Securely and Efficiently Computing the Hermite Normal Form of Integer Matrices via Cloud Computing

The prevalence of cloud computing greatly promotes the traditional computing paradigm. With the assistance of a cloud, the light-weight device can achieve computation-intensive tasks which may not be done on its own. While, computing the Hermite Normal Form (HNF), which is a standard form of integer matrices, not only is inescapable when solving the linear system of equations over integers, but also has lots of applications in many other fields, such as integer programming and lattice-based cryptography. However, the fast blowup phenomenon of intermediate numbers makes the HNF computation time-consuming. In this paper, we initialize the study of the cloud-assisted HNF computation and design an efficient outsourcing algorithm that enables the resource-constrained client to securely delegate this heavy computation to a resource-abundant yet maybe untrusted cloud server. The main idea involved in our algorithm is a novel matrix encryption method based on random permutation, unimodular matrix transformation and triangular matrix transformation, which makes our algorithm protect the client's input/output information with the one-way notion and enable the client to detect the cloud's deception with the optimal probability 1. Besides, rigorous theoretical analysis and extensive experimental evaluation validate the efficiency and the practical performance of our design, and the substantial client-side savings are remarkable as the problem size increases.