DOCSLIB.ORG

Fast Fourier Transform Algorithms with Applications Todd Mateer Clemson University, [email protected]

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Fast Fourier Transform Algorithms with Applications Todd Mateer Clemson University, Tmateer@Howardcc.Edu Clemson University TigerPrints All Dissertations Dissertations 8-2008 Fast Fourier Transform Algorithms with Applications Todd Mateer Clemson University, [email protected] Follow this and additional works at: https://tigerprints.clemson.edu/all_dissertations Part of the Applied Mathematics Commons Recommended Citation Mateer, Todd, "Fast Fourier Transform Algorithms with Applications" (2008). All Dissertations. 231. https://tigerprints.clemson.edu/all_dissertations/231 This Dissertation is brought to you for free and open access by the Dissertations at TigerPrints. It has been accepted for inclusion in All Dissertations by an authorized administrator of TigerPrints. For more information, please contact [email protected]. FAST FOURIER TRANSFORM ALGORITHMS WITH APPLICATIONS A Dissertation Presented to the Graduate School of Clemson University In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Mathematical Sciences by Todd Mateer August 2008 Accepted by: Dr. Shuhong Gao, Committee Chair Dr. Joel Brawley Dr. Neil Calkin Dr. Kevin James ABSTRACT This manuscript describes a number of algorithms that can be used to quickly evaluate a polynomial over a collection of points and interpolate these evaluations back into a polynomial. Engineers define the “Fast Fourier Transform” as a method of solving the interpolation problem where the coefficient ring used to construct the polynomials has a special multiplicative structure. Mathematicians define the “Fast Fourier Transform” as a method of solving the multipoint evaluation problem. One purpose of the document is to provide a mathematical treatment of the topic of the “Fast Fourier Transform” that can also be understood by someone who has an understanding of the topic from the engineering perspective. The manuscript will also introduce several new algorithms that efficiently solve the multipoint evaluation problem over certain finite fields and require fewer finite field operations than existing techniques. The document will also demonstrate that these new algorithms can be used to multiply polynomials with finite field coefficients with fewer operations than Sch¨onhage’s algorithm in most circumstances. A third objective of this document is to provide a mathematical perspective of several algorithms which can be used to multiply polynomials whose size is not a power of two. Several improvements to these algorithms will also be discussed. Finally, the document will describe several applications of the “Fast Fourier Transform” algorithms presented and will introduce improvements in several of these applications. In addition to polynomial multiplication, the applications of polynomial division with remainder, the greatest common divisor, decoding of Reed-Solomon error-correcting codes, and the computation of the coefficients of a discrete Fourier series will be addressed. ii DEDICATION I dedicate this work to my wife Jennifer and my children Nathan, Laura, Jonathan, and Daniel. In terms of our family, the progress of my graduate research program has been measured through a collection of fifty quarters produced by the United States mint over roughly the same ten year period while I completed my graduate studies. I look forward to placing the final quarter on our “doctor school” map at the end of this year (2008) when I anticipate being finished with several publications related to the research presented in this manuscript. I have really treasured this time to be at home with my family while my children were young and for their “company” and “support” while completing this project. Much of this dissertation was written with little children in my lap or by my side as my wife and I worked together to get through these first few years of parenthood. I consider this time to be more valuable than the degree for which this dissertation was written. iii ACKNOWLEDGMENTS There are many people who deserve recognition for their role in my education which has led me to this point in my academic career. I will attempt to be as complete as possible here, knowing that are likely several people that I have left out. Obviously, my