management of constraints in optimization problems abstract optimization problems are encountered in all scientific disciplines and in many aspects of everyday life application of established optimization methods assumes that an expert can state the optimization problem correctly unfortunately this is not the case in reality below we consider how to help an expert state and solve optimization problems the proposed technique splits constraints of an optimization problem into soft manageable and rigid constraints and modifies the soft constraints in an interactive mode with the expert the technique is illustrated with a numeric example devoted to the optimization of a real life nonlinear system

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

quasilinear multistep methods and variable step predictor corrector methods for neutral functional differential equations the theory of quasilinear multistep methods and predictor corrector methods for neutral functional differential equations is developed the theorem about the asymptotic behaviour of the global discretization error of quasilinear methods which forms a theoretical basis for step control strategy for predictor corrector methods for neutral equations is given some problems of implementation of these methods are also addressed and the results of numerical experiments are presented

shift register synthesis modulo m the berlekamp algorithm takes a sequence of elements from a field and finds the shortest linear recurrence or linear feedback shift register that can generate the sequence we present an algorithm which generalizes berlekamp s to the case when the elements of the sequence are integers modulo m where m is an arbitrary but known integer details will be published elsewhere

a fraction free matrix berlekamp massey algorithm abstract we describe a fraction free version of the matrix berlekamp massey algorithm the algorithm computes a minimal matrix generator of linearly generated square matrix sequences in an integral domain the algorithm performs all operations in the integral domain so all divisions performed are exact for scalar sequences the matrix algorithm specializes to a different algorithm than the algorithm currently in the literature this new scalar algorithm has smaller intermediate values than the known fraction free berlekamp massey algorithm

the numerical solution of volterra functional differential equations of neutral type in this paper methods of adams moulton type for neutral functional differential equations are investigated a general convergence theorem is presented and the order of convergence of these methods is examined some numerical examples are given

proof verification and the hardness of approximation problems we show that every language in np has a probablistic verifier that checks membership proofs for it using logarithmic number of random bits and by examining a constant number of bits in the proof if a string is in the language then there exists a proof such that the verifier accepts with probability 1 i e for every choice of its random string for strings not in the language the verifier rejects every provided proof with probability at least 1 2 our result builds upon and improves a recent result of arora and safra 1998 whose verifiers examine a nonconstant number of bits in the proof though this number is a very slowly growing function of the input length as a consequence we prove that no max snp hard problem has a polynomial time approximation scheme unless np p the class max snp was defined by papadimitriou and yannakakis 1991 and hard problems for this class include vertex cover maximum satisfiability maximum cut metric tsp steiner trees and shortest superstring we also improve upon the clique hardness results of feige et al 1996 and arora and safra 1998 and show that there exists a positive e such that approximating the maximum clique size in an n vertex graph to within a factor of n e is np hard

the space complexity analysis in the general number field sieve integer factorization the general number sieve is the most efficient algorithm for integer factorization it consists of polynomial selection sieving solving equations and finding square roots root lifting of polynomial is discussed in this paper the p adic evaluation provided by each root and the expected p value are also given then we gain the space complexity of sieving and building equations over the ring z 2 z

waveform relaxation methods for functional differential systems of neutral type abstract we investigate continuous time and discretized waveform relaxation iterations for functional differential systems of neutral type it is demonstrated that continuous time iterations converge linearly for neutral equations and superlinearly when the right hand side is independent of the history of the derivative of the solution the error bounds for discretized iterations are also obtained and some implementation aspects are discussed numerical results are presented which indicate a potential speedup of this technique as compared with the classical approach based on discrete variable methods

f2 lanczos revisited we present a new variant of the block lanczos algorithm for finding vectors in the kernel of a symmetric matrix over f2 our algorithm is at least as efficient as that of montgomery peter l montgomery a block lanczos algorithm for finding dependencies over gf 2 in advances in cryptology eurocrypt 95 saint malo 1995 lecture notes in comput sci vol 921 springer berlin 1995 pp 106 120 while the sequence of matrices wi constructed here have different algebraic properties that may be useful in eventually providing a provable upper bound on the time required to solve this problem namely our wi satisfy witwj 0 for i j as opposed to witawj 0 in 6

