Issuing New Shamir Secret Shares Using Multi-Party Computation

A deep technical exploration of Shamir's Secret Sharing (SSS) and how to extend it with multi-party computation to issue new shares without ever reconstructing the original secret. Covers polynomial interpolation fundamentals, Feldman's Verifiable Secret Sharing for trustless verification, and a detailed protocol for trustlessly inducting new shareholders. Includes a security proof analyzing honest and malicious inductee scenarios, practical use cases like SLIP39 share recovery, FROST threshold multisig repair, and repairable escrow schemes, plus discussion of alternative approaches and known limitations.