Secret Sharing via Modular Inverses and Prime Subsets
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Solution Overview
Problem
Conventional secret sharing schemes lack robustness in ensuring that only a threshold number of shares can reconstruct the secret, and they may allow unauthorized reconstruction by subsets of shares, especially when some shares are lost or absent, compromising security.
Innovation Solution
A method and system that distribute N shares of a secret by calculating multiplicative inverses of prime numbers, where each share contains a prime number and its inverse, allowing only a subset of shares to reconstruct the secret, enhancing security through the use of unique prime products and inverses.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If conventional secret sharing schemes are used, then the secret can be distributed among multiple parties, but the schemes lack robustness and may allow unauthorized reconstruction by subsets of shares
Solution Approach 1:
The secret is segmented into multiple shares using polynomial interpolation, where each share contains a portion of the secret information. The secret can only be reconstructed when a sufficient threshold of shares is combined, preventing unauthorized reconstruction by smaller subsets. This segmentation approach divides the secret into independent parts that lose meaning individually.
Solution Approach 2:
The invention changes the mathematical parameters used in secret sharing by employing finite field arithmetic and polynomial interpolation with carefully selected degrees. By adjusting the polynomial degree and field parameters, the scheme achieves robust security properties where exactly the required threshold is needed for reconstruction, eliminating vulnerabilities to subset attacks.
2Adaptability or versatility
If polynomial interpolation schemes are used, then the secret can be reconstructed by a subset of shares, but the amount of information held by each shareholder varies and complexity increases
Solution Approach 1:
Each shareholder receives a share with identical structural properties and information density. The polynomial interpolation method ensures that every share contains the same amount of information and has the same mathematical structure, making each share equally valuable and simplifying the distribution and management process.
Solution Approach 2:
The secret sharing scheme is designed to be universally applicable with fixed parameters that work for any number of participants and any threshold requirement. The polynomial-based approach provides a unified framework that handles different scenarios (different n and k values) through the same mathematical mechanism, enhancing versatility while maintaining simplicity.
Data Source
AI summary
A method and system distributes N shares of a secret among cooperating entities by calculating the multiplicative inverses of the secret. In one embodiment, a distributor selects N distinct prime numbers and forms unique subsets of the prime numbers, with each subset containing K of the N prime numbers (N>=K), where K is a threshold number of shares necessary to reconstruct the secret. The distributor calculates a product of the prime numbers in each subset, and, for each subset, calculates the multiplicative inverse of the secret modulo the product. A total of N shares are generated, with each share containing the multiplicative inverses and one of the prime numbers. The N shares are distributed to the cooperating entities for secret sharing.