Verifiable Computation Scheme Research Articles

An efficient polynomial-based verifiable computation scheme on multi-source outsourced data

With the development of cloud computing, users are more inclined to outsource complex computing tasks to cloud servers with strong computing capacity, and the cloud returns the final calculation results. However, the cloud is not completely trustworthy, which may leak the data of user and even return incorrect calculations on purpose. Therefore, it is important to verify the results of computing tasks without revealing the privacy of the users. Among all the computing tasks, the polynomial calculation is widely used in information security, linear algebra, signal processing and other fields. Most existing polynomial-based verifiable computation schemes require that the input of the polynomial function must come from a single data source, which means that the data must be signed by a single user. However, the input of the polynomial may come from multiple users in the practical application. In order to solve this problem, the researchers have proposed some schemes for multi-source outsourced data, but these schemes have the common problem of low efficiency. To improve the efficiency, this paper proposes an efficient polynomial-based verifiable computation scheme on multi-source outsourced data. We optimize the polynomials using Horner’s method to increase the speed of verification, in which the addition gate and the multiplication gate can be interleaved to represent the polynomial function. In order to adapt to this structure, we design the corresponding homomorphic verification tag, so that the input of the polynomial can come from multiple data sources. We prove the correctness and rationality of the scheme, and carry out numerical analysis and evaluation research to verify the efficiency of the scheme. The experimental indicate that data contributors can sign 1000 new data in merely 2 s, while the verification of a delegated polynomial function with a power of 100 requires only 18 ms. These results confirm that the proposed scheme is better than the existing scheme.

Libras: A Fair, Secure, Verifiable, and Scalable Outsourcing Computation Scheme Based on Blockchain

Libras: A Fair, Secure, Verifiable, and Scalable Outsourcing Computation Scheme Based on Blockchain

Protecting Function Privacy and Input Privacy in the Publicly Verifiable Outsourcing Computation of Polynomial Functions

With the prevalence of cloud computing, the outsourcing of computation has gained significant attention. Clients with limited computing power often outsource complex computing tasks to the cloud to save on computing resources and costs. In outsourcing the computation of functions, a function owner delegates a cloud server to perform the function’s computation on the input received from the user. There are three primary security concerns associated with this process: protecting function privacy for the function owner, protecting input privacy for the user and guaranteeing that the cloud server performs the computation correctly. Existing works have only addressed privately verifiable outsourcing computation with privacy or publicly verifiable outsourcing computation without input privacy or function privacy. By using the technologies of homomorphic encryption, proxy re-encryption and verifiable computation, we propose the first publicly verifiable outsourcing computation scheme that achieves both input privacy and function privacy for matrix functions, which can be extended to arbitrary multivariate polynomial functions. We additionally provide a faster privately verifiable method. Moreover, the function owner retains control over the function.

RPVC: A Revocable Publicly Verifiable Computation Solution for Edge Computing.

With publicly verifiable computation (PVC) development, users with limited resources prefer to outsource computing tasks to cloud servers. However, existing PVC schemes are mainly proposed for cloud computing scenarios, which brings bandwidth consumption or network delay of IoT devices in edge computing. In addition, dishonest edge servers may reduce resource utilization by returning unreliable results. Therefore, we propose a revocable publicly verifiable computation(RPVC) scheme for edge computing. On the one hand, RPVC ensures that users can verify the correct results at a small cost. On the other hand, it can revoke the computing abilities of dishonest edge servers. First, polynomial commitments are employed to reduce proofs’ length and generation speed. Then, we improve revocable group signature by knowledge signatures and subset covering theory. This makes it possible to revoke dishonest edge servers. Finally, theoretical analysis proves that RPVC has correctness and security, and experiments evaluate the efficiency of RPVC.

