Visual cryptography schemes with optimal pixel expansion

A visual cryptography scheme encodes a black and white secret image into n shadow images called shares which are distributed to the n participants. Such shares are such that only qualified subsets of participants can "visually" recover the secret image. Usually, the reconstructed image will be darker than the background of the image itself. In this paper we consider visual cryptography schemes satisfying the model introduced by Tzeng and Hu [A new approach for visual cryptography, Designs, Codes and Cryptography 27 (3) (2002) 207-227]. In such a model, the recovered secret image can be darker or lighter than the background. We prove a lower bound on the pixel expansion of the scheme and, for (2, n)-threshold visual cryptography schemes, we provide schemes achieving the bound. Our schemes improve on the ones proposed by Tzeng and Hu.