improved n term karatsuba like formulas in gf 2 it is well known that chinese remainder theorem crt can be used to construct efficient algorithms for multiplication of polynomials over gf 2 in this note we show how to select an appropriate set of modulus polynomials to obtain minimal number of multiplications

probabilistic checking of proofs a new characterization of np we give a new characterization of np the class np contains exactly those languages l for which membership proofs a proof that an input x is in l can be verified probabilistically in polynomial time using logarithmic number of random bits and by reading sublogarithmic number of bits from the proof we discuss implications of this characterization specifically we show that approximating clique and independent set even in a very weak sense is np hard

the serial test for congruential pseudorandom numbers generated by inversions two types of congruential pseudorandom number generators based on inversions were introduced recently we analyze the statistical independence properties of these pseudorandom numbers by means of the serial test the results show that these pseudorandom numbers perform satisfactorily under the serial test the methods of proof rely heavily on bounds for character sums such as the weil stepanov bound for character sums over finite fields

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

stability analysis of one step methods for neutral delay differential equations in this paper stability properties of one step methods for neutral functional differential equations are investigate stability regions are characterized for runge kutta methods with respect to the linear test equation begin gathered y left t right ay left t right by left t tau right cy left t tau right t geqq 0 hfill y left t right g left t right tau leqq t leqq 0 hfill end gathered 0 where a b andc are complex parameters in particular it is shown that everya stable collocation method for ordinary differential equations can be extended to a method for neutrals delay differential equations with analogous stability properties the so called np stable method we also investigate how the approximation to the derivative of the solution affects stability properties of numerical methods for neutral equations

routing in undirected graphs with constant congestion given an undirected graph g v e a collection s 1 t 1 ldots s k t k of k pairs of its vertices called demand pairs and an integer c the goal in the edge disjoint paths with congestion problem is to connect the maximum possible number of demand pairs by paths so that the maximum load on any edge called edge congestion does not exceed c we show an efficient randomized algorithm to route omega mathsf opt operatorname poly log k demand pairs with congestion at most 14 where mathsf opt is the maximum number of pairs that can be simultaneously routed on edge disjoint paths the algorithm in fact routes omega mathsf opt mathsf lp operatorname poly log k pairs where mathsf opt mathsf lp is the optimal value of the standard multicommodity flow relaxation for the problem the best previous efficient algorithm that routed omega mathsf opt operatorname poly log n pairs required congestion operatorname poly log log n and for the

on the discrepancy of gfsr pseudorandom numbers a new summation formula based on the orthogonal property of walsh functions is devised using this formula the k dimensional discrepancy of the generalized feedback shift register gfsr pseudorandom numbers is derived the relation between the discrepancy and k distribution of gfsr sequences is also obtained finally the definition of optimal gpsr pseudorandom number generators is introduced

euclid s algorithm and the lanczos method over finite fields this paper shows that there is a close relationship between the euclidean algorithm for polynomials and the lanczos method for solving sparse linear systems especially when working over finite fields it uses this relationship to account rigorously for the appearance of self orthogonal vectors arising in the course of the lanczos algorithm it presents an improved lanczos method which overcomes problems with self orthogonality and compares this improved algorithm with the euclidean algorithm

solving sparse linear systems of equations over finite fields using bit flipping algorithm let fq be the finite field with q elements we give an algorithm for solving sparse linear systems of equations over fq when the coefficient matrix of the system has a specific structure here called relatively connected this algorithm is based on a well known decoding algorithm for low density parity check codes called bit flipping algorithm we modify and extend this hard decision decoding algorithm the complexity of this algorithm is linear in terms of the number of columns n and the number of nonzero coefficients of the matrix per iteration the maximum number of iterations is bounded above by m the number of equations

computing the sign or the value of the determinant of an integer matrix a complexity survey computation of the sign of the determinant of a matrix and the determinant itself is a challenge for both numerical and exact methods we survey the complexity of existing methods to solve these problems when the input is an n n matrix a with integer entries we study the bit complexities of the algorithms asymptotically in n and the norm of a existing approaches rely on numerical approximate computations on exact computations or on both types of arithmetic in combination

on the equivalence between berlekamp s and euclid s algorithms corresp it is shown that berlekamp s iterative algorithm can be derived from a normalized version of euclid s extended algorithm simple proofs of the results given recently by cheng are also presented

