1. A method comprising: creating a plurality of shares for a secret, each share comprising a plurality of elements to represent a point on a multi-dimensional secret hyperplane that embeds the secret, wherein a number of dimensions in the multi-dimensional secret hyperplane corresponds to a minimum number of shares to reconstruct the secret; constructing, by a computer system, a plurality of sub-shares from one of the plurality of shares, wherein the plurality of sub-shares for the share define a set of secondary multi-dimensional hyperplanes that intersect at the point represented in the share; and distributing the sub-shares.