Constructions of a Visual Cryptography Scheme Using an Affine Resolvable BIBD

A visual cryptography scheme (VCS) is one of the secret sharing schemes for digital images. In the VCS, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex> images called shares are generated from a secret image and are distributed to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex> respective participants. In this paper, we first give a new construction of basis matrices of the VCS realizing the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(2, n)$</tex>-threshold access structure. We use a combinatorial design known as an affine resolvable BIBD. In the construction, the relative difference is almost equal to the optimal value, while the pixel expansion is almost half compared with an existing construction attaining the optimal value if we use a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(4l, 2l, 2l-1)$</tex> affine resolvable BIBD. We also give a construction of basis matrices using a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(4l, 2l, 2l-1)$</tex> affine resolvable BIBD for a certain access structure. In this access structure, participants are partitioned into disjoint groups of size two and the two participants belonging to any group can reproduce a secret image. In addition, any three participants from different groups can obtain no information on a secret image.