Showing papers in "Mathematics of Computation in 1987"

Elliptic curve cryptosystems

Abstract: We discuss analogs based on elliptic curves over finite fields of public key cryptosystems which use the multiplicative group of a finite field. These elliptic curve cryptosystems may be more secure, because the analog of the discrete logarithm problem on elliptic curves is likely to be harder than the classical discrete logarithm problem, especially over GF(2'). We discuss the question of primitive points on an elliptic curve modulo p, and give a theorem on nonsmoothness of the order of the cyclic subgroup generated by a global point.

The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods

Abstract: Mathematical and Computational Introduction The Euler Method and its Generalizations Analysis of Runge-Kutta Methods General Linear Methods Bibliography.

Speeding the Pollard and elliptic curve methods of factorization

Abstract: Since 1974, several algorithms have been developed that attempt to factor a large number N by doing extensive computations module N and occasionally taking GCDs with N. These began with Pollard's p 1 and Monte Carlo methods. More recently, Williams published a p + 1 method, and Lenstra discovered an elliptic curve method (ECM). We present ways to speed all of these. One improvement uses two tables during the second phases of p ? 1 and ECM, looking for a match. Polynomial preconditioning lets us search a fixed table of size n with n/2 + o(n) multiplications. A parametrization of elliptic curves lets Step 1 of ECM compute the x-coordinate of nP from that of P in about 9.3 1og2 n multiplications for arbitrary P.

Atlas of finite groups : maximal subgroups and ordinary characters for simple groups

Abstract: This atlas covers groups from the families of the classification of finite simple groups. Recently updated incorporating corrections

Numerical absorbing boundary conditions for the wave equation

Abstract: On developpe une theorie des approximations par differences des conditions aux limites absorbantes pour l'equation d'ondes scalaire a plusieurs dimensions d'espace

Computing in the Jacobian of a hyperelliptic curve

Abstract: In this paper we present algorithms, suitable for computer use, for computation in the Jacobian of a hyperelliptic curve. We present a reduction algorithm which is asymptotically faster than that of Gauss when the genus g is very large.

On the distribution of spacings between zeros of the zeta function

Abstract: Etude numerique de la distribution des espacements des zeros de la fonction zeta de Riemann. Observation de certains phenomenes inatendus

The numerical viscosity of entropy stable schemes for systems of conservation laws. I

Abstract: Discrete approximations to hyperbolic systems of conservation laws are studied. The amount of numerical viscosity present in such schemes, is quantified and related to their entropy stability by means of comparison. To this end, conservative schemes which are also entropy conservative are constructed. These entropy conservative schemes enjoy second-order accuracy; moreover, they admit a particular interpretation within the finite-element frameworks, and hence can be formulated on various mesh configurations. It is then shown that conservative schemes are entropy stable if and only if they contain more viscosity than the mentioned above entropy conservative ones.

TVB uniformly high-order schemes for conservation laws

Abstract: On decrit une procedure permettant d'obtenir des schemas reduisant la variation totale pour l'approximation des lois de conservation avec une grande precision meme aux points critiques. On traite des exemples numeriques

Multigrid solution of monotone second-order discretizations of hyperbolic conservation laws

Abstract: Construction des schemas aux differences monotones precis du second ordre pour les lois de conservation hyperboliques et solution multigrille des equations discretes a etat stationnaire correspondantes

A new collocation-type method for Hammerstein integral equations

Abstract: On presente une methode du type de collocation pour la resolution numerique des equations integrales d'Hammerstein. On etablit la convergence de l'approximation desiree vers la solution exacte

An a posteriori parameter choice for ordinary and iterated Tikhonov regularization of ill-posed problems leading to optimal convergence rates

Abstract: On propose un choix a posteriori du parametre pour la regularisation de Tikhonov ordinaire et iteree des problemes mal poses conduisant des vitesses de convergence optimales

Crosswind Smear and Pointwise Errors in Streamline Diffusion Finite Element Methods

