a 2 adic approach to the analysis of cyclic codes this paper describes how 2 adic numbers can be used to analyze the structure of binary cyclic codes and of cyclic codes defined over z sub 2 a a spl ges 2 the ring of integers modulo 2 sup a it provides a 2 adic proof of a theorem of mceliece that characterizes the possible hamming weights that can appear in a binary cyclic code a generalization of this theorem is derived that applies to cyclic codes over z sub 2 a that are obtained from binary cyclic codes by a sequence of hensel lifts this generalization characterizes the number of times a residue modulo 2 sup a appears as a component of an arbitrary codeword in the cyclic code the limit of the sequence of hensel lifts is a universal code defined over the 2 adic integers this code was first introduced by calderbank and sloane 1995 and is the main subject of this paper binary cyclic codes and cyclic codes over z sub 2 a are obtained from these universal codes by reduction modulo some power of 2 a special case of particular interest is cyclic codes over z sub 4 that are obtained from binary cyclic codes by means of a single hensel lift the binary images of such codes under the gray isometry include the kerdock preparata and delsart goethals codes these are nonlinear binary codes that contain more codewords than any linear code presently known fundamental understanding of the composition of codewords in cyclic codes over z sub 4 is central to the search for more families of optimal codes this paper also constructs even unimodular lattices from the hensel lift of extended binary cyclic codes that are self dual with all hamming weights divisible by 4 the leech lattice arises in this way as do extremal lattices in dimensions 32 through 48

two classes of quadratic apn binomials inequivalent to power functions this paper introduces the first found infinite classes of almost perfect nonlinear apn polynomials which are not carlet charpin zinoviev ccz equivalent to power functions at least for some values of the number of variables these are two classes of apn binomials from f2n to f2n for n divisible by 3 resp 4 we prove that these functions are extended affine ea inequivalent to any power function and that they are ccz inequivalent to the gold kasami inverse and dobbertin functions when n ges 12 this means that for n even they are ccz inequivalent to any known apn function in particular for n 12 20 24 they are therefore ccz inequivalent to any power function

monomial bent functions in this correspondence we focus on bent functions of the form f 2 n rarr f 2 where x rarr tr alphaxd the main contribution of this correspondence is that we prove that for n 4r r odd the exponent d 2r 1 2 allows the construction of bent functions this open question has been posed by canteaut based on computer experiments as a consequence for each of the well understood families of bent functions we now know an exponent d that yields to bent functions of the given type

differential properties of x mapsto x 2 t 1 we provide an extensive study of the differential properties of the functions x x2t 1 over bbf 2n for 1 t n we notably show that the differential spectra of these functions are determined by the number of roots of the linear polynomials x2t bx2 b 1 x where b varies in bbf 2n we prove a strong relationship between the differential spectra of x x2t 1 and x x2s 1 for s n t 1 as a direct consequence this result enlightens a connection between the differential properties of the cube function and of the inverse function we also determine the complete differential spectra of x x7 by means of the value of some kloosterman sums and of x x2t 1 for t n 2 n 2 1 n 2

secret sharing schemes from binary linear codes in principle every linear code can be used to construct a secret sharing scheme however in general determining the access structure of the scheme is very hard on the other hand finding error correcting codes that produce secret sharing schemes with efficient access structures is also difficult in this paper we study a set of minimal codewords for certain classes of binary linear codes and then determine the access structure of secret sharing schemes based on these codes furthermore we prove that the secret sharing schemes obtained are democratic in the sense that every participant is involved in the same number of minimal access sets

binary kloosterman sums modulo 256 and coefficients of the characteristic polynomial kloosterman sums are exponential sums on finite fields that have important applications in cryptography and coding theory we use stickelberger s theorem and the gross koblitz formula to determine the value of the binary kloosterman sum at a modulo 64 modulo 128 and modulo 256 in terms of coefficients of the characteristic polynomial of a

matrix rounding under the l p discrepancy measure and its application to digital halftoning we study the problem of rounding a real valued matrix into an integer valued matrix to minimize an lp discrepancy measure between them to define the lp discrepancy measure we introduce a family cal f of regions rigid submatrices of the matrix and consider a hypergraph defined by the family the difficulty of the problem depends on the choice of the region family cal f we first investigate the rounding problem by using integer programming problems with convex piecewise linear objective functions and give some nontrivial upper bounds for the lp discrepancy we propose laminar family for constructing a practical and well solvable class of cal f indeed we show that the problem is solvable in polynomial time if cal f is the union of two laminar families finally we show that the matrix rounding using l1 discrepancy for the union of two laminar families is suitable for developing a high quality digital halftoning software

computationally attractive reconstruction of bandlimited images from irregular samples an efficient method for the reconstruction of bandlimited images and the approximation of arbitrary images from nonuniform sampling values is developed the novel method is based on the observation that the reconstruction problem can be formulated as linear system of equations using two dimensional 2 d trigonometric polynomials where the matrix is of block toeplitz type with toeplitz blocks this system is solved iteratively by the conjugate gradient cg method we show that the use of so called adaptive weights in the establishment of the block toeplitz matrix can be seen as efficient preconditioning the superiority of the new method over conventional approaches is demonstrated by numerical experiments

