Communication efficient quantum secret sharing

A quantum secret sharing 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. In this paper we propose 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 a class of staircase codes proposed by Bitar and El Rouayheb. In a standard $\left(\right(k,n\left)\right)$ quantum threshold scheme, any subset of $k$ or more players can recover the secret. The quantum communication cost for reconstruction in such schemes is $k$ qudits for each secret qudit. Using the proposed construction, any subset of $d\ge k$ players can also collaborate to recover the secret with a communication cost of $d$ qudits for $d-k+1$ secret qudits. In other words, for the proposed schemes the quantum communication cost is only $\frac{d}{d-k+1}$ 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 $O\left(k\right)$. Further, when $n=2k-1$, the proposed schemes have optimal communication cost for secret reconstruction.