Traditional collaborative signature schemes face significant challenges in resisting quantum computing attacks, securing private keys in distributed architectures, and balancing operational efficiency, which are critical requirements for modern electronic and information systems like IoT, blockchain, and federated learning. This paper proposes P-CSNKS, a novel post-quotum collaborative signature scheme featuring a non-linear private key splitting technique. Unlike linear secret sharing, P-CSNKS partitions the master private key into multiple interdependent subkeys using multiplicative inverses and modular arithmetic, ensuring algebraic interdependencies prevent full key reconstruction even if attackers compromise sufficient shares. Simultaneously, the scheme embeds hash-based post-quantum signature components directly into the collaborative ECDSA signing workflow. This hybrid design maintains backward compatibility with standard ECDSA verification while establishing dual security layers: one for classical security and another providing provable existential unforgeability against quantum adversaries in the quantum random oracle model. Crucially, P-CSNKS achieves this quantum resistance without incurring prohibitive computational costs. Rigorous experimental evaluations demonstrate that P-CSNKS significantly outperforms lattice-based while also showing efficiency gains against hash-based scheme. The optimized algorithms for key generation, signing, and verification ensure lightweight performance suitable for latency-sensitive applications. Thus, P-CSNKS delivers enhanced security against both classical and quantum threats while meeting the stringent efficiency demands of next-generation distributed systems.
Long et al. (Thu,) studied this question.