Dealer-Based _(5,11) Threshold Quantum Secret Sharing Scheme: Design, Security, and Practical Considerations

We present DQ-(5,11), a dealer-based (5,11) threshold quantum secret sharing (QSS) scheme where any 5 of the 11 participants can find a dealer's secret and 4 or fewer get nothing about it. A key point is that, due to the Cleve-Gottesman-Lo no-cloning bound, there is only a (k,n) threshold scheme for any quantum secret if n < 2k. Since 11≥10, is a (5,11) quantity, the threshold scheme is impossible for quantum secrets, and in any (5,11) quantum secret sharing scheme there is a classical secret that is encrypted in a way that is protected by quantum communication. DQ-(5,11) is a combination of Shamir's polynomial sharing over GF₁₃ and a single-qudit reconstruction in dimension d=13. Each of the five cooperating participants in DQ-(5,11) performs one phase operation with the share of the quantum secret and a zero-sum mask so that the intermediate states will not have any partial sum. Decoy qudits from two mutually unbiased bases detect eavesdropping, and a dealer-published commitment lets participants check the reconstructed value. We achieve correctness of all 462 subsets of the network, information-theoretic secrecy against coalitions of up to four insiders and a requirement for the success of intercept-resend attacks to be (d+1)/2d)^m for m decoys (8.6×10^-9) for m=30). We use numerical data to verify the deterministic reconstruction for all 2,310 runs and show the effect of depolarizing noise (79.1% success at 5% noise at each stage) and channel loss. Furthermore, we compare our orbital-angular-momentum and time-bin qudit model to Trojan-horse countermeasures and other schemes.