Abstract: For a model convection-dominated singularly perturbed convection-diffusion prob- lem, it is shown that crosswind smear in the numerical streamline diffusion finite element method is minimized by introducing a judicious amount of artificial crosswind diffusion. The ensuing method with piecewise linear elements converges with a pointwise accuracy of almost h 5/4 under local smoothness assumptions. 1. Introduction. The streamline diffusion method is a finite element method for convection-dominated convection-diffusion problems which combines formal high accuracy with decent stability properties. The method was introduced in the case of stationary problems by Hughes and Brooks (7), cf. Raithby and Torrance (14) and Wahlbin (15) for earlier thoughts in this direction. The mathematical analysis of the method was started in Johnson and Navert (8) and continued with extensions to, e.g., time-dependent problems in Navert (12), Johnson, Navert and Pitkaranta (9) and Johnson and Saranen (10). In these papers local error estimates in L2 of order O(h k 1/2), in regions of smoothness, with piecewise polynomial finite elements of degree k, were derived, together with estimates stating, as a typical example, that in the zero diffusion limit a sharp discontinuity in the exact solution across a streamline will be captured in a numerical crosswind layer of width 0(h1/2), essentially. The purpose of the present paper is first to improve the result just mentioned on numerical crosswind smear to 0(h3/4). The improvement from 0(h1/2) to 0(h3/4) is obtained by adding a small amount, 0(h3/2), of artificial crosswind diffusion to the method. In the piecewise linear case (k = 1) this does not destroy the known O( h3/2) accuracy in L2 in smooth regions. Using our first result, we then obtain our second main result, localized pointwise error estimates of order 0(h5/4) in regions of smoothness. (The previously known best pointwise error estimate in the piecewise linear situation is 0(h1/2).) Another consequence is a global L,-estimate of order 0(h'/2) in the presence of typical crosswind and downwind singularities. We shall consider the model problem of finding u = u(x, y) such that

The multiple polynomial quadratic sieve

Abstract: A modification, due to Peter Montgomery, of Pomerance's Quadratic Sieve for factoring large integers is discussed along with its implementation. Using it, allows factorization with over an order of magnitude less sieving than the basic algorithm. It enables one to factor numbers in the 60-digit range in about a day, using a large minicomputer. The algorithm has features which make it well adapted to parallel implementation.

New convergence estimates for multigrid algorithms

Abstract: In this paper, new convergence estimates are proved for both symmetric and nonsymmetric multigrid algorithms applied to symmetric positive definite problems. Our theory relates the convergence of multigrid algorithms to a ''regularity and approximation'' parameter ..cap alpha.. epsilon (0, 1) and the number of relaxations m. We show that for the symmetric and nonsymmetric ..nu.. cycles, the multigrid iteration converges for any positive m at a rate which deteriorates no worse than 1-cj/sup -(1-//sup ..cap alpha..//sup )///sup ..cap alpha../, where j is the number of grid levels. We then define a generalized ..nu.. cycle algorithm which involves exponentially increasing (for example, doubling) the number of smoothings on successively coarser grids. We show that the resulting symmetric and nonsymmetric multigrid iterations converge for any ..cap alpha.. with rates that are independent of the mesh size. The theory is presented in an abstract setting which can be applied to finite element multigrid and finite difference multigrid methods.

The stability in _{} and ¹_{} of the ₂-projection onto finite element function spaces

Abstract: On etablit la stabilite de la L 2 -projection sur quelques espaces a elements finis V h , consideree comme une application dans L p et W p 1 , 1≤p≤∞, sous des hypotheses de regularite faibles

Primitive normal bases for finite fields

Abstract: It is proved that any finite extension of a finite field has a normal basis consisting of primitive roots. Introduction. Let q be a prime power, q > 1. We denote by F9 a finite field of q elements. It is well known that for every positive integer m there exists a normal basis of F m over F9, i.e., a basis of the form q~~~~ q 2 qP (aaq aq2 a9"1)with a E F m. It is also well known that the multiplicative group F? of Fqm is cyclic, i.e., that for some a E PQ we have F? = (an: n E Z}. Such an element a is called a primitive root of Fqm. Following Davenport [4] we call qq2 rn-i) a normal basis (a, a', a9 , .a a, a9 ) of Fq. over Fq a primitive normal basis if a is a primitive root of Fq. Carlitz [2], [3] proved in 1952 that for all sufficiently large qm there exists a primitive normal basis of F m over Fq. Davenport [4] proved in 1968 that a primitive normal basis exists for all m if q is prime. In the present paper this result is extended to the general case. THEOREM. For every prime power q > 1 and every positive integer m there exists a primitive normal basis of Fqm over Fq. Section 1 contains an exposition of certain results due to Ore [7] concerning the Galois module structure of finite fields. These lead to an alternative formulation of the theorem. In Section 2 we describe an improved version of the method of Carlitz and Davenport, which handles all but finitely many pairs (q, m). In Section 3 we determine which are the remaining pairs, and they are dealt with in Section 4. We denote the cardinality of a set S by #S, and the group of units of a ring R with 1 by R*. If f, g are polynomials in one variable, we mean by g I f that g divides f and is monic, i.e., has leading coefficient one. The same notation for divisibility is used for positive integers. Received March 5, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 11T30, 12E20.

