Matrix-Based Decoding for Lower Field Multiplication Load
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Solution Overview
Problem
Conventional error correcting coding techniques have high processing amounts, particularly in the field multiplication performed during the decoding process.
Innovation Solution
A decoding apparatus that reduces processing amounts by generating a ϕ-th order vector and an inverse matrix, allowing for efficient computation of plaintext elements through vector conversion and inverse matrix multiplication, utilizing a specific matrix structure and polynomial operations.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If conventional Reed-Solomon decoding is performed using the full inverse matrix A′−1, then complete plaintext recovery is achieved, but the processing amount of field multiplication becomes excessively high
Solution Approach 1:
The patent extracts only the necessary φ rows (corresponding to parity shares) from the full inverse matrix A′−1 to form a reduced inverse matrix. This extraction eliminates unnecessary computational operations while preserving the capability to recover the missing φ plaintext elements, directly reducing field multiplication processing amount without compromising decryption completeness
Solution Approach 2:
The patent segments the decoding process into two distinct phases: (1) recovering φ missing plaintext elements using only φ rows of the inverse matrix multiplied by φ parity shares, and (2) obtaining the remaining k-φ plaintext elements directly from k-φ available data shares. This segmentation avoids redundant computations and optimizes the overall decoding efficiency
2Manufacturing precision
If the full inverse matrix A′−1 is used for decoding, then all plaintext elements can be computed, but the computational complexity and processing time increase
Solution Approach 1:
The patent applies partial action by computing only what is necessary: φ rows of the inverse matrix are sufficient to recover φ missing plaintext elements. The remaining k-φ plaintext elements are obtained directly from available data shares without requiring matrix multiplication, thus achieving complete plaintext recovery with reduced computational effort and time
Data Source
AI summary
To reduce the processing amount of a field multiplication. a denotes a k-th order vector whose elements are a0, . . . , ak−1 (a0, . . . , ak−1∈GF(xq)). A denotes an n-by-k matrix formed by vertically connecting a identity matrix and a Vandermonde matrix. b denotes an n-th order vector obtained by multiplying the vector a and the matrix A whose elements are b0, . . . , bn−1 (b0, . . . , bn−1∈GF(xq)). A vector conversion part 11 generates a ϕ-th order vector b′ using ϕ elements bp0, . . . , bpϕ−1 of the vector b. An inverse matrix generation part 12 generates a ϕ-by-ϕ inverse matrix A′−1. A plaintext computation part 13 computes elements ae0, . . . , aeϕ−1 of the vector a by multiplying the vector b′ and the inverse matrix A′−1.