Secure outsourcing of modular exponentiations in cloud and cluster computing

A majority of services are deployed through outsourcing. Outsourcing computation allows resource-constrained clients to outsource their complex computation workloads to a powerful server which is rich of computation resources.

These service modes are realized in the way of outsourcing. The high cost parts, e.g. the cost of purchasing hardware and software, the cost of management and maintenance, are outsourced to cloud servers. Outsourcing service solves the resource-constrained problem for mobile devices. The complex computing tasks of clients whose computing resources are limited can be outsourced to the cloud server. Outsourcing computation provides a lot of convenience for people.

The server with powerful computing resources may obtain some sensitive information from the outsourced data. In outsourcing computation, the inputs and the outputs should not be revealed, the results need to be computed correctly, and the correctness of the results can be verified by clients. Since the user does not compute the outsourced data, when receiving the returned results, user has to do the verifiable computation . Server cannot cheat the user with a random value without computing the outsourced function. The most majority services of cloud computing are obtained by the way of outsourcing. Secure outsourcing of exponentiations was studied in many schemes [8,22,24], but most of the approaches focus on fixed-base exponentiation. Secure outsourcing for scientific computing and numerical calculation are firstly studied by Atallah et. In 2005, Hohenberger and Lysyanskaya defined the formal security of outsourcing, and proposed a classical algorithm for outsourcing of modular exponentiations based on precomputation and server-aided computation [14,27].

In 2008, Benjamin and Atallah constructed a verifiable secure outsourcing scheme for linear algebraic calculation by using semantic security based homomorphic encryption. In 2009, Gentry proposed a non-interactive fully homomorphic encryption scheme for verifiable outsourcing with input privacy. Proposed an outsourcing scheme of modular exponentiation with single untrusted program, but the checkability of the result is just 21. 1.2 Organization Some security definitions for outsourcing computation are given in the next section.

3 the proposed new outsourcing of modular exponentiations algorithm is given. 6 we give the application of our algorithm to secure outsourcing RSA encryption and ElGamal signature. Secret information, only available to M. Protected information, only available to M and E. Unprotected information, available to M, E, and U . We introduce the definitions for secure outsourcing of a cryptographic algorithm follows. 1: 3 New outsourcing algorithms of modular exponentiations In this section, we give a new outsourcing algorithm for modular exponentiations. 3.1 Algorithm of modular exponentiations Exp We provide outsourcing algorithm of modular exponentiations Exp which makes u a mod p in secure outsourcing in the one-malicious model. If not, M outputs "Error"; otherwise, M multiplies the real outputs of U1 and U2 to compute u a as w 1 v 2 r l r s = h a vl s = ua 3.2 Algorithm of simultaneous modular exponentiations SExp In the following, we give an algorithm of simultaneous modular exponentiations SExp which makes u a1 u b2 mod p in secure outsourcing in the one-malicious model.

Theorem 3 In the one-malicious model, the algorithms (M, (U1 , U2 are a 23 -checkable implementation of EXP. Proof Considering about the outsoutcing algorithm of u a , the outsourcing pairs are,, and, where, are outsourced to U1 , U2 respectively. Theorem 4 In the one-malicious model, the algorithms (M, (U1 , U2 are a 21 -checkable implementation of SEXP. Proof Considering about the outsoutcing algorithm of u a , the pairs,, and are outsourced to U1 , and,, and are outsourced to U2. The two non-colluding servers compute the same pairs and. In logical division phase and the outsourcing phase, the subroutine Rand needs to be invoked. Further more, another important advantage of our scheme is that the MInvs and invocation times of Rand keep invariant when the number of outsourcing simultaneous modular exponentiations increasing. 6 Time cost of u a1 u b2 , a is 500-bit long We applied our algorithm EXP to secure outsourcing RSA encryption.6 Application 6.1 Secure outsourcing of RSA encryption With the widespread use of communication network and electronic commerce, security and reliability of the transporting data become more and more important in cloud computing. 6.2 Secure outsourcing of ElGamal signature When data is stored in cloud servers, the clients cannot control it directly.

We applied our algorithm EXP and SEXP to secure outsourcing ElGamal signatures. The proposed secure outsourcing algotithm for ElGamal signature scheme consists of the following efficient algorithms.On input the verification key y, the message m, and the signature =, the outsourcer computes h(m) andverifies the signature as follows.Outsourcer outputs 1 if and only if y r r s = g h.

The Exp and SExp are only used for the signature verification.Atallah, M.J., Frikken, K.B.: Securely outsourcing linear algebra computations. Benjamin, D., Atallah, M.J.: Private and cheating-free outsourcing of algebraic computations. Gennaro, R., Gentry, C., Parno, B.: Non-interactive verifiable computing: Outsourcing computation to untrusted workers.

Question:

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