Homomorphic Encryption Bootstrapping via Complex Root-of-Unity Mapping
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Solution Overview
Problem
Existing homomorphic encryption schemes, such as the CKKS scheme, face challenges with noise accumulation, complex operations, high polynomial degrees, and inefficient throughput due to limited parallelization, especially when handling real and complex numbers, leading to decoding failures and increased bootstrapping times.
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 reduces errors by using symmetrical characteristics and conjugate complex operations to achieve efficient bootstrapping and encryption, enabling parallel processing of multiple ciphertexts.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If bootstrapping is performed using related art techniques, 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 transforms the input ciphertext from integer domain to complex root of unity domain, changing the parameter space in which bootstrapping operations are performed. This transformation enables the use of polynomial approximation methods that are more efficient and less error-prone than traditional approaches, directly resolving the contradiction between reliability and complexity.
Solution Approach 2:
The patent introduces complex root of unity data as an intermediary representation between the input ciphertext and the final output. This intermediary form allows for more stable and accurate polynomial operations during bootstrapping, reducing errors while maintaining operational feasibility.
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 dynamically adjusts the polynomial degree based on the specific function being approximated and the requirements of the bootstrapping operation. By using conjugate complex operations and exploiting symmetries, the method achieves high approximation accuracy with lower polynomial degrees than traditional methods, thereby reducing the number of operations and bootstrapping time.
Solution Approach 2:
The patent exploits the asymmetric properties of complex root of unity and conjugate complex relationships to optimize polynomial approximation. By recognizing and utilizing these asymmetric mathematical properties, the method achieves accurate function approximation with reduced computational complexity compared to symmetric polynomial approaches.
3Productivity
If parallelizing and vectorizing are limited, then implementation simplicity is maintained, but throughput becomes inefficient
Solution Approach 1:
The patent segments the bootstrapping process into distinct stages: transformation to complex root of unity domain, polynomial approximation, and transformation back to integer domain. Each stage can be independently parallelized and vectorized, enabling efficient throughput improvement without overwhelming implementation complexity. The modular structure allows for targeted optimization at each segment.
4Stability of the object's composition
If scale adjusting process is performed, then fixed point scale is maintained, but decoding failure probability increases
Solution Approach 1:
The patent performs scale adjustment operations with built-in error compensation mechanisms before the final decoding stage. By anticipating potential decoding failures and incorporating cushioning operations in advance, the method maintains fixed point scale stability while minimizing the probability of decoding failure.
Data Source
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.