Cloud computing is an internet based computing in which large groups of remote servers are networked together so as to allow the sharing of data processing tasks, centralized data storage, and online access to computer services or resources.
However, how to protect customers’ confidential data involved in the computations then becomes a major security concern. In this paper, we present a novel secure transmission scheme for solving systems of linear equations
(LE) in cloud computing. Our mechanism enables a customer to securely outsource the system of linear equations and find its solution by keeping both the sensitive input and output of the computation private.
System of linear equations, cloud computing.
Cloud computing is a model for delivering information technology services in which resources are retrieved from the internet through web-based tools and applications, rather than a direct connection to a server.
Data and software packages are stored in servers. However, cloud computing structure allows access to information as long as an electronic device has access to the web. This type of system allows employees to work remotely.
Let's say you're an executive at a large corporation.
Your particular responsibilities include making sure that all of your employees have the right hardware and software they need to do their jobs. Buying computers for everyone isn't enough -- you also have to purchase software or software licenses to give employees the tools they require. Whenever you have a new hire, you have to buy more software or make sure your current software license allows another user. It's so stressful.
Soon, there may be an alternative for executives like you. Instead of installing a suite of software for each computer, you'd only have to load one application. That application would allow workers to log into a Web-based service which hosts all the programs the user would need for his or her job. Remote machines owned by another company would run everything from e-mail to word processing to complex data analysis programs. It's called cloud computing, and it could change the entire computer industry.
Cloud Computing provides on-demand network access to a shared pool of configurable computing resources that can be rapidly deployed with great efficiency and minimal management overhead. Users can store their data in the cloud and there is a lot of personal information and potentially secure data that people store on their computers, and this information is now being transferred to the cloud. Here we must ensure the security of user’s data, which is in stored in the cloud. But the lack of physical control, or defined entrance and egress points, bring a whole host of security issues – data co-mingling, privileged user abuse, snapshots and backups, data deletion, data leakage, geographic regulatory requirements, cloud super-admins, and many more. Fortunately, experts agree that encryption is the unifying cloud security control, allowing you protect, control and comply.
A system of linear equations can be written as
Ax = b (1) where x is the n × 1 vector of unknowns, A is an n × n (nonsingular) coefficient matrix and b is an n ×1 right-hand side vector (so called constant terms). To transform the problem, the customer picks a random vector r

R n as his secret keying material. Then he rewrites Eq. (1) as A(x + r) = b + Ar.
Let y = x + r and b’= b + Ar, we have Ay = b’. To hide the coefficient matrix A, the customer would select a random invertible matrix Q that has the same dimension as A. Leftmultiplying Q to both sides of Ay = b’ would give us
A’y = b”
(2) where A’= QA and b”= Q (b + Ar). Clearly, as Q and r are chosen randomly and kept as secret, cloud has no way to know
(A; b; x), except the dimension of x.
The customer can then outsource

K
= (A’; b”) to the cloud, who solves

K
and sends back answer y. Once the correctness of y is verified, the customer can derive x via x = y
 r for the original problem

= (A;b).
We study in this section a straight-forward approach for encrypting the problem for direct solvers, and show that the local computation cost based on these techniques along may result in an unsatisfactory mechanism from the efficiency gain perspective. In the existing system the cloud customer use a random vector r

R n to transform the problem. Instead of that here cloud customer uses a random number r

R and to
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maintain the matrix operation the coefficient matrix A split into separate columns. This will help to make the operation a little more simple and understandable.
A system of linear equations can be written as
Ax = b (3) where x is the n × 1 vector of unknowns, A is an n × n (nonsingular) coefficient matrix and b is an n ×1 right-hand side vector (so called constant terms). We assume the system coefficient matrix A is a strictly diagonally dominant matrix. It helps ensure the non-singularity of A and the convergence of the iterative method use in cloud to find the solution of linear equations.
We describe the proposed system in three phases –
Problem transformation phase, Problem solving phase and
Result verification phase.
In the Problem Transformation phase, the customer selects a random invertible matrix Q that has the same dimension as A. Left multiplying Q to both sides of Ax = b would give us
A’x = b’ (4) where A’ = QA and b’ = Qb. It is clear that the solution of Eq.
(3) and Eq.(4) are same. So it will help to hide the coefficient matrix A. Let the first column of QA be P
1
, second column be P
2 and so on. To hide the solution x the customer would select a random vector r

R as his secret keying material. Then he rewrites the Eq. (4) as
A’(x + r) = b’ + r ( ∑ 𝑛 𝑖=1
𝑃 𝑖
Let y = x + r and b” = b + r ( ∑ 𝑛 𝑖=1
𝑃 𝑖
).
), we have
A’y = b”
(5)
Clearly, as Q and r are chosen randomly and kept as secret, cloud has no way to know (A; b; x), except the dimension of x.
Algorithm: Problem Transformation Phase
Data: original problem

= (A; b)
Result: transformed problem as shown in Eq. (5)
Begin
1.
select a random invertible matrix Q
2.
compute A’ = QA and b’ = Qb
3.
pick random r

R
4.
compute b” = b + r ( ∑ 𝑛 𝑖=1
𝑃 𝑖
) and
Return transformed problem

(A’, b”) y = x + r
The customer can then start the Problem Solving phase by outsourcing

= (A’; b”) to the cloud, who solves

by any iterative method and sends back answer y. Once the correctness of y is verified, in the Result Verification phase the customer can derive x via x = y

r for the original problem

= (A; b).
Focusing on the engineering and scientific computing problems, this proposed work investigates secure outsourcing for widely applicable systems of linear equations in cloud computing, which are among the most popular algorithmic and computational tools in various engineering disciplines that analyze and optimize real-world systems. Our proposed work has three phases- problem transformation, problem solving and result verification. In problem transformation, the cloud customer would select an invertible matrix and a random number to transform the LE problem into some encrypted form.
In problem solving, the cloud customer would use the encrypted form of LE to start the computation outsourcing process. In case of using the iterative methods, the protocol ends when the solution within the required accuracy is found. In result verification, the cloud customer would verify the encrypted result produced from the cloud server
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