 
         FIG.  3    is a flowchart illustrating a process of generating a secret key of a user for distributed decryption according to one embodiment of the present disclosure. 
         FIG.  4    is a flowchart illustrating a process of distributed description according to one embodiment of the present disclosure. 
         FIG.  5    is a flowchart illustrating a process of generating an evaluation key according to one embodiment of the present disclosure. 
         FIG.  6    is a flowchart illustrating a process of generating an evaluation key according to another embodiment of the present disclosure. 
         FIG.  7    is a block diagram for describing a computing environment including a computing device suitable to be used in exemplary embodiments. 
     
    
    
     Throughout the drawings and the detailed description, unless otherwise described, the same drawing reference numerals will be understood to refer to the same elements, features, and structures. The relative size and depiction of these elements may be exaggerated for clarity, illustration, and convenience. 
     DETAILED DESCRIPTION 
     The following description is provided to assist the reader in gaining a comprehensive understanding of the methods, apparatuses, and/or systems described herein. Accordingly, various changes, modifications, and equivalents of the methods, apparatuses, and/or systems described herein will be suggested to those of ordinary skill in the art. 
     Descriptions of well-known functions and constructions may be omitted for increased clarity and conciseness. Also, terms described in below are selected by considering functions in the embodiment and meanings may vary depending on, for example, a user or operator&#39;s intentions or customs. Therefore, definitions of the terms should be made on the basis of the overall context. The terminology used in the detailed description is provided only to describe embodiments of the present disclosure and not for purposes of limitation. Unless the context clearly indicates otherwise, the singular forms include the plural forms. It should be understood that the terms “comprises” or “includes” specify some features, numbers, steps, operations, elements, and/or combinations thereof when used herein, but do not preclude the presence or possibility of one or more other features, numbers, steps, operations, elements, and/or combinations thereof in addition to the description. 
       FIG.  1    is a diagram illustrating a configuration of an encryption system according to one embodiment of the present disclosure. 
     Referring to  FIG.  1   , the encryption system  100  according to one embodiment of the present disclosure includes a plurality of user terminals  110 ,  120 , and  130 . 
     Each of the plurality of user terminals  110 ,  120 , and  130  is a terminal used by a different user and may be, for example, a desktop personal computer (PC), a laptop PC, a smartphone, a phablet, or the like, but is not limited to a specific type of device as long as the device has a communication function and a data operation function using a wired/wireless network. 
     Hereinafter, it is assumed, for convenience of description, that there are three user terminals  110 ,  120 , and  130  included in the encryption system  100  as illustrated in  FIG.  1   , but the number of user terminals  110 ,  120 , and  130  may be two or four or more, unlike the example shown in  FIG.  1   . 
     In addition, hereafter, it is assumed that a first user terminal  110  is used by user  1 , a second user terminal  120  is used by user  2 , and a third user terminal  130  is used by user  3 . 
     Meanwhile, the encryption system  100  may perform generation of a common public key for a plurality of users, encryption using the common public key, and distributed decryption for a ciphertext encrypted using the common public key, on the basis of homomorphic encryption including the following four algorithms.
         Key generation algorithm (KeyGen): A key generation algorithm generates a public key used for encryption, an evaluation key for homomorphic evaluation, and a secret key used for decryption for a ciphertext encrypted using the public key.   Encryption algorithm (Enc): An Encryption algorithm generates a ciphertext for a plaintext using the public key.   Decryption algorithm (Dec): A decryption algorithm uses a secret key to decrypt a ciphertext encrypted using a public key.   Homomorphic evaluation algorithm (Eval): A homomorphic evaluation algorithm generates a ciphertext for evaluation results of plaintexts for each of a plurality ciphertexts by computing the plurality of ciphertexts, which are encrypted using the same public key, in an encrypted state. For example, the homomorphic evaluation algorithm may include a homomorphic addition algorithm in which ciphertext C of m and ciphertext C′ of m′, each of which is encrypted using a public key, are computed in an encrypted state so as to generate ciphertext C +  of m+m&#39; and a homomorphic multiplication algorithm in which ciphertext C of m and ciphertext C′ of m′ are computed in an encrypted state to generate ciphertext C* of m*m′. Meanwhile, in the embodiment of the present disclosure, homomorphic encryption used in the encryption system  100  is not necessarily limited to a specific type of homomorphic encryption as long as the following two conditions are satisfied.       

