Computation Of Polynomials Research Articles

Publicly Verifiable Computation of Polynomials Over Outsourced Data With Multiple Sources

Among all types of computations, the polynomial function evaluation is a fundamental, yet an important one due to its wide usage in the engineering and scientific problems. In this paper, we investigate publicly verifiable outsourced computation for polynomial evaluation with the support of multiple data sources. Our proposed verification scheme is universally applicable to all types of polynomial computations and allows the clients to outsource new data at any time. While the existing solutions only support the verification for polynomial evaluation over a single data source, i.e., all the inputs of the polynomial function are outsourced and signed by a single entity, our solution supports polynomial evaluations over multiple different data sources, which are more common and have wider applications, e.g., to assess the city air pollution, one needs to evaluate the environmental data uploaded from the multiple environmental monitor sites. In our proposed scheme, the verification cost for the client is independent with either the input size or the polynomial size so that it scales well in practice. We formally prove the correctness and soundness of our scheme and conduct numerical analysis and evaluation study to validate its high efficiency and scalability. The experimental results show that the data contributor signing 1000 new data only takes 2.1 s, and the verification of the delegated polynomial function takes only 22 ms, which is practically efficient for the real-world applications.

An experimental study of encrypted polynomial arithmetics for private set operations

Recent studies on the performance of private set operations have examined the use of homomorphic public-key encryption and the technique of representing sets as polynomials in a cryptographic model. These polynomial-based solutions require intensive polynomial arithmetic that exhibit quadratic computational complexity. In an effort to develop practical algorithms for private set operations, various well-known techniques such as Karatsuba’s algorithm or the fast Fourier transform (FFT) can be used to reduce the complexity of polynomial computations. The FFT appears to be the obvious best choice; however, the use of FFTs in the implementation of polynomial-based set operations may lead to certain subtle technical problems. These problems become particularly serious when the cardinality of the sets is dynamic. Furthermore, our experiment shows that Karatsuba’s algorithm delivers higher performance than the FFT in our application, provided that a reasonable response time is important. In addition, our experimental implementation demonstrates the heuristic bound on cardinality for which Karatsuba’s algorithm outperforms the FFT. This value can be used to determine the superior optimization techniques and settings in the deployment of private set operation schemes.

SAR images denoising using a novel stochastic diffusion wavelet scheme

The rapid processing of remote sensing (RS) images is crucial in real-time monitoring. However, the computation cost of RS is high and traditional methods are not effective. Cloud computing with its capability of parallel computing provides an effective service for executing RS processing. A cloud-integrated web platform for maintaining geographic information system (GIS) and RS application such as oil spill detection, meteorological monitoring through synthetic aperture radar (SAR) images is a fast-growing application. SAR RS helps also in studying land and sea-based phenomena, especially the capability of acquiring weather forecasts all-day. Wavelet transform is a very well-known tool for prime applications in time series, function estimation, and image analysis. In this work, an effective non-deterministic polynomial computation technique for noise mitigation of SAR Images which has its basis on Hybrid wavelet transform (WT) is proposed. Proper noise reduction parameters should be chosen while selecting wavelet transform for SAR images. So, in the proposed method Stochastic diffusion search (SDS) optimization algorithm is utilized for selecting the optimal noise reduction parameter thus leading better filtration performance. Effective choice of wavelet noise mitigation techniques includes wavelet function, decomposition levels, and threshold selection rules for improved noise reduction. Experimental results show that MSE, PSNR and standard deviation are enhanced with the proposed method.

Chaotic Maps-Based Strong Anonymous Authentication Scheme for Roaming Services in Global Mobility Networks

Traditional smart card and password based two-factor authentication schemes for roaming services in global mobility networks are based on public key or symmetric key cryptographic primitives. With the advancement of chaos-based cryptography, it has become an interesting topic to investigate if it can be used as an alternative tool for building mobile roaming authentication schemes that achieve higher efficiency and security. In this paper, we answer this question affirmatively by proposing a chaotic maps-based authentication scheme for roaming. The scheme supports authentication, security and strong anonymity using the applied pi calculus-based formal verification tool ProVerif. Besides, it applies the efficient Chebyshev polynomial computation for better performance.

A branch and bound algorithm for nonconvex quadratic optimization with ball and linear constraints

We suggest a branch and bound algorithm for solving continuous optimization problems where a (generally nonconvex) objective function is to be minimized under nonconvex inequality constraints which satisfy some specific solvability assumptions. The assumptions hold for some special cases of nonconvex quadratic optimization problems. We show how the algorithm can be applied to the problem of minimizing a nonconvex quadratic function under ball, out-of-ball and linear constraints. The main tool we utilize is the ability to solve in polynomial computation time the minimization of a general quadratic under one Euclidean sphere constraint, namely the so-called trust region subproblem, including the computation of all local minimizers of that problem. Application of the algorithm on sparse source localization problems is presented.