advisor Shuhong Gao and committee members deserve mention for being willing to mentor me through this process. This was an especially more challenging task given the fact that we were separated geographically during the entire writing of this dissertation. In fact, my advisor and I only saw each other two times in the three years it took to complete the research program. I would also like to thank Shuhong Gao for teaching a computer algebra class in 2001 which got me interested in this topic for my doctoral studies. I would also like to thank several anonymous reviewers who read over this entire manuscript several times. There are likely more places where the document can be improved and if there is an interest, I will make revisions to the manuscript as these places are pointed out to me. I would also like this opportunity to thank the teachers which taught me the subjects related to this research program. In particular, I would like to thank Rebecca Nowakowski, James Payne, Dale McIntyre, Yao-Huan Xu, Jenny Key, and Joel Brawley for equipping me with the algebra and complex variables background to take on this assignment. Also, Timothy Mohr, Frank Duda and Robert Mueller taught me the signals analysis which introduced me to the “engineering perspective” of the FFT. In classes taught by Joe Churm and Robert Jamison, I learned how the material of this dissertation is closely related to basic music theory. Through the instruction of Curt Frank, Jim Kendall, Fred Jenny, Michelle Claus, and James Peterson I learned iv the computer science and programming skills needed for this research project. Finally, I would like to thank my father Robert Mateer who taught me the trigonometry that is so fundamental to the topic of this dissertation. While I turned out to be an algebraist instead of following in his footsteps and making analysis (i.e. Calculus) my specialty, I still hope to become as good of a teacher as my father someday. ...but there is a God in heaven who reveals mysteries. Daniel 2:28a v TABLE OF CONTENTS Page TITLEPAGE.................................... i ABSTRACT .................................... ii DEDICATION ................................... iii ACKNOWLEDGMENTS ............................. iv LISTOFTABLES................................. x LISTOFFIGURES ................................ xi CHAPTER 1. INTRODUCTION............................. 1 1.1 The mathematician’s perspective . 3 1.2 Prerequisitemathematics . 4 1.3 Operationcountsofthealgorithms. 5 1.4 Multipoint polynomial evaluation . 7 1.5 Fast multipoint evaluation . 9 1.6 Lagrangian interpolation . 15 1.7 Fastinterpolation .......................... 17 1.8 Concludingremarks ......................... 23 2. MULTIPLICATIVE FAST FOURIER TRANSFORM ALGORITHMS 24 2.1 Thebitreversalfunction . 25 2.2 Classicalradix-2FFT ........................ 26 2.3 Twistedradix-2FFT ........................ 30 2.4 Hybridradix-2FFTs ........................ 35 2.5 Classicalradix-4FFT ........................ 36 2.6 Twistedradix-4FFT ........................ 41 2.7 Radix-8FFT............................. 43 2.8 Split-radixFFT ........................... 47 2.9 Modifiedsplit-radixFFT . 54 2.10 Theternaryreversalfunction . 59 2.11 Classicalradix-3FFT . 60 2.12 Twistedradix-3FFT ........................ 66 2.13 Hybridradix-3FFTs ........................ 73 vi Table of Contents (Continued) Page 2.14 Radix-3 FFT for symbolic roots of unity . 75 2.15 Concludingremarks . 77 3. ADDITIVE FAST FOURIER TRANSFORM ALGORITHMS . 78 3.1 VonzurGathen-GerhardadditiveFFT . 78 3.2 Wang-Zhu-CantoradditiveFFT . 87 3.3 ShiftedadditiveFFT ........................ 94 3.4 Gao’sadditiveFFT ......................... 99 3.5 AnewadditiveFFT......................... 108 3.6 Concludingremarks . 115 4. INVERSE FAST FOURIER TRANSFORM ALGORITHMS . 116 4.1 Classicalradix-2IFFT. 116 4.2 Twistedradix-2IFFT . 120 4.3 OthermultiplicativeIFFTs . 122 4.4 Wang-Zhu-Cantor additive IFFT . 129 4.5 AnewadditiveIFFT ........................ 132 4.6 Concludingremarks . 139 5. POLYNOMIAL MULTIPLICATION ALGORITHMS . 141 5.1 Karatsuba multiplication . 141 5.2 Karatsuba’s algorithm for other sizes . 145 5.3 FFT-based multiplication . 146 5.4 Sch¨onhage’salgorithm. 148 5.5 FFT-based multiplication using the new additive FFT algorithm 156 5.6 Comparison of the multiplication algorithms . 158 5.7 Concludingremarks . 