Linear Secret Sharing with Divisible Shares

Abstract

Shamir secret sharing is being considered in the broader context of linear secret sharing. It is shown that any Shamir scheme built over GF(q v ) can be converted into its linear equivalent defined over GF(q). A notion of uniform perfectness is introduced and it is proved that Shamir schemes built over GF(q v ) are not uniformly perfect. Probabilistic linear secret sharing is next studied and bounds on probability that the resulting secret sharing is uniformly perfect are given. The probabilistic arguments are later used to show that secret sharing with shift derived from Shamir scheme allows to achieve a secret sharing which is uniformly perfect.