a look ahead levinson algorithm for general toeplitz systems the authors present an extension of levinson s algorithm that is guaranteed to be weakly stable for a large class of general toeplitz matrices namely those that do not have many consecutive ill conditioned leading principal submatrices the new algorithm adapts itself to the given toeplitz matrix by skipping over all the ill conditioned leading principal submatrices encountered during the solution process this is done by a look ahead strategy that monitors the condition of the leading principal submatrices and if necessary switches to a block step of suitable size the overhead of the look ahead algorithm is typically small compared to the classical levinson algorithm and in addition a reliable condition number estimate is produced

hyperbent functions kloosterman sums and dickson polynomials this paper is devoted to the study of hyperbent functions in n variables i e bent functions which are bent up to a change of primitive roots in the finite field gf 2n our main purpose is to obtain an explicit trace representation for some classes of hyperbent functions we first exhibit an infinite class of monomial functions which is not hyperbent this result indicates that kloosterman sums on f2 m cannot be zero at some points for functions with multiple trace terms we express their spectra by means of dickson polynomials we then introduce a new tool to describe these hyperbent functions the effectiveness of this new method can be seen from the characterization of a new class of binomial hyperbent functions

nonbinary stabilizer codes over finite fields one formidable difficulty in quantum communication and computation is to protect information carrying quantum states against undesired interactions with the environment to address this difficulty many good quantum error correcting codes have been derived as binary stabilizer codes fault tolerant quantum computation prompted the study of nonbinary quantum codes but the theory of such codes is not as advanced as that of binary quantum codes this paper describes the basic theory of stabilizer codes over finite fields the relation between stabilizer codes and general quantum codes is clarified by introducing a galois theory for these objects a characterization of nonbinary stabilizer codes over fq in terms of classical codes over fq 2 is provided that generalizes the well known notion of additive codes over f4 of the binary case this paper also derives lower and upper bounds on the minimum distance of stabilizer codes gives several code constructions and derives numerous families of stabilizer codes including quantum hamming codes quadratic residue codes quantum melas codes quantum bose chaudhuri hocquenghem bch codes and quantum character codes the puncturing theory by rains is generalized to additive codes that are not necessarily pure bounds on the maximal length of maximum distance separable stabilizer codes are given a discussion of open problems concludes this paper

permutations of finite fields with prescribed properties classes of permutations of finite fields with various specific properties are often needed for applications we use a recent classification of permutation polynomials using their carlitz rank with advantage to produce examples of classes of permutations of f p for odd p which for instance are random have low differential uniformity prescribed cycle structure high polynomial degree large weight and large dispersion they are also easy to implement we indicate applications in coding and cryptography

how many weights can a cyclic code have upper and lower bounds on the largest number of weights in a cyclic code of given length dimension and alphabet are given an application to irreducible cyclic codes is considered sharper upper bounds are given for the special cyclic codes called here strongly cyclic whose nonzero codewords have period equal to the length of the code asymptotics are derived on the function gamma k q that is defined as the largest number of nonzero weights a cyclic code of dimension k over mathbb f q can have and an algorithm to compute it is sketched the nonzero weights in some infinite families of reed muller codes either binary or q ary as well as in the q ary hamming code are determined two difficult results of independent interest

parallel solution of symmetric positive definite systems with hyperbolic rotations abstract an algorithm based on hyperbolic rotations is presented for the solution of linear systems of equations ax b with symmetric positive definite coefficient matrix a forward elimination and backsubstitution are replaced by matrix vector multiplications rendering the method amenable to implementation on a variety of parallel and vector machines this method can be simplified and formulated without square roots if a is also toeplitz a systolic vlsi architecture implementing the resulting recurrence equations is more efficient than previously proposed pipelined toeplitz system solvers the hardware count becomes independent of the matrix size if its inverse is banded

parameters of goppa codes revisited we discuss parameters of goppa 1970 codes such as minimum distance covering radius distance distribution and generalized hamming weights by a variation on the exponential sums method and combinatorial arguments we sharpen known bounds on these parameters

a generalization of the levinson algorithm for hermitian toeplitz matrices with any rank profile the paper describes a recursive algorithm for solving hermitian toeplitz systems of linear equations without any restriction on the ranks of their nested toeplitz subsystems such a general algorithm is needed e g to obtain the eigenfilters for signal processing applications or to compute the inverse of a nondefinite toeplitz matrix the regular portion of the algorithm is made of the classical levinson recursion the singular portion requires solving some well defined systems of linear equations with gradient structure the dimension of each of these sytems equals the amplitude of the corresponding singularity

almost perfect nonlinear power functions on gf 2 n the niho case almost perfect nonlinear apn mappings are of interest for applications in cryptography we prove for odd n and the exponent d 22r 2r 1 where 4r 1 0modn that the power functions xd on gf 2n is apn the given proof is based on a new class of permutation polynomials which might be of independent interest our result supports a conjecture of niho stating that the power function xd is even maximally nonlinear or in other terms that the crosscorrelation function between a binary maximum length linear shift register sequences of degree n and a decimation of that sequence by d takes on precisely the three values 1 1 2 n 1 2

