solving homogeneous linear equations over gf 2 via block wiedemann algorithm we propose a method of solving large sparse systems of homogeneous linear equations over g f 2 the field with two elements we modify an algorithm due to wiedemann a block version of the algorithm allows us to perform 32 matrix vector operations for the cost of one the resulting algorithm is competitive with structured gaussian elimination in terms of time and has much lower space requirements it may be useful in the last stage of integer factorization we address here the problem of solving a large sparse system of homogeneous linear equations over gf 2 the field with two elements one important application which motivates the present work arises in integer factorization during the last stage of most integer factorization algorithms we are presented with a large sparse integer matrix and are asked to find linear combinations of the columns of this matrix which vanish modulo 2 for example 7 the matrix may have 100 000 columns with an average of 15 nonzero entries per column for this application we would like to obtain several solutions because a given solution will lead to a nontrivial factorization with probability 112 with n independent solutions our probability of finding a factorization rises to 1 2 structured gaussian elimination can be used 7 but as problems get larger it may become infeasible to store the matrices obtained in the intermediate stages of gaussian elimination the wiedemann algorithm 9 71 has smaller storage requirements one need only store a few vectors and an encoding of a sparse matrix not a dense matrix as occurs in gaussian elimination after fillin and it may have fewer computational steps since one takes advantage of the sparseness of the matrix but its efficiency is hampered by the fact that the algorithm acts on only one bit at a time in the present paper we work with blocks of vectors at a single time by treating 32 vectors at a time on a machine with 32 bit words we can perform 32 matrix vector products at once thus considerably decreasing the cost of indexing this can be viewed as a block wiedemann algorithm the main technical difficulty is in obtaining the correct generalization of the berlekamp massey algorithm to a block version namely a multidimensional version of the extended euclidean algorithm received by the editor november 20 1991 and in revised form july 24 1992 1991 mathematics subject classijication primary 15a33 11y05 11 04 15 04 1994 amencan mathernat cal soc ety 0025 571 8 94 1 oo 25 per page

on the complexity of multiplication in finite fields abstract in this paper we study the bilinear complexity of multiplying two arbitrary elements from an nth degree extension of a finite field f and the related problem of multiplying over f two polynomials of degree n 1 with indeterminate coefficients we derive a new linear lower bound and we describe an algorithm leading to a quasi linear upper bound

micropolar meets newtonian the rayleigh benard problem abstract we consider the rayleigh benard problem for the two dimensional boussinesq system for the micropolar fluid our main goal is to compare the value of the critical rayleigh number and estimates of the nusselt number and the fractal dimension of the global attractor with those values for the same problem for the classical navier stokes system our estimates reveal the stabilizing effects of micropolarity in comparison with the homogeneous navier stokes fluid in particular the critical rayleigh number for the micropolar model is larger than that for the navier stokes one and large micropolar viscosity may even prevent the averaged heat transport in the upward vertical direction the estimate of the fractal dimension of the global attractor for the considered problem is better than that for the same problem for the navier stokes system besides we obtain also other results which relate the heat convection in the micropolar fluids to the newtonian ones among them relations between the nusselt number and the energy dissipation rate and upper semicontinuous convergence of global attractors to the rayleigh benard attractor for the considered model

a note on grobner bases and berlekamp s algorithm it will be shown that it is possible to factorize square free polynomials over finite fields using grobner bases and a maple implementation of this method will be described

zero aliasing for modeled faults when using built in self test bist for testing vlsi circuits the circuit response to an input test sequence which may consist of thousands to millions of bits is compacted into a signature which consists of only tens of bits usually a linear feedback shift register lfsr is used for response compaction via polynomial division the compacting function is a many to one function and as a result some erroneous responses may be mapped to the same signature as the good response this is known as aliasing in this paper we deal with the selection of a feedback polynomial for the compacting lfsr such that an erroneous response resulting from any modeled fault is mapped to a signature that is different from that for the good response such lfsrs are called zero aliasing lfsrs only zero aliasing lfsrs with primitive or irreducible feedback polynomials are considered due to their suitability for bist test pattern generation

