On the Construction of (n, k)-schemes of Visual Cryptography Usinga Class of Linear Hash Functions Over a Binary Field
The paper explores the use of a set of hash functions for constructing a secret sharing scheme amongn participants based on the (k, k)-scheme M. Nahor and A. Shamir. Conditions are obtained for a set ofhash functions, in which it is possible to construct (k, n)-schemes where any coalition of power k or morecan restore a secret, and a coalition of lower power cannot restore the secret. In particular, the applicationof the class of linear hash functions is investigated. In general, this class does not allow us to construct a(k, n)-scheme, but it is possible to construct a (k, K, n)-scheme for which any k -1 and less participantscannot restore the secret, and any K and more can. For a class of linear hash functions, sufﬁcient conditionsare obtained for K, in which the coalition of power K and more can restore the secret. In a particular case,a secret sharing scheme for eight participants was devel oped, based on the (4, 4)-scheme of M. Naor andA. Shamir using a class of linear hash functions. It is shown that for any 4-power coalition the secret can berestored, that is, the constructed scheme is a (8, 4)-scheme. The (8, 4)-scheme constructed in particularis characterized by a shorter length of shares than the (8, 4)-scheme constructed in accordance with thealgorithm proposed by M. Naor and A. Shamir.