Secret Sharing Method Using GF(xq) Matrix Arithmetic
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Solution Overview
Problem
Shamir's secret sharing technique has high computational complexity, and XOR-based secret sharing lacks parameter flexibility, limiting its applicability and security flexibility.
Innovation Solution
A secret sharing method using an n-by-k matrix defined by specific formulas, incorporating random number generation, share generation, selection, and inverse-matrix calculation to reduce computational complexity and enhance flexibility in k and n values.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If Shamir's secret sharing is used, then information-theoretic security is achieved, but computational complexity increases significantly
Solution Approach 1:
The patent changes the mathematical parameters from general field multiplications to polynomial arithmetic over GF(xq), where computations are performed modulo an irreducible polynomial. This parameter change reduces the computational complexity from (k-1)n field multiplications to polynomial operations that are computationally more efficient while maintaining the same security level.
2Device complexity
If XOR-based secret sharing is used, then computational complexity is reduced, but parameter flexibility is limited
Solution Approach 1:
The patent creates a universal secret sharing scheme based on polynomial arithmetic over GF(xq) that can accommodate any k and n values satisfying n≥2k-1. The matrix A defined in the patent provides a unified framework that works for different threshold values k and total shares n, making the system multi-functional and adaptable to various security requirements unlike XOR-based schemes.
3Reliability
If higher security level (larger k) is implemented, then security is improved, but computational complexity increases
Solution Approach 1:
The patent replaces the mechanical field multiplication operations in Shamir's scheme with polynomial arithmetic operations over GF(xq). This substitution maintains the security level determined by k while reducing the computational burden through more efficient polynomial operations, particularly using the properties of irreducible polynomials to simplify multiplication and inversion.
Data Source
AI summary
Computational complexity is reduced in accordance with given k and n. A random number generation unit 12 generates random numbers r0 to rk−2 ϵGF(xq). A share generation unit 14 generates shares b0 to bn−1 by calculating a product of a vector a=(r0, . . . , rk−2, s), having the random numbers r0 to rk−2 and plaintext s ϵGF(xq) as its elements, and a matrix A. A share selection unit 15 generates a vector b′=(bp0, . . . , bpk−1) having, as its elements, k shares bp0 to bpk−1 selected from the shares b0 to bn−1. An inverse-matrix generation unit 16 generates an inverse matrix A′−1 of a k-degree square matrix having the p0-th to pk−1-th rows of the matrix A. A plaintext calculation unit 17 restores the plaintext s by multiplying the k-th row of the inverse matrix A′−1 and the vector b′.