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

implicit explicit one leg methods for nonlinear stiff neutral equations abstract in this paper by adapting the underlying implicit explicit imex one leg methods cf 1 2 a class of extended imex one leg eieol methods are suggested for solving nonlinear stiff neutral equations snes it is proven under some suitable conditions that eieol methods are d convergent of order 2 and stable for nonlinear snes several numerical examples are given to testify the obtained theoretical results and the computational effectiveness of eieol methods moreover a comparison with the fully implicit one leg methods is presented which shows that eieol methods have the higher computational efficiency

stability of runge kutta methods for delay integro differential equations we study stability of runge kutta rk methods for delay integro differential equations with a constant delay on the basis of the linear equation du dt lu t mu t k t t u d where l m k are constant complex matrices in particular we show that the same result as in the case k 0 koto bit 34 1994 262 267 holds for this test equation i e every a stable rk method preserves the delay independent stability of the exact solution whenever a step size of the form h m is used where m is a positive integer

the serial test for pseudo random numbers generated by the linear congruential method we consider the linear congruential method for pseudo random number generation and establish effective criteria for the choice of parameters in this method which guarantee statistical almost independence of successive pseudo random numbers applications to numerical integration are also discussed

edge disjoint paths in moderately connected graphs we study the edge disjoint paths edp problem in undirected graphs given a graph g with n nodes and a set mathcal t of pairs of terminals connect as many terminal pairs as possible using paths that are mutually edge disjoint this leads to a variety of classic np complete problems for which approximability is not well understood we show a polylogarithmic approximation algorithm for the undirected edp problem in general graphs with a moderate restriction on graph connectivity we require the global minimum cut of g to be omega log 5n previously constant or polylogarithmic approximation algorithms were known for trees with parallel edges expanders grids grid like graphs and most recently even degree planar graphs these graphs either have special structure e g they exclude minors or have large numbers of short disjoint paths our algorithm extends previous techniques in that it applies to graphs with high diameters and asymptotically large minors

the exact determination of rectangle discrepancy for linear congruential pseudorandom numbers l algorithme presente peut etre utilise dans les cas de grands generateurs avec un temps de calcul tres raisonnable

additive preconditioning and aggregation in matrix computations we combine our novel svd free additive preconditioning with aggregation and other relevant techniques to facilitate the solution of a linear system of equations and other fundamental matrix computations our analysis and experiments show the power of our algorithms guide us in selecting most effective policies of preconditioning and aggregation and provide some new insights into these and related subjects compared to the popular svd based multiplicative preconditioners our additive preconditioners are generated more readily and for a much larger class of matrices furthermore they better preserve matrix structure and sparseness and have a wider range of applications e g they facilitate the solution of a consistent singular linear system of equations and of the eigenproblem

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

periodic orbits of the ensemble of sinai arnold cat maps and pseudorandom number generation we propose methods for constructing high quality pseudorandom number generators rngs based on an ensemble of hyperbolic automorphisms of the unit two dimensional torus sinai arnold map or cat map while keeping a part of the information hidden the single cat map provides the random properties expected from a good rng and is hence an appropriate building block for an rng although unnecessary correlations are always present in practice we show that introducing hidden variables and introducing rotation in the rng output accompanied with the proper initialization dramatically suppress these correlations we analyze the mechanisms of the single cat map correlations analytically and show how to diminish them we generalize the percival vivaldi theory in the case of the ensemble of maps find the period of the proposed rng analytically and also analyze its properties we present efficient practical realizations for the rngs and check our predictions numerically we also test our rngs using the known stringent batteries of statistical tests and find that the statistical properties of our best generators are not worse than those of other best modern generators

fast matrix decomposition in f2 in this work an efficient algorithm to perform a block decomposition for large dense rectangular matrices with entries in f 2 is presented matrices are stored as column blocks of row major matrices in order to facilitate rows operation and matrix multiplications with blocks of columns one of the major bottlenecks of matrix decomposition is the pivoting involving both rows and column exchanges since row swaps are cheap and column swaps are order of magnitude slower the number of column swaps should be reduced as much as possible here an algorithm that completely avoids the column permutations is presented an asymptotically fast algorithm is obtained by combining the four russian algorithm and the recursion with the strassen algorithm for matrix matrix multiplication moreover optimal parameters for the tuning of the algorithm are theoretically estimated and then experimentally verified a comparison with the state of the art public domain software sage shows that the proposed algorithm is generally faster

