On the Pixel Expansion of Hypergraph Access Structures in Visual Cryptography Schemes

Abstract. In a visual cryptography scheme, a secret image is encoded into n shares, in the form of transparencies. The shares are then dis-tributed to n participants. Qualified subsets of participants can recover the secret image by superimposing their transparencies, but non-qualified subsets of participants have no information about the secret image. Pixel expansion, which represents the number of subpixels in the encoding of the secret image, should be as small as possible. Optimal schemes are those that have the minimum pixel expansion. In this paper we study the pixel expansion of hypergraph access structures and introduce a number of upper bounds on the pixel expansion of special kinds of access struc-tures. Also we demonstrate the minimum pixel expansion of induced matching hypergraph is sharp when every qualified subset is exactly one edge with odd size. Furthermore we explain that the minimum pixel ex-pansion of every graph access structure Pn is exactly n+12 . It indicates the lower bound mentioned in [4] is sharp