multiplicative complexity of polynomial multiplication over finite fields let m q n denote the number of multiplications required to compute the coefficients of the product of two polynomials of degree n over a q element field by means of bilinear algorithms it is shown that m q n n 3 n o n in particular if q 2 n l q 1 we establish the tight bound m q n 3 n 1 q 2 the technique we use can be applied to analysis of algorithms for multiplication of polynomials modulo a polynomial as well

the ranks of m times n times mn 2 tensors it is shown that there is essentially only one m times n times mn 2 tensor of rank mn 1 it is also proved that except for this tensor all m times n times p tensors with p leqq mn 2 have rank at most mn 2 the main tool is kronecker s theory of matrix pencils which has already been applied directly by ja ja siam j comput 8 1979 pp 443 462 to study the ranks of m times n times 2 tensors we show that each nondegenerate m times n times mn 2 tensor is determined by a related m times n times 2 tensor and apply the kronecker theory to this related tensor

connections on discrete fibre bundles a new approach to gauge fields on a discrete space time is proposed in which the fundamental object is a discrete version of a principal fibre bundle if the bundle is twisted the gauge fields are topologically non trivial automatically

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

on the number of distinct functional graphs of affine linear transformations over finite fields abstract we study the number of non isomorphic functional graphs of affine linear transformations from f q n to itself and we prove upper and lower bounds on this quantity as n as a corollary to our result we prove bounds on the number of conjugacy classes in the symmetric group s q n that intersect agln q

characterization of division algebras of minimal rank and the structure of their algorithm varieties the purpose of this paper is to show that every finite dimensional division algebra of minimal rank i e of minimal complexity with respect to the noncommutative model of computation is a finite simple field extension moreover we investigate the structure of the variety of optimal algorithms for the computation of the multiplication in such fields modulo the isotropy group of the problem

topology of lattice gauge fields non abelian gauge fields on a four dimensional hypercubic lattice with small action density tr u dot p for su 2 gauge fields are shown to carry an integer topological chargeq which is invariant under continuous deformations of the field a concrete expression forq is given and it is verified thatq reduces to the familiar chern number in the classical continuum limit

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

cantor versus harley optimization and analysis of explicit formulae for hyperelliptic curve cryptosystems hyperelliptic curves hec look promising for cryptographic applications because of their short operand size compared to other public key schemes the operand sizes seem well suited for small processor architectures where memory and speed are constrained however the group operation has been believed to be too complex and thus hec have not been used in this context so far in recent years a lot of effort has been made to speed up group operation of genus 2 hec in this paper we increase the efficiency of the genus 2 and genus 3 hyperelliptic curve cryptosystems hecc for certain genus 3 curves we can gain almost 80 percent performance for a group doubling this work not only improves gaudry and harley s algorithm but also improves the original algorithm introduced by cantor 1987 contrary to common belief we show that it is also practical for certain curves to use cantor s algorithm to obtain the highest efficiency for the group operation in addition we introduce a general reduction method for polynomials according to karatsuba we implemented our most efficient group operations on pentium and arm microprocessors

on the direct sum conjecture abstract we prove the direct sum conjecture for various sets of systems of bilinear forms our results depend on a priori knowledge of the complexity of at least one of the direct summands and its underlying algebraic structure we also briefly survey some previous results concerning the complexity and structure of minimal algorithms for various direct sum systems

on the validity of the direct sum conjecture the direct sum conjecture states that the multiplicative complexity of disjoint sets of bilinear computations is the sum of their separate multiplicative complexities this conjecture is known to hold for only a few specialized cases in this paper we establish its validity for large classes of computations one such class can be defined as follows let s 1 be a set of r m times n bilinear forms and let s 2 be a different set of s p times q bilinear forms then if 2 in r m n s p q we show that the direct sum conjecture holds over any field the proof involves some nontrivial facts from linear algebra and relies on the theory of invariant polynomials this result also settles the multiplicative complexity of pairs of bilinear forms over any field with large enough cardinality it is also shown that the direct sum conjecture is true for the case when r mn 2

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

multiplicative complexity of vector valued boolean functions abstract we consider the multiplicative complexity of boolean functions with multiple bits of output studying how large a multiplicative complexity is necessary and sufficient to provide a desired nonlinearity for so called circuits we show that there is a tight connection between error correcting codes and circuits computing functions with high nonlinearity combining this with known coding theory results we show that functions with n inputs and n outputs with the highest possible nonlinearity must have at least 2 32n and gates we further show that one cannot prove stronger lower bounds by only appealing to the nonlinearity of a function we show a bilinear circuit computing a function with almost optimal nonlinearity with the number of and gates being exactly the length of such a shortest code additionally we provide a function which for general circuits has multiplicative complexity at least 2 n 3 finally we study the multiplicative complexity of almost all functions we show that every function with n bits of input and m bits of output can be computed using at most 2 5 1 o 1 m 2 n and gates

