Colored visual cryptography without color darkening

In a visual cryptography scheme a secret image is encoded into n shares, in the form of transparencies. The shares are then distributed to n participants. Qualified subsets of participants can recover the secret image by superimposing their transparencies. Non-qualified subsets of participants have no information about the secret image. In this paper we consider the case when the secret image is a colored image. Most of the previous work on colored visual cryptography allows the superposition of pixels having the same color assuming that the resulting pixel still has the same color. This is not what happens in reality since when superimposing two pixels of the same color one gets a darker version of that color, which effectively is a different color. Superimposing many pixels of the same color might result in a so dark version of the color that the resulting pixel might not be distinguishable from a black pixel. In this paper we propose a model where the reconstruction has to guarantee that the reconstructed secret pixel has the exact same color of the original one and not a darker version of it. We consider (k,n)-threshold schemes where a qualified set of participants consists of any k participants. We provide a general construction for any k, 2≤k≤n and a construction for the special case k=2. We also prove lower bounds on the pixel expansion (which is a measure of the goodness of the scheme) for the cases k=2 and k=n. The lower bounds match the pixel expansion of the schemes provided in these two cases, thus proving that our schemes are optimal with respect to the pixel expansion. We also provide an upper bound on the contrast of (k,n)-threshold schemes and (2,n)-threshold schemes with optimal contrast.