160 6. TRUNCATED FAST FOURIER TRANSFORM ALGORITHMS . 162 6.1 AtruncatedFFTalgorithm. 164 6.2 An inverse truncated FFT algorithm . 167 6.3 Illustration of truncated FFT algorithms . 175 6.4 Truncated algorithms based on roots of xN 1 .......... 177 6.5 Truncated algorithms based on roots of xN − x .......... 179 6.6 Concludingremarks . .− . 180 7. POLYNOMIAL DIVISION WITH REMAINDER . 181 7.1 Classical division . 182 vii Table of Contents (Continued) Page 7.2 Newtondivision ........................... 183 7.3 Newton divison using the multiplicative FFT . 188 7.4 Newton divison for finite fields of characteristic 2 . 192 7.5 Concludingremarks . 194 8. THE EUCLIDEAN ALGORITHM . 195 8.1 TheEuclideanAlgorithm . 195 8.2 The Extended Euclidean Algorithm . 199 8.3 Normalized Extended Euclidean Algorithm . 209 8.4 The Fast Euclidean Algorithm . 210 8.5 Algorithm improvements due to the Fast Fourier Transform . 226 8.6 Concludingremarks . 231 9. REED-SOLOMON ERROR-CORRECTING CODES . 234 9.1 Systematic encoding of Reed-Solomon codewords . 235 9.2 A transform of the Reed-Solomon
Load more

Recommended publications
  • Polynomial Evaluation Engl2
    Alexander Rostovtsev, Alexey Bogdanov and Mikhail Mikhaylov St. Petersburg state Polytechnic University [email protected] Secure evaluation of polynomial using privacy ring homomorphisms Method of secure evaluation of polynomial y = F(x1, …, xk) over some rings on untrusted computer is proposed. Two models of untrusted computer are considered: passive and active. In passive model untrusted computer correctly computes polyno- mial F and tries to know secret input (x1, …, xk) and output y. In active model un- trusted computer tries to know input and output and tries to change correct output y so that this change cannot be determined. Secure computation is proposed by using one-time privacy ring homomorphism /n ® /n[z]/(f(z)), n = pq, generated by trusted computer. In the case of active model secret check point v = F(u1, …, uk) is used. Trusted computer generates poly- nomial f(z) = (z - t)(z + t), t Î /n and input Xi(z) Î /n[z]/(f(z)) such that Xi(t) º xi (mod n) for passive model, and f(z) = (z - t1)(z - t2)(z - t3), ti Î /n and input Xi(z) Î /n[z]/(f(z)) such that Xi(t1) º xi (mod n), Xi(t2) º ui (mod n) for active model. Untrusted computer computes function Y(z) = F(X1(z), …, Xk(z)) in the ring /n[z]/(f(z)). For passive model trusted computer determines secret output y º Y(t) (mod n). For active model trusted computer checks that Y(t2) º v (mod n), then deter- mines correct output y º Y(t1) (mod n).
    [Show full text]
  • Generalized Kakeya Sets for Polynomial Evaluation and Faster Computation of Fermionants
    Generalized Kakeya sets for polynomial evaluation and faster computation of fermionants Andreas Björklund1, Petteri Kaski2, and Ryan Williams3 1 Department of Computer Science, Lund University, Lund, Sweden 2 Department of Computer Science, Aalto University, Helsinki, Finland 3 Department of Electrical Engineering and Computer Science & CSAIL, MIT, Cambridge, MA, USA Abstract We present two new data structures for computing values of an n-variate polynomial P of degree at most d over a finite field of q elements. Assuming that d divides q − 1, our first data structure relies on (d + 1)n+2 tabulated values of P to produce the value of P at any of the qn points using O(nqd2) arithmetic operations in the finite field. Assuming that s divides d and d/s divides q − 1, our second data structure assumes that P satisfies a degree-separability condition and relies on (d/s+1)n+s tabulated values to produce the value of P at any point using O(nqssq) arithmetic operations. Our data structures are based on generalizing upper-bound constructions due to Mockenhaupt and Tao (2004), Saraf and Sudan (2008), and Dvir (2009) for Kakeya sets in finite vector spaces from linear to higher-degree polynomial curves. As an application we show that the new data structures enable a faster algorithm for com- puting integer-valued fermionants, a family of self-reducible polynomial functions introduced by Chandrasekharan and Wiese (2011) that captures numerous fundamental algebraic and combin- atorial invariants such as the determinant, the permanent, the number of Hamiltonian cycles in a directed multigraph, as well as certain partition functions of strongly correlated electron systems in statistical physics.
