Threshold visual cryptography schemes with specified whiteness levels of reconstructed pixels

In 1994, Naor and Shamir introduced an unconditionally secure method for en coding black and white images. This method, known as a threshold visual cry ptography scheme (VCS), has the benefit of requiring no cryptographic compu tation on the part of the decoders. In a (k, n)-VCS, a share, in the form o f a transparency, is given to n users. Any k users can recover the secret s imply by stacking transparencies, but k-1 users can gain no information abo ut the secret whatsoever. In this paper, we first explore the issue of contrast, by demonstrating tha t the current definitions are inadequate, and by providing an alternative d efinition. This new definition motivates an examination of minimizing pixel expansion subject to fixing the VCS parameters h and l. New bounds on pixe l expansion are introduced, and connections between these bounds are examin ed. The best bound presented is tighter than any previous bound. An analysi s of connections between (2, n) schemes and designs such as BIBD's, PBD's, and (r, lambda)-designs is performed. Also, an integer linear program is pr ovided whose solution exactly determines the minimum pixel expansion of a ( 2, n)-VCS with specified h and l.