Polynomial-Based Secret Sharing with Reconstruction Verification
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Solution Overview
Problem
Existing secret sharing schemes face challenges in detecting cheating when a cheater holds k or more pieces of distributed information, and the size of the distributed information is significantly larger than the secret information, making it difficult to verify reconstruction results without participating in the process.
Innovation Solution
A distributed information generation and reconstruction system that uses (k−1)-degree polynomials to distribute and reconstruct secret information, with a method to verify the integrity of the reconstruction process by checking if f(t) = g(0), reducing the size of the distributed information and enhancing cheating detection probability.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If a (k-1)-degree polynomial is used to distribute secret information, then the secret information can be completely reconstructed by collecting k or more pieces of distributed information, but the size of the distributed information becomes significantly larger than the secret information itself
Solution Approach 1:
The patent divides the secret information into multiple segments and distributes them through separate polynomials. Instead of using one large polynomial that embeds all secret information, the system uses multiple (k-1)-degree polynomials where each polynomial handles a portion of the secret information. This segmentation reduces the degree and size of each individual polynomial while maintaining the ability to reconstruct the complete secret information when k or more distributed information pieces are collected.
Solution Approach 2:
The patent introduces a verification dimension by generating both distributed information for secret reconstruction and verification information for integrity checking. This adds a new dimension to the system where verification can be performed independently of reconstruction, allowing detection of cheating without requiring additional reconstruction operations. The verification polynomial g(x) operates in parallel with the secret polynomial f(x), creating a multi-dimensional information structure.
2Reliability
If verification mechanisms are added to detect cheating in distributed information, then the probability of detecting tampering increases, but the complexity of the system increases
Solution Approach 1:
The patent merges the verification function with the existing distributed information structure. The verification polynomial g(x) is constructed using the same random coefficients and evaluation points as the secret polynomial f(x), integrating verification into the existing framework. The verification information is distributed alongside the secret information through the same channels, eliminating the need for separate verification infrastructure and reducing overall system complexity.
Solution Approach 2:
The system enables self-verification where the distributed information itself contains the means to verify its integrity. The verification polynomial g(x) is derived from the same random parameters used to generate f(x), allowing any participant to verify the authenticity of distributed information using only locally stored verification data and the received distributed information, without requiring external verification authorities or complex additional mechanisms.
3Quantity of substance
If the distributed information size is reduced by embedding secret information in higher-degree coefficients, then the distributed information size decreases, but information on the secret information leaks from (k-1) or less pieces of distributed information
Solution Approach 1:
The patent segments the secret information into multiple portions and assigns each portion to a separate (k-1)-degree polynomial. By dividing the secret information into segments and distributing them through multiple polynomials, the system maintains security because any (k-1) or fewer distributed information pieces from any single polynomial reveal only a segment of the secret information, not the complete secret. The full secret information is only reconstructed when k or more pieces are collected from the segmented polynomials.
Solution Approach 2:
The patent performs preliminary embedding of secret information segments into multiple separate polynomials before distribution. Each polynomial f_i(x) is pre-configured with a specific segment of the secret information in its coefficients, and verification polynomials g_i(x) are pre-configured with corresponding verification data. This preliminary segmentation and distribution structure ensures that no single polynomial contains the complete secret information, preventing information leakage while maintaining efficient distributed information sizes.
Data Source
AI summary
Disclosed is a distributed information generation apparatus which generates a polynomial F in which secret information s is embedded to output distributed secret information, generates a polynomial G in which an output when a fixed value t is entered into the polynomial F is embedded as the secret information, and generates distributed distributed secret information. The reconstruction apparatus receives k sets of the distributed secret information to generate a polynomial F′, and receives k sets of the distributed distributed secret information to generate a polynomial G′. When a value embedded in the polynomial G′ as the secret information and an output obtained by entering the fixed value into the polynomial F′ is equal, the distributed information generation apparatus outputs the embedded value. When the embedded value is not equal to the output, the distributed information generation apparatus outputs information indicating detection of cheating.