Accelerated Homomorphic Ciphertext Division with Newton Iteration
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Solution Overview
Problem
Existing homomorphic encryption (HE) schemes struggle with complex and time-consuming computations, particularly inversion operations like division, which can reveal sensitive information when decrypting numerator and denominator.
Innovation Solution
A method for performing operations on homomorphically encrypted data using modified Newton iteration recurrence relations and minimax polynomial approximations to iteratively improve the accuracy of computations while keeping data encrypted, reducing the number of homomorphic multiplications required.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If division is performed by decrypting numerator and denominator, then computational accuracy is improved, but data security deteriorates
Solution Approach 1:
The patent introduces an intermediary polynomial approximation function that operates on encrypted values to produce encrypted results without decryption. This mediator enables division-like computation while maintaining encryption, resolving the contradiction between accuracy and security by providing a third pathway that avoids the harmful decryption step entirely.
Solution Approach 2:
The patent replaces the mechanical decryption-computation-encryption process with a homomorphic polynomial evaluation system. Instead of decrypting data to perform division, the system uses polynomial approximations that can be evaluated directly on encrypted inputs, substituting the traditional mechanical approach with a cryptographic-mathematical hybrid that preserves security while achieving computational goals.
2Object-affected harmful factors
If standard homomorphic encryption is used for division, then data security is maintained, but computational speed deteriorates
Solution Approach 1:
The patent applies preliminary action by pre-computing and storing polynomial approximation coefficients that can be rapidly evaluated during division operations. These pre-prepared mathematical tools enable fast homomorphic computation without requiring complex real-time calculations, thus maintaining security while significantly improving computational speed.
Solution Approach 2:
The patent changes the parameter of computational approach from direct homomorphic division to polynomial approximation evaluation. By transforming the mathematical operation into a different form (polynomial evaluation instead of direct division), the system achieves both security and speed improvements, as polynomial evaluations are computationally more efficient while maintaining homomorphic properties.
3Measurement precision
If polynomial approximation degree is increased, then approximation accuracy is improved, but computational complexity deteriorates
Solution Approach 1:
The patent applies partial action by using a polynomial approximation degree that is sufficient for achieving acceptable accuracy without going excessive. Rather than using the highest possible degree polynomial, the system selects a moderate degree that provides adequate precision while keeping computational complexity manageable, balancing accuracy requirements with practical computational constraints.
Data Source
AI summary
Methods and systems for performing an operation on at least one homomorphically encrypted ciphertext, the method include determining, by a computing device, a value that is an initial approximation of a result of the operation on the at least one homomorphically encrypted ciphertext; and iteratively improving, by the computing device, the value using a recurrence relation wherein a number of iterations is determined based on a predetermined accuracy to minimize an approximation error.