FHE Bootstrapping FFT Reduction via Precomputed Phase Vectors
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Solution Overview
Problem
Current bootstrapping processes for Fully Homomorphic Encryption (FHE) require a large number of computationally expensive Fast Fourier Transform (FFT) operations, which impede efficient encryption and decryption in security applications.
Innovation Solution
The proposed method reduces the number of FFT operations by performing FFTs only during initialization and eliminating the need for FFTs during subsequent iterations, using a phase vector W(a) to represent the output of FFT operations, thereby minimizing computational resources and time.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If a large number of FFT operations are performed during bootstrapping, then the accuracy and security of FHE are maintained, but the computational overhead and processing time increase significantly
Solution Approach 1:
The patent pre-calculates and stores the FFT of the bootstrap key in a lookup table during an initialization phase. This preliminary action allows subsequent bootstrapping operations to retrieve pre-computed values instead of performing FFT operations in real-time, significantly reducing processing time while maintaining the same computational accuracy and security properties
Solution Approach 2:
The patent pre-computes phase vectors W(a) and stores them in advance. By performing these computationally intensive FFT operations beforehand and caching the results, the system eliminates the need to repeat them during actual bootstrapping operations, thereby reducing operational overhead without compromising reliability
2Reliability
If multiple FFT operations are performed for each iteration of bootstrapping, then the correctness of ciphertext transformation is ensured, but the computational resources and processing power required increase
Solution Approach 1:
The patent performs FFT operations on the bootstrap key during an initialization phase and stores the results in a lookup table. This preliminary computation eliminates the need to repeatedly perform FFT operations during subsequent bootstrapping iterations, significantly reducing the computational processing power required while ensuring transformation correctness through the use of pre-validated mathematical relationships
Solution Approach 2:
The patent uses pre-computed phase vectors W(a) that represent the output of FFT operations. By copying and storing these pre-calculated values instead of re-computing them, the system maintains mathematical correctness while reducing the processing power needed during actual bootstrapping operations
3Measurement precision
If FFT operations are performed during each iteration of bootstrapping, then the noise elimination accuracy is maintained, but the computational complexity and time consumption increase
Solution Approach 1:
The patent pre-calculates phase vectors W(a) that encode the necessary transformation information for noise elimination. By performing these computations in advance and storing them for retrieval, the system maintains the mathematical precision required for accurate noise elimination while significantly reducing the computational complexity during actual bootstrapping iterations
Solution Approach 2:
The patent introduces pre-computed phase vectors W(a) as intermediary structures that bridge the gap between the bootstrap key and the ciphertext. These intermediaries contain pre-processed information that enables accurate noise elimination without requiring complex real-time FFT operations, thereby reducing computational complexity while preserving precision
Data Source
AI summary
A system and method to reduce fast Fourier transforms (FFT) required for bootstrapping in a Fully Homomorphic Encryption process. Ciphertext is separated into a vector of n samples. A fast Fourier transfer (FFT) is performed over a first vector of the samples. An FFT is performed for each of n polynomial terms multiplied by a bootstrap key. A point wise multiplication of each of the FFT outputs of the FFTs of the polynomial terms and the output of the FFT over the vector of the n samples is performed. The result of the FFT over the vector of the n samples is added to the results of the set of pointwise multiplications. An inverse FFT (IFFT) is performed on the FFT over the vector of n samples and the accumulated results of the point-wise multiplications to obtain a bootstrapping result of the ciphertext.