     Condition (1): As shown in Equation 1 below, evaluation ‘+ pk ’ between public keys and evaluation ‘+ sk ’ between secret keys which can generate new valid public key-secret key pair (pk new , sk new ) from n (here, n is an integer greater than or equal to 2) public key-secret key pairs (pk 1 , sk 1 ), . . . , and (pk n , sk n ) may be defined.
 
( pk   new   ,sk   new )=( pk   1 + pk  . . . + pk   pk   n   ,sk   1 + sk  . . . + sk   sk   n )  [Equation 1]
 
     Condition (2): Equation 2 below is established for a ciphertext C encrypted through an encryption algorithm using a new public key pk new .
 
Dec( sk   new   ,C )=Dec( sk   1   ,C )+ . . . +Dec( sk   n   ,C )  [Equation 2]
 
     Here, Dec(sk,C) denotes a result of decrypting a ciphertext C through a decryption algorithm using a secret key sk. 
       FIG.  2    is a flowchart illustrating a process of generating a common public key according to one embodiment of the present disclosure. 
     Referring to  FIG.  2   , a first user terminal  110  generates a public key pk 1  and a secret key sk 1  of user  1  ( 201 ) and provides the generated public key pk 1  to a second user terminal  120  and a third user terminal  130  ( 202  and  203 ). 
     In this case, the first user terminal  110  may generate the public key pk 1  and the secret key sk 1  using a key generation algorithm of homomorphic encryption. 
     The second user terminal  120  generates a public key pk 2  and a secret key sk 2  of user  2  ( 204 ) and provides the generated public key pk 2  to the first user terminal  110  and the third user terminal  130  ( 205  and  206 ). 
     In this case, the second user terminal  120  may generate the public key pk 2  and the secret key sk 2  using a key generation algorithm of homomorphic encryption. 
     The third user terminal  130  generates a public key pk 3  and a secret key sk 3  of user  3  ( 207 ) and provides the generated public key pk 3  to the first user terminal  110  and the second user terminal  120  ( 208  and  209 ). 
     In this case, the third user terminal  130  may generate the public key pk 3  and the secret key sk 3  using a key generation algorithm of homomorphic encryption. 
     Then, the first user terminal  110 , the second user terminal  120 , and the third user terminal  130  generate a common public key pk c  using pk 1 , pk 2 , and pk 3 , respectively ( 210 ,  211 , and  212 ). 
     In this case, the common public key pk c  may be generated using Equation 3 below.
 
 pk   c   =pk   1 + pk  . . . + pk   pk   N   [Equation 3]
 
     In Equation 3, N denotes the number of users involved in generating a common public key pk c  (hereinafter, N will be used in the same sense), and N=3 in the example illustrated in  FIG.  2   . 
     Meanwhile, a common secret key sk c  capable of decrypting a ciphertext encrypted through an encryption algorithm of homomorphic encryption using the common public key pk c  may be defined as Equation 4 below.
 
 sk   c   =sk   1 + sk  . . . + sk   sk   N   [Equation 4]
 
