Constructions and analysis of some efficient \(t\)-\((k,n)^*\)-visual cryptographic schemes using linear algebraic techniques

In this paper we put forward an efficient construction, based on linear algebraic technique, of a \(t\)-\((k,n)^*\)-visual cryptographic scheme (VCS) for monochrome images in which \(t\) participants are essential in a \((k,n)\)-VCS. The scheme is efficient in the sense that it only requires solving a system of linear equations to construct the required initial basis matrices. To make the scheme more efficient, we apply the technique of deletion of common columns from the initial basis matrices to obtain the reduced basis matrices. However finding exact number of common columns in the initial basis matrices is a challenging problem. In this paper we deal with this problem. We first provide a construction and analysis of \(t\)-\((k,n)^*\)-VCS. We completely characterize the case of \(t\)-\((n-1,n)^*\)-VCS, \(0 \le t \le n-1\), by finding a closed form of the exact number of common columns in the initial basis matrices and thereby deleting the common columns to get the exact value of the reduced pixel expansion and relative contrast of the efficient and simple scheme. Our proposed closed form for reduced pixel expansion of \((n-1,n)\)-VCS matches with the numerical values of the optimal pixel expansions for every possible values of \(n\) that exist in the literature. We further deal with the \((n-2,n)\)-VCS and resolve an open issue by providing an efficient algorithm for grouping the system of linear equations and thereby show that our proposed algorithm works better than the existing scheme based on the linear algebraic technique. Finally we provide a bound for reduced pixel expansion for \((n-2,n)\)-VCS and numerical evidence shows it achieves almost optimal pixel expansion.