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.
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.
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.
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.
Choi et al. (TCC 2013) introduced the notion of multi-client verifiable computation (MVC) in which a set of clients outsource to an untrusted server the computation of a function f over their collective inputs in a sequence of time periods. In that work, the authors defined and realized multi-client verifiable computation satisfying soundness against a malicious server and privacy against the semi-honest corruption of a single client. Very recently, Goldwasser et al. (Eurocrypt 2014) provided an alternative solution relying on multi-input functional encryption. Here we conduct a systematic study of MVC, with the goal of satisfying stronger security requirements. We begin by introducing a simulation-based notion of security that provides a unified way of defining soundness and privacy, and automatically captures several attacks not addressed in previous work. We then explore the feasibility of achieving this notion of security. Assuming no collusion between the server and the clients, we demonstrate a protocol for multi-client verifiable computation that achieves strong security in several respects. When server-client collusion is possible, we show (somewhat surprisingly) that simulation-based security cannot be achieved in general, even assuming semi-honest behavior.
Web-servers are a dominant medium of modern communication and thus process vast amounts of highly sensitive, private data. Social networks, online auctions, and all kinds of cloud services constitute popular examples of online services complying with this paradigm. Due to many alarming scandals regarding the misuse of private data, IT security solutions protecting this sensitive data (for example by hiding it through encryption) are indispensable. Ideally, web-servers should be able to compute on private data without seeing it - we refer to this as the "Secure Outsourcing of Computation". Most notably, the cryptographic primitive of Homomorphic Encryption (HE) enables the web-servers to process data in the encrypted domain, which thereby hides/protects all private data as it is never available in the clear. More precisely, HE is a type of encryption that allows for the computation of certain functionalities on encrypted data without knowledge of the decryption key. Depending on the HE scheme, various functionalities can be evaluated in the encrypted domain, ranging from additions or multiplications only (Group Homomorphic Encryption, GHE), to virtually any computable functionality (Fully Homomorphic Encryption, FHE). We provide a clean foundational framework for HE that permits security characterizations for a large class of HE schemes and shows strong similarities between GHE and FHE. We study these characterizations in order to assess the limits of HE, and show that GHE is obsolete in the quantum world, meaning that any GHE scheme can be broken by a quantum computer. Together with our newly constructed universal blueprint that encompasses virtually any GHE scheme, we know exactly how these schemes must be designed and what the underlying hardness assumptions must look like. To some extent, this rounds off the topic of GHE both in the standard and quantum model of computation. In the standard model (which is presently assumed to be the most realistic model of computation), GHE is still highly attractive as it provides very efficient solutions to a variety of practical problems. On the basis of our foundational framework, we construct new GHE schemes with unique features and show their employability in practical applications. Most importantly, we use the special features of one of these schemes and develop a novel technique to achieve a practically efficient solution to the problem of securely outsourcing computations. Our construction allows multiple users to send encryptions of their private data under their own keys to a distrusted web-server that can then, on behalf of the users, evaluate any dynamically chosen computable function on these inputs in the encrypted domain (even across encryptions under different public keys) without learning anything at all, and without interacting with the users.
Systems of linear equations, as a fundamental topic in the development of the computational sciences, have been well studied in the scientific community. In this paper, we investigate secure outsourcing for large-scale systems of linear equations, which is considered as a prohibitively expensive computation for the clients with limited computational resources. Based on Chen's work, we propose a verifiable algorithm for secure outsourcing of large-scale systems of linear equations in the case of no solution under the fully malicious model. We uniquely utilize linear programming, which is another well studied scientific computing problem, to help the client to verify the correctness of results returned by the server. Compared with the state-of-the-art algorithm (Chen et al' scheme), the proposed algorithm is suitable not only for any nonsingular dense matrix A, but for a set of singular matrixes that lead the system of linear equations has no solution. Besides, the computational complexity for the verification of no solution in our scheme is still O (n2). Therefore, our proposed algorithm is more suitable for real applications.
The rapid development of cloud computing has probably benefited each of us. However, the privacy risks brought by untrusty cloud servers arise the attention of more and more people and legislatures. In the last two decades, plenty of works seek the way of outsourcing various specific tasks while ensuring the security of private data. Although the addition and multiplication are enough for implementing any functions, the direct utilization of existing schemes like homomorphic encryption will lead to significant efficiency and accuracy loss, which is not suitable for outsourcing computation tasks. The tasks to be outsourced are endless, however, the involved calculations are similar. In this paper, inspired by additive secret sharing and multiplicative secret sharing technologies, we construct a series of novel protocols which support the common secure calculations on numbers (e.g., basic elementary functions) or matrices (e.g., solve eigenvectors) in arbitrary n number of servers (n≥2), and the n-party protocols ensure the security of the original data even if n−1 servers collude. All protocols we designed only need constant interaction rounds, and we demonstrate them under universally composability security. We believe that these protocols can provide a new basic tool for actual outsourced tasks.