     However, according to the embodiment of the present disclosure, it is possible to generate a plaintext for a ciphertext encrypted using the common public key pk c , without using the common secret key sk c , through distributed decryption as described below, and thus, unlike the common public key pk c , none of the user terminals  110 ,  120 , and  130  generate the common secret key sk c . 
       FIG.  3    is a flowchart illustrating a process of generating a secret key of a user for distributed decryption according to one embodiment of the present disclosure. 
     Procedures illustrated in  FIG.  3    may be performed after the common public key pk c  in accordance with  FIG.  2    is generated. 
     Referring to  FIG.  3   , the first user terminal  110  generates a share sk 1,1  of user  1 , a share sk 1,2  of user  2 , and a share sk 1,3  of user  3  with respect to the secret key sk 1  of user  1  ( 301 ). 
     Hereinafter, the first user terminal  110  provides sk 1,2  to the second user terminal  120  ( 302 ) and provides sk 1,3  to the third user terminal  130  ( 303 ). 
     The second user terminal  120  generates a share sk 2,1  of user  1 , a share sk 2,2  of user  2 , and a share sk 2,3  of user  3  with respect to the secret key sk 2  of user  2  ( 304 ). 
     Then, the second user terminal  120  provides sk 2,1  to the first user terminal  110  ( 305 ) and provides sk 2,3  to the third user terminal  130  ( 306 ). 
     The third user terminal  130  generates a share sk 3,1  of user  1 , a share sk 3,2  of user  2 , and a share sk 3,3  of user  3  with respect to the secret key sk 3  of user  3  ( 307 ). 
     Then, the third user terminal  130  provides sk 3,2  to the second user terminal  120  ( 308 ) and provides sk 3,1  to the first user terminal ( 309 ). 
     Thereafter, the first user terminal  110  generates a new secret key sk 1   new  of user  1  using the shares sk 1,1 , sk 2,1 , and sk 3,1  of user  1  for the secret key sk 1  of user  1 , the secret key sk 2  of user  2 , and the secret key sk 3  of user  3 , respectively ( 310 ). 
     In addition, the second user terminal  120  generates a new secret key sk 2   new  of user  2  using the shares sk 1,2,  sk 2,2,  and sk 3,2  of user  2  for the secret key sk 1  of user  1 , the secret key sk 2  of user  2 , and the secret key sk 3  of user  3 , respectively ( 311 ). 
     Also, the third user terminal  130  generates a new secret key sk 3   new  of user  3  using the shares sk 1,3 , sk 2,3 , and sk 3,3  of user  3  for the secret key sk 1  of user  1 , the secret key sk 2  of user  2 , and the secret key sk 3  of user  3 , respectively ( 312 ). 
     According to one embodiment, in operations  301 ,  304 , and  307  of  FIG.  3   , each of the user terminals  110 ,  120 , and  130  may generate sk 0 , sk 1,2 , and sk 1,3  such that a secret key sk i  of user i can be generated using a predetermined number or more of shares among shares sk i,1 , sk i,2 , and sk i,3  for the secret key sk i  of user i (here, i=1, 2, . . . , N) of each of the user terminals  110 ,  120 , and  130 . 
     Specifically, the secret key sk i  of user i among N users involved in generating a common public key pk c  may be generated using t shares (here, t is an integer and 1&lt;t≤N) among shares sk i,1 , sk i,2 , . . . , and sk i,N  of each of the N users, as shown in Equation 5 below.
 
 sk   i   =a   i,1 + sk  . . . + sk   a   t   sk   i,t   [Equation 5]
 
     In Equation 5, each of the coefficients a 1 , . . . , a N  multiplied to a share of each of the users for a secret key sk i  may be a fixed value predetermined for each user. 
     To this end, each of the user terminals  110 ,  120 , and  130  may generate shares sk i,1 , sk i,2 , and sk i,3  for the secret key sk i  of user i using Shamir&#39; s secret sharing which uses the secret key sk i  of user i as secret information. 
     According to one embodiment of the present disclosure, in operations  310  to  312 , each of the users  110 ,  120 , and  130  may generate a new secret key sk i   new  of user i using Equation 6 below.
 
 sk   i   new   =sk   1,i + sk  . . . + sk   sk   N,i   [Equation 6]
 
     Meanwhile, according to Equations 4 to 6 described above, the common secret key sk c  may satisfy such a relation as Equation 7. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
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                                   , 
                                   1 
                                 
                               
                             
                             ⁢ 
                             
                               + 
                               sk 
                             
                             ⁢ 
                             ⋯ 
                           
                         
                       
                     
                   
                   
                     
                         
                     
                     
                       
                         
                             
                           ⁢ 
                           
                             
                               + 
                               sk 
                             
                             ⁢ 
                             
                               
                                 a 
                                 t 
                               
                               ⁢ 
                               
                                 sk 
                                 
                                   1 
                                   , 
                                   t 
                                 
                               
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           + 
                           sk 
                         
