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IEEE Transactions on Cloud Computing

How to Securely and Efficiently Solve the Large-Scale Modular System of Linear Equations on the Cloud

Chengliang Tian, Jia Yu, Panpan Meng, Guoyan Zhang, Weizhong Tian, Yan Zhang

Published Jun 3, 2024• 2 citations
Publisher

Abstract

Cloud-assisted computation empowers resource-constrained clients to efficiently tackle computationally intensive tasks by outsourcing them to resource-rich cloud servers. In the current era of Big Data, the widespread need to solve large-scale modular linear systems of equations ($\mathcal {LMLSE}$) of the form$\mathbf {A}\mathbf {x}\equiv \mathbf {b}\;{\rm mod}\;{q}$poses a significant challenge, particularly for lightweight devices. This paper delves into the secure outsourcing of$\mathcal {LMLSE}$under a malicious single-server model and, to the best of our knowledge, introduces the inaugural protocol tailored to this specific context. The cornerstone of our protocol lies in the innovation of a novel matrix encryption method based on sparse unimodular matrix transformations. This novel technique bestows our protocol with several key advantages. First and foremost, it ensures robust privacy for all computation inputs, encompassing$\mathbf {A},\mathbf {b}, q$, and the output$\mathbf {x}$, as validated by thorough theoretical analysis. Second, the protocol delivers optimal verifiability, enabling clients to detect cloud server misbehavior with an unparalleled probability of 1. Furthermore, it boasts high efficiency, requiring only a single interaction between the client and the cloud server, significantly reducing local-client time costs. For an$m$-by-$n$matrix$\mathbf {A}$, a given parameter$\lambda =\omega (\log q)$, and$\rho =2.371552$, the time complexity is diminished from$O(\max \lbrace m n^{\rho -1}, m^{\rho -2} n^{2}\rbrace \cdot (\log q)^{2})$to$O((mn+m^{2})\lambda \log q+mn(\log q)^{2})$. The comprehensive results of our experimental performance evaluations substantiate the protocol's practical efficiency and effectiveness.

DOI
10.1109/tcc.2024.3408240

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Abstract Solving large-scale modular system of linear equations ( LMSLE\mathcal {LMSLE}LMSLE ℒℳSℒE ) is pervasive in modern computer and communication community, especially in the fields of coding theory and cryptography. However, it is computationally overloaded for lightweight devices arisen in quantity with the dawn of the things of internet (IoT) era. As an important form of cloud computing services, secure computation outsourcing has become a popular topic. In this paper, we design an efficient outsourcing scheme that enables the resource-constrained client to find a solution of the LMSLE\mathcal {LMSLE}LMSLE ℒℳSℒE with the assistance of a public cloud server. By utilizing affine transformation based on sparse unimodular matrices, our scheme has three merits compared with previous work: 1) Our scheme is efficiency/security-adjustable. Our encryption method is dynamic, and it can balance the security and efficiency to match different application scenarios by skillfully control the number of unimodular matrices. 2) Our scheme is versatile. It is suit for genericm-by-ncoefficient matrixA, no matter it is square or not. 3) Our scheme satisfies public verifiability and achieves the optimal verification probability. It enables any verifier which is not necessarily the client to verify the correctness of the results returned from the cloud server with probability 1. Finally, theoretical analysis and comprehensive experimental results confirm our scheme’s security and high efficiency.
Outsourcing computation to cloud server has recently become popular in cloud computing. Cloud computing technologies enable clients with limited computational resources to outsource their massive computations to powerful cloud servers. Outsourcing computation has some new concerns, such as outsourced data and result privacy, verifiability and efficiency. Matrix multiplication is one of the most basic computational problems. In this paper, we are motivated to design a secure and efficient protocol for outsourcing the massive matrix multiplication computations to the cloud server. The existing works are mostly under amortized model that have an expensive phase as offline mode. Our proposed scheme does not have any expensive phase. So the client can outsource its matrices in online mode in which the efficiency of outsourcing will be increased. In our scheme, the client encrypts two matrices and sends them to a semi-honest cloud server. The cloud server computes matrix multiplication and a proof. After that, it sends the encrypted result to the client and finally the client checks the validity of the computation and decrypts the result. Our proposed scheme achieves privacy protection of outsourced data and multiplication result, unforgeability of proof, verification and high efficiency.
We give improved protocols for the secure and private outsourcing of linear algebra computations, that enable a client to securely outsource expensive algebraic computations (like the multiplication of large matrices) to a remote server, such that the server learns nothing about the customer's private input or the result of the computation, and any attempted corruption of the answer by the server is detected with high probability. The computational work performed at the client is linear in the size of its input and does not require the client to locally carry out any expensive encryptions of such input. The computational burden on the server is proportional to the time complexity of the current practically used algorithms for solving the algebraic problem (e.g., proportional to n3 for multiplying two n x n matrices). The improvements we give are: (i) whereas the previous work required more than one remote server and assumed they do not collude, our solution works with a single server (but readily accommodates many, for improved performance); (ii) whereas the previous work required a server to carry out expensive cryptographic computations (e.g., homomorphic encryptions), our solution does not make use of any such expensive cryptographic primitives; and (iii) whereas in previous work collusion by the servers against the client revealed to them the client's inputs, our scheme is resistant to such collusion. As in previous work, we maintain the property that the scheme enables the client to detect any attempt by the server(s) at corruption of the answer, even when the attempt is collusive and coordinated among the servers.
The highly development of Internet and cloud computing technology makes it possible to share computing power, storage and data. Cloud computing makes users with limited computing resources be able to carry out large computational tasks with the help of cloud. but security is a major concern that hinders the widespread use of outsourcing computing. We focus on outsourcing large matrix multiplication in this paper. Firstly, we exploit Chinese Remainder Theorem(CRT) to construct a large matrix multiplication outsourcing protocol which is highly efficient in the malicious cloud model. Then formal security proof and extensive performance evaluation show that our protocol achieves a higher security level i.e., information-theoretic security for input privacy and adaptive chosen ciphertext attack (CCA2) security for output privacy). Compared with previous works, users in the proposed protocol are able to verify the results received from the cloud with tiny error rate.
We give protocols for the secure and private outsourcing of linear algebra computations, that enable a client to securely outsource expensive algebraic computations (like the multiplication of huge matrices) to two remote servers, such that the servers learn nothing about the customer's private input or the result of the computation,and any attempted corruption of the answer by the servers is detected with high probability. The computational work done locally by the client is linear in the size of its input and does not require the client to carry out locally any expensive encryptions of such input.The computational burden on the servers is proportional to the time complexity of the current practically used algorithms for solving the algebraic problem (e.g., proportional to n3for multiplying two ntimesn matrices). If the servers were to collude against the client,then they would only find out the client's private inputs, but they would not be able to corrupt the answer without detection by the client.
With the advent of cloud computing, individuals and companies alike are looking for opportunities to leverage cloud resources not only for storage but also for computation. Nevertheless, the reliance on the cloud to perform computation raises the unavoidable challenge of how to assure the correctness of the delegated computation. In this regard, we introduce two cryptographic protocols for publicly verifiable computation that allow a lightweight client to securely outsource to a cloud server the evaluation of high-degree univariate polynomials and the multiplication of large matrices. Similarly to existing work, our protocols follow the amortized verifiable computation approach. Furthermore, by exploiting the mathematical properties of polynomials and matrices, they are more efficient and give way to public delegatability. Finally, besides their efficiency, our protocols are provably secure under well-studied assumptions.