Polynomial Evaluation Research Articles

Oblivious polynomial evaluation and oblivious neural learning

We study the problem of oblivious polynomial evaluation (OPE). There are two parties, Alice who has a polynomial P, and Bob who has an input x. The goal is for Bob to compute P ( x ) in such a way that Alice learns nothing about x and Bob learns only what can be inferred from P ( x ) . Previously existing protocols were based on some newly-invented intractability assumptions that have not been well studied, so one may have doubts about the security of these protocols. In this paper, we propose OPE protocols which are only based on the standard primitive oblivious transfer, and still our protocols are more efficient in several natural cases. Our protocols can also be easily modified to handle multi-variate polynomials and polynomials over floating-point numbers. As an application, we study the problem of oblivious neural learning, where one party has a neural network and the other, with some training set, wants to train the neural network in an oblivious way. We provide a protocol for this problem, which is based on our protocol for OPE.

High-speed function approximation using a minimax quadratic interpolator

A table-based method for high-speed function approximation in single-precision floating-point format is presented in this paper. Our focus is the approximation of reciprocal, square root, square root reciprocal, exponentials, logarithms, trigonometric functions, powering (with a fixed exponent p), or special functions. The algorithm presented here combines table look-up, an enhanced minimax quadratic approximation, and an efficient evaluation of the second-degree polynomial (using a specialized squaring unit, redundant arithmetic, and multioperand addition). The execution times and area costs of an architecture implementing our method are estimated, showing the achievement of the fast execution times of linear approximation methods and the reduced area requirements of other second-degree interpolation algorithms. Moreover, the use of an enhanced minimax approximation which, through an iterative process, takes into account the effect of rounding the polynomial coefficients to a finite size allows for a further reduction in the size of the look-up tables to be used, making our method very suitable for the implementation of an elementary function generator in state-of-the-art DSPs or graphics processing units (GPUs).

Polynomial evaluation and interpolation on special sets of points

We give complexity estimates for the problems of evaluation and interpolation on various polynomial bases. We focus on the particular cases when the sample points form an arithmetic or a geometric sequence, and we discuss applications, respectively, to computations with linear differential operators and to polynomial matrix multiplication.

Establishing pairwise keys in distributed sensor networks

Pairwise key establishment is a fundamental security service in sensor networks; it enables sensor nodes to communicate securely with each other using cryptographic techniques. However, due to the resource constraints on sensor nodes, it is not feasible to use traditional key management techniques such as public key cryptography and key distribution center (KDC). A number of key predistribution techniques have been proposed for pairwise key establishment in sensor networks recently. To facilitate the study of novel pairwise key predistribution techniques, this paper develops a general framework for establishing pairwise keys between sensor nodes using bivariate polynomials. This paper then proposes two efficient instantiations of the general framework: a random subset assignment key predistribution scheme, and a hypercube-based key predistribution scheme. The analysis shows that both schemes have a number of nice properties, including high probability, or guarantee to establish pairwise keys, tolerance of node captures, and low storage, communication, and computation overhead. To further reduce the computation at sensor nodes, this paper presents an optimization technique for polynomial evaluation, which is used to compute pairwise keys. This paper also reports the implementation and the performance of the proposed schemes on MICA2 motes running TinyOS, an operating system for networked sensors. The results indicate that the proposed techniques can be applied efficiently in resource-constrained sensor networks.

Generation and evaluation of orthogonal polynomials in discrete Sobolev spaces II: numerical stability

In this paper, we concern ourselves with the determination and evaluation of polynomials that are orthogonal with respect to a general discrete Sobolev inner product, that is, an ordinary inner product on the real line plus a finite sum of atomic inner products involving a finite number of derivatives. In a previous paper we provided a complete set of formulas to compute the coefficients of this recurrence. Here, we study the numerical stability of these algorithms for the generation and evaluation of a finite series of Sobolev orthogonal polynomials. Besides, we propose several techniques for reducing and controlling the rounding errors via theoretical running error bounds and a carefully chosen recurrence.

Generation and evaluation of orthogonal polynomials in discrete Sobolev spaces I: algorithms

In this paper, we study theoretically the determination and evaluation of polynomials that are orthogonal with respect to a general discrete Sobolev inner product, that is, an ordinary inner product on the real line plus a finite sum of atomic inner products involving a finite number of derivatives. This Sobolev inner product has the property that the orthogonal polynomials with respect to it satisfy a linear recurrence relation of fixed order. We provide a complete set of formulas to compute the coefficients of this recurrence. Besides, we study the determination of the Fourier–Sobolev coefficients of a finite approximation of a function and the numerical evaluation of the resulting finite series at a general point.