optimal bounds on the dimension of the attractor of the navier stokes equations abstract in this article we derive optimal upper bounds on the dimension of the attractor for the navier stokes equations in two dimensional domains these bounds fully agree with the lower bounds obtained by babin and vishik 1983 see also ghidaglia and temam and liu 1993 as in babin and vishik 1983 we consider here elongated domains and leaving the density of volume forces and the viscosity fixed we let the shape ratio of the domain become large so that the grashof number is large the estimates derived here are based on the general methods for estimating attractors dimensions as in constantin et al 1988 on a new version of the lieb thirring inequalities for elongated domains and on techniques developed for such domains in rs 1993 and temam and ziane 1996 at the end of the article we also give some partial results in the three dimensional case for which we need a physical assumption on the reynolds number introduced in ghidaglia and temam

bosonic representations of yangian double with on the basis of the rtt ttr formalism we introduce the quantum double of the yangian for with a central extension the gauss decomposition of the t matrices gives us the so called drinfel d generators using these generators we present some examples of both finite and infinite dimensional representations that are quite natural deformations of their corresponding affine counterpart

solving sparse linear equations over finite fields a coordinate recurrence method for solving sparse systems of linear equations over finite fields is described the algorithms discussed all require o n 1 omega n 1 log k n 1 field operations where n 1 is the maximum dimension of the coefficient matrix omega is approximately the number of field operations required to apply the matrix to a test vector and the value of k depends on the algorithm a probabilistic algorithm is shown to exist for finding the determinant of a square matrix also probabilistic algorithms are shown to exist for finding the minimum polynomial and rank with some arbitrarily small possibility of error

inversion of two level circulant matrices over zp abstract we consider the problem of inverting block circulant with circulant blocks bccb matrices with entries over the field z p this problem arises in the study of of two dimensional linear cellular automata since the standard reduction to diagonal form by means of fft has some drawbacks when working over z p we solve this problem by transforming it into the equivalent problem of inverting a circulant matrix with entries over a suitable ring r we show that a bccb matrix of size mn can be inverted in o mn c m n operations in z p where c is a low degree polynomial in log m and log n

the black box niederreiter algorithm and its implementation over the binary field the most time consuming part of the niederreiter algorithm for factoring univariate polynomials over finite fields is the computation of elements of the nullspace of a certain matrix this paper describes the so called black box niederreiter algorithm in which these elements are found by using a method developed by wiedemann the main advantages over an approach based on gaussian elimination are that the matrix does not have to be stored in memory and that the computational complexity of this approach is lower the black box niederreiter algorithm for factoring polynomials over the binary field was implemented in the c programming language and benchmarks for factoring high degree polynomials over this field are presented these benchmarks include timings for both a sequential implementation and a parallel implementation running on a small cluster of workstations in addition the wan algorithm which was recently introduced is described and connections between implementation aspects of wan s and niederreiter s algorithm are given

boundary conditions for integrable quantum systems a new class of boundary conditions is described for quantum systems integrable by means of the quantum inverse scattering r matrix method the method proposed allows the author to treat open quantum chains with appropriate boundary terms in the hamiltonian the general considerations are applied to the xxz and xyz models the nonlinear schrodinger equation and toda chain

subquadratic time factoring of polynomials over finite fields new probabilistic algorithms are presented for factoring univariate polynomials over finite fields the algorithms factor a polynomial of degree n over a finite field of constant cardinality in time o n 1 815 previous algorithms required time n 2 o 1 the new algorithms rely on fast matrix multiplication techniques more generally to factor a polynomial of degree n over the finite field f q with q elements the algorithms use o n 1 815 log q arithmetic operations in f q the new baby step giant step techniques used in our algorithms also yield new fast practical algorithms at super quadratic asymptotic running time and subquadratic time methods for manipulating normal bases of finite fields