                         ⁢ 
                         ⋯ 
                       
                     
                   
                   
                     
                         
                     
                     
                       
                           
                         ⁢ 
                         
                           
                             + 
                             sk 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   a 
                                   1 
                                 
                                 ⁢ 
                                 
                                   sk 
                                   
                                     N 
                                     , 
                                     1 
                                   
                                 
                               
                               ⁢ 
                               
                                 + 
                                 sk 
                               
                               ⁢ 
                               ⋯ 
                             
                           
                         
                       
                     
                   
                   
                     
                         
                     
                     
                       
                           
                         ⁢ 
                         
                           
                             
                               + 
                               sk 
                             
                             ⁢ 
                             
                               a 
                               t 
                             
                           
                           ⁢ 
                           
                             sk 
                             
                               N 
                               , 
                               t 
                             
                           
                         
                         ) 
                       
                     
                   
                   
                     
                       
                           
                         = 
                       
                     
                     
                       
                           
                         ⁢ 
                         
                           
                             
                               sk 
                               1 
                             
                             ⁢ 
                             
                               + 
                               sk 
                             
                             ⁢ 
                             ⋯ 
                             ⁢ 
                             
                               + 
                               sk 
                             
                             ⁢ 
                             
                               sk 
                               N 
                             
                           
                           = 
                           
                             sk 
                             c 
                           
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     7 
                   
                   ] 
                 
               
             
           
         
       
     
       FIG.  4    is a flowchart illustrating a process of distributed description according to one embodiment of the present disclosure. 
     Referring to  FIG.  4   , the first user terminal  110  generates a result p 1 =Dec(sk 1   new , C) of partially decrypting a ciphertext C for a plaintext m, which is encrypted using the common public key pk c , by using sk 1   new , ( 401 ). 
     Then, the first user terminal  110  request the second user terminal  120  and the third user terminal  130  to partially decrypt the ciphertext C ( 402  and  403 ). 
     Then, the second user terminal  120  generates a result p 2 =Dec(sk 2   new , C) of partially decrypting the ciphertext C using sk 2   new  ( 404 ) and the third user terminal  130  generates a result p 3 =Dec(sk 3   new , C) of partially decrypting the ciphertext C using sk 3   new  ( 405 ). 
     Meanwhile, in each of operations  410 ,  404 , and  405 , the partial decryption may be performed through an encryption algorithm of homomorphic encryption using sk i   new  as a decryption key. 
     Then, the second user terminal  120  and the third user terminal  130  each provide the generated partial decryption result p 2  and p 3  to the first user terminal  110  ( 406  and  407 ). 
     Then, the first user terminal  110  generates a plaintext m for the ciphertext C using p 1 , p 2 , and p 3  ( 408 ). In this case, the first user terminal  110  may generate the plaintext m for the ciphertext C through linear combination of p 1 , p 2 , and p 3  as shown in Equation 8 below.
 
 m=a   1   p   1   + . . . +a   t   p   t   =a   1 Dec( sk   1   new   ,C )+ . . . + a   t Dec( sk   t   new   ,C )  [Equation 8]
 
     According to Equations 2 and 7 described above, Equation 8 may satisfy such a relation as Equation 9 below. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               a 
                               1 
                             
                             ⁢ 
                             
                               p 
                               1 
                             
                           
                           + 
                           ⋯ 
                           + 
                           
                             
                               a 
                               t 
                             
                             ⁢ 
                             
                               p 
                               t 
                             
                           
                         
                         ⁢ 
                           
                         = 
                       
                     
                     
                       
                           
                         ⁢ 
                         
                           
                             
                               a 
                               1 
                             
                             ⁢ 
                             
                               Dec 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     sk 
                                     1 
                                     new 
                                   
                                   , 
                                   C 
                                 
                                 ) 
                               
                             
                           
                           + 
                           ⋯ 
                         
                       
                     
                   
                   
                     
                         
                     
                     
                       
                           
                         ⁢ 
                         
                           
                             + 
                             
                               a 
                               t 
                             
                           
                           ⁢ 
                           
                             Dec 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   sk 
                                   t 
                                   new 
                                 