an efficient deterministic test for kloosterman sum zeros we propose a simple deterministic test for deciding whether or not a non zero element a in mathbb f 2 n or mathbb f 3 n is a zero of the corresponding kloosterman sum over these fields and analyse its complexity the test seems to have been overlooked in the literature for binary fields the test has an expected operation count dominated by just two mathbb f 2 n multiplications when n is odd with a slightly higher cost for even extension degrees making its repeated invocation the most efficient method to date to find a non trivial kloosterman sum zero in these fields the analysis depends on the distribution of sylow p subgroups in two corresponding families of elliptic curves which we prove using a theorem due to howe

some aspects of circulant preconditioners this paper studies circulant approximations of hermitian toeplitz matrices that are solutions of certain minimization problems relative to the frobenius norm the lub 1 or the lub infty norms the examinations supplement the family of the so called optimal and superoptimal preconditioners that have been proposed for the preconditioned conjugate gradient method for solving toeplitz systems t n x b for the new frobenius norm approximations it is shown that they can be computed in o n log n arithmetic operations and that the eigenvalues of the preconditioned linear equations are asymptotically clustered around 1 if t n is a leading principal submatrix of an infinite toeplitz matrix connected with a positive function of the wiener class

formally biorthogonal polynomials and a look ahead levinson algorithm for general teoplitz systems abstract systems of linear equations with toeplitz coefficient matrices arise in many important applications the classical levinson algorithm computes solutions of toeplitz systems with only o n 2 arithmetic operations as compared to o n 3 operations that are needed for solving general linear systems however the levinson algorithm in its original form requires that all leading principal submatrices be nonsingular in this paper an extension of the levinson algorithm to general toeplitz systems is presented the algorithm uses look ahead to skip over exactly singular as well as ill conditioned leading submatrices and at the same time it still fully exploits the toeplitz structure in our derivation of this algorithm we make use of the intimate connection of toeplitz matrices with formally biorthogonal polynomials in particular the occurence of singular or ill conditioned submatrices corresponds to a breakdown or near breakdown in the standard recurrence relations for biorthogonal polynomials we present new general recurrence relations that connect successive pairs in any given subsequence of all existing formally biorthogonal polynomials these recurrences then immediately lead to the proposed look ahead levinson algorithm for solving toeplitz systems implementation details for this algorithm and operation counts are given numerical experiments for toeplitz systems with ill conditioned submatrices are reported

a stabilized superfast solver for indefinite hankel systems abstract we present a stabilized superfast solver for indefinite hankel systems whose size is a power of 2 the hankel system is transformed into a loewner system which is solved by using an inversion formula for loewner matrices this explicit formula for the inverse of a loewner matrix contains certain parameters that are computed by solving two linearized rational interpolation problems on the unit circle the heart of our hankel solver is a superfast algorithm to solve these interpolation problems this algorithm is stabilized via pivoting iterative improvement and by giving the so called difficult interpolation points an adequate treatment we have implemented our algorithm in fortran 90 numerical examples illustrate the effectiveness of our approach

construction of highly nonlinear 1 resilient boolean functions with optimal algebraic immunity and provably high fast algebraic immunity in 2013 tang carlet and tang ieee tit 59 1 653 664 2013 presented two classes of boolean functions the functions in the first class are unbalanced and the functions in the second one are balanced both of those two classes of functions have high nonlinearity high algebraic degree optimal algebraic immunity and high fast algebraic immunity however they are not 1 resilient which represents a drawback for their use as filter functions in stream ciphers in this paper we first propose a large family of 1 resilient boolean functions having high lower bound on nonlinearity optimal algebraic immunity and optimal algebraic degree that is meeting the siegenthaler bound most notably we can mathematically prove that every function in n variables belonging to this family has fast algebraic immunity no less than n 6 which is the first time that an infinite family of 1 resilient functions with provably high fast algebraic immunity has been invented furthermore we exhibit a subclass of the family which has higher lower bound on nonlinearity than all the known 1 resilient functions with potentially optimal algebraic immunity and potentially high fast algebraic immunity

a look ahead algorithm for the solution of general hankel systems the solution of systems of linear equations with hankel coefficient matrices can be computed with onlyo n 2 arithmetic operations as compared too n 3 operations for the general cases however the classical hankel solvers require the nonsingularity of all leading principal submatrices of the hankel matrix the known extensions of these algorithms to general hankel systems can handle only exactly singular submatrices but not ill conditioned ones and hence they are numerically unstable in this paper a stable procedure for solving general nonsingular hankel systems is presented using a look ahead technique to skip over singular or ill conditioned submatrices the proposed approach is based on a look ahead variant of the nonsymmetric lanczos process that was recently developed by freund gutknecht and nachtigal we first derive a somewhat more general formulation of this look ahead lanczos algorithm in terms of formally orthogonal polynomials which then yields the look ahead hankel solver as a special case we prove some general properties of the resulting look ahead algorithm for formally orthogonal polynomials these results are then utilized in the implementation of the hankel solver we report some numerical experiments for hankel systems with ill conditioned submatrices