Computing Polynomials Using Unipolar Stochastic Logic

This article addresses subtraction and polynomial computations using unipolar stochastic logic. Stochastic computing requires simple logic gates, and stochastic logic--based circuits are inherently fault tolerant. Thus, these structures are well suited for nanoscale CMOS technologies. It is well known that an AND gate and a multiplexer can be used to implement stochastic unipolar multiplier and adder, respectively. Although it is easy to realize multiplication and scaled addition, implementation of subtraction is nontrivial using unipolar stochastic logic. Additionally, an accurate computation of subtraction is critical for the implementation of polynomials with negative coefficients in stochastic unipolar representation. This work, for the first time, demonstrates that instead of using well-known Bernstein polynomials, stochastic computation of polynomials can be implemented by using a stochastic subtractor and factorization. Three major contributions are given in this article. First, two approaches are proposed to compute subtraction in stochastic unipolar representation. In the first approach, the subtraction operation is approximated by cascading multilevels of OR and AND gates. The accuracy of the approximation is improved with the increase in the number of stages. In the second approach, the stochastic subtraction is implemented using a multiplexer and a stochastic divider. This approach requires more hardware complexity due to the use of a linear-feedback shift register and a counter for division. Second, computation of polynomials in stochastic unipolar format is presented using scaled addition and proposed stochastic subtraction. Third, we propose stochastic computation of polynomials using factorization. Stochastic implementations of first- and second-order factors are presented for different locations of polynomial roots. From experimental results, it is shown that the proposed stochastic logic circuits require less hardware complexity than the previous stochastic polynomial implementation using Bernstein polynomials.

Generation of Long Perfect Gaussian Integer Sequences

Recently, the perfect Gaussian integer sequences have been widely used in modern wireless communication systems, such as code division multiple access and orthogonal frequency-division multiplexing systems. This letter presents two different methods to generate the long perfect Gaussian integer sequences with ideal periodic auto-correlation functions. The key idea of the proposed methods is to use a short perfect Gaussian integer sequence together with the polynomial or trace computation over an extension field to construct a family of the long perfect Gaussian integer sequences. The period of the resulting long sequences is not a multiple of that of the short sequence, which has not been investigated so far. Compared with the already existing methods, the proposed methods have three significant advantages that a single short perfect Gaussian integer sequence is employed, the long sequences consist of two distinct Gaussian integers, and their energy efficiency is monotone increasing.

Incentivize Multi-Class Crowd Labeling Under Budget Constraint

Crowdsourcing systems allocate tasks to a group of workers over the Internet, which have become an effective paradigm for human-powered problem solving, such as image classification, optical character recognition, and proofreading. In this paper, we focus on incentivizing crowd workers to label a set of multi-class labeling tasks under strict budget constraint. We properly profile the tasks’ difficulty levels and workers’ quality in crowdsourcing systems, where the collected labels are aggregated with sequential Bayesian approach. To stimulate workers to undertake crowd labeling tasks, the interaction between workers and the platform is modeled as a reverse auction. We reveal that the platform utility maximization could be intractable, for which an incentive mechanism that determines the winning bid and payments with polynomial-time computation complexity is developed. Moreover, we theoretically prove that our mechanism is truthful, individually rational, and budget feasible. Through extensive simulations, we demonstrate that our mechanism utilizes budget efficiently to achieve high platform utility with polynomial computation complexity.

Exponential Enhancement of the Efficiency of Quantum Annealing by Non-Stoquastic Hamiltonians

Non-stoquastic Hamiltonians have both positive and negative signs in off-diagonal elements in their matrix representation in the standard computational basis and thus cannot be simulated efficiently by the standard quantum Monte Carlo method due to the sign problem. We describe our analytical studies of this type of Hamiltonians with infinite-range non-random as well as random interactions from the perspective of possible enhancement of the efficiency of quantum annealing or adiabatic quantum computing. It is shown that multi-body transverse interactions like $XX$ and $XXXXX$ with positive coefficients appended to a stoquastic transverse-field Ising model render the Hamiltonian non-stoquastic and reduce a first-order quantum phase transition in the simple transverse-field case to a second-order transition. This implies that the efficiency of quantum annealing is exponentially enhanced, because a first-order transition has an exponentially small energy gap (and therefore exponentially long computation time) whereas a second-order transition has a polynomially decaying gap (polynomial computation time). The examples presented here represent rare instances where strong quantum effects, in the sense that they cannot be efficiently simulated in the standard quantum Monte Carlo, have analytically been shown to exponentially enhance the efficiency of quantum annealing for combinatorial optimization problems.