multiplication of polynomials modulo xn let n be positive integers with email protected 2n 1 let r be an arbitrary nontrivial ring not necessarily commutative and not necessarily having a multiplicative identity and r x be the polynomial ring over r in this paper we give improved upper bounds on the minimum number of multiplications needed to multiply two arbitrary polynomials of degree at most n 1 modulo x n over r moreover we introduce a new complexity notion the minimum number of multiplications needed to multiply two arbitrary polynomials of degree at most n 1 modulo x over r this new complexity notion provides improved bounds on the minimum number of multiplications needed to multiply two arbitrary polynomials of degree at most n 1 modulo x n over r

on a class of primary algebras of minimal rank abstract denote the rank bilinear multiplicative complexity of an associative k algebra a by r a it is well known that r a 2dim a t where t is the number of maximal two sided ideals of a a is said to be of minimal rank iff r a 2dim a t this paper is concerned with the structure of primary algebras t 1 of minimal rank we show that a primary k algebra is of minimal rank if there exists a pair 1 x 2 x n 1 y 2 y n of k bases of a such that for all i j x i y j ky i ky j at least for local algebras this sufficient condition is also necessary and leads to a complete description of the structure of local algebras of minimal rank as an application we investigate the ranks of the 5 dimensional local algebras furthermore it turns out that k 2 2 is the only primary nonlocal algebra satisfying the abovementioned sufficient condition

multiplication of polynomials over finite fields the authors prove the 2 5 n o n lower bound on the number of multiplications divisions required to compute the coefficients of the product of two polynomials of degree n over a finite field by means of straight line algorithms

tensor decompositions and applications this survey provides an overview of higher order tensor decompositions their applications and available software a tensor is a multidimensional or n way array decompositions of higher order tensors i e n way arrays with n geq 3 have applications in psycho metrics chemometrics signal processing numerical linear algebra computer vision numerical analysis data mining neuroscience graph analysis and elsewhere two particular tensor decompositions can be considered to be higher order extensions of the matrix singular value decomposition candecomp parafac cp decomposes a tensor as a sum of rank one tensors and the tucker decomposition is a higher order form of principal component analysis there are many other tensor decompositions including indscal parafac2 candelinc dedicom and paratuck2 as well as nonnegative variants of all of the above the n way toolbox tensor toolbox and multilinear engine are examples of software packages for working with tensors

on a boundary value problem for delay differential equations of population dynamics and chebyshev approximation in this paper a boundary value problem for delay differential equations of population dynamics is considered we obtain approximate solutions by using chebyshev polynomial series and newton raphson s procedure and give the error estimation the method of the error estimation has been obtained in an existence theorem proved by a part of the authors we carry out some numerical experiments by a computer language matlab

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

classification of all the minimal bilinear algorithms for computing the coefficients of the product of two polynomials modulo a polynomial part 1 the algebra guue a q u a abstract in this paper we classify all the minimal bilinear algorithms for computing the coefficients of n 1i 0 xiui n 1i 0 yiui mod q u l where deg q u j jl n and q u is irreducible over g is studied the case where l 1 was studied in 8 for l 1 the main results are that we have to distinguish between two cases j 1 and j 1 the case where j 1 was studied in 1 for j 1 it is shown that up to equivalence every minimal 2n 1 multiplications bilinear algorithm for computing the coefficients of n 1i 0 xiui n 1i 0 yiui mod un is done either by first computing the coefficients of n 1i 0 xiui n 1i 0 yiui and then reducing them modulo un or by first computing the coefficients n 2i 0 xiui n 1i 0 yiui and then reducing them modulo un and adding xn 1y0un 1 or by first computing the coefficients n 2i 0 xiui n 2i 0 yiui and then reducing them modulo un and adding xn 1y0 x0yn 1 un 1

multiplicative complexity of direct sums of quadratic systems abstract we consider the quadratic complexity of certain sets of quadratic forms we study classes of direct sums of quadratic forms for these classes of problems we show that the complexity of one direct sum is the sum of the complexities of the summands and that every minimal quadratic algorithm for computing the direct sums is a direct sum algorithm

monotonically convergent algorithms for symmetric tensor approximation abstract reduced rank approximations to symmetric tensors find use in data compaction and in multi user blind source separation we derive iterative algorithms which feature monotonic convergence to a minimum of a frobenius norm approximation criterion for a certain rank r tucker product version of the approximation problem the approach exploits the gradient inequality for convex functions to establish monotonic convergence while sparing the cumbersome step size analysis required from a manifold gradient approach it likewise overcomes some limitations of symmetric versions of alternating least squares the computational load per iteration amounts to computing an unfolded matrix and a qr decomposition

the computation of characteristic classes of lattice gauge fields agl p c valued lattice gauge fieldu on a simplicial complex determines a principalgl p c bundle if the plaquette products are sufficiently small with respect to the maximum distortion coefficient of the transporters a representative cocyclecq for theqth chern class of can be computed on each 2q simplex by takingcq to be the intersection number of a certain singular 2q cubem with a schubert type variety q in the space of allp p matrices this reduces to the solution of polynomial equations with coefficients coming fromu and thus avoids numerical integration or cooling type procedures an application of this method is suggested for the computation of the topological charge of ansu 3 valued lattice gauge field on a 4 complex