                                 , 
                                 C 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                           
                         = 
                       
                     
                     
                       
                           
                         ⁢ 
                         
                           Dec 
                           ( 
                           
                             
                               
                                 a 
                                 1 
                               
                               ⁢ 
                               
                                 sk 
                                 1 
                                 new 
                               
                             
                             ⁢ 
                             
                               + 
                               sk 
                             
                             ⁢ 
                             ⋯ 
                           
                         
                       
                     
                   
                   
                     
                         
                     
                     
                       
                         
                             
                           ⁢ 
                           
                             
                               
                                 
                                   + 
                                   sk 
                                 
                                 ⁢ 
                                 
                                   a 
                                   t 
                                 
                               
                               ⁢ 
                               
                                 sk 
                                 t 
                                 new 
                               
                             
                             , 
                             C 
                           
                           ) 
                         
                         = 
                       
                     
                   
                   
                     
                         
                     
                     
                       
                           
                         ⁢ 
                         
                           
                             Dec 
                             ⁡ 
                             
                               ( 
                               
                                 sk 
                                 , 
                                 C 
                               
                               ) 
                             
                           
                           = 
                           m 
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     9 
                   
                   ] 
                 
               
             
           
         
       
     
     Therefore, it is possible to decrypt the ciphertext C without generating a common public key sk c  for decrypting the ciphertext C. 
     Hereinafter, detailed embodiments using illustrative homomorphic encryption will be described. 
     Specifically, illustrative homomorphic encryption may consist of the following four algorithms.
         Key generation algorithm (KeyGen): A key generation algorithm generates a public key pk, a secret key sk, an evaluation key evk for a homomorphic multiplication operation using Equations 10 to 12, respectively.
 
 sk =(1, s )∈ R   2   [Equation 10]
 
 pk =( b,a )=(− as+e,a )∈ R   q   2   [Equation 11]
 
 evk =( b′,a ′)=(− a′s+e′+s   2   ,a ′)∈ R   Pq   2   [Equation 12]
       

     Here, R denotes a polynomial ring in which R=Z[X]/(X N +1), R q  denotes a quotient ring in which R q =R/qR (here, q is an arbitrary integer), s and a are arbitrary elements of R, e denotes a very small error value, which is an element of R, a′ denotes an arbitrary element of R Pq , and P denotes a sufficiently large integer.
         Encryption algorithm (Enc): An encryption algorithm generates a ciphertext C for a plaintext m that is an element of R using Equation 13 below.
 
 C =Enc( pk,m )=( C   0   ,C   1 )+ v ( b,a )+( m+e   0   ,e   1 )+( vb+m+e   0   ,va+e   1 )  [Equation 13]
       

     Here, v denotes a very small arbitrary element of R, and e 0  and e 1  denote very small error values, which are elements of R.
         Decryption algorithm (Dec): A decryption algorithm generates a plaintext m for a ciphertext C using a dot product of the ciphertext C and a secret key sk, as shown in Equation 14 below.
 
 p =Dec( sk,C )= C   0   +C   1   s=m+e ∈R   q   [Equation 14]
       

     Here, if an error value e is sufficiently small compared to the plain text m, then p may be considered an approximate of m.
         Homomorphic evaluation algorithm (Eval): A homomorphic evaluation algorithm supports a homomorphic addition operation in which ciphertext C of m and ciphertext C′ of m′, each of which is encrypted using a public key pk, are computed in an encrypted state so as to generate ciphertext C +  of m+m′ and a homomorphic multiplication operation in which ciphertext C of m and ciphertext C′ of m′ are computed in an encrypted state to generate ciphertext C* of m*m′. Here, the homomorphic addition operation does not require an evaluation key evk, unlike the homomorphic multiplication operation.       

