Discrete Galois Transform for Homomorphic Encryption Polynomial Multiplication
Find Innovative SolutionsGenerate Solutions
Solution Overview
Problem
Current methods for computing on encrypted data in fields like cryptography and data encryption are inefficient, particularly in handling large integer numbers and polynomial multiplication, which is essential for operations like fully homomorphic encryption.
Innovation Solution
The method employs discrete Galois transforms and negacyclic convolution to efficiently multiply polynomials using arbitrary prime numbers, enabling computations on encrypted data without decryption, by determining discrete Galois transforms, point-wise products, and inverse discrete Galois transforms to obtain polynomial products.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If conventional polynomial multiplication methods are used in homomorphic encryption, then the computation can be performed on encrypted data, but the computational efficiency is poor and the process is time-consuming
Solution Approach 1:
The patent transforms the polynomial multiplication problem into a different parameter space by applying Discrete Galois Transforms. By changing the representation domain from time/spatial domain to frequency domain, the multiplication operation is converted into point-wise multiplication, dramatically improving computational efficiency from O(n²) to O(n log n) complexity
Solution Approach 2:
The patent replaces the conventional mechanical polynomial multiplication process with a mathematical transformation approach using Galois fields. Instead of directly multiplying polynomials coefficient-by-coefficient, the system substitutes this mechanical process with algebraic transformations in finite fields, achieving faster computation through mathematical elegance rather than brute-force calculation
2Reliability
If polynomial multiplication is performed on encrypted data using traditional methods, then security is maintained, but the noise growth in the encryption process increases significantly
Solution Approach 1:
By transforming the computation into the Galois transform domain, the patent changes the operational parameters of the encryption system. This parameter transformation allows multiplication to be performed as simple point-wise operations on transformed coefficients, which generates less noise compared to traditional polynomial multiplication, thereby maintaining reliability while reducing harmful noise growth
Solution Approach 2:
The Discrete Galois Transform acts as an intermediary mechanism between the plaintext polynomial multiplication and the encrypted domain computation. This intermediary transformation enables the system to perform multiplication operations in a domain that is more friendly to homomorphic encryption, reducing noise accumulation while preserving security properties
Data Source
AI summary
Efficient polynomial multiplication for Accelerated Fully Homomorphic Encryption (FHE). An efficient method for large integer and polynomial multiplication in a ring using negacyclic convolution and discrete Galois transform with arbitrary primes is described. The method is adapted to work with arbitrary primes that support Gaussian arithmetic. Dealing with non-Gaussian primes gives rise to another problem of how to find primitive roots of unity and of (i). An efficient solution to find those roots of interest is provided.