Linear Interpolation Secret Sharing via Hyperplane Points
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Solution Overview
Problem
Existing secret sharing schemes lack secure methods for distributing and reconstructing secrets among cooperating entities, particularly in scenarios where some shares may be lost or holders are absent, without compromising security.
Innovation Solution
A method and system using linear interpolation to distribute shares of a secret among entities, where shares are created as points on a K-dimensional secret hyperplane, allowing reconstruction with a subset of shares, and sub-shares are distributed to ensure security and availability.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If secret sharing schemes distribute shares among participants, then the secret can be reconstructed when required number of shares are combined, but individual shares must remain useless on their own which complicates the distribution and reconstruction process
Solution Approach 1:
The secret is segmented into multiple shares using linear interpolation, where each share represents a point on a K-dimensional secret hyperplane. The segmentation is achieved by forming linear equations with the secret and random elements, creating K shares that individually contain no useful information but collectively can reconstruct the secret through linear combination.
Solution Approach 2:
Linear interpolation serves as an intermediary mathematical mechanism that bridges the secret and its distributed shares. The linear equations act as mediators, transforming the secret into share points on a hyperplane while maintaining the property that any K points can uniquely determine the hyperplane and thus reconstruct the secret.
2Adaptability or versatility
If secret shares are distributed to multiple entities, then availability is improved in case of loss or absence, but security is compromised if any single entity gains access to sufficient shares
Solution Approach 1:
Each share is given local quality by being a distinct point on the secret hyperplane with unique coordinates derived from linear interpolation. No single share (local portion) contains the complete secret information, but any K shares together provide the necessary information to reconstruct the secret, achieving both availability and security.
Solution Approach 2:
The security parameter is changed by using linear interpolation in K-dimensional space, where the secret is embedded as a hyperplane. The parameters of the linear equations (coefficients and constants) are changed through random element selection, ensuring that individual shares have different information content while maintaining the ability to reconstruct the secret when K shares are combined.
3Reliability
If linear interpolation is used to create shares as points on a K-dimensional secret hyperplane, then secure reconstruction is enabled with subset of shares, but the mathematical complexity of forming and solving linear equations increases
Solution Approach 1:
The mechanical process of secret sharing is substituted with linear algebra operations. Instead of physical distribution methods, the patent uses linear interpolation and matrix operations to create and reconstruct shares. The linear equations replace traditional mechanical secret sharing mechanisms, enabling automated and scalable implementation.
Solution Approach 2:
The patent changes the parameter space by working in K-dimensional linear algebra space rather than traditional secret sharing spaces. The secret is transformed into parameters of a linear hyperplane equation, and shares are generated by evaluating this equation at different points. This parameter transformation simplifies the reconstruction process to solving linear equations, which is computationally efficient.
Data Source
AI summary
A method and system distributes shares of a secret among cooperating entities using linear interpolation. In one embodiment, a linear equation is formed using the secret and random elements. The linear equation represents a K-dimensional secret hyperplane, where K is the number of shares to reconstruct the secret. Shares of the secrets are created, with each share containing a point on the secret hyperplane. The shares are then distributed to cooperating entities for secret sharing.