weight distributions of some binary primitive cyclic codes let s and k be integers such that s is a divisor of 2 sup k 1 let g x be a divisor of x sup s 1 over f sub 2 and let spl pi x be a primitive polynomial of degree k over f sub 2 we consider the binary cyclic code c of length n 2 sup k 1 generated by x sup n 1 g x spl pi x for special cases we determine the weight distribution of c by using the weights of the cyclic code of length s generated by x sup s 1 g x

constructing differentially 4 uniform permutations over bbf _ 2 2k via the switching method many block ciphers use permutations defined on f 22k with low differential uniformity high nonlinearity and high algebraic degree as their s boxes to provide confusion it is well known that for a function on f 2n the lowest differential uniformity is 2 and the functions achieving this lower bound are called almost perfect nonlinear apn functions however due to the lack of knowledge on apn permutations on f 22k differentially 4 uniform permutations are usually chosen as s boxes for example the currently endorsed advanced encryption standard chooses one such function the multiplicative inverse function as its s box by a recent survey on differentially 4 uniform permutations over f 22k there are only five known infinite families of such functions and most of them have small algebraic degrees in this paper we apply the powerful switching method to discover many ccz inequivalent infinite families of such functions on f 22k with optimal algebraic degree where k is an arbitrary positive integer this greatly expands the list of differentially 4 uniform permutations and hence provide more choices for the s boxes furthermore lower bounds for the nonlinearity of the functions obtained in this paper are presented and they imply that some infinite families have high nonlinearity

the moments of a kloosterman sum and the weight distribution of a zetterberg type binary cyclic code in this correspondence we give the moments of a kloosterman sum over fq in terms of the frequencies of weights in the binary zetterberg code of length q 1 which are known by the work of schoof and van der vlugt the method is illustrated by giving explicit formulae for the moments up to the tenth moment as a corollary the weight distribution of a zetterberg type binary cyclic code is obtained

an extended berlekamp massey algorithm for the inversion of toeplitz matrices the berlekamp massey algorithm bma which solves special toeplitz systems of linear equations is extended to an algorithm for inverting toeplitz matrices the original bma itself leads to one row of the inverse of the corresponding toeplitz matrix the other rows are derived using the same operations that are central to the original bma two alternatives for the extended bma are presented the first includes the usual bma without any changes and the second simplifies the structure by some modifications also in the original bma part both versions follow a tree like structure if the branches of the tree are implemented in parallel the time demand would be nearly the same as for the usual bma in contrast to other toeplitz algorithms only slight modifications must be incorporated to handle singular submatrices

classes of quadratic apn trinomials and hexanomials and related structures a method for constructing differentially 4 uniform quadratic hexanomials has been recently introduced by j dillon we give various generalizations of this method and we deduce the constructions of new infinite classes of almost perfect nonlinear quadratic trinomials and hexanomials from f22m to f22m we check for m 3 that some of these functions are ccz inequivalent to power functions

artin schreier curves exponential sums and coding theory abstract this is a survey of some results recently obtained on the distribution of the weights of some classical linear codes on the one hand such as the dual of the melas code and the geometric bch codes discovered by goppa subfield subcodes of goppa codes on the other hand these results depend upon the properties of certain algebraic curves defined over a finite field and the associated exponential sums such as the kloosterman sums

semibent functions from dillon and niho exponents kloosterman sums and dickson polynomials kloosterman sums have recently become the focus of much research most notably due to their applications in cryptography and coding theory in this paper we extensively investigate the link between the semibentness property of functions in univariate forms obtained via dillon and niho functions and kloosterman sums in particular we show that zeros and the value four of binary kloosterman sums give rise to semibent functions in even dimension with maximum degree moreover we study the semibentness property of functions in polynomial forms with multiple trace terms and exhibit criteria involving dickson polynomials

on spectra of bch codes derives an estimate for the error term in the binomial approximation of spectra of bch codes this estimate asymptotically improves on the bounds by sidelnikov 1971 kasami et al 1985 and sole 1990

recursive lower bounds on the nonlinearity profile of boolean functions and their applications the nonlinearity profile of a boolean function i e the sequence of its minimum hamming distances nlr f to all functions of degrees at most r for r ges 1 is a cryptographic criterion whose role against attacks on stream and block ciphers has been illustrated by many papers it plays also a role in coding theory since it is related to the covering radii of reed muller codes we introduce a method for lower bounding its values and we deduce bounds on the second order nonlinearity for several classes of cryptographic boolean functions including the welch and the multiplicative inverse functions used in the s boxes of the advanced encryption standard aes in the case of this last infinite class of functions we are able to bound the whole profile and we do it in an efficient way when the number of variables is not too small this allows showing the good behavior of this function with respect to this criterion as well

large p groups of automorphisms of algebraic curves in characteristic p abstract let s be a p subgroup of the k automorphism group aut x of an algebraic curve x of genus g 2 and p rank defined over an algebraically closed field k of characteristic p 3 nakajima 27 proved that if 2 then s p p 2 g 1 if equality holds x is a nakajima extremal curve we prove that if s p 2 p 2 p 1 g 1 then one of the following cases occurs i 0 and the extension k x k x s completely ramifies at a unique place and does not ramify elsewhere ii s p and x is an ordinary curve of genus g p 1 iii x is an ordinary nakajima extremal curve and k x is an unramified galois extension of a function field of a curve given in ii iii x is an ordinary nakajima extremal curve and k x is an unramified galois extension of a function field of a curve given in ii there are exactly p 1 subgroups m of s such that k x k x m is such a galois extension moreover if some of them is an abelian extension then s has maximal nilpotency class the full k automorphism group of any nakajima extremal curve is determined and several infinite families of nakajima extremal curves are constructed by using their pro p fundamental groups

