A Construction of the (4, n)-Threshold Visual Cryptography Scheme Using a 3-Design

A (θ, n)-threshold visual cryptography scheme ((θ, n)-VCS) is one of secret sharing schemes for black-white images. In the (θ, n)-VCS, a dealer generates n black-white images called shares from a secret image by using two Boolean matrices called basis matrices. In this paper we propose a new construction of the basis matrices of the (4, n)-VCS using a combinatorial structure known as the 3-design. We establish validity of the obtained basis matrices and give formulas of the pixel expansion m and the relative difference α_n of superimposed n shares. This construction can yield basis matrices with a much smaller pixel expansion than existing ones.