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

efficient maximum likelihood decoding of ldpc codes over the binary erasure channel we propose an efficient maximum likelihood ml decoding algorithm for decoding low density parity check ldpc codes over the binary erasure channel bec we also analyze the computational complexity of the proposed algorithm

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

a polylogarithmic approximation algorithm for edge disjoint paths with congestion 2 in the edge disjoint paths with congestion problem edpwc we are given an undirected n vertex graph g a collection m s1 t1 sk tk of pairs of vertices called demand pairs and an integer c the goal is to connect the maximum possible number of the demand pairs by paths so that the maximum edge congestion the number of paths sharing any edge is bounded by c when the maximum allowed congestion is c 1 this is the classical edge disjoint paths problem edp the best current approximation algorithm for edp achieves an o n approximation by rounding the standard multi commodity flow relaxation of the problem this matches the n lower bound on the integrality gap of this relaxation we show an o poly log k approximation algorithm for edpwc with congestion c 2 by rounding the same multi commodity flow relaxation this gives the best possible congestion for a sub polynomial approximation of edpwc via this relaxation our results are also close to optimal in terms of the number of pairs routed since edpwc is known to be hard to approximate to within a factor of log n 1 c 1 for any constant congestion c prior to our work the best approximation factor for edpwc with congestion 2 was o n3 7 and the best algorithm achieving a polylogarithmic approximation required congestion 14

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

extended backward differentiation methods in the numerical solution of the neutral volterra integro differential equations the numerical solution of first order volterra integro differential equations of neutral type with continuous kernel is considered preliminary results from the application of extended backward differentiation methods are given and comparisons are made with adams bashforth adams moulton predictor corrector methods and collocation methods

a nonlinear congruential pseudorandom number generator with power of two modulus a nonlinear congruential pseudorandom number generator is studied where the modulus is a power of two investigation of this generator was suggested by knuth 7 a simple necessary and sufficient condition is given for this generator to have the maximal period length

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

analysis of coppersmith s block wiedemann algorithm for the parallel solution of sparse linear systems by using projections by a block of vectors in place of a single vector it is possible to parallelize the outer loop of iterative methods for solving sparse linear systems we analyze such a scheme proposed by coppersmith for wiedemann s coordinate recurrence algorithm which is based in part on the krylov subspace approach we prove that by use of certain randomizations on the input system the parallel speed up is roughly by the number of vectors in the blocks when using as many processors our analysis is valid for fields of entries that have sufficiently large cardinality our analysis also deals with an arising subproblem of solving a singular block toeplitz system by use of the theory of toeplitz like matrices

factoring multivariate polynomials via partial differential equations a new method is presented for factorization of bivariate polynomials over any field of characteristic zero or of relatively large characteristic it is based on a simple partial differential equation that gives a system of linear equations as in berlekamp s and niederreiter s algorithms for factoring univariate polynomials the dimension of the solution space of the linear system is equal to the number of absolutely irreducible factors of the polynomial to be factored and any basis for the solution space gives a complete factorization by computing gcd s and by factoring univariate polynomials over the ground field the new method finds absolute and rational factorizations simultaneously and is easy to implement for finite fields local fields number fields and the complex number field the theory of the new method allows an effective hilbert irreducibility theorem thus an efficient reduction of polynomials from multivariate to bivariate

new hardness results for routing on disjoint paths in the classical node disjoint paths sf ndp problem the input consists of an undirected n vertex graph g and a collection mathcal m s 1 t 1 ldots s k t k of pairs of its vert

adams methods for neutral functional differential equations in this paper adams type methods for the special case of neutral functional differential equations are examined it is shown thatk step methods maintain orderk 1 for sufficiently small step size in a sufficiently smooth situation however when these methods are applied to an equation with a non smooth solution the order of convergence is only one some computational considerations are given and numerical experiments are presented

