Resultant-based verifiable threshold secret sharing for secure AI–blockchain ecosystems: towards quantum-safe cryptography

Introduction This paper introduces two novel verifiable (t,n)-threshold secret sharing schemes utilising the resultant of bivariate polynomials. Methods Both schemes leverage the evaluation and scaling properties of resultants, facilitating share generation as well as secure verification processes capable of identifying adversarial participants. One scheme uses efficient numerical shares derived from resultant evaluations, while the other distributes polynomial (symbolic) shares offering enhanced security and potential advantages for use in post-quantum cryptography applications. An algorithm is also provided to assist the share distributor (the dealer) in constructing suitable input bivariate polynomials for these resultant-based schemes. Results In addition to the two novel schemes, this study includes complete mathematical proofs, presented as theorems and lemmas, which formally verify the correctness of the polynomial construction algorithm and establish the correctness and verifiability (including adversary identification) of both proposed schemes. Discussion These resultant-based approaches expand the design space of secret sharing, enabling efficient, flexible, and provably verifiable schemes with guaranteed adversary detection (excluding events of negligible probability). The proposed schemes are particularly relevant to secure AI–blockchain convergence, where distributed trust, auditability, privacy-preserving computation, and protection of sensitive cryptographic material are critical requirements.