determining the 2 sylow subgroup of an elliptic curve over a finite field in this paper we describe an algorithm that outputs the order and the structure including generators of the 2 sylow subgroup of an elliptic curve over a finite field to do this we do not assume any knowledge of the group order the results that lead to the design of this algorithm are of inductive type then a right choice of points allows us to reach the end within a linear number of successive halvings the algorithm works with abscissas so that halving of rational points in the elliptic curve becomes computing of square roots in the finite field efficient methods for this computation determine the efficiency of our algorithm

generalized artin mumford curves over finite fields let fqfq be the finite field of order q phq ph with p 2p 2 prime and h 1h 1 and let fq be a subfield of fqfq from any two q linearized polynomials l1 l2 f q t of degree q we construct an ordinary curve x l1 l2 x l1 l2 of genus g q 1 2g q 1 2 which is a generalized artin schreier cover of the projective line p1p1 the automorphism group of x l1 l2 x l1 l2 over the algebraic closure f q of fqfq contains a semidirect product of an elementary abelian p group of order q2q2 by a cyclic group of order q 1 we show that for l1 l2l1 l2 is the full automorphism group aut x l1 l2 aut x l1 l2 over f q for l1 l2l1 l2 there exists an extra involution and aut x l1 l1 aut x l1 l1 with a dihedral group of order 2 q 1 containing two different choices of the pair l1 l2 l1 l2 may produce birationally isomorphic curves even for l1 l2l1 l2 we prove that any curve of genus q 1 2 q 1 2 whose f q automorphism group contains an elementary abelian subgroup of order q2q2 is birationally equivalent to x l1 l2 x l1 l2 for some separable q linearized polynomials l1 l2l1 l2 of degree q we produce an analogous characterization in the special case l1 l2l1 l2 this extends a result on the artin mumford curves due to arakelian and korchmaros 1

a quantitative version of the absolute subspace theorem the celebrated subspace theorem of w m schmidt 12 says the following subspace theorem let l1 ln be linearly independent linear forms in n variables with real or complex algebraic coefficients suppose 0 consider the inequality 1 1 l1 x ln x 0 put 1 5 q c x r li x qi 1 i n then the parametric subspace theorem can be stated as follows parametric subspace theorem let c be a fixed tuple satisfying 1 3 let 0 then there are finitely many proper linear subspaces t1 tb of q n such that for any q which is sufficiently large there exists a subspace ti t1 tb with 1 6 q q c z ti

the spectra of super optimal circulant preconditioned toeplitz systems the solutions of hermitian positive definite toeplitz systems a n x b by the preconditioned conjugate gradient method are studied the preconditioner called the super optimal preconditioner is the circulant matrix t n that minimizes i c n 1 a n f over all circulant matrices c n the convergence rate is known to be governed by the distribution of the eigenvalues of t n 1 a n for n by n toeplitz matrix a n with entries being fourier coefficients of a positive function in the wiener class the asymptotic behaviour of the eigenvalues of the preconditioned matrix t n 1 a n is found as n increases and it is proved that they are clustered around one

the weights of the orthogonals of the extended quadratic binary goppa codes starting from results on elliptic curves and kloosterman sums over the finite field ge 2 sup t the authors determine the weights of the orthogonals of some binary linear codes the melas code of length the irreducible cyclic binary code of length 2 sup t 1 and the extended binary goppa codes defined by polynomials of degree two

on almost perfect nonlinear functions over mmb f_2 n we investigate some open problems on almost perfect nonlinear apn functions over a finite field of characteristic 2 we provide new characterizations of apn functions and of apn permutations by means of their component functions we generalize some results of nyberg 1994 and strengthen a conjecture on the upper bound of nonlinearity of apn functions we also focus on the case of quadratic functions we contribute to the current works on apn quadratic functions by proving that a large class of quadratic functions cannot be apn

hyperbent functions via dillon like exponents this paper is devoted to hyperbent functions with multiple trace terms including binomial functions via dillon like exponents we show how the approach developed by mesnager to extend the charpin gong family which was also used by wang and coworkers to obtain another similar extension fits in a much more general setting to this end we first explain how the original restriction for charpin gong criterion can be weakened before generalizing the mesnager approach to arbitrary dillon like exponents afterward we tackle the problem of devising infinite families of extension degrees for which a given exponent is valid and apply these results not only to reprove straightforwardly the results of mesnager and wang and coworkers but also to characterize the hyperbentness of several new infinite classes of boolean functions we go into full details only for a few of them but provide an algorithm and the corresponding software to apply this approach to an infinity of other new families finally we compare the asymptotic and practical performances of different characterizations including these in terms of hyperelliptic curves and actually build hyperbent functions in cases which could not be attained through naive computations of exponential sums

