Encryption processing method, encryption processing device, and computer-readable recording medium storing program for encryption processing

1. An encryption processing method for authenticating a user at a client terminal, the method comprising:

converting a first vector using a first polynomial representation to acquire a first polynomial, the first vector corresponding to first authentication information;

converting an expression using a second polynomial representation to acquire a second polynomial, the expression being obtained based on a second vector and a random number, the second vector corresponding to second authentication information;

converting the random number using at least one of the first polynomial representation and the second polynomial representation to acquire a random number polynomial;

encrypting the first polynomial, the second polynomial, and the random polynomial using a homomorphic encryption scheme to acquire the encrypted first polynomial, the encrypted second polynomial, and the encrypted random polynomial;

transmitting the encrypted first polynomial, the encrypted second polynomial, and the encrypted random polynomial;

receiving the encrypted first polynomial, the encrypted second polynomial, and the encrypted random polynomial;

decrypting the received encrypted first polynomial, the received encrypted second polynomial, and the received encrypted random polynomial, to generate decrypted first polynomial, decrypted second polynomial, and decrypted random polynomial, respectively;

determining a first resolved value of the decrypted first polynomial;

determining a second resolved value of the decrypted second polynomial;

calculating a difference between the first resolved value and the second resolved value; and

authenticating a user when the difference is less than a threshold.

2. The encryption processing method according to claim 1 , wherein the expression being obtained based on the second vector and the random number is identical to a sum of the first vector and the random number.

3. The encryption processing method according to claim 1 , wherein the expression being obtained based on the second vector and the random number is identical to a product of the second vector and the random number.

4. The encryption processing method according to claim 1 , further comprising:

generating the random number using a password input.

5. The encryption processing method according to claim 1 , wherein

the first polynomial representation is formed so that components of the first vector are arranged in an ascending order of power and the second polynomial representation is formed so that components of the second vector are arranged in a descending order of power, or

the first polynomial representation is formed so that the components of the first vector are arranged in the descending order of power and the second polynomial representation is formed so that the components of the second vector are arranged in the ascending order of power.

6. An encryption processing device for authenticating a user, the encryption processing device comprising:

a memory; and

at least one processor coupled to the memory and configured to:

convert a first vector using a first polynomial representation to acquire a first polynomial, the first vector corresponding to first authentication information;

convert an expression obtained based on a second vector and a random number using a second polynomial representation to acquire a second polynomial, the second vector corresponding to second authentication information,

convert the random number using at least one of the first polynomial representation and the second polynomial representation to acquire a random number polynomial, and

encrypt the first polynomial, the second polynomial, and the random polynomial using a homomorphic encryption scheme to acquire the encrypted first polynomial, the encrypted second polynomial, and the encrypted random polynomial;

transmit the encrypted first polynomial, the encrypted second polynomial, and the encrypted random polynomial;

receive the encrypted first polynomial, the encrypted second polynomial, and the encrypted random polynomial;

decrypt the received encrypted first polynomial, the received encrypted second polynomial, and the received encrypted random polynomial, to generate decrypted first polynomial, decrypted second polynomial, and decrypted random polynomial, respectively;

determine a first resolved value of the decrypted first polynomial;

determine a second resolved value of the decrypted second polynomial;

calculate a difference between the first resolved value and the second resolved value; and

authenticate a user when the difference is less than a threshold.

7. The encryption processing device according to claim 6 , wherein the expression being obtained based on the second vector and the random number is identical to a sum of the first vector and the random number.

8. The encryption processing device according to claim 6 , wherein the expression being obtained based on the second vector and the random number is identical to a product of the second vector and the random number.

9. The encryption processing device according to claim 6 , wherein the at least one processor is further configured to:

generate the random number using a password input from outside of the encryption processing device.

10. The encryption processing device according to claim 6 , wherein the at least one processor is further configured to:

form the first polynomial representation so that components of the first vector are arranged in an ascending order of power and the second polynomial representation so that components of the second vector are arranged in a descending order of power, or

form the first polynomial representation so that the components of the first vector are arranged in the descending order of power and the second polynomial representation so that the components of the second vector are arranged in the ascending order of power.

11. A non-transitory computer-readable recording medium having stored therein a program for causing a computer to authenticate a user, the process comprising:

converting a first vector using a first polynomial representation to acquire a first polynomial, the first vector corresponding to first authentication information;

converting an expression obtained based on a second vector and a random number using a second polynomial representation to acquire a second polynomial, the second vector corresponding to second authentication information;

converting the random number using at least one of the first polynomial representation and the second polynomial representation to acquire a random number polynomial;

encrypting the first polynomial, the second polynomial, and the random polynomial using a homomorphic encryption scheme to acquire the encrypted first polynomial, the encrypted second polynomial, and the encrypted random polynomial;

decrypting the received encrypted first polynomial, the received encrypted second polynomial, and the received encrypted random polynomial, to generate decrypted first polynomial, decrypted second polynomial, and decrypted random polynomial, respectively;

determining a first resolved value of the decrypted first polynomial;

determining a second resolved value of the decrypted second polynomial;

calculating a difference between the first resolved value and the second resolved value; and

authenticating a user when the difference is less than a threshold.

12. The non-transitory computer-readable recording medium having stored therein the program for causing the computer to execute the process for encryption according to claim 11 , wherein the expression being obtained based on the second vector and the random number is identical to a sum of the first vector and the random number.

13. The non-transitory computer-readable recording medium having stored therein the program for causing the computer to execute the process for encryption according to claim 11 , wherein the expression being obtained based on the second vector and the random number is identical to a product of the second vector and the random number.

14. The non-transitory computer-readable recording medium having stored therein the program for causing the computer to execute the process for encryption according to claim 11 , the processes further comprising:

generating the random number using a password input.

15. The non-transitory computer-readable recording medium having stored therein the program for causing the computer to execute the process for encryption according to claim 11 , wherein

the first polynomial representation is formed so that components of the first vector are arranged in an ascending order of power and the second polynomial representation is formed so that components of the second vector are arranged in a descending order of power, or

the first polynomial representation is formed so that the components of the first vector are arranged in the descending order of power and the second polynomial representation is formed so that the components of the second vector are arranged in the ascending order of power.