Two-Variable NTT Decomposition for Lower-Twiddle Polynomial Multiplication
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Solution Overview
Problem
Existing encryption algorithms, particularly those based on prime factorization and homomorphic ring encryption, face computational complexity and inefficiencies, especially when applied to polynomial multiplication, which can be exacerbated by quantum computing threats.
Innovation Solution
Implementing a two-variable number theoretic transform (2NTT) that decomposes matrices using two roots for fast Fourier transforms (FFTs) to reduce the number of twiddle factors required, allowing for efficient polynomial multiplication in encrypted data.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If traditional homomorphic encryption operations are performed on encrypted data, then security is maintained, but computational complexity increases significantly
Solution Approach 1:
The patent segments the large N-point NTT operation into multiple smaller stages by decomposing the input into matrices of dimensions a×b where N=ab. This segmentation allows the computation to be divided into manageable blocks that can be processed more efficiently, reducing the overall computational complexity while maintaining security through the preserved mathematical properties of the segmented transforms.
Solution Approach 2:
The patent transitions from a one-dimensional NTT approach to a two-dimensional matrix-based approach. By organizing data into matrices and applying row-wise and column-wise transforms with different roots (α and β), the system exploits additional dimensional structure to reduce computational complexity. This dimensional change enables more efficient memory access patterns and parallel processing while maintaining the cryptographic security guarantees.
2Productivity
If traditional NTT with single root is used, then transform is applied, but number of twiddle factors increases memory requirements
Solution Approach 1:
The patent segments the twiddle factor storage requirement by associating different roots (α for rows, β for columns) with different dimensions of the matrix. This segmentation allows the system to reuse smaller sets of twiddle factors across multiple transform operations, significantly reducing the total memory space required to store precomputed twiddle factors while maintaining transform efficiency.
Solution Approach 2:
The patent makes the twiddle factor sets universal by designing them to be reused across different stages of the transform. The row-wise twiddle factors based on root α are reused for all rows, and column-wise twiddle factors based on root β are reused for all columns. This multi-functionality reduces the overall quantity of twiddle factors needed compared to traditional approaches that require unique factors for each position.
3Ease of operation
If polynomial multiplication is performed directly in time domain, then simplicity is maintained, but computational time increases
Solution Approach 1:
The patent substitutes the mechanical process of direct polynomial multiplication with a transformed-domain approach using 2D NTT. By converting polynomials to the frequency domain via matrix-based transforms, performing element-wise multiplication, and then applying inverse transforms, the system replaces the O(n²) mechanical multiplication process with an O(n log n) transform-based approach, significantly reducing computational time while maintaining operational simplicity through standardized transform routines.
Data Source
AI summary
Embodiments of the present application provide a system, a device, and a method for a two-variable number theoretic transform. A matrix having the two-variable number theoretic transform may be applied to a matrix having dimensions a×b, where a=2x and b=2y for x≥y. The two-variable number theoretic transform includes two stages: decomposition by rows and row-wise fast Fourier transforms (FFTs), with twiddle factors computed based on a first root α satisfying αb=−1 mod p; and decomposition by columns and column-wise FFTs, with twiddle factors computed based on a second root β satisfying β2a=2 or −2 mod p. Since each of the stages uses a different root, the number of twiddle factors is reduced.