Publicly Verifiable Outsourcing Computation for QR Decomposition Based on Blockchain

In the Big Data Era, outsourcing computation has become increasingly significant as it supplies computation resources for clients with limited resources. However, there are still many security challenges such as payment fairness, privacy protection, and verification. In this paper, we propose a secure publicly verifiable outsourcing computation scheme for the large-scale matrix QR decomposition. In the proposed scheme, client can pay for outsourcing services through blockchain-based payment system which achieves the payment fairness. Moreover, to protect privacy, both permutation matrix and block diagonal matrix are applied in encryption process. Meanwhile, to achieve the public verification, the computational complexity is reduced by using the matrix digest technology. It is worth mentioning that our scheme is provable and secure under the co-CDH assumption.

Privacy-preserving verifiable delegation of polynomial and matrix functions

AbstractOutsourcing computation has gained significant popularity in recent years due to the development of cloud computing and mobile services. In a basic outsourcing model, a client delegates computation of a function f on an input x to a server. There are two main security requirements in this setting: guaranteeing the server performs the computation correctly, and protecting the client’s input (and hence the function value) from the server. The verifiable computation model of Gennaro, Gentry and Parno achieves the above requirements, but the resulting schemes lack efficiency. This is due to the use of computationally expensive primitives such as fully homomorphic encryption (FHE) and garbled circuits, and the need to represent f as a Boolean circuit. Also, the security model does not allow verification queries, which implies the server cannot learn if the client accepts the computation result. This is a weak security model that does not match many real life scenarios. In this paper, we construct efficient (i.e., without using FHE, garbled circuits and Boolean circuit representations) verifiable computation schemes that provide privacy for the client’s input, and prove their security in a strong model that allows verification queries. We first propose a transformation that provides input privacy for a number of existing schemes for verifiable delegation of multivariate polynomial f over a finite field. Our transformation is based on noisy encoding of x and keeps x semantically secure under the noisy curve reconstruction (CR) assumption. We then propose a construction for verifiable delegation of matrix-vector multiplication, where the delegated function f is a matrix and the input to the function is a vector. The scheme uses PRFs with amortized closed-form efficiency and achieves high efficiency. We outline applications of our results to outsourced two-party protocols.

Multi-matrix verifiable computation

The problem of securely outsourcing computation to cloud servers has attracted a large amount of attention in recent years. The verifiable computation of Gennaro, Gentry, Parno (Crypto'10) allows a client to verify the server's computation of a function with substantially less time than performing the outsourced computation from scratch. In a multi-function model (Parno, Raykova, Vaikuntanathan; TCC'12) of verifiable computation, the process of encoding function and the process of preparing input are decoupled such that any client can freely submit a computation request on its input, without having to generate an encoding of the function in advance. In this paper, we propose a multi-matrix verifiable computation scheme that allows the secure outsourcing of the matrix functions over a finite field. Our scheme is outsourceable. When it is used to outsource $m$ linear functions, the scheme is roughly $m$ times faster and has less communication cost than the previously best known scheme by Fiore and Gennaro (CCS'12), both in the client-side computation and in the server-side computation. We also show the cost saving with detailed implementations.

Practical Verifiable Computation by Using a Hardware-Based Correct Execution Environment

The verifiable computation paradigm has been studied extensively as a means to verifying the result of outsourced computation. In said scheme, the verifier requests computation from the prover and verifies the result by checking the output and proof received from the prover. Although they have great potential for various critical applications, verifiable computations have not been widely used in practice, because of their significant performance overhead. Existing cryptography-based approaches incur significant overhead, because a cryptography-based mathematical frame needs to be constructed, which prevents deviation from the correct computation. The proposed approach is to reduce the overhead by trusting the computing hardware platform where the computation is outsourced. If one trusts the hardware to do the computation, the hardware can take the place of the cryptographic computing frame, thereby guaranteeing correct computation. The key challenge of this approach is to define what exactly the hardware should guarantee for verifiable computation. For this, we introduce the concept of Correct Execution Environment (CEE), which guarantees instruction correctness and state preservation. We prove that these two requirements are satisfactory conditions for a correct output. By employing a CEE, the verifiable computation scheme can be simplified, and its overhead can be reduced drastically. The presented experimental results demonstrate that the execution time is approximately 1.7 million times faster and the verification time over 50 times faster than a state-of-the-art cryptographic approach.