threshold accepting a general purpose optimization algorithm appearing superior to simulated annealing abstract a new general purpose algorithm for the solution of combinatorial optimization problems is presented the new threshold accepting method is even simpler structured than the wellknown simulated annealing approach the power of the new algorithm is demonstrated by computational results concerning the traveling salesman problem and the problem of the construction of error correcting codes moreover deterministic versions of the new heuristic turn out to perform nearly equally well consuming only a fraction of the computing time of the stochastic versions as an example the deterministic threshold accepting method yields very near to optimum tours for the famous 442 cities traveling salesman problem of grotschel within 1 to 2 s of cpu time

almost perfect nonlinear power functions on gf 2 sup n the welch case we summarize the state of the classification of almost perfect nonlinear apn power functions x sup d on gf 2 sup n and contribute two new cases to prove these cases we derive new permutation polynomials the first case supports a well known conjecture of welch stating that for odd n 2m 1 the power function x sup 2m 3 is even maximally nonlinear or in other terms that the crosscorrelation function between a binary maximum length linear shift register sequence of degree n and a decimation of that sequence by 2 sup m 3 takes on precisely the three values 1 1 spl plusmn 2 sup m 1

factoring integers with elliptic curves this paper is devoted to the description and analysis of a new algorithm to factor positive integers it depends on the use of elliptic curves the new method is obtained from pollard s p 1 method proc cambridge philos soc 76 1974 521 528 by replacing the multiplicative group by the group of points on a random elliptic curve it is conjectured that the algorithm determines a non trivial divisor of a composite number n in expected time at most k p log n 2 where p is the least prime dividing n and k is a function for which log k x 2 o 1 log x log log x for x x o in the worst case when n is the product of two primes of the same order of magnitude this is

look up table based halftoning algorithm screening is a low complexity halftoning algorithm that has been widely used in many applications however screen design requires that the stacking property be obeyed this constraint limits the texture quality at each gray level we present a look up table based halftoning algorithm for which the stacking constraint is not necessarily satisfied but the binary patterns for individual levels are still correlated the binary patterns are designed level by level using the direct binary search method the algorithm improves halftone image quality compared with screening

niho type cross correlation functions via dickson polynomials and kloosterman sums suppose that n 2k is even we study the cross correlation function between two m sequences for niho type decimations d 2 sup k 1 s 1 we develop a new technique to study the value distribution of these cross correlation functions which makes use of dickson polynomials as a first application we derive here the distribution of the six valued cross correlation function for s 3 and odd k up to a term which depends on kloosterman sums in addition applying simpler methods we prove a theorem providing niho type decimations with four valued cross correlation functions and their distribution we conjecture that the latter result actually covers all such decimations

schur algorithms for hermitian toeplitz and hankel matrices with singular leading principal submatrices it is shown how a simple matrix algebra procedure can be used to induce schur type algorithms for the solution of certain toeplitz and hankel linear systems of equations when given levinson durbin algorithms for such problems the algorithm of p delsarte et al 1985 for hermitian toeplitz matrices in the singular case is used to induce a schur algorithm for such matrices an algorithm due to g heinig and k rost 1984 for hankel matrices in the singular case is used to induce a schur algorithm for such matrices the berlekamp massey algorithm is viewed as a kind of levinson durbin algorithm and so is used to induce a schur algorithm for the minimal partial realization problem the schur algorithm for hermitian toeplitz matrices in the singular case is shown to be amenable to implementation on a linearly connected parallel processor array of the sort considered by kung and hu 1983 and in fact generalizes their result to the singular case

new spatial measure for dispersed dot halftoning assuring good point distribution in any density as a core component of dispersed dot halftoning this paper focuses on the definition of new measures for giving or measuring good point distributions in a plane by defining good point distributions from a purely geometric viewpoint of circle packing it is shown that the energy defined by a certain strong convex function satisfies the necessary conditions for obtaining good point distributions in any point density by minimizing the energy the energy with such the conditions are mathematically plain and there are no obscure parameters the theory is also significantly motivated by a requirement of the adjustability to discrete spaces and it is shown that the conditions actually work well also in the spaces as an application by using technically simple methods dispersed dot halftone masks are designed and goodness of point distributions of masks are estimated

bent and hyper bent functions in polynomial form and their link with some exponential sums and dickson polynomials bent functions are maximally nonlinear boolean functions with an even number of variables they were introduced by rothaus in 1976 for their own sake as interesting combinatorial objects but also because of their relations to coding theory reed muller codes and applications in cryptography design of stream ciphers they have attracted a lot of research specially in the last 15 years the class of bent functions contains a subclass of functions introduced by youssef and gong in 2001 the so called hyper bent functions whose properties are still stronger and whose elements are still rarer than bent functions bent and hyper bent functions are not classified a complete classification of these functions is elusive and looks hopeless so it is important to design constructions in order to know as many of hyper bent functions as possible this paper is devoted to the constructions of bent and hyper bent boolean functions in polynomial forms we survey and present an overview of the constructions discovered recently we extensively investigate the link between the bentness property of such functions and some exponential sums involving dickson polynomials and give some conjectures that lead to constructions of new hyper bent functions

