Efficient and Secure Threshold Encryption Schemes with Universal Zero-Knowledge Proofs: Foundations and Implementations

Abstract

This paper presents a comprehensive examination of advanced cryptographic primitives and their instantiations, focusing on signature schemes, zero-knowledge proofs, and threshold encryption schemes. The foundational properties required for a secure and complete digital signature scheme are first outlined, with an emphasis on existential unforgeability. Zero-knowledge proofs are then examined in detail, including definitions of completeness, zero-knowledge, soundness, and simulation extractability, along with a discussion of universal versus non-universal proof systems. The core contribution lies in the design and analysis of a threshold encryption scheme based on Shamir’s secret sharing and a CLT encryption framework. The construction of a (t, P)-threshold encryption scheme is defined, introducing the concept of partial decryption simulability. The framework ensures that any (t + 1)-sized subset of parties can decrypt the ciphertext while maintaining security against adversarial attempts. Additionally, the implementation of universal zero-knowledge proof systems is discussed, highlighting the trade-offs between universal and specific SRS-based proofs. The instantiation of pseudorandom functions and their weak robustness properties is also examined, ensuring secure key management and resistance to adversarial key collisions. Through detailed analysis and construction, this work provides a solid foundation for building secure cryptographic systems with efficient threshold encryption and zero-knowledge proofs, contributing to the advancement of cryptographic protocols and their applications in secure communications and data protection.