Secret Share Recovery With Cheater Identification and Smaller Shares

The patent applies preliminary action by generating verification information (syndrome values) along with the secret shares during the share generation phase. This verification information is prepared in advance to enable future detection of falsified shares without adding complexity to the recovery process itself. The syndrome values are computed using random polynomials and stored with the shares, so that when recovery is attempted, the verification can be performed immediately without additional computational overhead.

The patent introduces syndrome values as an intermediary element that mediates between the secret shares and the verification process. These syndrome values act as a bridge that allows the system to verify the authenticity of shares without directly examining the secret information itself. The syndrome values are computed from the shares using random polynomials and serve as a separate verification layer that detects falsification without interfering with the core secret sharing mechanism.

The patent replaces the mechanical verification process (simple check whether recovery succeeded) with a mathematical verification system based on Reed-Solomon error-correcting codes. Instead of merely detecting that recovery failed, the system uses polynomial-based syndrome computation to mathematically identify which specific shares are falsified. This substitution transforms the verification from a binary success/failure outcome to a precise identification of corrupted elements.

The patent changes the verification parameter from a simple recovery status indicator to syndrome values computed using random polynomials. By introducing these polynomial-based parameters, the system gains the ability to not only detect falsification but to specifically identify which shares are corrupted. The syndrome values serve as new parameters that encode information about the authenticity of each share, enabling precise identification of falsified shares rather than just detecting recovery failure.

The patent applies partial action by generating syndrome values using only t+1 randomly selected points instead of requiring full polynomial verification across all shares. This partial verification approach is sufficient to identify falsified shares with high probability while significantly reducing the computational and storage overhead. The system performs exactly enough verification to achieve the security goal without the excessive complexity of full polynomial reconstruction and verification for all possible share combinations.

The patent changes the share structure by adding compact syndrome values rather than expanding shares with full polynomial information. The syndrome values serve as condensed parameters that capture the essential verification information in a space-efficient manner. By using polynomial-based syndrome computation instead of traditional verification methods, the system achieves high-probability falsified share identification while keeping the additional overhead minimal compared to the total share size.