new classes of almost bent and almost perfect nonlinear polynomials new infinite classes of almost bent and almost perfect nonlinear polynomials are constructed it is shown that they are affine inequivalent to any sum of a power function and an affine function

efficient embeddings of trees in hypercubes the boolean hypercube is a particularly versatile network for parallel computing it is well known that multi dimensional grid machines can be simulated on a hypercube with no communications overhead in this paper we show that every bounded degree tree can be simulated on the hypercube with constant communications overhead our proof in fact shows that every bounded degree graph with an 0 1 separator can be embedded in a hypercube of the same size with dilation and congestion both 0 1 we prove also that not all bounded degree graphs can be efficiently embedded within the hypercube

an algorithm for solving discrete time wiener hopf equations based upon euclid s algorithm an algorithm for solving a discrete time wiener hopf equation is presented based upon euclid s algorithm the discrete time wiener hopf equation is a system of linear inhomogeneous equations with a given toeplitz matrix m a given vector b and an unknown vector lambda such that m lambda b the algorithm is able to find a solution of the discrete time wiener hopf equation for any type of toeplitz matrices except for the all zero matrix while the levinson algorithm and the trench algorithm are not available when at least one of the principal submatrices of the toeplitz matrix m is singular the algorithm gives a solution if one exists even when the toeplitz matrix m is singular while the brent gustavson yun algorithm only states that the toeplitz matrix m is singular the algorithm requires o t 2 arithmetic operations for t unknowns in the sense that the number of multiplications or divisions is directly proportional to t 2 like the levinson and trench algorithms furthermore a faster algorithm is also presented based upon the half greatest common divisor algorithm and hence it requires o t log 2 t arithmetic operations like the brent gustavson yun algorithm

diophantine equations with binomial coefficients and perturbations of symmetric boolean functions this paper presents a study of perturbations of symmetric boolean functions in particular it establishes a connection between exponential sums of these perturbations and diophantine equations of the form sum l 0 n binom n l x l 0 where x j belongs to some fixed bounded subset gamma of mathbb z the concepts of trivially balanced symmetric boolean function and sporadic balanced boolean function are extended to this type of perturbations an observation made by canteaut and videau for symmetric boolean functions of fixed degree is extended to be specific it is proved that excluding the trivial cases balanced perturbations of fixed degree do not exist when the number of variables grows some sporadic balanced perturbations are presented finally a beautiful but unexpected identity between exponential sums for perturbations of two different symmetric boolean functions is also included in this work

a look ahead bareiss algorithm for general toeplitz matrices the bareiss algorithm is one of the classical fast solvers for systems of linear equations with toeplitz coefficient matrices the method takes advantage of the special structure and it computes the solution of a toeplitz system of order n with only o n 2 arithmetic operations instead of o n 3 operations however the original bareiss algorithm requires that all leading principal submatrices be nonsingular and the algorithm is numerically unstable if singular or ill conditioned submatrices occur in this paper an extension of the bareiss algorithm to general toeplitz systems is presented using look ahead techniques the proposed algorithm can skip over arbitrary blocks of singular or ill conditioned submatrices and at the same time it still fully exploits the toeplitz structure implementation details and operations counts are given and numerical experiments are reported we also discuss special versions of the proposed look ahead bareiss algorithm for hermitian indefinite toeplitz systems and banded toeplitz systems

on the values of kloosterman sums given a prime p and a positive integer n we show that the shifted kloosterman sums sigmaxisinf p npsi x alphaxpn 2 sigmaxisinf p npsi x alphax 1 1 alphaisinf pn where psi is a nontrivial additive character of a finite field fpn of pn elements do not vanish if alpha belongs to a small subfield fpm sube fpn this complements recent results of p charpin and g gong which in turn were motivated by some applications to bent functions

computing the l power torsion of an elliptic curve over a finite field the algorithm we develop outputs the order and the structure including generators of the l sylow subgroup of the group of rational points of an elliptic curve defined over a finite field to do this we do not assume any knowledge of the group order we are able to choose points in such a way that a linear number of successive l divisions leads to generators of the subgroup under consideration after the computation of a couple of polynomials each division step relies on finding rational roots of polynomials of degree l we specify in complete detail the case l 3 when the complexity of each trisection is given by the computation of cubic roots in finite fields

codes with the same coset weight distributions as the z sub 4 linear goethals codes we study the coset weight distributions of the family of z sub 4 linear goethals like codes of length n 2 sup m 1 m spl ges 3 odd constructed by helleseth kumar and shanbhag see designs codes and cryptography vol 17 no 1 3 p 246 62 1999 these codes have the same lee weight distribution as the z sub 4 linear goethals code spl gscr sub 1 and therefore taking into account the result of hammons kumar sloane calderbank and sole the binary images of all these codes by the gray map have the same weight distribution as the binary goethals code we prove that all these codes have the same coset weight distributions as the z sub 4 linear goethals code constructed by hammons kumar sloane calderbank and sole see ibid vol 40 p 301 19 march 1994 the cosets of weight four is the most difficult case in order to find the number of codewords of weight four in a coset of weight four we have to solve a nonlinear system of equations over the galois field gf 2 sup m such a system the degree of one of the equations depends on k we prove that the distribution of solutions to such a system does not depend on k and therefore coincides with the case k 1 considered earlier by helleseth and zinoviev see designs codes and cryptography vol 17 no 1 3 p 246 62 1999 for k 1 we solved this system in the following sense for all cases of cosets of weight four we have either an exact expression or an expression in terms of the kloosterman sums