     Meanwhile, if it is defined that pk+ pk pk′=(b+b′,a), sk+ sk sk′=(1,s+s&#39;) for two public key-secret key pairs (pk=(b,a), sk=(1,s)) and (pk′=(b′,a), sk=(1,s′)) generated using the above-described illustrative key generation algorithm of homomorphic encryption, the above illustrative homomorphic encryption satisfies the above-described conditions (1) and (2). 
     Therefore, the encryption system  100  may perform the procedures in accordance with  FIGS.  2  to  4    using the above-described illustrative homomorphic encryption. 
     Specifically, in operations  201 ,  205 , and  207  in the flowchart shown in  FIG.  2   , each of the user terminals  110 ,  120 , and  130  may generate a public key pk i =(b i ,a) and a secret key sk i =(1,s i ) of user i of each of the user terminals  110 ,  120 , and  130  using the key generation algorithm of the illustrative homomorphic encryption described above. 
     Also, in operations  210 ,  211 , and  212 , each of the user terminals  110 ,  120 , and  130  may generate a common public key pk c  using Equation 15 below.
 
 pk   c   =pk   1 + pk  . . . + pk   pk   N =( b   1   + . . . +b   N   ,a )=( b,a )  [Equation 15]
 
     Meanwhile, a common secret key sk c  corresponding to the common public key pk c  that satisfies Equation 15 may be defined as below.
 
 sk   c   =sk   1 + sk  . . . + sk   sk   N =(1 ,s   1   + . . . +s   N )=(1 ,s )  [Equation 16]
 
     In addition, referring to Equations 10 to 13, an evaluation key for a homomorphic multiplication operation for a ciphertext encrypted using the common public key pk c  that satisfies Equation 15 may have the same form as a ciphertext of s 2 =(s 1   + . . . +s   N ) 2 =Σ i s i   2 +Σ i Σ j≠i s i s j . 
     Thus, according to one embodiment of the present disclosure, each of the user terminals  110 ,  120 , and  130  of the encryption system  100  may generate an evaluation key evk through evaluation key generation procedures shown in  FIG.  5  or  6   . 
     The evaluation key generation procedures shown in  FIG.  5    may be performed after the common public key pk c  in accordance with  FIG.  2    is generated. 
     Referring to  FIG.  5   , first, each of the user terminals  110 ,  120 , and  130  generates a ciphertext C i  for a secret key sk i  of the user of each of the user terminals  110 ,  120 , and  130  using the common public key pk c  ( 501 ,  502 , and  503 ). 
     Here, the ciphertext C i  may be a ciphertext obtained by encrypting s i  with the common public key pk c  on the basis of the secret key sk i =(1,s i ) of the user of each of the user terminals  110 ,  120 , and  130 . 
     Then, each of the user terminals  110 ,  120 , and  130  provides the generated ciphertext C i  to the other user terminals ( 504  to  509 ). 
     Thereafter, each of the user terminals  110 ,  120 , and  130  may generate an evaluation key share evk i  of the user from the ciphertext C i  and a ciphertext C j  (here, j≠i) using a homomorphic addition operation on the basis of the secret key sk i  of the user of each of the user terminals  110 ,  120 , and  130  ( 510 ,  511 , and  512 ). 
     In this case, the evaluation key share evk i  of the user may be the same as a ciphertext (i.e., s i   2 +Σ j≠i s i s j ) obtained by encrypting s i   2 +Σ j≠i s i s j  using the common public key plcv. 
     Specifically, each of the user terminals  110 ,  120 , and  130  may repetitively perform the homomorphic addition operation for the cipher text Ci to generate a ciphertext for s i   2  and may repetitively perform the homomorphic addition operation for the ciphertext C j  to generate a ciphertext for Σ j≠i s i s j . Also, each of the user terminals  110 ,  120 , and  130  may perform the homomorphic addition operation between the ciphertext for s i   2  and the ciphertext for Σ j≠i s i s j  to generate a ciphertext for s i   2 +Σ j≠i s i s j . 
     Then, each of the user terminals  110 ,  120 , and  130  provides the evaluation key share evk i  of the user to the other user terminals ( 513  to  518 ). 
     Thereafter, each of the user terminals  110 ,  120 , and  130  uses the evaluation key share evk i  of the user, which is generated by the user terminal itself, and the evaluation key shares evk j  of the other users received from the other user terminals to generate an evaluation key evk for a homomorphic multiplication operation for the ciphertext encrypted using the common public key pk c  ( 519 ,  520 , and  521 ). 
     Specifically, each of the user terminals  110 ,  120 , and  130  may generate the evaluation key evk through the homomorphic addition operation between the evaluation key share evk i  of the user, which is generated by the user terminal itself, and the evaluation key shares evk j  of the other users received from the other user terminals. Here, the evaluation key evk may be the same as a ciphertext obtained by encrypting Σ i s i   2 +Σ i Σ j≠i s i s j  with the common public key pk c . 
     Meanwhile, the evaluation key generation procedures shown in  FIG.  6    may be performed after the public key pk i  and the secret key sk i  of the user for generating the common public key pk c  in accordance with  FIG.  2    are generated. 
     Referring to  FIG.  6   , first, each of the user terminals  110 ,  120 , and  130  generates a ciphertext C′ i  for the secret key sk i  of the user using the public key pk i  of the user ( 601 ,  602 , and  603 ). 
     Here, the ciphertext C′ i  may be a ciphertext obtained by encrypting s i  using the public key pk i  of the user on the basis of the secret key sk i =(1,s i ) of the user of each of the user terminals  110 ,  120 , and  130  and the public key pk i =(b i , a)=(−a·s i +e i , a) of the user. That is, according to the above Equation 13, the ciphertext C′ i  may satisfy Equation 17 below.
 