    [Show full text]
  • Factoring Polynomials Over Finite Fields
    Factoring Polynomials over Finite Fields More precisely: Factoring and testing irreduciblity of sparse polynomials over small finite fields Richard P. Brent MSI, ANU joint work with Paul Zimmermann INRIA, Nancy 27 August 2009 Richard Brent (ANU) Factoring Polynomials over Finite Fields 27 August 2009 1 / 64 Outline Introduction I Polynomials over finite fields I Irreducible and primitive polynomials I Mersenne primes Part 1: Testing irreducibility I Irreducibility criteria I Modular composition I Three algorithms I Comparison of the algorithms I The “best” algorithm I Some computational results Part 2: Factoring polynomials I Distinct degree factorization I Avoiding GCDs, blocking I Another level of blocking I Average-case complexity I New primitive trinomials Richard Brent (ANU) Factoring Polynomials over Finite Fields 27 August 2009 2 / 64 Polynomials over finite fields We consider univariate polynomials P(x) over a finite field F. The algorithms apply, with minor changes, for any small positive characteristic, but since time is limited we assume that the characteristic is two, and F = Z=2Z = GF(2). P(x) is irreducible if it has no nontrivial factors. If P(x) is irreducible of degree r, then [Gauss] r x2 = x mod P(x): 2r Thus P(x) divides the polynomial Pr (x) = x − x. In fact, Pr (x) is the product of all irreducible polynomials of degree d, where d runs over the divisors of r. Richard Brent (ANU) Factoring Polynomials over Finite Fields 27 August 2009 3 / 64 Counting irreducible polynomials Let N(d) be the number of irreducible polynomials of degree d. Thus X r dN(d) = deg(Pr ) = 2 : djr By Möbius inversion we see that X rN(r) = µ(d)2r=d : djr Thus, the number of irreducible polynomials of degree r is ! 2r 2r=2 N(r) = + O : r r Since there are 2r polynomials of degree r, the probability that a randomly selected polynomial is irreducible is ∼ 1=r ! 0 as r ! +1.
    [Show full text]
  • The Fast Fourier Transform and Applications to Multiplication
    The Fast Fourier Transform and Applications to Multiplication Analysis of Algorithms Prepared by John Reif, Ph.D. Topics and Readings: - The Fast Fourier Transform • Reading Selection: • CLR, Chapter 30 Advanced Material : - Using FFT to solve other Multipoint Evaluation Problems - Applications to Multiplication Nth Roots of Unity • Assume Commutative Ring (R,+,·, 0,1) • ω is principal nth root of unity if – ωk ≠ 1 for k = 1, …, n-1 – ωn = 1, and n-1 ∑ω jp = 0 for 1≤ p ≤ n j=0 2πi/n • Example: ω = e for complex numbers Example of nth Root of Unity for Complex Numbers ω = e 2 π i / 8 is the 8th root of unity Fourier Matrix #1 1 … 1 $ % n−1 & %1 ω … ω & M (ω) = %1 ω 2 … ω 2(n−1) & n % & % & % n−1 (n−1)(n−1) & '1 ω … ω ( ij so M(ω)ij = ω for 0 ≤ i, j<n #a0 $ % & given a =% & % & 'a n-1 ( Discrete Fourier Transform T Input a column n-vector a = (a0, …, an-1) Output an n-vector which is the product of the Fourier matrix times the input vector DFTn (a) = M(ω) x a "f0 # $ % = $ % where $ % &fn-1 ' n-1 f = a ik i ∑ kω k=0 Inverse Fourier Transform -1 -1 DFTn (a) = M(ω) x a 1 Theorem M(ω)-1 = ω -ij ij n proof We must show M(ω)⋅M(ω)-1 = I 1 n-1 1 n-1 ∑ωikω -kj = ∑ω k(i-j) n k=0 n k=0 $0 if i-j ≠ 0 = % &1 if i-j = 0 n-1 using identity ∑ω kp = 0, for 1≤ p < n k=0 Fourier Transform is Polynomial Evaluation at the Roots of Unity T Input a column n-vector a = (a0, …, an-1) T Output an n-vector (f0, …, fn-1) which are the values polynomial f(x)at the n roots of unity "f0 # DFT (a) = $ % where n $ % $ % &fn-1 ' i fi = f(ω ) and n-1 f(x) = a x j ∑ j
    [Show full text]
  • Arxiv:0911.1393V5 [Cs.CC] 1 Jul 2013 Author’S Addresses: C
