Two matrices for Blakley's secret sharing scheme

The secret sharing scheme was invented by Adi Shamir and George Blakley independently in 1979. In a (k, n)-threshold linear secret sharing scheme, any k-out-of-n participants could recover the shared secret, and any less than k participants could not recover the secret. Shamir's secret sharing scheme is more popular than Blakley's even though the former is more complex than the latter. The reason is that Blakley's scheme lacks determined, general and suitable matrices. In this paper, we present two matrices that can be used for Blakley's secret sharing system. Compared with the Vandermonde matrix used by Shamir's scheme, the elements in these matrices increase slowly. Furthermore, we formulate the optimal matrix problem and find the lower bound of the minimal maximized element for k=2 and upper bound of the minimal maximized element of matrix for given k.

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