A Knot Polynomial Invariant for Analysis of Topology of RNA Stems and Protein Disulfide Bonds

Knot polynomials have been used to detect and classify knots in biomolecules. Computation of knot polynomials in DNA and protein molecules have revealed the existence of knotted structures, and provided important insight into their topological structures. However, conventional knot polynomials are not well suited to study RNA molecules, as RNA structures are determined by stem regions which are not taken into account in conventional knot polynomials. In this study, we develop a new class of knot polynomials specifically designed to study RNA molecules, which considers stem regions. We demonstrate that our knot polynomials have direct structural relation with RNA molecules, and can be used to classify the topology of RNA secondary structures. Furthermore, we point out that these knot polynomials can be used to model the topological effects of disulfide bonds in protein molecules.

Chemical reaction networks for computing logarithm.

Living cells constantly process information from their living environment. It has recently been shown that a number of cell signaling mechanisms (e.g. G protein-coupled receptor and epidermal growth factor) can be interpreted as computing the logarithm of the ligand concentration. This suggests that logarithm is a fundamental computation primitive in cells. There is also an increasing interest in the synthetic biology community to implement analog computation and computing the logarithm is one such example. The aim of this article is to study how the computation of logarithm can be realized using chemical reaction networks (CRNs). CRNs cannot compute logarithm exactly. A standard method is to use power series or rational function approximation to compute logarithm approximately. Although CRNs can realize these polynomial or rational function computations in a straightforward manner, the issue is that in order to be able to compute logarithm accurately over a large input range, it is necessary to use high-order approximation that results in CRNs with a large number of reactions. This article proposes a novel method to compute logarithm accurately in CRNs while keeping the number of reactions in CRNs low. The proposed method can create CRNs that can compute logarithm to different levels of accuracy by adjusting two design parameters. In this article, we present the chemical reactions required to realize the CRNs for computing logarithm. The key contribution of this article is a novel method to create CRNs that can compute logarithm accurately over a wide input range using only a small number of chemical reactions.

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.

Symbolic–numeric computation of orthogonal polynomials and Gaussian quadratures with respect to the cardinal B-spline

The first 60 coefficients in the three-term recurrence relation for monic polynomials orthogonal with respect to cardinal B-splines φm as the weight functions on [0, m] (m ∈ ℕ) are obtained in a symbolic form. They enable calculation of parameters, nodes, and weights, in the corresponding Gaussian quadrature up to 60 nodes. The efficiency of these Gaussian quadratures is shown in some numerical examples. Finally, two interesting conjectures are stated.

Supplier selection and order allocation with green criteria: An MCDM and multi-objective optimization approach

This research provides a decision-making tool to solve a multi-period green supplier selection and order allocation problem. The tool contains three integrated components. First, fuzzy TOPSIS (technique for order of preference by similarity to ideal solution) is used to assign two preference weights to each potential supplier according to two sets of criteria taken separately: traditional and green. Second, top management uses an analytic hierarchy process (AHP) to assign a global importance weight to each of the two sets of criteria based on the strategy of the company and independently of the potential suppliers. Third, for each supplier, the preference weight obtained from fuzzy TOPSIS regarding the traditional criteria is then multiplied by the global importance weight of the set of traditional criteria. The same is done for the green criteria. The two combined preference weights obtained for each supplier are then used in addition to total cost to select the best suppliers and to allocate orders using multi-period bi-objective and multi-objective optimization. The mathematical models are solved using the weighted comprehensive criterion method and the branch-and-cut algorithm. The approach of this research has a major advantage: it provides top management with flexibility in giving more or less importance weight to green or traditional criteria regardless of the number of criteria in each category through the use of AHP, which reduces the effect of the number of criteria on the preference weight of the suppliers. Contrary to the case in which each supplier is evaluated on the basis of all criteria at the same time, the proposed approach would not necessarily result in a supplier with poor green performance being ranked among the best for a situation in which the number of green criteria is smaller than the number of traditional criteria. In this case, the final ranking would mainly depend on the global weight of the green criteria set given by the top management using AHP as well as on the ranking of the supplier in terms of green criteria obtained from fuzzy TOPSIS. Extensive numerical experiments are conducted in which the bi-objective and multi-objective models are compared and the effect of the separation between green and traditional criteria is investigated. The results show that the two optimization approaches provide very close solutions, which leads to a preference for the bi-objective approach because of its lower computation time. Moreover, the results confirm the flexibility of the proposed approach and show that combining all criteria in one set is a special case. Finally, a time study is performed, which shows that the bi-objective integer linear programming model has a polynomial computation time.