On the Lánczos method for solving symmetric linear systems with several right-hand sides

Abstract: This paper analyzes a few methods based on the Lanczos algorithm for solving large sparse symmetric linear systems with several right-hand sides. The methods examined are suitable for the situation when the right sides are not too different from one another, as is often the case in time-dependent or parameter-dependent problems. We propose a theoretical error bound for the approximation obtained from a projection process onto a Krylov subspace generated from processing a previous right-hand side.

On the convergence of a finite element method for a nonlinear hyperbolic conservation law

Abstract: We consider a space-time finite element discretization of a time-dependent nonlinear hyperbolic conservation law in one space dimension (Burgers' equation). The finite element method is higher-order accurate and is a Petrov-Galerkin method based on the so-called streamline diffusion modification of the test functions giving added stability. We first prove that if a sequence of finite element solutions converges boundedly almost everywhere (as the mesh size tends to zero) to a function u, then u is an entropy solution of the conservation law. This result may be extended to systems of conservation laws with convex entropy in several dimensions. We then prove, using a compensated compactness result of Murat-Tartar, that if the finite element solutions are uniformly bounded then a subsequence will converge to an entropy solution of Burgers' equation. We also consider a further modification of the test functions giving a method with improved shock capturing. Finally, we present the results of some numerical experiments.

A note on elliptic curves over finite fields

Abstract: Let E be an elliptic curve over a finite field k and let E(k) be the group of k-rational points on E. We evaluate all the possible groups E(k) where E runs through all the elliptic curves over a given fixed finite field k.

Class numbers of the simplest cubic fields

Abstract: Using the \"simplest cubic fields\" of D. Shanks, we give a modified proof and an extension of a result of Uchida, showing how to obtain cyclic cubic fields with class number divisible by n, for any n. Using 2-descents on elliptic curves, we obtain precise information on the 2-Sylow subgroups of the class groups of these fields. A theorem of H. Heilbronn associates a set of quartic fields to the class group. We show how to obtain these fields via these elliptic curves. In [10], D. Shanks discussed a family of cyclic cubic fields and showed that they could be regarded as the cubic analogues of the real quadratic fields Q(\\a2 + 4). These fields had previously appeared in the work of H. Cohn [4], who used them to produce cubic fields of even class number. Later, they appeared in the work of K. Uchida [12], who showed that for each n there are infinitely many cubic fields with class number divisible by n. In the following we first give another proof of Uchida's result and extend the techniques to handle some new cases. In the second part of the paper we study the relationship between elliptic curves and the 2-part of the class group, interpreting and extending the work of Cohn. 1. The Simplest Cubic Fields. Let m > 0 be an integer such that m & 3 mod 9. Let K be the cubic field defined by the irreducible (over Q) polynomial f(X) = X3 A mX2 -(mA 3)X + 1. The discriminant of f(X) is D2 = (m2 + 3m + 9)2 (note that m # 3 mod9 implies D # 0 mod27). Let p be the negative root of f(X). Then p' = 1/(1 p) and p\" = 1 1/p are the other two roots, so K = Q(p) is a cyclic cubic field. Note that p, p', p\" are units; in fact, p, p' are independent, hence generate a subgroup of finite index in the full group of units of K. Since -m2 < p < -m 1 < 0 < p' < 1 < p\" < 2, it follows easily that all 8 combinations of signs may be obtained from units and their conjugates; hence, every totally positive unit is a square and the narrow and wide class numbers are equal. Received January 3, 1986. 1980 Mathematics Subject Classification. Primary 12A30, 12A50, 14K.07. \"Research partially supported by NSF and the Max Planck Institut, Bonn. 371 ©1987 American Mathematical Society 0025-5718/87 $1.00 + $.25 per page License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 372 LAWRENCE C. WASHINGTON Let a = —1 A p p2. Then a ( í Q) is a root of

Large Integral Points on Elliptic Curves

Abstract: To ml, friend Dan Shanks Abstract. We describe several methods which permit one to search for big integral points on certain elliptic curves, i.e., for integral solutions (x, ,') of certain Diophantine equations of the form V2 = X3 + ax + b (a, b E Z) in a large range Ixi, Iy I < B, in time polynomial in loglogB. We also give a number of individual examples and of parametric families of examples of specific elliptic curves having a relatively large integral point.

Asymptotic boundary conditions and numerical methods for nonlinear elliptic problems on unbounded domains

Abstract: We present a derivation and implementation of asymptotic boundary conditions to be imposed on ''artificial'' boundaries for nonlinear elliptic boundary value problems on semi-infinite ''cylindrical'' domains. A general theory developed by the authors in (11) is applied to establish the existence of exact boundary conditions and then to obtain useful approximations to them. The derivation is based on the Laplace transform solution of the linearized problem at infinity. We discuss the incorporation of the asymptotic boundary conditions into a finite-difference scheme and present the results of numerical experiments on the solution of the Bratu problem in a two-dimensional stepped channel. We also touch on certain problems concerning the existence of solutions of this problem on infinite domains and conjecture on the behavior of the critical parameter value with respect to changes in the domain. Some numerical evidence supporting the conjecture is given.