 C   i ′=( v·b   i   +s   i   +e   0   ,v·a+e   1 )=(− a′·s   i   +e   i   ′+s   i   ,a ′)  [Equation 17]
 
     Then, each of the user terminals  110 ,  120 , and  130  provides the generated ciphertext C′ i  to the other user terminals ( 604  to  609 ). 
     Then, each of the user terminals  110 ,  120 , and  130  generates an intermediate evaluation key evk 0  using ciphertexts C′ i  and C′ j  (here, j≠i) ( 610 ,  611 , and  612 ). 
     In this case, each of the user terminals  110 ,  120 , and  130  may perform a homomorphic addition operation between the ciphertexts C′ i  and C′ j  to generate the intermediate evaluation key evk 0 . Specifically, when the ciphertexts C′ i  and C′ j  each satisfy the above Equation 17, the homomorphic addition operation may be performed through an addition operation between the ciphertexts, and as a result, the intermediate evaluation key evk 0  becomes the same as a ciphertext obtained by encrypting s (i.e., Σ i=1   N s i ) in the common secret key sk c =(1,s) using the common public key pk c =(b, a)=(−a·s+e,a) as shown in Equation 18 below. 
     
       
         
           
             
               
                 
                   
                     evk 
                     0 
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         N 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         C 
                         i 
                         ′ 
                       
                     
                     = 
                     
                       
                         ( 
                         
                           
                             
                               
                                 - 
                                 
                                   a 
                                   ′ 
                                 
                               
                               · 
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   N 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   s 
                                   i 
                                 
                               
                             
                             + 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 N 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 e 
                                 i 
                                 ′ 
                               
                             
                             + 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 N 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 s 
                                 i 
                               
                             
                           
                           , 
                           
                             a 
                             ′ 
                           
                         
                         ) 
                       
                       = 
                       
                         ( 
                         
                           
                             
                               
                                 - 
                                 
                                   a 
                                   ′ 
                                 
                               
                               · 
                               s 
                             
                             + 
                             e 
                             + 
                             s 
                           
                           , 
                           
                             a 
                             ′ 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     18 
                   
                   ] 
                 
               
             
           
         
       
     
