In this paper, color images are encrypted and subsequently damage occlusion is made to the encrypted figures, with different sizes; the intention is to simulate an attack. In this research, two aspects are discussed, namely: the first is to encrypt images with quality; that is, the figures encrypted pass randomness tests proposed in this paper. The second aspect deals with the problem of recovering the encrypted figure information when it has been damaged. To retrieve information from encrypted images, the enryption of images is carried out in two steps: in the first a permutation is applied to the entire image and the second uses the AES cryptosystem with variable permutations. To perform this task an algorithm is used that utilizes the pi number to generate the permutations. To improve the sharpness of the deciphered figures with damage two filters are applied; median and average. To measure the degree of improvement in the damaged images two tests are proposed; the first is the correlation coefficient between adjacent pixels in the horizontal, vertical and diagonal directions. The second is based on the information entropy.

Seja C uma desordem (família de Sperner) e seja A sua matriz de incidência. Se o sistema linear x > 0; x A < 1 tem a propriedade do arredondamento inteiro, fornecemos a descrição do módulo canônico e do a-invariante de certos subaneis monomiais associados a C. Se a desordem é um grafo conexo, descreve-se quando o supra-mencionado sistema linear tem a propriedade do arredondamento inteiro em termos combinatórios e algébricos, usando a teoria dos grafos e a teoria das álgebras de Rees. Como consequência, mostra-se que a álgebra de Rees estendida do ideal de arestas de um grafo bipartido é um anel de Gorenstein se e somente se o grafo é de altura pura.

Let C be a clutter and let A be its incidence matrix. If the linear system x > 0; x A < 1 has the integer rounding property, we give a description of the canonical module and the a-invariant of certain normal subrings associated to C. If the clutter is a connected graph, we describe when the aforementioned linear system has the integer rounding property in combinatorial and algebraic terms using graph theory and the theory of Rees algebras. As a consequence we show that the extended Rees algebra of the edge ideal of a bipartite graph is Gorenstein if and only if the graph is unmixed.