    0 Most Tensor Problems are NP-Hard CHRISTOPHER J. HILLAR, Mathematical Sciences Research Institute LEK-HENG LIM, University of Chicago We prove that multilinear (tensor) analogues of many efficiently computable problems in numerical linear algebra are NP-hard. Our list here includes: determining the feasibility of a system of bilinear equations, deciding whether a 3-tensor possesses a given eigenvalue, singular value, or spectral norm; approximating an eigenvalue, eigenvector, singular vector, or the spectral norm; and determining the rank or best rank-1 approximation of a 3-tensor. Furthermore, we show that restricting these problems to symmetric tensors does not alleviate their NP-hardness. We also explain how deciding nonnegative definiteness of a symmetric 4-tensor is NP-hard and how computing the combinatorial hyperdeterminant of a 4-tensor is NP-, #P-, and VNP-hard. We shall argue that our results provide another view of the boundary separating the computa- tional tractability of linear/convex problems from the intractability of nonlinear/nonconvex ones. Categories and Subject Descriptors: G.1.3 [Numerical Analysis]: Numerical Linear Algebra General Terms: Tensors, Decidability, Complexity, Approximability Additional Key Words and Phrases: Numerical multilinear algebra, tensor rank, tensor eigenvalue, tensor singular value, tensor spectral norm, system of multilinear equations, hyperdeterminants, symmetric ten- sors, nonnegative definite tensors, bivariate matrix polynomials, NP-hardness, #P-hardness, VNP-hardness, undecidability, polynomial time approximation schemes ACM Reference Format: Hillar, C. J., and Lim, L.-H. 2012. Most tensor problems are NP-hard J. ACM 0, 0, Article 0 (June 2013), 38 pages. DOI = 10.1145/0000000.0000000 http://doi.acm.org/10.1145/0000000.0000000 1.
    [Show full text]
  • Polynomial Evaluation and Interpolation on Special Sets of Points
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Complexity 21 (2005) 420–446 www.elsevier.com/locate/jco Polynomial evaluation and interpolation on special sets of points Alin Bostan∗, Éric Schost Laboratoire STIX, École polytechnique, 91128 Palaiseau, France Received 31 January 2004; accepted 6 September 2004 Available online 10 February 2005 Abstract We give complexity estimates for the problems of evaluation and interpolation on various polyno- mial 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. © 2005 Elsevier Inc. All rights reserved. Keywords: Polynomial evaluation and interpolation; Transposition principle; Polynomial matrix multiplication; Complexity 1. Introduction Let k be a field and let x = x0,...,xn−1 be n pairwise distinct points in k. Given arbitrary values v = v0,...,vn−1 in k, there exists a unique polynomial F in k[x] of degree less than n such that F(xi) = vi, for i = 0,...,n− 1. Having fixed a basis of the vector space of polynomials of degree at most n − 1, interpolation and evaluation questions consist in computing the coefficients of F on this basis from the values v, and conversely. ∗ Corresponding author. E-mail addresses: [email protected] (A. Bostan), [email protected] (É. Schost). 0885-064X/$ - see front matter © 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jco.2004.09.009 A.
    [Show full text]
  • Some Parallel Algorithms for Integer Factorisation
    Some Parallel Algorithms for Integer Factorisation Richard P. Brent Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1 3QD, UK [email protected] http://www.comlab.ox.ac.uk/ oucl/people/richard.brent.html Abstract. Algorithms for finding the prime factors of large composite numbers are of practical importance because of the widespread use of public key cryptosystems whose security depends on the presumed diffi- culty of the factorisation problem. In recent years the limits of the best integer factorisation algorithms have been extended greatly, due in part to Moore’s law and in part to algorithmic improvements. It is now rou- tine to factor 100-decimal digit numbers, and feasible to factor numbers of 155 decimal digits (512 bits). We describe several integer factorisa- tion algorithms, consider their suitability for implementation on parallel machines, and give examples of their current capabilities. 1 Introduction Any positive integer N has a unique prime power decomposition α α α 1 2 ··· k N = p1 p2 pk (p1 <p2 < ··· <pk primes, αj > 0). This result is easy to prove, but the standard proof gives no hint of an efficient algorithm for computing the prime power decomposition. In order to compute it, we need – 1. An algorithm to test if an integer N is prime. 2. An algorithm to find a nontrivial factor f of a composite integer N. Given these components there is a simple recursive algorithm to compute the prime power decomposition. Fortunately or unfortunately, depending on one’s point of view, problem 2 is generally believed to be hard.