Gaussian-type Quadrature Rules for Müntz Systems

A method for constructing the generalized Gaussian quadrature rules for Muntz polynomials on $(0,1)$ is given. Such quadratures possess several properties of the classical Gaussian formulae (for polynomial systems), such as positivity of the weights, rapid convergence, etc. They can be applied to the wide class of functions, including smooth functions, as well as functions with end-point singularities, such as those in boundary-contact value problems, integral equations with singular kernels, complex analysis, potential theory, etc. The constructive method is based on an application of orthogonal Muntz polynomials introduced by Badalyan [{\it Akad. Nauk Armyan. SSR. Izv. Fiz.-Mat. Estest. Tehn. Nauk}, 8 (1955), pp. 1--28; 9 (1956), pp. 3--22 (Russian; Armenian summary)] and studied intensively by Borwein, Erdelyi, and Zhang [{\it Trans. Amer. Math. Soc}., 342 (1994), pp. 523--542], as well as by Milovanovic on a numerical procedure for evaluation of such polynomials with high precision [{\it Muntz orthogonal polynomials and their numerical evaluation}, in Applications and Computation of Orthogonal Polynomials, Internat. Ser. Numer. Math. 131, W. Gautschi, G. H. Golub, and G. Opfer, eds., Birkhauser, Basel, 1999, pp. 179--194]. The method is numerically stable. Some numerical examples are included.

Effective computation of optimal stability polynomials

The construction of stable explicit multistage Runge-Kutta methods in 1950–1960 stumbled over a certain extremal problem for polynomials. The solution to this problem is known as the optimal stability polynomial and its computation is notoriously difficult. We propose a new method for the effective evaluation of optimal stability polynomials which is based on the explicit analytical representation of the solution. The main feature of the method is its independence of the computational complexity of the degree of the solution.

Distributed oblivious function evaluation and its applications

This paper is about distributed oblivious function evaluation (DOFE). In this setting one party (Alice) has a function f(x), and the other party (Bob) with an input α wants to learn f(α) in an oblivious way with the help of a set of servers. What Alice should do is to share her secret function f(x) among the servers. Bob obtains what he should get by interacting with the servers. This paper proposes the model and security requirements for DOFE and analyzes three distributed oblivious polynomial evaluation protocols presented in the paper.

On the complexities of multipoint evaluation and interpolation

We compare the complexities of multipoint polynomial evaluation and interpolation. We show that, over a field of characteristic zero, both questions have equivalent complexities, up to a constant number of polynomial multiplications.

Reconstructing noisy polynomial evaluation in residue rings

Let q > 1 be an integer and let a and b be elements of the residue ring Z q of integers modulo q. We show how, when given a polynomial f ∈ Z q [ X ] and approximations to v 0 , v 1 ∈ Z q such that v 1 ≡ f ( v 0 ) mod q one can recover v 0 and v 1 efficiently. This result has direct applications to predicting the polynomial congruential generator: a sequence ( v n ) of pseudorandom numbers defined by the relation v n + 1 ≡ f ( v n ) mod q for some polynomial f ∈ Z q [ X ] . The applications lead to analogues of results known for the linear congruential generator x n + 1 ≡ a x n + b mod q , although the results are much more restrictive due to nonlinearity of the problem.

On improving the accuracy of Horner?s and Goertzel?s algorithms

It is known that Goertzel's algorithm is much less numerically accurate than the Fast Fourier Transform (FFT)(Cf. \cite{gen:69}). In order to improve accuracy we propose modifications of both Goertzel's and Horner's algorithms based on the divide-and-conquer techniques. The proof of the numerical stability of these two modified algorithms is given. The numerical tests in Matlab demonstrate the computational advantages of the proposed modifications. The appendix contains the proof of numerical stability of Goertzel's algorithm of polynomial evaluation.

Oblivious polynomial evaluation

The problem of two-party oblivious polynomial evaluation (OPE) is studied, where one party (Alice) has a polynomial P(x) and the other party (Bob) with an input x wants to learn P(x) in such an oblivious way that Bob obtains P(x) without learning any additional information about P except what is implied by P(x) and Alice does not know Bob's input x. The former OPE protocols are based on an intractability assumption except for OT protocols. In fact, evaluating P(x) is equivalent to computing the product of the coefficient vectors (a 0 ,..., a n ) and (1,..., xn). Using this idea, an efficient scale product protocol of two vectors is proposed first and then two OPE protocols are presented which do not need any other cryptographic assumption except for OT protocol. Compared with the existing OPE protocol, another characteristic of the proposed protocols is the degree of the polynomial is private. Another OPE protocol works in case of existence of untrusted third party.