a new algorithm for factoring polynomials over finite fields in f t there are two standard methods one is due to e berlekamp the other appears to be a folk method both are described by d knuth in 3 pp 381 397 further improvements for special values of q are given by r moenck in 5j see also e berlekamp 1 note that both methods as described in the references apply only when d 1 however straightforward modifications described below allow d 1 both methods use the calculation of resultants or equivalently the solution of linear equations to reduce the problem to finding the roots of a polynomial which has all of its roots in f when p is very small probabilistic methods are used an improvement to the folk method method along with a more explicit calculation of the work required has recently been given by m rabin 6 we present here a new probabilistic method which when combined with the above algorithms avoids the need for both resultants and linear equations it leads to algorithms which are conceptually simpler than previous methods moreover it works equally well for all finite fields f regardless of the magnitude of q when used for factoring a quadratic x2 a it reduces to berlekamp s algorithm other standard algorithms for factoring quadratics are due to d h lehmer 4 and d shanks 7 our algorithm is also suitable for finding solutions of polynomial equations over finite fields

primary decomposition of zero dimensional ideals over finite fields gao shuhong clemson univ dept math sci clemson sc 29634 usa wan daqing univ calif irvine dept math irvine ca 92697 usa wang mingsheng acad sinica inst software informat secur lab beijing 100080 peoples r china

better circuits for binary polynomial multiplication we develop a new and simple way to describe karatsuba like algorithms for multiplication of polynomials over mathbb f 2 f 2 we restrict the search of small circuits to a class of circuits we call symmetric bilinear these are circuits in which and gates only compute functions of the form sum i in s a i cdot sum i in s b i quad quad s subseteq lbrace 0 ldots n 1 rbrace i s a i i s b i s 0 n 1 these techniques yield improved recurrences for m kn m k n the number of gates used in a circuit that multiplies two kn k n term polynomials for k 4 5 6 k 4 5 6 and 7 we built and verified the circuits for n n term binary polynomial multiplication for values of n n of practical interest circuits for n n up to 100 are posted at http cs www cs yale edu homes peralta circuitstuff binpolmult tar gz

factoring polynomials over finite fields using differential equations and normal bases the deterministic factorization algorithm for polynomials over finite fields that was recently introduced by the author is based on a new type of linearization of the factorization problem the main ingredients are differential equations in rational function fields and normal bases of field extensions for finite fields of characteristic 2 it is known that this algorithm has several advantages over the classical berlekamp algorithm we develop the algorithm in a general framework and we show that it is feasible for arbitrary finite fields in the sense that the linearization can be achieved in polynomial time

pseudorandom vector generation by the multiple recursive matrix method pseudorandom vectors are of importance for parallelized simulation methods in this paper we carry out an in depth analysis of the multiple recursive matrix method for the generation of uniform pseudorandom vectors which was introduced in an earlier paper of the author we study in particular the periodicity properties the lattice structure and the behavior under the serial test for sequences of pseudorandom vectors generated by this method

an upper bound on the attractor dimension of a 2d turbulent shear flow in lubrication theory abstract we consider a two dimensional navier stokes shear flow there exists a unique global in time solution of the considered problem as well as the global attractor for the associated semigroup our aim is to estimate from above the dimension of the attractor in terms of given data and geometry of the domain of the flow first we obtain a kolmogorov type bound on the time averaged energy dissipation rate independent of viscosity at large reynolds numbers then we establish a version of the lieb thirring inequality for a class of functions defined on the considered non rectangular flow domain this research is motivated by a problem from lubrication theory

substitutions for linear shift register sequences and the factorization algorithms of berlekamp and niederreiter abstract an algorithm is presented yielding substitutions finite automata for linear shift register sequences over finite fields some computational applications of these substitutions are discussed and connections to both berlekamp s and niederreiter s factorization algorithms polynomials over finite fields are established