commutative algebras of minimal rank abstract if a is a finite dimensional associative k algebra with unit then its rank r a i e its bilinear complexity is never less than 2dim a m a a is said to be of minimal rank if r a 2dim a m a in this paper we determine for infinite perfect fields k all commutative k algebras of minimal rank roughly speaking these algebras are built up from simply generated structures which annihilate each other furthermore we indicate how this result can be used to obtain new lower bounds for the rank of specific commutative algebras

topological charge of lattice gauge fields using recently derived explicit formulae for the 2 and 3 cochains in su 2 gauge theory we are able to integrate the chern simons density analytically we arrive in su 2 at a local algebraic expression for the topological charge which is the sum of local winding numbers associated with the corners lattice points of the cells covering the manifold plus contributions from possible isolated gauge singularities which manifest themselves as vortices in the 1 2 or 3 cochains among others we consider hypercubic geometry i e covering the manifold by hypercubes which is of particular interest to lattice monte carlo applications finally we extend our results to su 3 gauge theory

partial and total matrix multiplication in 1979 considerable progress was made in estimating the complexity of matrix multiplication here the new techniques and recent results are presented based upon the notion of approximate rank and the observation that certain patterns of partial matrix multiplication some of the entries of the matrices may be zero can efficiently be utilized to perform multiplication of large total matrices by combining pan s trilinear technique with a strong version of our compression theorem for the case of several disjoint matrix multiplications it is shown that multiplication of n times n matrices over arbitrary fields is possible in time o n beta where beta is a bit smaller than 3 ln 52 ln 110 approx 2 522

direct sums of bilinear algorithms abstract we consider the bilinear complexity of certain sets of bilinear forms we study a class of direct sums of bilinear forms for this class of problems we show that the bilinear complexity of one direct sum is the sum of the bilinear complexities of the summands and that every minimal bilinear algorithm for computing the direct sums is a direct sum algorithm we also exhibit some sets of bilinear forms which belong to this class

maximal bilinear complexity and codes abstract the connection between bilinear complexity and error correcting codes discovered by brockett and dobkin in 1973 yields lower bounds on the maximal ranks of tensors with a given shape the resulting bounds are linear and thus interesting only for unbalanced shapes like n n 2 and n n n 2 k with k n as an example for odd n the maximal rank of n n 2 tensors is larger over z 2 than over an algebraic closure of z 2

an upper bound on the complexity of multiplication of polynomials modulo a power of an irreducible polynomial let q2 n k denote the minimum number of multiplications required to compute the coefficients of the product of two degree n k 1 polynomials modulo the kth power of an irreducible polynomial of degree n over the q2 element field bbf q2 it is shown that for all odd q and all n 1 2 liminfk q2 n k k n 2 1 1 q 2 for the proof of this upper bound we show that for an odd prime power q all algebraic function fields in the garcia stichtenoth tower over bbf q2 have places of all degrees and apply a chudnovsky like algorithm for multiplication of polynomials modulo a power of an irreducible polynomial

on the algorithmic complexity of associative algebras abstract the complexity l a of a finite dimensional associative algebra a is the number of non scalar multiplications divisions of an optimal algorithm to compute the product of two elements of the algebra we show l a 2ddim a t where t is the number of maximal two sided ideals of a

lattice gauge fields principal bundles and the calculation of topological charge su2 valued lattice gauge fields are studied on a 4 dimensional simplicial lattice ifu has sufficiently small plaquette products then there is a unique principal su2 bundle admitting transition functions defined on the intersections of adjacent dual cells which take values within 8 ofu an algorithm is explicitly given which associates an integer to everyu off a certain set of measure zero this algorithm only involves evaluation of 4 4 determinants and the solution of quadratic equations whenu is as above the integer produced is the second chern number of i e the topological charge ofu

maximal rank of m x n x mn k tensors it is shown that the maximal rank of m times n times mn k tensors with k leqq min m 1 2 2 n 1 2 2 is greater than mn 4 sqrt 2k o 1

an abstract framework in the numerical solution of boundary value problems for neutral functional differential equations we consider the numerical solution of boundary value problems for general neutral functional differential equations the problems are restated in an abstract form and then a general discretization of the abstract form is introduced and a convergence analysis of this discretization is developed

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

factorization of multivariate polynomials over finite fields we present a probabilistic algorithm that finds the irreducible factors of a bivariate polynomial with coefficients from a finite field in time polynomial in the input size i e in the degree of the polynomial and log cardinality of field the algorithm generalizes to multivariate polynomials and has polynomial running time for densely encoded inputs a deterministic version of the algorithm is also discussed whose running time is polynomial in the degree of the input polynomial and the size of the field

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