Analytic Evaluation of Factorial Polynomials for Nonautonomous Matrices

We make use of the previously derived solution for a system of two coupled finite difference equations with variable coefficients to obtain explicit expressions for factorial polynomials (Pochhammers) of nonautonomous matrices. We present combinatorial, recursive, and graphical methods for the evaluation of these expressions.

Polynomial values, the linking form and unknotting numbers

We show how the signed evaluations of link polynomials can be used to calculate unknotting numbers. We use the Jones-Rong value of the Brandt-Lickorish-Millett-Ho polynomial Q to calculate the unknotting numbers of 8_{16}, 9_{49} and 6 further new entries in Kawauchi's tables. Another method is developed by applying and extending the linking form criterion of Lickorish. This leads to several conjectured relations between the Jones value of Q and the linking form.

The Cell Probe Complexity of Succinct Data Structures

In the cell probe model with word size 1 (the bit probe model), a static data structure problem is given by a map f : {0,1}^n * {0,1}^m -> {0,1}, where {0,1}^n is a set of possible data to be stored, {0,1}^m is a set of possible queries (for natural problems, we have m << n) and f(x,y) is the answer to question y about data x.<br /> <br />A solution is given by a representation phi : {0,1}^n -> {0,1}^s and a query algorithm q so that q(phi(x), y) = f(x,y). The time t of the query algorithm is the number of bits it reads in phi(x).<br /> <br />In this paper, we consider the case of <em>succinct</em> representations where s = n + r for some <em>redundancy</em> r << n. For a boolean version of the problem of polynomial evaluation with preprocessing of coefficients, we show a lower bound on the redundancy-query time trade-off of the form <br />(r + 1) t >= Omega(n/log n).<br /> In particular, for very small redundancies r, we get an almost optimal lower bound stating that the query algorithm has to inspect almost the entire data structure (up to a logarithmic factor). We show similar lower bounds for problems satisfying a certain combinatorial property of a coding theoretic flavor. Previously, no omega(m) lower bounds were known on t in the general model for explicit functions, even for very small redundancies.<br /> <br />By restricting our attention to <em>systematic</em> or <em>index</em> structures phi satisfying phi(x) = x · phi*(x) for some map phi* (where · denotes concatenation) we show similar lower bounds on the redundancy-query time trade-off for the natural data structuring problems of Prefix Sum and Substring Search.

Fast Parallel Algorithm for Polynomial Evaluation

We present a new efficient parallel algorithm for polynomial evaluation based on a previously introduced divide-and-conquer method for solving linear recurrence systems with constant coefficients, which is formulated in terms of the level 1 BLAS (Basic Linear Algebra Subprograms) routine AXPY. We also discuss its platform-independent implementation with OpenMP and finally present the results of experiments performed on a dual processor Pentium III computer running under Linux operating system with Altas as an efficient implementation of BLAS. The sequential version of the algorithm is up to three times faster than the Horner's scheme.

Stability of parallel algorithms for polynomial evaluation

In this paper, we analyse the stability of parallel algorithms for the evaluation of polynomials written as a finite series of orthogonal polynomials. The basic part of the computation is the solution of a triangular tridiagonal linear system. This fact allows us to present a more detailed analysis. The theoretical results show that the parallel algorithms are almost as stable as their sequential counterparts for practical applications. Extensive numerical experiments confirm the theoretical conclusions.

Explicit fluxes and productions for large systems of the moment method based on extended thermodynamics

The moment method of kinetic theory solves Boltzmann’s equation approximately via an infinite hierarchy of transfer equations for the moments of the distribution function. Extended thermodynamics furnishes the moment method with a rational constitutive theory. Since more and more moment equations are needed to describe extreme non-equilibrium processes, there is need for an algorithmical derivation of large explicit moment equations. This paper presents detailed techniques and formulas which are needed to implement a numerical equation generator. This includes tensorial conversion formulas as well as the core equations of the constitutive theory. In the last part of the paper, the special case of a one dimensional process is discussed. In such a case, only one generic polynomial evaluation needs to be implemented, whereas the coefficients may be easily calculated a priori.

A unified rounding error bound for polynomial evaluation

We present a unified rounding error bound for polynomial evaluation. The bound presented here takes the same general form for the evaluation of a polynomial written in any polynomial basis when the evaluation algorithm can be expressed as a linear recurrence or a first-order linear matrix recurrence relation. Examples of these situations are: Horner's algorithm in the evaluation of power series, Clenshaw's and Forsythe's algorithms in the evaluation of orthogonal polynomial series, de-Casteljau's algorithm for Bernstein polynomial series, the modification of Clenshaw's algorithms in the evaluation of Szegő polynomial series, and so on.