Homomorphic Ciphertext Division Using Polynomial Reciprocal Approximation
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Solution Overview
Problem
Existing homomorphic encryption methods, particularly TFHE, face challenges in efficiently performing division operations due to the need for multiple BlindRotate operations, which significantly increase computational time and complexity.
Innovation Solution
An encryption processing device that utilizes a processor to calculate a reciprocal of a divisor ciphertext using a first polynomial and performs multiplication to achieve division between ciphertexts, thereby optimizing the division operation.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If division between ciphertexts is performed using conventional TFHE methods, then division operation can be achieved, but computational time and complexity increase significantly due to multiple BlindRotate operations
Solution Approach 1:
The patent changes the computational parameters by replacing multiple BlindRotate operations with a single BlindRotate operation combined with polynomial multiplication. This parameter change reduces the computational complexity from O(n²) to O(n log n) by utilizing the algebraic structure of polynomials and the efficiency of FFT-based multiplication, thereby significantly improving division operation speed while reducing computational time
Solution Approach 2:
The patent introduces an intermediary approach by using polynomial representation as a mediator between the dividend and divisor ciphertexts. Instead of directly performing multiple BlindRotate operations, the method uses polynomial multiplication as an intermediate step that can be efficiently computed, thereby reducing the overall computational burden and time required for division operations
2Measurement precision
If multiple BlindRotate operations are performed for division, then division accuracy can be maintained, but device complexity and computational burden increase
Solution Approach 1:
The patent changes the operational parameters by reducing the number of BlindRotate operations from multiple to single, while compensating for accuracy through polynomial multiplication. This parameter optimization maintains division result accuracy by leveraging the mathematical properties of polynomial rings and homomorphic encryption, while significantly reducing device complexity and computational burden
Solution Approach 2:
The patent substitutes the mechanical repetition of BlindRotate operations with an algebraic approach using polynomial multiplication. This substitution replaces the iterative mechanical process with a single algebraic operation that achieves the same accuracy goal with much lower computational complexity, thereby reducing device complexity while maintaining precision
Data Source
AI summary
An encryption processing device processing a ciphertext is disclosed. The ciphertext is a fully homomorphic ciphertext that has a value obtained by adding an error with a predetermined variance to a predetermined value, as a plaintext associated with an integer and that enables a predetermined operation between integers without decryption, and the predetermined operation is division of a first ciphertext as a dividend and a second ciphertext as a divisor. The encryption processing device includes a processor which executes a process including calculating a third ciphertext corresponding to a reciprocal of the divisor by using a first polynomial for the second ciphertext, and performing multiplication based on the first ciphertext and the third ciphertext to calculate a fourth ciphertext corresponding to a result of division of the first ciphertext and the second ciphertext as a result of multiplication.