Evaluation Of Multivariate Polynomials Research Articles

HepLib: A C++ library for high energy physics (version 1.1)

HepLib is a C++ Library for computations in High Energy Physics, it builds on top of other well-established libraries or programs, including GiNaC, Fermat, Form, Fire, etc., its first version has been released in Comput. Phys. Commun. 265 (2021) 107982. Here we provide a minor upgraded version with parts of obligated aspects removed and many new features added, the main feature in this new version is to use Flint for the simplification on multi-variate polynomial, so one can replace the process-based parallelism by the thread-based parallelism for the related polynomial evaluations to achieve a higher performance. New version program summaryProgram title:HepLib 1.1CPC Library link to program files:https://doi.org/10.17632/jspjxy3z66.2Developer's repository link:https://github.com/HepLib/srcLicensing provisions: : GPLv3Programming language:C++Journal reference of previous version: Comput. Phys. Commun. 265 (2021) 107982Does the new version supersede the previous version?: YesExternal libraries and programs: : GiNaC [2], Qgraf [3], Flint [4], Qhull [5], Form [6], Fermat [7], Fire [8]Nature of problem: There are lots of independent packages or programs in high energy physics, which are developed in different programming languages, here we provide an integrated interface in C++ language, to generate Feynman diagrams/amplitudes, perform algebraic simplifications on Dirac/Color matrixes and reduce scalar integrals to master integrals. The evaluation of master integrals is the core part for the high-order calculations, one has to resolve ultraviolet/infrared divergence within dimensional regularization and the singularities inside the integration domain while performing multiple dimensional numerical integrations.Solution method:HepLib uses Qgraf to generate the Feynman diagrams/aptitudes, Form to perform the related computations on Dirac/Color matrix, Fire for the IBP reduction. A C++ implementation of sector decomposition method is used to extract both ultraviolet and infrared singularity of the integral, the contour deformation is adopted to avoid the singularity inside the integration domain, a parallelized numerical integration is chosen to achieve high efficiency or performance.Reasons for the new version: Fix bugs from GiNaC and implement new features.Summary of revisions: First, we have improved the installation procedure, now one can install the whole package with a single script install.sh, and furthermore one can install all libraries into a distinct directory, so the libraries will not affect the others which are installed on the user's computer. We have upgraded the GiNaC to the latest version with a few bugs fixed in the official site and removed the dependence on the libraries MinUits [9] and Cuba [10], which are used in performing sector decomposition in the previous version. To get high performance on the evaluation of multi-variate polynomials, we introduce Flint in the new version, and one can convert the expression from GiNaC to Flint, and vice versa. Finally, we also made adaptations to the codes and examples to match the new version.

Amortized multi-point evaluation of multivariate polynomials

The evaluation of a polynomial at several points is called the problem of multi-point evaluation. Sometimes, the set of evaluation points is fixed and several polynomials need to be evaluated at this set of points. Several efficient algorithms for this kind of “amortized” multi-point evaluation have been developed recently for the special cases of bivariate polynomials or when the set of evaluation points is generic. In this paper, we extend these results to the evaluation of polynomials in an arbitrary number of variables at an arbitrary set of points. We prove a softly linear complexity bound when the number of variables is fixed. Our method relies on a novel quasi-reduction algorithm for multivariate polynomials, that operates simultaneously with respect to several orderings on the monomials.

Fast amortized multi-point evaluation

The efficient evaluation of multivariate polynomials at many points is an important operation for polynomial system solving. Kedlaya and Umans have recently devised a theoretically efficient algorithm for this task when the coefficients are integers or when they lie in a finite field. In this paper, we assume that the set of points where we need to evaluate is fixed and “sufficiently generic”. Under these restrictions, we present a quasi-optimal algorithm for multi-point evaluation over general fields. We also present a quasi-optimal algorithm for the opposite interpolation task.

Cross Subspace Alignment Codes for Coded Distributed Batch Computation

The goal of coded distributed computation is to efficiently distribute a computation task, such as matrix multiplication, N-linear computation, or multivariate polynomial evaluation, across S servers through a coding scheme, such that the response from any R servers ( R is called the recovery threshold) is sufficient for the user to recover the desired computed value. Current state-of-art approaches are based on either exclusively matrix-partitioning (Entangled Polynomial (EP) Codes for matrix multiplication), or exclusively batch processing (Lagrange Coded Computing (LCC) for N-linear computations or multivariate polynomial evaluations). We present three related classes of codes, based on the idea of Cross-Subspace Alignment (CSA) which was introduced originally in the context of secure and private information retrieval. CSA codes are characterized by a Cauchy-Vandermonde matrix structure that facilitates interference alignment along Vandermonde terms, while the desired computations remain resolvable along the Cauchy terms. These codes are shown to unify, generalize and improve upon the state-of-art codes for distributed computing. First we introduce CSA codes for matrix multiplication, which yield LCC codes as a special case, and are shown to outperform LCC codes in general in download-limited settings. While matrix-partitioning approaches (EP codes) for distributed matrix multiplication have the advantage of flexible server computation latency, batch processing approaches (CSA, LCC) have significant advantages in communication costs as well as encoding and decoding complexity per matrix multiplication. In order to combine the benefits of these approaches, we introduce Generalized CSA (GCSA) codes for matrix multiplication that bridge the extremes of matrix-partitioning and batch processing approaches and demonstrate synergistic gains due to cross subspace alignment. Finally, we introduce N-CSA codes for N-linear distributed batch computations and multivariate batch polynomial evaluations. N-CSA codes include LCC codes as a special case, and are in general capable of outperforming LCC codes in download-constrained settings by upto a factor of N. Generalizations of N-CSA codes to include X-secure data and B-byzantine servers are also provided.