multicriteria analysis tools in real life problems applied optimization problems such as design identification design of controlled systems operational development of prototypes analysis of large scale systems and forecasting from observational data are multicriteria problems in essence construction of the feasible solution set is of primary importance in the above problems the definition of a feasible solution set is usually considered to be the skill of a designer even though this skill is essential it is by no means sufficient for the correct statement of the problem there are many antagonistic performance criteria and all kinds of constraints in these problems therefore it is quite difficult to correctly determine the feasible set as a result ill posed problems are solved and optimal solutions are searched for far from where they should be as a consequence the optimization results have no practical meaning in this work we propose methods and tools that will assist the designer in defining the feasible solution set correctly

efficient matrix preconditioners for black box linear algebra abstract the main idea of the black box approach in exact linear algebra is to reduce matrix problems to the computation of minimum polynomials in most cases preconditioning is necessary to obtain the desired result here good preconditioners will be used to ensure geometrical algebraic properties on matrices rather than numerical ones so we do not address a condition number we offer a review of problems for which algebraic preconditioning is used provide a bestiary of preconditioning problems and discuss several preconditioner types to solve these problems we present new conditioners including conditioners to preserve low displacement rank for toeplitz like matrices we also provide new analyses of preconditioner performance and results on the relations among preconditioning problems and with linear algebra problems thus improvements are offered for the efficiency and applicability of preconditioners the focus is on linear algebra problems over finite fields but most results are valid for entries from arbitrary fields

fast matrix rank algorithms and applications we consider the problem of computing the rank of an m n matrix a over a field we present a randomized algorithm to find a set of r rank a linearly independent columns in o vav p r field operations where vav denotes the number of nonzero entries in a and

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

convergence analysis of the solution of retarded and neutral delay differential equations by continuous numerical methods we have recently developed a generic approach to solving retarded and neutral delay differential equations ddes the approach is based on the use of an explicit continuous runge kutta formula and employs defect control the approach can be applied to problems with state dependent delays and vanishing delays in this paper we determine convergence properties for the numerical solution associated with methods that implement this approach we first analyze such properties for retarded ddes and then the analysis is extended to the case of neutral ddes ndes such an extension is particularly important since ndes have received little attention in the literature the main result we establish is that the global error of the numerical solution can be efficiently and reliably controlled by directly monitoring the magnitude of the associated defect our analysis can also be applied directly to other numerical methods which use defect error control based on runge kutta formulas for ddes

the lattice structure of pseudo random vectors generated by matrix generators abstract in many areas of science the problems treated by monte carlo simulations become more and more complex and more extensive because of that generators like linear congruential matrix generators are needed which produce enormously many pseudo random numbers in order to assess stochastical properties of the generated pseudo random vectors the lattice structure of these matrix generators is studied here

fast evaluation of logarithms in fields of characteristic two a method for determining logarithms in gf 2 n is presented its asymptotic running time is o exp cn 1 3 log 2 3 n for a small constant c while by comparison adleman s scheme runs in time o exp c n 1 2 log 1 2 n the ideas give a dramatic improvement even for moderate sized fields such as gf 2 127 and make barely possible computations in fields of size around 2 400 the method is not applicable to gf q for a large prime q

approximating disjoint path problems using packing integer programs in a packing integer program we are given a matrix a and column vectors b c with nonnegative entries we seek a vector x of nonnegative integers which maximizes c t x subject to ax leq b the edge and vertex disjoint path problems together with their unsplittable flow generalization are np hard problems with a multitude of applications in areas such as routing scheduling and bin packing these two categories of problems are known to be conceptually related but this connection has largely been ignored in terms of approximation algorithms we explore the topic of approximating disjoint path problems using polynomial size packing integer programs motivated by the disjoint paths applications we introduce the study of a class of packing integer programs called column restricted we develop improved approximation algorithms for column restricted programs a result that we believe is of independent interest additional approximation algorithms for disjoint paths are presented that are simple to implement and achieve good performance when the input has a special structure

a double large prime variation for small genus hyperelliptic index calculus in this article we examine how the index calculus approach for computing discrete logarithms in small genus hyperelliptic curves can be improved by introducing a double large prime variation two algorithms are presented the first algorithm is a rather natural adaptation of the double large prime variation to the intended context on heuristic and experimental grounds it seems to perform quite well but lacks a complete and precise analysis our second algorithm is a considerably simplified variant which can be analyzed easily the resulting complexity improves on the fastest known algorithms computer experiments show that for hyperelliptic curves of genus three our first algorithm surpasses pollard s rho method even for rather small field sizes