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 for secret reconstruction and q is a power m of an odd prime p; multiplying 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 generated shares and N is greater than K; and generating N shares of the secret for distribution among cooperating entities, each of the N shares including a component of the N-tuple.