Verifiable outsourcing computation for modular exponentiation from shareable functions

Verifiable computation (VC) enables a computationally weak client to outsource the computation of a function f to a more powerful but untrusted server. The server returns an answer that allows the client to recover the correct result, accompanied by a proof of correctness. The basic requirement of a VC scheme is efficiency, i.e., the time consuming in preparing its input and verifying the server’s proof must be less than the time required to compute the function on its own. In this paper, we propose a novel outsourcing scheme for modular exponentiation from shareable functions, which is secure even in the presence of adaptive adversaries. Compared with previous works, our algorithm is superior in both efficiency and checkability. In order to keep the exponent privacy, our algorithm invokes the less subroutine, which accounts for the largest proportion of the time spent. The client can detect the error with overwhelming probability. We then gave a comparation that demonstrates the efficiency of the proposed outsourcing algorithm and scheme. Finally, we propose a generic scheme for shareable function family, and analyze the concrete security.

Verifiable Subgraph Matching With Cryptographic Accumulators in Cloud Computing

Due to the rapid development of social networks, bio-informatics, internet-of-things etc. , subgraph matching query can be found in various applications. Meanwhile, the increasing popularity of storing graph data in the cloud drives demands for graph query processing on a remote cloud server. However, the query results in this scenario may not be guaranteed to be correct, especially when the cloud service provider (CSP) is malicious or compromised by some adversaries, for example, a CSP might omit some edges of the graph so that its search cost would be substantially reduced. Besides, various software bugs and unintended errors are also inevitable. All current generic verifiable computation (VC) schemes applied in this scenario are not only too impractical to be implemented but also need a lot of space to locally store their auxiliary data. To that end, we have put forth both public and designated verification schemes which focus on subgraph matching problems for outsourced graph data. They utilize a modified cryptographic primitive called accumulator to realize fast verification and low local storage overhead. In addition to the two main constructions, we have proposed an optimization to make the scheme more applicable, namely, supporting dynamic updates of the graph. At last, rigorous security proofs and efficiency analysis are given, which justify that our proposed schemes are secure and efficient, satisfying the requirements of general verifiable computation protocols.

Verifiable outsourced computation over encrypted data

In recent years, cloud computing has become the most popular and promising service platform. A cloud user can outsource its heavy computation overhead to a cloud service provider (CSP) and let the CSP make the computation instead. In order to guarantee the correctness of the outsourced processing (e.g., machine learning and data mining), a proof should be provided by the CSP in order to make sure that the processing is carried out properly. On the other hand, from the security and privacy points of view, users will always encrypt their sensitive data first before they are outsourced to the CSP rather than sending the raw data directly. However, processing and verifying of encrypted data computation has always been a challenging problem. Homomorphic Encryption (HE) has been proposed to tackle this task on computations over encrypted data and ensure the confidentiality of the data. However, original HE cannot provide an efficient approach to verify the correctness of computation over encrypted data that is processed by CSP. In this paper, we propose a verifiable outsourced computation scheme over encrypted data with the help of fully homomorphic encryption and polynomial factorization algorithm. Our scheme protects user data security in outsourced processing and allows public verification on the computation result processed by CSP with zero knowledge. We then prove the security of our scheme and analyze its performance by comparing it with some latest related works. Performances analysis shows that our scheme reduces the overload of both the cloud users and the verifier.

