Optimal XOR Based (2,n)-Visual Cryptography Schemes

A (2, n) Visual Cryptography Scheme (VCS) is a kind of secret sharing scheme, where n participants share a secret image, and any two of them can recover the secret image visually without any cryptographic knowledge and computation devices, but any one of them cannot get any information about the secret image other than the size of the secret image. This paper studies the \((2,n)-VCS_{XOR}\), and shows the smallest (optimal) pixel expansion of such scheme, and the largest possible contrast for the \((2,n)-VCS_{XOR}\) given their optimal pixel expansion. It also shows the largest (optimal) contrast of the \((2,n)-VCS_{XOR}\), and the smallest possible pixel expansion of such schemes given their optimal contrast. The results of this paper show that the \((2,n)-VCS_{XOR}\) can achieve smaller pixel expansion and larger contrast than that of \((2,n)-VCS_{OR}\). It also shows that the construction of the basis matrix of optimal contrast \((2,n)-VCS_{XOR}\) is equivalent to the construction of binary codes when they reach the maximum capability, and the construction of a specific class of optimal contrast \((2,n)-VCS_{XOR}\) for \(n=2^{k}-1\) is given.