On the Theory of Quantum Secret Sharing

In a classical secret sharing scheme, some sensitive classical data is distributed among a number of people such that certain sufficiently large sets of people can access the data, but smaller sets can gain no information about the shared secret. For instance, a possible application is to share the key for a joint checking account shared by many people. No individual is able to withdraw money, but sufficiently large groups can use the account. One particularly symmetric variety of secret sharing scheme is called a threshold scheme. A (k, n) classical threshold scheme has n shares, of which any k are sufficient to reconstruct the secret, while any set of k − 1 or fewer shares has no information about the secret. Blakely [1] and Shamir [2] showed that threshold schemes exist for all values of k and n with n ≥ k. It is also possible to consider more general secret sharing schemes which have an asymmetry between the power of the different shares. For instance, one might consider a scheme with four shares A, B, C, and D. Any set containing A, B, and C or A and D can reconstruct the secret, but any other set of shares has no information. In this example, the presence of A is essential to reconstructing the secret, but not sufficient — A needs the help of either D or both B and C. This particular scheme can be constructed by taking a (5, 7) threshold scheme, and assigning 3 shares to A, 2 to D, and 1 to each of B and C, but other schemes exist which cannot