An improved scheme for outsourced computation with attribute‐based encryption

SummaryWith the wide deployment of cloud computing, outsourcing complicated computational tasks to cloud service providers has attracted much attention. An increasing number of clients with computationally constrained devices choose to outsource their heavy tasks to cloud servers to reduce the computational overhead in local. However, how to preserve the integrity of computational results becomes a challenge since commercial cloud servers are not trusted. Public verifiability is an effective mechanism to allow clients to verify the integrity of the results returned by the servers. Because the results are sensitive in many applications, it raises the problem of privacy leakage in the public verification process. In this paper, we propose an efficient verifiable computation scheme while keeping output privacy. The proposed scheme achieves blind verifiability such that the verifiers who have not the additional information (retrieve key) can verify the integrity of the result without learning the result. Furthermore, by combining with (k,n)‐threshold sharing, our scheme allows the clients jointly learn the results.

Confidentiality-Preserving Publicly Verifiable Computation Schemes for Polynomial Evaluation and Matrix-Vector Multiplication

With the development of cloud services, outsourcing computation tasks to a commercial cloud server has drawn attention of various communities, especially in the Big Data era. Public verifiability offers a flexible functionality in real circumstance where the cloud service provider (CSP) may be untrusted or some malicious users may slander the CSP on purpose. However, sometimes the computational result is sensitive and is supposed to remain undisclosed in the public verification phase, while existing works on publicly verifiable computation (PVC) fail to achieve this requirement. In this paper, we highlight the property of result confidentiality in publicly verifiable computation and present confidentiality-preserving public verifiable computation (CP-PVC) schemes for multivariate polynomial evaluation and matrix-vector multiplication, respectively. The proposed schemes work efficiently under the amortized model and, compared with previous PVC schemes for these computations, achieve confidentiality of computational results, while maintaining the property of public verifiability. The proposed schemes proved to be secure, efficient, and result-confidential. In addition, we provide the algorithms and experimental simulation to show the performance of the proposed schemes, which indicates that our proposal is also acceptable in practice.

Cryptanalysis of Morillo–Obrador polynomial delegation schemes

Verifiable computation (VC) allows a client to outsource (delegate) the computation of a function f on an input x to a server and then verify the server's results with substantially less time than computing f(x) from scratch. The security of VC requires no efficient adversary can persuade the client to accept any wrong results. Morillo and Obrador (PST 2013) proposed three VC schemes for outsourcing the computation of polynomial functions and claimed that all schemes are secure under the decisional subgroup membership assumption. The authors show a simple attack against the security of their first scheme and then extend the attack to the other two schemes. Morillo and Obrador (PST 2013) also claimed that their third scheme keeps the client's input private under the square root assumption. The authors show that this is not true under the standard definition of input privacy. In particular, a curious server can extract the client's input x, if the x is not too large. The authors' results show that Morillo-Obrador schemes cannot be used in the polynomial delegation.

New publicly verifiable computation for batch matrix multiplication

With the prevalence of cloud computing, the resource constrained clients are trended to outsource their computation-intensive tasks to the cloud server. Although outsourcing computation paradigm brings many benefits for both clients and cloud server, it causes some security challenges. In this paper, we focus on the outsourcing computation of matrix multiplication, and propose a new publicly verifiable computation scheme for batch matrix multiplication. Different from traditional matrix computation outsourcing model, the outsourcing task of our scheme is to compute MXi for group of clients, where Xi is a private matrix chosen by different clients and M is a public matrix given by a data center beforehand. Based on the two techniques of privacy-preserving matrix transformation and matrix digest, our scheme can protect the secrecy of the client’s private matrix Xi and dramatically reduce the computation cost in both the key generation and the computing phases. Security analysis shows that the proposed scheme can achieve the desired security properties under the co-computational Diffie-Hellman assumption. We also provide the experimental evaluation that demonstrates the efficiency of our scheme.

Comments on “Verifiable and Exculpable Outsourced Attribute-Based Encryption for Access Control in Cloud Computing”

Recently in IEEE Transactions on Dependable and Secure Computing (TDSC) (doi: 10.1109/TDSC.2015.2499755), Ma et al. proposed a new construction of attribute-based encryption (ABE) which can outsource the complicated encryption task to Encryption Service Provider (ESP) in a verifiable manner. Despite the authors claimed that the results of the outsourced encryption can be checked by the user, we show that Ma et al.’s proposal fails to provide the verifiability property for outsourced encryption, the most essential security goal that a verifiable computation scheme should achieve. Specifically, by giving concrete attacks, we demonstrate that the ESP can return forged intermediate ciphertext to the user without being detected.

