A method and system distributes N shares of a secret among cooperating entities using hyperplanes over GF(q), such that the secret can be reconstructed from K of the N shares (where K≦N). In one embodiment, the method constructs a K-tuple that contains the secret and elements of GF(q), where q is a power m of an odd prime p. The method further multiplies the K-tuple by a matrix of size (N×K) to produce an N-tuple using arithmetic defined on GF(q). Thus, N shares of the secret are generated, with each of the N shares including a component of the N-tuple.