Quantum Secret Sharing [1]

((k,n)) threshold scheme

The simple case described above can be extended similarly to that done in CSS by Shamir [4] and Blakley [5] via a thresholding scheme. In the ((k,n)) threshold scheme (double parentheses denoting a quantum scheme), Alice splits her secret key (quantum state) into n shares such that any k≤n shares are required to extract the full information but k-1 or less shares cannot extract any information about Alice’s key.

The number of users needed to extract the secret is bounded by n/2 < k ≤ n. Consider for n ≥ 2k, if a ((k,n)) threshold scheme is applied to two disjoint sets of k in n, then two independent copies of Alice’s secret can be reconstructed. This of course would violate the no-cloning theorem and is why n must be less than 2k.

As long as a ((k,n)) threshold scheme exists, a ((k,n-1)) threshold scheme can be constructed by simply discarding one share. This method can be repeated until k=n.

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“Experimental demonstration of quantum secret sharing” [3] was the first experimental demonstration of QSS in 2001.

Quantum entanglement for secret sharing and secret splitting [2] is a similar scheme developed shortly after Hillery et al’s GHZ scheme, using Bell states instead of GHZ states.