立體魔術方塊在資訊隱藏的應用

立體魔術方塊是指「將三維空間中立方體的邊長等分劃分為相同長度所構成的分割方塊的組合」。我們將灰階影像中相鄰的三像素所構成的三元數對(x1,x2,x3)視為立體魔術方塊在三維空間中X,Y,Z三個座標軸上的邊，則我們可以模仿EMD(Exploiting Modification Direction)的概念，定義擷取函數並將每一像素加減k的關係與立體魔術方塊所構成的分割方塊做一對應。換句話說，我們可以將一個(2k+1)3進制的機密訊息藏入(x1,x2,x3)這個三元數對且其中每一個像素至多加減k。以上機密訊息的藏入過程中，每一像素至多加減k的對應關係是唯一的，所以機密訊息的取出可以保証其正確性。而且，此對應關係的存在也代表了機密訊息藏入量與藏入後的影像品質的最佳化。除此之外，我們可以將三元數對擴展至n元數對，亦保有以上的良好性質。

In this study, a magic cube is a 3-dimensional cube, and each side segment into equal length partitions. Suppose (x1,x2,x3) is a triple neighboring pixel group in a grayscale image, and each pixel corresponding to the values of X,Y and Z axis in 3-dimensional space. We define the extraction function as the EMD(Exploiting Modification Direction) method does. Therefore, we can define an extraction function between the values of X, Y and Z of 3-dimensional space of the magic cube and the variations of the pixels in this triple group. In other word, we can embed one n3-ary notation secret digit into this triple group, and for each pixel in this triple group the value at most increases or decreases by k. As we mention above, this mapping relationship of the value at most increases or decreases by k in the embedding procedure is one-to-one. Therefore, we can extract the secret digits correctly. In addition, this mapping relationship also shows a optimize results for the payload and the quality of stego image. Besides, we can extend triple group to n-t