Modular Polynomial Multiplication for R-LWE Cryptosystems
Find Innovative SolutionsGenerate Solutions
Solution Overview
Problem
The complexity of modular polynomial multiplication in ring-learning with errors (R-LWE) and fully homomorphic encryption schemes, particularly in large coefficient operations, is not efficiently addressed by existing methods, leading to high computational costs and resource requirements.
Innovation Solution
Integrating modular reduction into the Karatsuba polynomial multiplication process by applying it to intermediate segment products instead of the final product, allowing for additional substructure sharing and reducing the number of coefficient additions needed to assemble the result.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If modular reduction is applied to the final product after Karatsuba multiplication, then the correctness of the result is ensured, but the computational complexity and number of operations remain high
Solution Approach 1:
The patent applies modular reduction to intermediate segment products before the final assembly, rather than waiting until the end. This preliminary action reduces the size of numbers that need to be handled in subsequent operations, thereby reducing overall computational complexity while maintaining correctness.
Solution Approach 2:
The patent divides the polynomial multiplication into segments and applies modular reduction to each segment product individually. This segmentation allows the reduction operation to be performed on smaller, more manageable pieces rather than the entire large product, reducing the computational burden.
2Device complexity
If the Karatsuba scheme is used to reduce polynomial multiplication complexity, then the number of coefficient multiplications is reduced, but the number of coefficient additions increases
Solution Approach 1:
The patent changes the parameter of when modular reduction is applied - from after the final product to during the assembly process of segment products. This parameter change allows the reduction to happen at optimal points where it minimizes the total number of additions required.
Solution Approach 2:
The patent merges the modular reduction operation with the Karatsuba multiplication process itself, integrating reduction into the assembly of segment products. This merging allows the reduction to be performed concurrently with the multiplication process, optimizing the balance between multiplications and additions.
3Productivity
If modular reduction is applied to intermediate segment products, then the number of coefficient additions is reduced by 13-17%, but the algorithm complexity increases
Solution Approach 1:
By performing modular reduction on intermediate segment products before final assembly, the algorithm reduces the number of additions needed in the final steps. This preliminary reduction action, though adding an extra operation step, significantly reduces the overall computational burden and is justified by the 13-17% reduction in coefficient additions.
Data Source
AI summary
A method includes receiving a first polynomial and a second polynomial, both of order n−1 and forming d polynomial segments from both the first polynomial and the second polynomial such that each polynomial segment is of order (n/d)−1. The polynomial segments of the first polynomial and the d polynomial segments of the second polynomial are used to form segment products. Each segment product is divided into a first polynomial substructure of order n/d and a second polynomial substructure of order (n/d)−1. A first polynomial substructure containing the first n/d coefficients of a product of the first polynomial and the second polynomial is summed with a second polynomial substructure to form a sum substructure. The sum substructure is used multiple times to determine coefficients of a polynomial representing the modulo xn+1 of the product of the first polynomial and the second polynomial.