Efficient Secure and Verifiable Outsourcing of Matrix Multiplications

Abstract

With the emergence of cloud computing services, a resource-constrained client can outsource its computationally-heavy tasks to cloud providers. Because such service providers might not be fully trusted by the client, the need to verify integrity of the returned computation result arises. The ability to do so is called verifiable delegation or verifiable outsourcing. Furthermore, the data used in the computation may be sensitive and it is often desired to protect it from the cloud throughout the computation. In this work, we put forward solutions for verifiable outsourcing of matrix multiplications that favorably compare with the state of the art. Our goal is to minimize the cost of verifying the result without increasing overhead associated with other aspects of the scheme. In our scheme, the cost of verifying the result of computation uses only a single modulo exponentiation and the number of modulo multiplications linear in the size of the output matrix. This cost can be further reduced to avoid all cryptographic operations if the cloud is rational. A rational cloud is neither honest nor arbitrarily malicious, but rather economically motivated with the sole purpose of maximizing its monetary reward. We extend our core constructions with several desired features such as data protection, public verifiability, and computation chaining.

Efficient Techniques for Publicly Verifiable Delegation of Computation

“…Table 3 4 depicts a comparison of our proposal for matrix multiplication with the solution proposed by Fiore and Gennaro [11] and Zhang and Blanton [21].…”

“…Polynomial preparation d`1 exp in G1 2 pairing d-SBDH Yes [17] 2d n pairings nm exp in G1 n pairings co-CDH No [11] 2pn`mq exp in G1 n exp in GT DLin Zhang and Blanton 1 pairing n exp in G1 nm exp in G2 n pairings M-DDH Yes [21] m exp in G2 pn`1q exp in GT XDH pn`1q exp in GT Our scheme 2nm exp in G1 m exp in GT nm exp in G1 2 pairings co-CDH Yes m pairings n exp in G1 Table 4: Comparison of computation complexity with existing work for matrix multiplication correct computation [17], and uses polynomial commitments [16] to construct these signatures. Besides public verifiability, this solution implements public delegatability.…”

“…However, only the client which outsourced the matrix can submit input vectors to the outsourced multiplication, hence their constructions do not meet the public delegatability requirement. Zhang and Blanton [21] present a construction for publicly delegatable and verifiable outsourcing of matrix multiplication that uses mathematical properties of matrices instead of algebraic PRFs. Unlike our work, the public verifiable scheme suggested in [21] does not transfer the matrix M to the server during Setup (whose purpose is reduced to generating the public parameters).…”

“…Zhang and Blanton [21] present a construction for publicly delegatable and verifiable outsourcing of matrix multiplication that uses mathematical properties of matrices instead of algebraic PRFs. Unlike our work, the public verifiable scheme suggested in [21] does not transfer the matrix M to the server during Setup (whose purpose is reduced to generating the public parameters). Instead, the problem generation phase prepares the matrix and the input vector for the delegation.…”

Enabling Efficient and Secure Outsourcing of Large Matrix Multiplications

“…All of these schemes can be divided into two categories. One is the calculation of general function [5] [6] [11] [12], and the other is the calculation of specific function [7] [8] [13] [14] [15] . In [5], Gennaro et al presented an outsourced computing scheme that is based on Yao's garbled circuits [16] and the fully homomorphism encryption (FHE).…”