primitive normal polynomials over finite fields in this note we significantly extend the range of published tables of primitive normal polynomials over finite fields for each pn 1050 with p 97 we provide a primitive normal polynomial of degree n over fp moreover each polynomial has the minimal number of nonzero coefficients among all primitive normal polynomials of degree n over fp the roots of such a polynomial generate a primitive normal basis of fpn over fp and so are of importance in many computational problems we also raise several conjectures concerning the distribution of such primitive normal polynomials including a refinement of the primitive normal basis theorem

new boundary conditions for integrable lattices new boundary conditions for classical integrable nonlinear lattices of the xxx type such as the heisenberg chain and the toda lattice are presented these integrable extensions are formulated in terms of a generic xxx heisenberg magnet interacting with two additional spins at each end of the chain the construction uses the most general rank 1 ansatz for the 2 2 l operator satisfying the reflection equation algebra with rational r matrix the associated quadratic algebra is shown to be that of dynamical symmetry for the a1 and bc2 calogero moser problems other physical realizations of our quadratic algebra are also considered

univariate polynomial factorization over finite fields abstract this paper is a tutorial introduction to univariate polynomial factorization over finite fields we recall the classical methods that induced most factorization algorithms berlekamp s and the cantor zassenhaus ones and some refinements which can be applied to these methods explicit algorithms are presented in a form suitable for almost immediate implementation we give a detailed description of an efficient implementation of the cantor zassenhaus algorithm used in the release 2 of the axiom computer algebra system

on a new factorization algorithm for polynomials over finite fields a new deterministic factorization algorithm for polynomials over finite fields was recently developed by the first author the bottleneck in this algorithm is the last stage in which the irreducible factors of the polynomial are derived from the solutions of a system of linear equations an efficient approach to the last stage was designed by the second author for the case of finite fields of characteristic 2 in this paper we describe a different approach to the last stage which works for arbitrary fields of positive characteristic in particular we obtain in this way an acceleration of the factorization algorithm of the first author which makes this algorithm polynomial time for fixed characteristic

attractor dimension estimates for two dimensional shear flows abstract we study the large time behavior of boundary and pressure gradient driven incompressible fluid flows in elongated two dimensional channels with emphasis on estimates for their degrees of freedom i e the dimension of the attractor for the solutions of the navier stokes equations for boundary driven shear flows and flux driven channel flows we present upper bounds for the degrees of freedom of the form c re 3 2 where c is a universal constant denotes the aspect ratio of the channel length width and re is the reynolds number based on the channel width and the imposed outer velocity scale for fixed pressure gradient driven channel flows we obtain an upper bound of form c re2 where c is another universal positive constant and the reynolds number is based on a velocity defined by the infimum over all possible trajectories of the time averaged mass flux per unit channel width we discuss these results in terms of physical arguments based on small length scales in turbulent flows

statistical independence properties of pseudorandom vectors produced by matrix generators abstract the uniformity test and the serial test are standard statistical tests for equidistribution and statistical independence respectively in sequences of uniform pseudorandom numbers we introduce analogues of these tests for sequences of uniform pseudorandom vectors and we apply these tests to matrix generators for uniform pseudorandom vector generation the results show that these generators behave well under these tests provided that the matrix in the generation algorithm is chosen carefully the essential condition is that a certain figure of merit attached to the matrix be large we also prove a general theorem which guarantees the existence of matrices with large figure of merit

unification of variational principles for turbulent shear flows the background method of doering constantin and the mean fluctuation formulation of howard busse abstract the recent discovery of a new variational formulation for bounding turbulent flow quantities has rekindled interest in the whole field of producing rigorous inequality results directly from the navier stokes equations initiated by howard and busse in the 1960s the new background method introduced by doering and constantin 1992 1994 1995 1996 for bounding turbulent viscous dissipation appears to differ so fundamentally from that of howard and busse that any connection between the two methods has been fascinatingly unclear recently highly suggestive results obtained asymptotically kerswell 1997 have indicated that the variational problems produced for both cases of shear flow and convection may be equivalent here we confirm this for the shear flow case by uncovering the common underlying functional which both approaches seek to make stationary busse s upper bound problem is found to be the optimal dual or complementary variational problem to that which emerges from the doering constantin approach this ensures that busse s best bound exactly coincides with that available within the doering constantin formalism for all reynolds numbers

