1. A computer-implemented method for distributing a secret, the method comprising: constructing a K-tuple that contains the secret and elements of GF(q), wherein K is the number of shares necessary for secret reconstruction and q is a power m of an odd prime p; multiplying, by a processor, the K-tuple by a matrix of size (N×K) to produce an N-tuple using arithmetic defined on GF(q), wherein N is the number of shares to be generated and N is greater than K; and generating N shares of the secret for distribution among cooperating entities, wherein generating N shares of the secret further comprises forming each of the N shares by including a row of the (N×K) matrix and a corresponding component of the N-tuple in each of the N shares, and wherein each share represents an equation for a K-dimensional hyperplane over GF(q).