Encryption method and apparatus based on homomorphic encryption using composition of functions

1. An encryption method using homomorphic encryption, the encryption method comprising:

generating a ciphertext by encrypting data; and

bootstrapping the ciphertext by performing a modular reduction based on a composition of a function for a modulus corresponding to the ciphertext,

wherein the bootstrapping comprises bootstrapping the ciphertext by approximating the modular reduction based on the function and an inverse function of the function.

2. The encryption method of claim 1 , wherein the bootstrapping of the ciphertext by approxi mating the modular reduction based on the function and the inversefunction of the function comprises:

obtaining an approximate polynomial of the function;

obtaining an approximate polynomial of the inverse function; and

approximating the modular reduction based on a composite function of the approximate polynomial of the function and the approximate polynomial of the inverse function.

3. The encryption method of claim 2 , wherein the obtaining of the approximate polynomial of the function comprises obtaining, in response to the function being a trigonometric function, a double-angle function of the trigonometric function by applying the double-angle formula to the trigonometric function.

4. The encryption method of claim 2 , wherein the obtaining of the approximate polynomial of the function comprises:

determining one or more reference points based on a degree of the approximate polynomial;

obtaining an arbitrary polynomial based on the one or more reference points; and

generating the approximate polynomial of the function based on one or more extreme points selected from the arbitrary polynomial.

5. The encryption method of claim 4 , wherein the obtaining of the arbitrary polynomial comprises:

obtaining a piecewise continuous function that passes through the one or more reference points; and

obtaining the arbitrary polynomial by generating a polynomial such that absolute values of errors between the polynomial and the piecewise continuous function at the one or more reference points are a specific value.

6. The encryption method of claim 5 , wherein the obtaining of the arbitrary polynomial by generating the polynomial comprises obtaining the arbitrary polynomial by generating a polynomial such that a first error at a first reference point included in the one or more reference points and a second error at a second reference point adjacent to the first reference point are different in sign, and absolute values of the first and second errors are the specific value.

7. The encryption method of claim 4 , wherein the generating of the approximate polynomial based on the one or more extreme points selected from the arbitrary polynomial comprises:

obtaining candidate points whose absolute values are greater than or equal to a specific value among extreme points of errors between the arbitrary polynomial and a piecewise continuous function that passes through the one or more reference points;

selecting target points from among the candidate points, where the number of target points is based on the degree of the approximate polynomial; and

generating the approximate polynomial based on the target points.

8. The encryption method of claim 1 , wherein the function comprises a trigonometric function.

9. A non-transitory computer-readable storage medium storing instructions that, when executed by a processor, cause the processor to perform the encryption method of claim 1 .

10. An encryption apparatus using homomorphic encryption, the encryption apparatus comprising:

a processor configured to generate a ciphertext by encrypting data, and to bootstrap the ciphertext by performing a modular reduction based on a composition of a function for a modulus corresponding to the ciphertext; and

a memory configured to store instructions to be executed by the processor,

wherein the processor is configured to bootstrap the ciphertext by approximating the modular reduction based on the function and an inverse function of the function.

11. The encryption apparatus of claim 10 , wherein the processor is configured to:

obtain an approximate polynomial of the function,

obtain an approximate polynomial of the inverse function, and

approximating the modular reduction based on a composite function of the approximate polynomial of the function and the approximate polynomial of the inverse function.

12. The encryption apparatus of claim 11 , wherein the processor is configured to obtain, in response to the function being a trigonometric function, a double-angle function of the trigonometric function by applying the double-angle formula to the trigonometric function.

13. The encryption apparatus of claim 11 , wherein the processor is configured to:

determine one or more reference points based on a degree of the approximate polynomial,

obtain an arbitrary polynomial based on the one or more reference points, and

generate the approximate polynomial of the function based on one or more extreme points selected from the arbitrary polynomial.

14. The encryption apparatus of claim 13 , wherein the processor is configured to:

obtain a piecewise continuous function that passes through the one or more reference points, and

obtain the arbitrary polynomial by generating a polynomial such that absolute values of errors between the polynomial and the piecewise continuous function at the one or more reference points are a specific value.

15. The encryption apparatus of claim 14 , wherein the processor is configured to obtain the arbitrary polynomial by generating a polynomial such that a first error at a first reference point included in the one or more reference points and a second error at a second reference point adjacent to the first reference point are different in sign, and absolute values of the first and second errors are the specific value.

16. The encryption apparatus of claim 13 , wherein the processor is configured to:

obtain candidate points whose absolute values are greater than or equal to a specific value among extreme points of errors between the arbitrary polynomial and a piecewise continuous function that passes through the one or more reference points,

select target points from among the candidate points, where the number of target points is based on the degree of the approximate polynomial, and

generate the approximate polynomial based on the target points.

17. The encryption apparatus of claim 10 , wherein the function comprises a trigonometric function.