    [Show full text]
  • Some Integer Factorization Algorithms Using Elliptic Curves
    Some Integer Factorization Algorithms using Elliptic Curves Richard P. Brent Computer Sciences Laboratory Australian National University 24 September 1985 Revised 10 Dec. 1985 Republished 7 Nov. 1998 Abstract Lenstra’s integer factorization algorithm is asymptotically one of the fastest known al- gorithms, and is also ideally suited for parallel computation. We suggest a way in which the algorithm can be speeded up by the addition of a second phase. Under some plausible assumptions, the speedup is of order log(p), where p is the factor which is found. In practice the speedup is significant. We mention some refinements which give greater speedup, an alternative way of implementing a second phase, and the connection with Pollard’s “p − 1” factorization algorithm. 1 Introduction Recently H.W. Lenstra Jr. proposed a new integer factorization algorithm, which we shall call “Lenstra’s algorithm” or the “one-phase elliptic curve algorithm” [17]. Under some plausible assumptions Lenstra’s algorithm finds a prime factor p of a large composite integer N in expected time q T1(p) = exp (2 + o(1)) ln p ln ln p , (1.1) where “o(1)” means a term which tends to zero as p → ∞. Previously algorithms with run- ning time exp p(1 + o(1)) ln N ln ln N were known [27]. However, since p2 ≤ N, Lenstra’s algorithm is comparable in the worst case and often much better, since it often happens that 2 ln p ln N. The Brent-Pollard “rho” algorithm [5] is similar to Lenstra’s algorithm in that its expected 1/2 1/2 running time depends on p, in fact it is of order p .
    [Show full text]
  • Polynomial Evaluation and Side Channel Analysis Claude Carlet, Emmanuel Prouff
    Polynomial Evaluation and Side Channel Analysis Claude Carlet, Emmanuel Prouff To cite this version: Claude Carlet, Emmanuel Prouff. Polynomial Evaluation and Side Channel Analysis. TheNew Codebreakers, 9100, Springer, pp.315 - 341, 2016, Lecture Notes in Computer Science, 978-3-662- 49300-7. 10.1007/978-3-662-49301-4_20. hal-01399573 HAL Id: hal-01399573 https://hal.archives-ouvertes.fr/hal-01399573 Submitted on 19 Nov 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Polynomial Evaluation and Side Channel Analysis Claude Carlet1 and Emmanuel Prouff2;3 1 LAGA, UMR 7539, CNRS, Department of Mathematics, University of Paris XIII and University of Paris VIII [email protected] 2 ANSSI, FRANCE [email protected] 3 POLSYS, UMR 7606, LIP6, Sorbonne Universities, UPMC University Paris VI Abstract. Side Channel Analysis (SCA) is a class of attacks that ex- ploits leakage of information from a cryptographic implementation dur- ing execution. To thwart it, masking is a common countermeasure. The principle is to randomly split every sensitive intermediate variable oc- curring in the computation into several shares and the number of shares, called the masking order, plays the role of a security parameter.