Class groups of number fields: numerical heuristics

Abstract: Extending previous work of H. W. Lenstra, Jr. and the first author, we give quantitative conjectures for the statistical behavior of class groups and class numbers for every type of field of degree less than or equal to four (given the signature and the Galois group of the Galois closure). The theoretical justifications for these conjectures will appear elsewhere, but the agreement with the existing tables is quite good. 1. Introduction and Notations. In (3), H. W. Lenstra, Jr., and the first author developed a method for conjecturing quantitative results on class groups of quadratic fields and cyclic extensions of prime degree. In a forthcoming paper (4) we shall show that this technique can be extended to a much wider class of number fields, and also to relative extensions. The aim of the present paper is to rapidly make available the numerical conjec- tures obtained, for people not really interested in our heuristic reasoning or not wanting to wait for (4) to appear. Hence, apart from a total lack of justifications for the conjectures that we present, this paper is essentially self-contained. The plan is as follows. In the rest of this section we present the notations used in the sequel. Some of them being nonstandard (and in general differing from the notations of (3)), we urge the reader to read the notations carefully before applying the conjectures. In the next section we present templates for the subsequent conjectures, and then the conjectures themselves, illustrated by numerical examples, first for their own sake, and second as a double check for the reader to understand the templates. These conjectures are given for all types of fields of degree less than or equal to four. In the final section we comment on the consistency of the conjectures with existing tables (which is quite good). Combinatorial Notations: * If X is a set, {XI denotes its cardinality.

How to implement the spectral transformation

Abstract: The general, linear eigenvalue equations (H X M)z = 0, where H and M are real symmetric matrices with M positive semidefinite, must be transformed if the Lanczos algorithm is to be used to compute eigenpairs (X, z). When the matrices are large and sparse (but not diagonal) some factorization must be performed as part of the transformation step. If we are interested in only a few eigenvalues X near a specified shift, then the spectral transformation of Ericsson and Ruhe [1] proved itself much superior to traditional methods of reduction. The purpose of this note is to show that a small variant of the spectral transformation is preferable in all respects. Perhaps the lack of symmetry in our formulation deterred previous investigators from choosing it. It arises in the use of inverse iteration. A second goal is to introduce a systematic modification of the computed Ritz vectors, which improves the accuracy when M is ill-conditioned or singular. We confine our attention to the simple Lanczos algorithm, although the first two sections apply directly to the block algorithms as well. 1. Overview. This contribution is an addendum to the paper by Ericsson and Ruhe [1] and also [7]. The value of the spectral transformation is reiterated in a later section. Here we outline our implementation of this transformation. The equation to be solved, for an eigenvalue A and eigenvector z, is (1) (H XM)z = 0, H and M are real symmetric n X n matrices, and M is positive semidefinite. A practical instance of (1) occurs in dynamic analysis of structures, where H and M are the stiffness and mass matrices, respectively. We assume that a linear combination of H and M is positive definite. It then follows that all eigenvalues X are real. In addition, one has a real scalar a, distinct from any eigenvalue, and we seek a few eigenvalues A close to a, together with their eigenvectors z. Ericsson and Ruhe replace (1) by a standard eigenvalue equation (2) [C(H aM) -CT v]y = 0, where C is the Choleski factor of M; M = CTC and y = Cz. If M is singular then so is C, but fortunately the eigenvector z can be recovered from y via z = (H aM) -ICTy. Of course, there is no intention to invert (H a M) explicitly. The Received May 14, 1984; revised December 20, 1985. 1980 Mathematics Subject Classification. Primary 65F15. * This research was supported in part by the AFOSR contract F49620-84-C-0090. The third author was also supported in part by the Swedish Natural Science Research Council. * *The paper was written while this author was visiting the Center for Pure and Applied Mathematics, University of California, Berkeley, California 94720. ?1987 American Mathematical Society 0025-5718/87 $1.00 + $.25 per page

The Discrete Galerkin Method for Integral Equations

Abstract: A general theory is given for discretized versions of the Galerkin method for solving Fredholm integral equations of the second kind. The discretized Galerkin method is obtained from using numerical integration to evaluate the integrals occurring in the Galerkin method. The theoretical framework that is given parallels that of the regular Galerkin method, including the error analysis of the superconvergence of the iterated Galerkin and discrete Galerkin solutions. In some cases, the iterated discrete Galerkin solution is shown to coincide with the Nystrom solution with the same numerical integration method. The paper concludes with applications to finite element Galerkin methods.