Ciphertext Bootstrapping via Root-of-Unity Polynomial Approximation
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Solution Overview
Problem
Existing homomorphic encryption schemes, such as the Cheon-Kim-Kim-Song (CKKS) scheme, face limitations in noise accumulation, complex operations, high polynomial degrees, and inefficiencies in bootstrapping processes, particularly when handling real and complex numbers, leading to errors and decreased throughput.
Innovation Solution
An electronic device and method that transforms integer data to complex root of unity data, performs polynomial operations in a ciphertext state, and applies conjugate complex operations to reduce errors and improve efficiency, allowing for parallel processing and accurate function evaluation without decoding.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If bootstrapping is performed using related art methods, then noise can be removed and ciphertext can be re-encrypted, but operations become quite complex and errors become significant
Solution Approach 1:
The patent introduces an intermediary polynomial approximation method that bridges the gap between ciphertext operations and function evaluation. By using polynomial approximations of target functions, the system can perform complex operations through simpler polynomial evaluations, reducing operational complexity while maintaining accuracy. The polynomial serves as a mediator that transforms difficult homomorphic operations into more manageable polynomial arithmetic.
Solution Approach 2:
The patent changes key parameters of the bootstrapping process by using optimized polynomial degrees and modified decryption procedures. Instead of using high-degree polynomials that increase complexity, the patent employs carefully selected lower-degree polynomials that maintain approximation accuracy while reducing computational burden. Parameter optimization in the polynomial selection and noise management resolves the contradiction between accuracy and complexity.
2Measurement precision
If polynomial degree is increased for function approximation, then approximation accuracy improves, but bootstrapping time becomes longer due to number of operations increasing
Solution Approach 1:
The patent applies partial action by using polynomial approximations that are sufficient but not excessive for the required accuracy. Instead of using high-degree polynomials that provide more accuracy than needed, the patent selects polynomial degrees that provide just enough approximation quality for the application, thereby reducing the number of operations and bootstrapping time while maintaining acceptable accuracy levels.
Solution Approach 2:
The patent optimizes polynomial degree parameters to find the sweet spot between accuracy and computational efficiency. By carefully selecting polynomial degrees based on the specific function being approximated and the required precision, the system achieves good approximation accuracy without the computational overhead of excessively high-degree polynomials, thus reducing bootstrapping time.
3Productivity
If parallelizing and vectorizing are limited, then polynomial degree can be kept manageable, but throughput and bit unit processing efficiency decrease
Solution Approach 1:
The patent segments the bootstrapping process into independent parallelizable stages: polynomial evaluation, noise calculation, and ciphertext re-encryption. By dividing the overall process into discrete segments that can be executed in parallel, the system increases throughput efficiency without requiring excessive parallelization infrastructure. Each segment can be vectorized independently, improving productivity while keeping individual operation complexity manageable.
Solution Approach 2:
The patent creates a universal bootstrapping framework that can handle multiple functions and operations through a single polynomial approximation mechanism. This multi-functional approach allows the same infrastructure to serve various computational needs, improving overall throughput efficiency without requiring separate specialized systems for each operation, thereby avoiding the complexity of extensive parallelization.
4Adaptability or versatility
If CKKS scheme is used for processing real numbers and complex data, then vector calculations can be performed, but noise accumulates and decoding becomes impossible as scale collapses
Solution Approach 1:
The patent uses polynomial approximation as an intermediary mechanism that enables function evaluation on encrypted data without directly collapsing the ciphertext scale. The polynomial evaluation serves as a mediator that transforms the problematic direct operation on encrypted real/complex numbers into a series of manageable polynomial arithmetic operations that preserve ciphertext validity while achieving the desired computational functionality.
Solution Approach 2:
The patent dynamically adjusts scale parameters during the bootstrapping process to prevent collapse. By monitoring and adjusting the ciphertext scale parameter in response to noise accumulation, the system maintains ciphertext validity even after multiple operations on real and complex data. Parameter adaptation ensures that the CKKS scheme continues to function reliably for vector calculations without scale collapse.
Data Source
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AI summary
An electronic device and a controlling method thereof are disclosed. Specifically, the electronic device performs bootstrapping for an input ciphertext encrypted with integer data, obtains an intermediate ciphertext by transforming the integer data to complex root of unity data, obtains, based on the intermediate ciphertext, a polynomial for approximating a pre-defined function in a complex root of unity, and obtains, by inputting the complex root of unity data in the polynomial to approximately perform an operation corresponding to the function in a ciphertext state, an output ciphertext comprising a function value corresponding to the integer data.