Arbitrary \(d~(\ge 2)\)-level \((t,n)\) threshold quantum secret reconstruction scheme using symmetric entanglements

Secret Sharing schemes are very much well-developed in classical cryptography. This paper introduces a novel Secret Sharing scheme that leverages entanglement for secure communication. While our protocol initially focuses on a single reconstructor, it offers the flexibility to dynamically change the reconstructor without compromising the reconstruction security of the shared secret. Traditional Secret Sharing schemes often require secure channels for transmitting secret shares to the reconstructor, which can be costly and complex. In contrast, our proposed protocol eliminates the need for secure channels, significantly reducing implementation overhead. Our proposed scheme introduces a secret reconstruction method for \(d \ge 2\), expanding upon previous works that primarily focused on \(d > 2.\) Our work provides a unified framework that bridges the gap between the cases \(d = 2\) and \(d > 2.\) We carefully analyze the conditions under which each case achieves its highest level of security, utilizing newly developed concepts, termed Perfectly Symmetric, Almost Symmetric, and queryless or Vacuously Symmetric entanglements. By eliminating the need for Quantum Fourier Transform and Inverse Quantum Fourier Transform, which were commonly used in previous schemes, we simplify the proposed protocol and potentially improve its efficiency. We thoroughly analyze the correctness and security of our proposed scheme, ensuring its reliability and resistance to certain quantum attacks. Finally, we propose a detailed comparison with the previous works.