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.

Matrix methods for the simplicial Bernstein representation and for the evaluation of multivariate polynomials

In this paper, multivariate polynomials in the Bernstein basis over a simplex (simplicial Bernstein representation) are considered. Two matrix methods for the computation of the polynomial coefficients with respect to the Bernstein basis, the so-called Bernstein coefficients, are presented. Also matrix methods for the calculation of the Bernstein coefficients over subsimplices generated by subdivision of the standard simplex are proposed and compared with the use of the de Casteljau algorithm. The evaluation of a multivariate polynomial in the power and in the Bernstein basis is considered as well. All the methods solely use matrix operations such as multiplication, transposition, and reshaping; some of them rely also on the bidiagonal factorization of the lower triangular Pascal matrix or the factorization of this matrix by a Toeplitz matrix. The latter one enables the use of the Fast Fourier Transform hereby reducing the amount of arithmetic operations.

Oblivious evaluation of multivariate polynomials

One of the fundamental problems of multi-party computation is Oblivious Polynomial Evaluation. In that problem, that was introduced by Naor and Pinkas, Alice has a polynomial P (x) and Bob has a point α. The goal is to allow Bob to compute P (α) so that Alice remains oblivious of α and Bob of P (x), apart from what is implied by P (α) and α. We introduce the multivariate version of this problem, where x and α are vectors, and offer an efficient secure protocol. In addition, we discuss several applications that may be solved efficiently using oblivious multivariate polynomial evaluation, such as private linear algebraic computations and private support vector machines (SVM). MSC 2010 classification. 94A60 Cryptography

On the evaluation of multivariate polynomials over finite fields

A method to evaluate multivariate polynomials over a finite field is described and its multiplicative complexity is discussed.

Fast Polynomial Factorization and Modular Composition

We obtain randomized algorithms for factoring degree $n$ univariate polynomials over $\mathbb{F}_q$ requiring $O(n^{1.5 + o(1)}\,{\rm log}^{1+o(1)} q+ n^{1 + o(1)}\,{\rm log}^{2+o(1)} q)$ bit operations. When ${\rm log}\, q < n$, this is asymptotically faster than the best previous algorithms [J. von zur Gathen and V. Shoup, Comput. Complexity, 2 (1992), pp. 187-224; E. Kaltofen and V. Shoup, Math. Comp., 67 (1998), pp. 1179-1197]; for ${\rm log}\, q \ge n$, it matches the asymptotic running time of the best known algorithms. The improvements come from new algorithms for modular composition of degree $n$ univariate polynomials, which is the asymptotic bottleneck in fast algorithms for factoring polynomials over finite fields. The best previous algorithms for modular composition use $O(n^{(\omega + 1)/2})$ field operations, where $\omega$ is the exponent of matrix multiplication [R. P. Brent and H. T. Kung, J. Assoc. Comput. Mach., 25 (1978), pp. 581-595], with a slight improvement in the exponent achieved by employing fast rectangular matrix multiplication [X. Huang and V. Y. Pan, J. Complexity, 14 (1998), pp. 257-299]. We show that modular composition and multipoint evaluation of multivariate polynomials are essentially equivalent, in the sense that an algorithm for one achieving exponent $\alpha$ implies an algorithm for the other with exponent $\alpha + o(1)$, and vice versa. We then give two new algorithms that solve the problem near-optimally: an algebraic algorithm for fields of characteristic at most $n^{o(1)}$, and a nonalgebraic algorithm that works in arbitrary characteristic. The latter algorithm works by lifting to characteristic 0, applying a small number of rounds of multimodular reduction, and finishing with a small number of multidimensional FFTs. The final evaluations are reconstructed using the Chinese remainder theorem. As a bonus, this algorithm produces a very efficient data structure supporting polynomial evaluation queries, which is of independent interest. Our algorithms use techniques that are commonly employed in practice, in contrast to all previous subquadratic algorithms for these problems, which relied on fast matrix multiplication.