    [Show full text]
  • Polynomial Evaluation Schemes
    MATHEMATICS OF COMPUTATION, VOLUME 29, NUMBER 132 OCTOBER 1975, PAGES 1046-1050 Polynomial Evaluation Schemes By A. C. R. Newbery* Abstract. An attempt is made to define a polynomial evaluation algorithm that is more resistant to accumulated round-off error than the schemes of Horner and Clenshaw under conditions of floating-point arithmetic. An algorithm is presented which generally compares favorably with both. Some suggestions are made, which could plausibly lead to substantial further improvements. In [1] it was established that the total effect of rounding error for Horner's (nested multiplication) evaluation scheme is bounded by (e + no) P(\a\)/(1 - no), where the arithmetic is performed in floating-point, e is the smallest number which significantly adds to 1, a = e(2 + e), and P(x) = 'Ln)\pr\xr. It was noted that both the theoretical and the empirically observed errors vary steeply with the magnitude of the argument. In view of this, it seems appropriate to look into the possibility of replacing a given polynomial evaluation problem by an equivalent one for which the argument has a smaller magnitude. We are presupposing a situation in which the over- head costs (for transforming one problem into another) may be neglected, e.g. where the polynomial approximates an elementary function on a library tape. Let the given problem be to evaluate P(x) = E^ p,xr at any argument in [-1,1]. It was observed in [1] that any finite-domain problem can be scaled into this form. For convenience we assume n = 2n '. (The analysis for odd-degree polynomials is only trivially different.) First, we split P into its even and odd components P(x) = E(x2) + x4>(x2), where E and 4> are polynomials of degree n' ,n' — 1 in the argument x2\ secondly, by writing x2 = xk + t with \t\ < J4, we can rewrite E(x2) and <ï>(x2)as Ê(t) and q>(r) with |r| < lk.
    [Show full text]
  • Fast Polynomial Factorization and Modular Composition∗
    Fast polynomial factorization and modular composition∗ Kiran S. Kedlayay Christopher Umansz MIT Caltech June 13, 2011 Abstract We obtain randomized algorithms for factoring degree n univariate polynomials over Fq requiring O(n1:5+o(1) log1+o(1) q + n1+o(1) log2+o(1) q) bit operations. When log q < n, this is asymptotically faster than the best previous algorithms (von zur Gathen & Shoup (1992) and Kaltofen & Shoup (1998)); for log q ≥ 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 poly- nomials, which is the asymptotic bottleneck in fast algorithms for factoring polynomials over finite fields. The best previous algorithms for modular composition use O(n(!+1)=2) field operations, where ! is the exponent of matrix multiplication (Brent & Kung (1978)), with a slight improvement in the exponent achieved by employing fast rectangular matrix multiplication (Huang & Pan (1997)). We show that modular composition and multipoint evaluation of multivariate polynomials are essen- tially equivalent, in the sense that an algorithm for one achieving exponent α implies an algorithm for the other with exponent α + o(1), and vice versa. We then give two new algorithms that solve the prob- lem near-optimally: an algebraic algorithm for fields of characteristic at most no(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.
    [Show full text]
  • Oblivious Polynomial Evaluation and Secure Set-Intersection from Algebraic Prfs∗
    Oblivious Polynomial Evaluation and Secure Set-Intersection from Algebraic PRFs∗ Carmit Hazayy Abstract In this paper we study the two fundamental functionalities oblivious polynomial evaluation in the exponent and set-intersection, and introduce a new technique for designing efficient secure protocols for these problems (and others). Our starting point is the [BGV11] technique (CRYPTO 2011) for verifiable delegation of polynomial evaluations, using algebraic PRFs. We use this tool, that is useful to achieve verifiability in the outsourced setting, in order to achieve privacy in the standard two-party setting. Our results imply new simple and efficient oblivious polynomial evaluation (OPE) protocols. We further show that our OPE protocols are readily used for secure set-intersection, implying much simpler protocols in the plain model. As a side result, we demonstrate the usefulness of algebraic PRFs for various search functionalities, such as keyword search and oblivious transfer with adaptive queries. Our protocols are secure under full simulation-based definitions in the presence of malicious adversaries. Keywords: Efficient Secure Computation, Oblivious Polynomial Evaluation, Secure Set-Intersection, Committed Oblivious PRF ∗A preliminary version of this paper appeared in the proceedings of TCC 2015 [Haz15]. yFaculty of Engineering, Bar-Ilan University, Israel. Email: [email protected]. Research partially supported by a grant from the Israel Ministry of Science and Technology (grant No. 3-10883). 1 Introduction Efficient secure two-party computation. Secure two-party computation enables two parties to mutually run a protocol that computes some function f on their private inputs, while preserving a number of security properties. Two of the most important properties are privacy and correctness.
    [Show full text]