Abstract. Linear algebra plays an important role in computer science, especially in cryptography.Numerous cryptog-raphic protocols, scientific computations, and numerical computations are based on linear algebra. Many linear algebra tasks can be reduced to some core problems, such as matrix multiplication, determinant of matrix and the characteristic polynomial of matrix. However, it is difficult to execute these tasks independently for client whose computation abilities are weaker than polynomial-time computational ability. Cloud Computing is a novel eco-nomical paradigm which provides powerful computational resources that enables resources-constrained client to outsource their mass computing tasks to the cloud. In this paper, we propose a new verifiable and secure outsourcing protocol for the problem of computing the characteristic polynomial and eigenvalues of matrix. These protocols are not only effi-cient and secure, but also unnecessary for any cryptographic assumption.
Abstract. Linear algebra plays an important role in computer science, especially in cryptography.Numerous cryptog-raphic protocols, scientific computations, and numerical computations are based on linear algebra. Many linear algebra tasks can be reduced to some core problems, such as matrix multiplication, determinant of matrix and the characteristic polynomial of matrix. However, it is difficult to execute these tasks independently for client whose computation abilities are weaker than polynomial-time computational ability. Cloud Computing is a novel eco-nomical paradigm which provides powerful computational resources that enables resources-constrained client to outsource their mass computing tasks to the cloud. In this paper, we propose a new verifiable and secure outsourcing protocol for the problem of computing the characteristic polynomial and eigenvalues of matrix. These protocols are not only effi-cient and secure, but also unnecessary for any cryptographic assumption.
Linear algebra plays an important role in computer science, especially in cryptography. Numerous cryptographic protocols and scientific computations are based on linear algebra, which can be reduced to some core problems, such as matrix multiplication, determinant and the characteristic polynomial of a matrix. However, it is difficult to carry out these expensive computations independently for resource-limited cloud users. Outsourced computation, a service provided by cloud computing, enables a resources-constrained client to outsource his mass computing tasks to the cloud. In this paper, we use data hiding technique to design a secure and verifiable outsourcing protocol for computing the characteristic polynomial and eigenvalues of a matrix. Our protocols achieve several desired features, such as data privacy, verifiability and efficiency. Moreover, no cryptographic assumption is needed in our protocols.
Matrix calculation plays an important role in both scientific computation and cryptography. Many cryptographic protocols, scientific computations, and numerical computations are based on matrix calculation. However, it is difficult to finish matrix calculation independently for these customers whose computation abilities are limited. Cloud Computing has a great deal of computational resources, which enable customers with limited computational resources to outsource their mass computing to the cloud. In this paper we design secure, verifiable, and practical outsourcing protocols for matrix calculation, including matrix multiplication, computing the determinant and inverse of a matrix. Compared with those existing outsourcing protocols, our protocols have obvious improvement concerning both efficiency and security. Furthermore, no cryptographic assumption is needed in our protocols. Finally, we give two applications for our outsourcing protocols; one is to construct an outsourcing protocol for solving the large-scale systems of linear equations, and the other is to design an outsourcing protocol for realizing a cryptosystem based on error-correction code.
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.
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.
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.
Matrix calculation, which has been considered as one of the most expensive operations, plays an important role in most current cryptosystems. In this paper, we propose three new and secure outsourcing schemes of matrix calculation by using one untrusted cloud server. The first one is for matrix multiplication, the second is for the inverse of a matrix, and the third is for the determinant of a matrix. Compared with other proposed schemes, our schemes are superior in efficiency owing to the reduction of user cost. In particular, the user's computation cost in our schemes is at most 1/4, 1/4, 7/16 of the corresponding schemes in the state-of-the-art algorithms, respectively. In addition, the efficiency and security of our schemes are analysed and proved.