Generalized oblivious transfer by secret sharing

The notion of Generalized Oblivious Transfer (GOT) was introduced by Ishai and Kushilevitz (Proceeding of ISTCS97, IEEE Computer Society, pp 174---184, 1997). In a GOT protocol, Alice holds a set U of messages. A decreasing monotone collection of subsets of U defines the retrieval restrictions. Bob is allowed to learn any permissable subset of messages from that collection, but nothing else, while Alice must remain oblivious regarding the selection that Bob made. We propose a simple and efficient GOT protocol that employs secret sharing. We compare it to another secret sharing based solution for that problem that was recently proposed in Shankar et al. (Proceeding of ICDCN08, LNCS 4904, pp 304---309, 2008). In particular, we show that the access structures that are realized by the two solutions are related through a duality-type relation that we introduce here. We show that there are examples which favor our solution over the second one, while in other examples the contrary holds. Two applications of GOT are considered--priced oblivious transfer, and oblivious evaluation of multivariate polynomials.

Evaluation algorithms for multivariate polynomials in Bernstein–Bézier form

The evaluation of multivariate polynomials of n variables in Bernstein–Bézier form is considered. A forward error analysis for the corresponding de Casteljau algorithm and the VS algorithm is performed. We also include algorithms that simultaneously evaluate the polynomial and provide “a posteriori” error bounds, without increasing significantly the computational cost. The sharpness of our running error bounds is shown in the case of trivariate polynomials.

On the Multivariate Horner Scheme

We present an efficient version of the Horner scheme for the evaluation of multivariate polynomials and study its stability properties. In particular, we show that the backward error of evaluation is bounded by a quantity that is linear in the total degree of the polynomial, which itself is usually significantly smaller than the number of operations involved in the evaluation process. Moreover, we show that the same type of estimates holds also for the forward error, provided that the polynomial basis is chosen suitably.

A unified approach to evaluation algorithms for multivariate polynomials

We present a unified framework for most of the known and a few new evaluation algorithms for multivariate polynomials expressed in a wide variety of bases including the Bernstein-Bézier, multinomial (or Taylor), Lagrange and Newton bases. This unification is achieved by considering evaluation algorithms for multivariate polynomials expressed in terms of L-bases, a class of bases that include the Bernstein-Bézier, multinomial, and a rich subclass of Lagrange and Newton bases. All of the known evaluation algorithms can be generated either by considering up recursive evaluation algorithms for L-bases or by examining change of basis algorithms for L-bases. For polynomials of degree n n in s s variables, the class of up recursive evaluation algorithms includes a parallel up recurrence algorithm with computational complexity O ( n s + 1 ) O(n^{s+1}) , a nested multiplication algorithm with computational complexity O ( n s log ⁡ n ) O(n^s \log n) and a ladder recurrence algorithm with computational complexity O ( n s ) O(n^s) . These algorithms also generate a new generalization of the Aitken-Neville algorithm for evaluation of multivariate polynomials expressed in terms of Lagrange L-bases. The second class of algorithms, based on certain change of basis algorithms between L-bases, include a nested multiplication algorithm with computational complexity O ( n s ) O(n^s) , a divided difference algorithm, a forward difference algorithm, and a Lagrange evaluation algorithm with computational complexities O ( n s ) O(n^s) , O ( n s ) O(n^s) and O ( n ) O(n) per point respectively for the evaluation of multivariate polynomials at several points.

Evaluation of multivariate polynomials and their derivatives

An extension of Horner’s algorithm to the evaluation of m-variate polynomials and their derivatives is obtained. The schemes of computation are represented by trees because this type of graph describes exactly in which order the computations must be done. Some examples of algorithms for one and two variables are given.

Evaluation of Multivariate Polynomials and Their Derivatives

An extension of Horner’s algorithm to the evaluation of m-variate polynomials and their derivatives is obtained. The schemes of computation are represented by trees because this type of graph describes exactly in which order the computations must be done. Some examples of algorithms for one and two variables are given.

On the Parallel Evaluation of Multivariate Polynomials

We prove that any multivariate polynomial P of degree d that can be computed with $C(P)$ multiplications-divisions can be computed in $O(\log d \cdot \log C(P))$ parallel steps and $O(\log d)$ parallel multiplicative steps.

On the evaluation of multivariate polynomials and their derivatives

LetH be a polynomial inn>1 variables over the fields of real or complex numbers. An algorithm is presented here for the simultaneous evaluation ofH and its first and second (F-) derivativesH′ andH″, or of any combination ofH,H′,H″. The evaluations ofH alone or ofH andH′ together are of the same order inn andd whered is the degree ofH, while the computation ofH,H′, andH″ isd times this order. The process takes account of the sparsity pattern ofH by using a tree structure induced by the nonzero coefficients. It also allows for simultaneous operation with several polynomials with the same sparsity pattern. The data structure for the method is rather simple in nature and can be adapted easily to specific types of polynomials. Several possible implementations and their complexity are discussed.