Attribute-based multi-function verifiable computation

Verifiable Computation (VC) allows resource-restricted clients to delegate expensive computations to more powerful servers, and then to verify the correctness of the results. A number of VC schemes have been constructed since it was introduced. However, existing VC schemes support verifying and retrieving results either only by the client (privately) or by anybody (publicly). This one-or-all mode of verification does not fill the need in practical applications, for example, in healthcare applications, some users with special attributes can be authorized to verify and retrieve the results and other users without special attributes cannot be authorized such rights.In this work, we introduce and formalize the notion of attribute-based multi-function verifiable computation (AM-VC). It provides fine-grained access control to the computation results. Specifically, it labels the encoded functions and inputs with policies, as well as associates the secret key of each receiver with attributes. In this way, only the receiver whose attributes satisfy both a function’s policy and an input’s policy can verify and retrieve the corresponding computation result. In addition, executing multiple functions on one encoded input and executing one function on multiple encoded inputs are allowed. We also formally model the security properties of an AM-VC scheme and propose a secure construction.

Secure Collaborative Publicly Verifiable Computation

Publicly Verifiable Computation (PVC) enables computationally weak trusted sources to outsource several computations to some more powerful public untrusted clouds. On issuing a query, the public cloud replies the result of the function evaluation with a witness vouching for correctness of computation. This primitive requires high efficiency and public verifiability. However, existing PVC constructions all request trusted sources to know delegated function beforehand, and thus it fails to meet diverse requirements, especially outsourced target unknown need to be jointly computed among different entities in a privacy-preserving manner. To strengthen current PVC’s flexibility, we proposed a new primitive called Secure Collaborative PVC ( ${\mathcal {SCPVC}}$ ), where TTP is responsible only for initializing system parameter and publishing some information in its bulletin. After some rounds, the public cloud owns lots of functions outsourced in PVC ways. The private cloud works out an algebraic operation structure ${\mathcal {L}}$ , which involves some functions provided by public cloud and himself. Based on ${\mathcal {L}}$ , they jointly perform the protocol to generate the target function. At the end of the protocol, the public cloud obtains target function while not disclosing respective secrets. Due to the misbehavior of the public cloud, this mechanism allows the private cloud to check the integrity of target function and any client to verify the correctness of results. Our scheme without jointly computing is a typical existing PVC scheme. Therefore, our protocol is compatible with the prevailing publicly verifiable computation Scheme. Before investigating ${\mathcal {SCPVC}}$ , we tailored two secure two-party polynomial computation protocols using 1-out-of- ${l}$ Oblivious Transfer protocol as the main building block to $ {\mathcal {SCPVC}}$ . More preciously, polynomial multiplication protocol transforms two polynomials multiplication into another two addition such that the result of sum is equal to the result of multiplication. Similarly, polynomial addition protocol is as same as multiplication protocol converts two polynomials addition into another two multiplication.

New verifiable outsourced computation scheme for an arbitrary function

We study the construction of efficient verifiable outsourced computation for arbitrary functions. We improve previous verifiable outsourced computation schemes for arbitrary functions by using hybr...

New verifiable outsourced computation scheme for an arbitrary function

We study the construction of efficient verifiable outsourced computation for arbitrary functions. We improve previous verifiable outsourced computation schemes for arbitrary functions by using hybrid encryption based on fully homomorphic encryption and introducing multi-free gates in circuits, which can efficiently outsource the computation of an arbitrary function f on arbitrary input x to an untrusted cloud server, and reduce the one-time security error and the client's computation cost. Moreover, our scheme can greatly reduce online communication cost. Finally, we prove the security and the efficiency of our scheme.