Homomorphic Encryption Bootstrapping Using Chebyshev Polynomials
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Solution Overview
Problem
Current fully homomorphic encryption methods face challenges in efficiently bootstrapping ciphertexts due to noise accumulation, which requires effective noise processing to maintain privacy and operational efficiency, especially in modulus reduction operations that are not arithmetic in nature.
Innovation Solution
The method determines an approximate polynomial using Chebyshev polynomials for modulus reduction, optimizing the polynomial degree and sample points to minimize errors through L2-norm minimization, allowing for efficient homomorphic evaluation and reduced level loss in bootstrapping.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If traditional bootstrapping algorithms are used for homomorphic encryption, then security is maintained, but computational overhead and runtime are high
Solution Approach 1:
The patent changes the parameters of the polynomial approximation by optimizing the degree of the polynomial and the sampling points. This allows achieving the same security level with reduced computational complexity, thereby improving runtime efficiency while maintaining security requirements for bootstrapping in homomorphic encryption
Solution Approach 2:
The patent replaces traditional mechanical/computational bootstrapping algorithms with a mathematical approximation approach using polynomials. By substituting the complex traditional bootstrapping mechanism with a polynomial-based approximation system, the patent achieves faster computation while preserving the essential security properties
2Measurement precision
If polynomial degree is increased to reduce approximation error, then accuracy improves, but computational complexity increases
Solution Approach 1:
The patent optimizes the polynomial degree parameter to find the optimal balance between approximation accuracy and computational complexity. By carefully selecting the polynomial degree and sampling points, the patent achieves sufficient accuracy without unnecessarily increasing polynomial complexity, thus resolving the trade-off between precision and complexity
3Measurement precision
If more samples are used for polynomial approximation, then approximation accuracy improves, but processing time increases
Solution Approach 1:
The patent optimizes the sampling parameters including the number of samples, their distribution, and locations. By carefully tuning these sampling parameters, the patent achieves high approximation accuracy with a reduced number of samples, thereby minimizing processing time while maintaining the required precision for secure bootstrapping
Data Source
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AI summary
A method and apparatus for processing a ciphertext based on homomorphic encryption. The method includes determining an approximate polynomial corresponding to a modulus reduction for bootstrapping a ciphertext based on samples extracted from the modulus reduction, and bootstrapping the ciphertext based on the approximate polynomial.