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Constructions and analysis of some efficient --visual cryptographic schemes using linear algebraic techniques

机译：使用线性代数技术构建和分析某些有效的可视密码方案

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摘要

In this paper we put forward an efficient construction, based on linear algebraic technique, of a --visual cryptographic scheme (VCS) for monochrome images in which participants are essential in a -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 --VCS. We completely characterize the case of --VCS, , 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 -VCS matches with the numerical values of the optimal pixel expansions for every possible values of that exist in the literature. We further deal with the -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 -VCS and numerical evidence shows it achieves almost optimal pixel expansion.
机译：在本文中，我们提出了一种基于线性代数技术的有效的单色图像视觉加密方案（VCS），其中参与者对于-VCS是必不可少的。该方案在仅需要求解线性方程组以构造所需的初始基础矩阵的意义上是有效的。为了使该方案更有效，我们应用了从初始基础矩阵中删除公共列的技术，以获得简化的基础矩阵。但是，在初始基础矩阵中找到确切数目的公共列是一个难题。在本文中，我们处理这个问题。我们首先提供--VCS的构造和分析。我们通过在初始基础矩阵中找到确切数量的公共列的闭合形式来完全表征--VCS的情况，从而删除这些公共列以获得减少的像素扩展和有效对比度的相对对比度的确切值和简单的方案。对于减少的-VCS像素扩展，我们提出的封闭形式与文献中存在的每种可能值的最佳像素扩展的数值相匹配。我们进一步处理-VCS并通过提供一种用于对线性方程组进行分组的有效算法来解决一个悬而未决的问题，从而证明我们提出的算法比基于线性代数技术的现有方案更好地工作。最终，我们为-VCS提供了减少像素扩展的界限，数值证据表明，它几乎可以实现最佳像素扩展。

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