In this paper, a $(k,n,m)$-intragroup and intergroup secret image sharing (I${}^2$SIS) scheme for the multi-group (i.e., $m$-group) environment is investigated. $(2^m-1)$ secret images are shared within groups (intragroups) and between groups (intergroups). Differing from existing SIS methods, any $k$ or more qualified participants among different groups (i.e., intergroup) can collaboratively recover the respective secret image via the homomorphic Lagrange interpolation. The proposed scheme consists of three major procedures: preprocessing, sharing, and recovering. In the preprocessing procedure, an approach is designed to partition the coefficients in the $m$ $(k-1)$-degree polynomials for sharing the $(2^m-1)$ secret images. In addition, an optimal algorithm for the preprocessing is given to let all the percentages of recovering secret images be the same as possible. Shadows belonging to all participants in the $m$ groups are generated in the sharing procedure. In the recovering procedure, the secret images can be reconstructed in intragroups and intergroups. Moreover, experiments and comparisons are demonstrated to show the merits of the proposed scheme.

在本文中，一个 $(克,n,米)$-组内和组间秘密图像共享（我${}^2$SIS）多组方案（即， $米$-group) 环境进行调查。 $(2^{米}-1)$秘密图像在组内（组内）和组之间（组间）共享。与现有的 SIS 方法不同，任何$克$不同组（即组间）中的或更多合格的参与者可以通过同态拉格朗日插值协同恢复各自的秘密图像。提议的方案包括三个主要过程：预处理、共享和恢复。在预处理过程中，设计了一种方法来划分系数$米$ $(克-1)$-次多项式用于共享 $(2^{米}-1)$秘密图像。此外，给出了预处理的最佳算法，使所有恢复秘密图像的百分比尽可能相同。属于所有参与者的影子$米$组是在共享过程中生成的。在恢复过程中，可以在组内和组间重建秘密图像。此外，实验和比较证明了所提出方案的优点。