     Then, each of the user terminals  110 ,  120 , and  130  generates an evaluation key share evk i  of the user using the secret key sk i  and the intermediate evaluation key evk 0  of the user of each of the user terminals  110 ,  120 , and  130  ( 613 ,  614 , and  615 ). 
     Here, the evaluation key share evk i  of user i may be the same as a ciphertext obtained by encrypting s i   2 +s i Σ j≠i s j  with the common public key pk c  and may be generated by multiplying the intermediate evaluation key evk 0  by s i . 
     Then, each of the user terminals  110 ,  120 , and  130  provides the evaluation key share evk i  to the other user terminals ( 616  to  621 ). 
     Then, each of the user terminals  110 ,  120 , and  130  uses the evaluation key share of the user, which is generated by the user terminal itself, and the evaluation key shares evk j  of the other users received from the other user terminals to generate an evaluation key evk for a homomorphic multiplication operation for a ciphertext encrypted using the common public key pk c  ( 622 ,  623 , and  624 ). 
     Specifically, each of the user terminals  110 ,  120 , and  130  may generate the evaluation key evk through a homomorphic addition operation between the evaluation key share evk i  of the user, which is generated by the user terminal itself, and the evaluation key shares evk j  of the other users received from the other user terminals. Here, the evaluation key evk may be the same as a ciphertext obtained by encrypting Σ i s i   2 +Σ i Σ j≠i s i s j  with the common public key plcv. 
     Meanwhile, in the flowcharts illustrated in  FIGS.  2  to  6   , the above decryption process is described as being divided into a plurality of operations. However, it should be noted that at least some of the operations may be performed in different order or may be combined into fewer operations or further divided into more operations. In addition, some of the operations may be omitted, or one or more extra operations, which are not illustrated, may be added to the flowchart and be performed. 
       FIG.  7    is a block diagram for describing a computing environment including a computing device suitable to be used in exemplary embodiments. In the illustrated embodiments, each of the components may have functions and capabilities different from those described hereinafter and additional components may be included in addition to the components described herein. 
     The illustrated computing environment  10  includes a computing device  12 . In one embodiment, the computing device  12  may be one or more components included in each of the user terminals  110 ,  120 , and  130 . 
     The computing device  12  may include at least one processor  14 , a computer-readable storage medium  16 , and a communication bus  18 . The processor  14  may cause the computing device  12  to operate according to the above-described exemplary embodiment. For example, the processor  14  may execute one or more programs stored in the computer-readable storage medium  16 . The one or more programs may include one or more computer executable commands, and the computer executable commands may be configured to, when executed by the processor  14 , cause the computing device  12  to perform operations according to the exemplary embodiment. 
     The computer readable storage medium  16  is configured to store computer executable commands and program codes, program data and/or information in other suitable forms. The program  20  stored in the computer readable storage medium  16  may include a set of commands executable by the processor  14 . In one embodiment, the computer readable storage medium  16  may be a memory (volatile memory, such as random access memory (RAM), non-volatile memory, or a combination thereof), one or more magnetic disk storage devices, optical disk storage devices, flash memory devices, storage media in other forms capable of being accessed by the computing device  12  and storing desired information, or a combination thereof. 
     The communication bus  18  connects various other components of the computing device  12  including the processor  14  and the computer readable storage medium  16 . 
     The computing device  12  may include one or more input/output interfaces  22  for one or more input/output devices  24  and one or more network communication interfaces  26 . The input/output interface  22  and the network communication interface  26  are connected to the communication bus  18 . The input/output device  24  may be connected to other components of the computing device  12  through the input/output interface  22 . The illustrative input/output device  24  may be a pointing device (a mouse, a track pad, or the like), a keyboard, a touch input device (a touch pad, a touch screen, or the like), an input device, such as a voice or sound input device, various types of sensor devices, and/or a photographing device, and/or an output device, such as a display device, a printer, a speaker, and/or a network card. The illustrative input/output device  24 , which is one component constituting the computing device  12 , may be included inside the computing device  12  or may be configured as a device separate from the computing device  12  and be connected to the computing device  12 . 
     According to the embodiments of the present disclosure, management authority for a common secret key that corresponds to a common public key is distributed to all users who have cooperated in generating the common public key and decryption for a ciphertext encrypted using the common public key is allowed only when the minimum number of users who have cooperated in generating the common public key agree, so that unauthorized data leakage due to leakage of the common secret key can be prevented and safe management of the secret key is enabled among users who lack mutual trust. 
     A number of examples have been described above. Nevertheless, it will be understood that various modifications may be made. For example, suitable results may be achieved if the described techniques are performed in a different order and/or if components in a described system, architecture, device, or circuit are combined in a different manner and/or replaced or supplemented by other components or their equivalents. Accordingly, other implementations are within the scope of the following claims.