Statistical learning based fully homomorphic encryption on encrypted data

Statistical learning has been widely used in many fields, such as science, engineering and finance, to extract important patterns, trends, and understand “what the data say”. Privacy of statistical learning, i.e., user and sensitive data, is significant problem of performing computation, especially outsourcing cloud computing. Some fully homomorphic encryption schemes can achieve computation on the encrypted data, but most of them are lack of efficiency. Fully homomorphic encryption based on learning with errors over rings (RLWE) supports a finite number of addition and multiplication on the encrypted data, thus can be viewed as the polynomial computation in cyclotomic fields. Computation on the encrypted data can be converted into computation associated with polynomial. So, fully homomorphic encryption from RLWE is very efficient relative to other schemes. Our contribution includes two aspects. Firstly, we show a scheme to represent the training and testing data for statistical learning. The proposed scheme firstly transforms the data into integer and then encodes them into polynomial so that the encryption, decryption and homomorphic operation can be performed efficiently. We also carefully choose the parameters of fully homomorphic encryption from RLWE to meet the requirement of efficiency. User only needs to upload the encrypted data to cloud server, and then the server trains and tests the encrypted data, returns the analysis and prediction results to user. Secondly, we present a comparison scheme on the encrypted data for statistical learning algorithms, which is security on the known plain-text attack and ciphertext only attack model.

Minimal consistent DFA revisited

We address the classic problem of polynomial computation of a minimal finite-state representation compatible with two finite input sets that contain those strings that have to be accepted and rejected. We extend the previously defined uniformly-complete sample and prove that it is a special case of the sufficient condition we propose.

Efficient computation of the characteristic polynomial of a threshold graph

An efficient algorithm is presented to compute the characteristic polynomial of a threshold graph. Threshold graphs were introduced by Chvátal and Hammer, as well as by Henderson and Zalcstein in 1977. A threshold graph is obtained from a one vertex graph by repeatedly adding either an isolated vertex or a dominating vertex, which is a vertex adjacent to all the other vertices. Threshold graphs are special kinds of cographs, which themselves are special kinds of graphs of clique-width 2. We obtain a running time of O(nlog2⁡n) for computing the characteristic polynomial, while the previously fastest algorithm ran in quadratic time. We improve the running time drastically in the case where there is a small number of alternations between 0's and 1's in the sequence defining a threshold graph.

Computation of Hurwitz spaces and new explicit polynomials for almost simple Galois groups

We compute the first explicit polynomials with Galois groups $G=P\Gamma L_3(4)$, $PGL_3(4)$, $PSL_3(4)$ and $PSL_5(2)$ over $\mathbb{Q}(t)$. Furthermore we compute the first examples of totally real polynomials with Galois groups $PGL_2(11)$, $PSL_3(3)$, $M_{22}$ and $Aut(M_{22})$ over $\mathbb{Q}$. All these examples make use of families of covers of the projective line ramified over four or more points, and therefore use techniques of explicit computations of Hurwitz spaces. Similar techniques were used previously e.g. by Malle, Couveignes, Granboulan and Hallouin. Unlike previous examples, however, some of our computations show the existence of rational points on Hurwitz spaces that would not have been obvious from theoretical arguments.

Improving the Efficiency of Stochastic Vehicle Routing: A Partial Lagrange Multiplier Method

More realistic than deterministic vehicle routing, stochastic vehicle routing considers uncertainties in traffic. Its two representative optimization models are the probability tail (PT) and the stochastic shortest path problem with delay excess penalty (SSPD), which can be approximately solved by expressing them as mixed-integer linear programming (MILP) problems. The traditional method to solve these two MILP problems, i.e., branch and bound (B&B), suffers from exponential computation complexity because of integer constraints. To overcome this computation inefficiency, we propose a partial Lagrange multiplier method. It benefits from the total unimodularity of the incidence matrix in the models, which guarantees an optimal integer solution by only solving a linear programming (LP) problem. Thus, this partial Lagrange multiplier problem can be further solved using the subgradient method, and the proposed method can guarantee polynomial computation complexity. Moreover, we theoretically prove the convergence and the efficiency, which are also assessed by the experiments on three different scales of graphs (road networks): small scale, medium scale, and large scale. More importantly, the experimental results on the Beijing road network with real traffic data show that our method can efficiently solve the time-dependent routing problem. Additionally, the implementation on the navigation system based on the Singapore road network further confirms that our method can be applied to efficiently solve the real-world stochastic vehicle routing problem.

Byzantine Agreement in Expected Polynomial Time

We address the problem of Byzantine agreement, to bring processors to agreement on a bit in the presence of a strong adversary. This adversary has full information of the state of all processors, the ability to control message scheduling in an asynchronous model, and the ability to control the behavior of a constant fraction of processors that it may choose to corrupt adaptively. In 1983, Ben-Or proposed an algorithm for solving this problem with expected exponential communication time. In this article, we improve that result to require expected polynomial communication time and computation time. Like Ben-Or’s algorithm, our algorithm uses coinflips from individual processors to repeatedly try to generate a fair global coin. We introduce a method that uses spectral analysis to identify processors that have thwarted this goal by flipping biased coins.