Practical and Secure Outsourcing Algorithms of Matrix Operations Based on a Novel Matrix Encryption Method

Abstract

With the recent growth and commercialization of cloud computing, outsourcing computation has become one of the most important cloud services, which allows the resource-constrained clients to efficiently perform large-scale computation in a pay-per-use manner. Meanwhile, outsourcing large scale computing problems and computationally intensive applications to the cloud has become prevalent in the science and engineering computing community. As important fundamental operations, large-scale matrix multiplication computation (MMC), matrix inversion computation (MIC), and matrix determinant computation (MDC) have been frequently used. In this paper, we present three new algorithms to enable secure, verifiable, and efficient outsourcing of MMC, MIC, and MDC operations to a cloud that may be potentially malicious. The main idea behind our algorithms is a novel matrix encryption/decryption method utilizing consecutive and sparse unimodular matrix transformations. Compared to previous works, this versatile technique can be applied to many matrix operations while achieving a good balance between security and efficiency. First, the proposed algorithms provide robust confidentiality by concealing the local information of the entries in the input matrices. Besides, they also protect the statistic information of the original matrix. Moreover, these algorithms are highly efficient. Our theoretical analysis indicates that the proposed algorithms reduce the time overhead on the client side from O(n 2.3728639 ) to O(n 2 ). Finally, the extensive experimental evaluations demonstrate the practical efficiency and effectiveness of our algorithms.INDEX TERMS Cloud computing, outsourcing computation, matrix operations, privacy, efficiency.

Publicly verifiable and efficiency/security-adjustable outsourcing scheme for solving large-scale modular system of linear equations

“…any verifier (not necessarily the client) can assess the correctness of the server's results. The main technique involved in our scheme essentially can be treated as an improvement of Zhang et al 's technique [22] and Chen et al 's technique [23], which makes our scheme superior in the following three aspects: 1) Our scheme provides an adjustable encryption method based on successively sparse unimodular matrix transformations. That is, we can adjust the number of sparse unimodular matrices in the scheme to balance security and efficiency to match the different application scenarios.…”

“…gcd(det(A T A), q) = 1). Also, the existing outsourcing algorithms [22,28] for matrix inversion only detect the cloud server's misbehavior with a certain probability and thus do not achieve the optimal verifiability 1. Another direction is along the way of securely outsourcing LSLE [23,29].…”

“…Different from the encryption method in [23] which encrypts A by only two sparse matrices, we blind the matrix A by dense enough matrices. Inspired by the method introduced in [22], we present a dynamic and adjustable method to encrypt the original matrix by using a series of sparse and unimodular matrix transformations. That is, the A in Eq.…”

“…the number of zeros. Recently, Zhang et al [22] presented a novel matrix encryption method based on consecutive sparse and unimodular matrix transformation, and then exploited this method to design a new MIC outsourcing algorithm over finite fields. Their algorithm favorably balances the security and efficiency, i.e.…”

“…In this case, we can realize the securely outsourcing of (LMSLE) by designing an efficient method to outsource the inverse of A modulo q. Although many researches have considered the outsourcing of large-scale matrix inversion computation [20][21][22], they focused on matrices over the real field R or the finite field F q and the probability of verifiability can not achieve the optimal 1. Furthermore, the existing methods only designed for a square and invertible matrix A, while here we consider a general m-by-n matrix A.…”