quantum current algebras associated with rational r matrix abstract we study quantum current algebra a r associated with the rational r matrix of gl n and we give explicit formulae for the elements of its center at the critical level due to etingof kazhdan s construction the level c vacuum module v c r for the algebra a r possesses a quantum vertex algebra structure for any complex number c we prove that any module for the quantum vertex algebra v c r is naturally equipped with a structure of restricted a r module of level c and vice versa

hbar adic quantum vertex algebras and their modules this is a paper in a series to study vertex algebra like structures arising from various algebras including quantum affine algebras and yangians in this paper we study notions of hbar adic nonlocal vertex algebra and hbar adic weak quantum vertex algebra slightly generalizing etingof kazhdan s notion of quantum vertex operator algebra for any topologically free mathbb c lbrack lbrack hbar rbrack rbrack module w we study hbar adically compatible subsets and hbar adically mathcal s local subsets of end w x x 1 we prove that any hbar adically compatible subset generates an hbar adic nonlocal vertex algebra with w as a module and that any hbar adically mathcal s local subset generates an hbar adic weak quantum vertex algebra with w as a module a general construction theorem of hbar adic nonlocal vertex algebras and hbar adic quantum vertex algebras is obtained as an application we associate the centrally extended double yangian of mathfrak s mathfrak l 2 to hbar adic quantum vertex algebras

center of the quantum affine vertex algebra in type a abstract we consider the quantum vertex algebra associated with the double yangian in type a as defined by etingof and kazhdan we show that its center is a commutative associative algebra and construct algebraically independent families of topological generators of the center at the critical level

factorization of polynomials and some linear algebra problems over finite fields abstract the recently developed algorithm of niederreiter for the factorization of polynomials over finite fields provides a linearization of the factorization problem via differential equations in rational function fields we show that the range of applicability of this algorithm can be extended by various techniques such as the application of normal bases and the use of hasse teichmuller derivatives we also discuss topics from pseudorandom vector generation and combinatorial linear algebra that lead to interesting linear algebra problems over finite fields

connections between the algorithms of berlekamp and niederreiter for factoring polynomials over fq abstract in this paper we determine an explicit isomorphism between the solution spaces of berlekamp s and niederreiter s algorithm for factoring polynomials over finite fields this can be used for the implementation of a procedure combining the benefits of both methods

algebraic structures related to reflection equations quadratic algebras related to the reflection equations are introduced they are quantum group comodule algebras the quantum group fq gl 2 is taken as the example the properties of the algebras centre representation realizations real forms fusion procedure etc as well as the generalizations are discussed

an acceleration of the niederreiter factorization algorithm in characteristic 2 a new deterministic factorization algorithm for polynomials over finite fields was recently developed by niederreiter the bottleneck in this algorithm is the last stage in which the irreducible factors of the polynomial are derived from the solutions of a system of linear equations in this paper we consider finite fields of characteristic 2 and we show that in this case there is a more efficient approach to the last stage of the niederreiter algorithm which speeds up the algorithm considerably

inversion of circulant matrices over z m in this paper we consider the problem of inverting an n n circulant matrix with entries over zm we show that the algorithm for inverting circulants based on the reduction to diagonal form by means of fft has some drawbacks when working over zm we present three different algorithms which do not use this approach our algorithms require different degrees of knowledge of m and n and their costs range roughly from n log n log log n to n log2 n log log n log m operations over zm we also present an algorithm for the inversion of finitely generated bi infinite toeplitz matrices the problems considered in this paper have applications to the theory of linear cellular automata