A quantum secret sharing (QSS) scheme is a cryptographic protocol by which a dealer can share a secret among a group of n players so that only certain subsets of players can recover the secret by collaboration. This work proposes communication efficient quantum threshold secret sharing schemes. They minimize the amount of quantum communication required to reconstruct the secret when more than the necessary number of players collaborate. They are based on Staircase codes proposed by Bitar and El Rouayheb.



In a standard ((k,n)) QSS scheme, any subset of k or more players can recover the secret. The quantum communication cost for reconstruction in such schemes is k qudits (quantum digits) for each qudit in the secret. Using the proposed construction, any subset of d ≥ k players can also collaborate to recover the secret with a communication cost of d/(d-k+1) qudits for every secret qudit. For d > k, proposed schemes are communication efficient with respect to standard schemes and when d = 2k-1, the quantum communication cost is reduced by a factor O(k). Further, the proposed schemes have optimal communication cost for secret reconstruction.