Communication Efficient Quantum Secret Sharing

TL;DR

This paper introduces a class of quantum secret sharing schemes that significantly reduce communication costs during secret reconstruction by allowing non-minimal authorized sets, leveraging staircase codes.

Contribution

It proposes communication efficient quantum secret sharing schemes based on staircase codes, reducing communication overhead compared to standard schemes, and proves their optimality.

Findings

Reduces communication complexity by a factor of O(k) for certain parameters.

Allows recovery of a secret with fewer communicated qudits than standard schemes.

Achieves optimal communication cost for secret reconstruction.

Abstract

In the standard model of quantum secret sharing, typically, one is interested in minimal authorized sets for the reconstruction of the secret. In such a setting, reconstruction requires the communication of all the shares of the corresponding authorized set. If we allow for non-minimal authorized sets, then we can trade off the size of the authorized sets with the amount of communication required for reconstruction. Based on the staircase codes, proposed by Bitar and El Rouayheb, we propose a class of quantum threshold secret sharing schemes that are also communication efficient. We call them $((k,2k-1,d))$ communication efficient quantum secret sharing schemes where $k\leq d\leq2k-1$. Using the proposed construction, we can recover a secret of $d-k+1$ qudits by communicating $d$ qudits whereas using the standard $((k,2k-1))$ quantum secret sharing requires $k(d-k+1)$ qudits to be communicated. In other words, to share a secret of one qudit, the standard quantum secret sharing requires $k$ qudits whereas the proposed schemes communicate only $\frac{d}{d-k+1}$ qudits per qudit in the communication complexity. Proposed schemes can reduce communication overheads by a factor $O(k)$ with respect to standard schemes, when $d$ equals $2k-1$. Further, we show that our schemes have optimal communication cost for secret reconstruction.