stable row recurrences for the pade table and generically superfast lookahead solvers for non hermitian toeplitz systems abstract we present general recurrences for the pade table that allow us to skip ill conditioned pade approximants while we proceed along a row of the table in conjunction with a certain inversion formula for toeplitz matrices these recurrences form the basis for fast algorithms for solving non hermitian toeplitz systems under the assumption that the lookahead step size i e the number of successive skipped approximants remains bounded we give both o n 2 and o n log 2 n algorithms which are presumably weakly stable with little additional work still in o n 2 operations we can also obtain a decomposition of the toeplitz matrix t according to tr ld where r is upper triangular l is unit lower triangular and d is block diagonal the relation to continued fractions is also discussed

symmetric gaussian elimination for cauchy type matrices with application to positive definite toeplitz matrices the problem of solving linear equations with a toeplitz matrix t appears in many applications often t is positive definite but ill conditioned with many small eigenvalues in this case fast and superfast algorithms may show a very poor behavior or even break down in recent papers the transformation of a toeplitz matrix into a cauchy type matrix is proposed the resulting new linear equations can be solved in o n 2 operations using standard pivoting strategies which leads to very stable fast methods also for ill conditioned systems the basic tool is the formulation of gaussian elimination for matrices with low displacement rank in this paper we will transform a hermitian toeplitz matrix t into a cauchy type matrix b ftf rm h by applying the fourier transform we will prove some useful properties of b and formulate a symmetric gaussian elimination algorithm for positive definite b using the symmetry and persymmetry of t we can reduce the total costs of this algorithm compared with unsymmetric gaussian elimination for complex hermitian t the complexity of the new algorithm is then nearly the same as for the schur algorithm furthermore it is possible to include some strategies for ill conditioned positive definite matrices that are well known in optimization numerical examples show that this new algorithm is fast and reliable

cyclic codes and reducible additive equations we prove a weil serre type bound on the number of solutions of a class of reducible additive equations over finite fields using the trace representation of cyclic codes this enables us to write a general estimate for the weights of cyclic codes we extend wolfmann s weight bound to a larger classes of cyclic codes in particular our result is applicable to any cyclic code over fp and fp 2 where p is an arbitrary prime examples indicate that our bound performs very well against the bose chaudhuri hocquenghem bch bound and that it yields the exact minimum distance in some cases

on integer values of kloosterman sums this paper considers rational integer values of kloosterman sums over finite fields of characteristic p 3 we shall prove two main results the first one is a congruence relation satisfied by possible integer values one consequence is that there are no kloosterman zeroes in the case of characteristic p 3 which generalizes recent works by shparlinski moisio and lisonek on this subject this in turn implies that there are no dillon type bent functions in the case p 3 thus answering a question posed recently by helleseth and kholosha our other main result states that the kloosterman sum obtains an integer value at a point if and only if the same sum lifted to any extension field remains an integer

further optimized look ahead recurrences for adjacent rows in the pade table and toeplitz matrix factorizations abstract in a recent paper we introduced a new look ahead algorithm for recursively computing pade approximants this algorithm generates a subsequence of the pade approximants on two adjacent rows defined by fixed numerator degree of the pade table its two basic versions reduce to the classical levinson and schur algorithms if no look ahead is required in this paper we show that the computational overhead of the look ahead steps in the o n2 versions of the look ahead levinson and the look ahead schur type algorithm can be further reduced if the algorithms are used to solve toeplitz systems of equations tx b then the corresponding block ldu decompositions of t 1 or t respectively can be found with less computational effort than with any other look ahead algorithm available today

on the second order nonlinearities of some bent functions the rth order nonlinearity of boolean functions plays a central role against several known attacks on stream and block ciphers it plays also an important role in coding theory since its maximum equals the covering radius of the rth order reed muller code but it is difficult to calculate and even to bound in this paper we show lower bounds on the second order nonlinearity of two subclasses of well known bent functions we first improve a known lower bound on the second order nonlinearity of the simplest partial spread bent functions whose nonlinearity profile has been bounded by the second author this improvement allows obtaining a better bound for the whole profile we subsequently give a lower bound on the second order nonlinearity of some infinite class of maiorana mcfarland m m bent functions which generalizes a result by gangopadhyay et al

embedding multidimensional grids into optimal hypercubes abstract let g and h be graphs with v h v g and f v g v h a one to one map of their vertices let dilation f max dist h f x f y x y e g where dist h v w is the distance between vertices v and w of h now let b g h min f dilation f over all such maps f the parameter b g h is a generalization of the classic and well studied bandwidth of g defined as b g p n where p n is the path on n points and n v g let a 1 a 2 a k be the k dimensional grid graph with integer values 1 through a i in the i th coordinate in this paper we study b g h in the case when g a 1 a 2 a k and h is the hypercube q n of dimension n log 2 v g the hypercube of smallest dimension having at least as many points as g our main result is that b a 1 a 2 a k q n 3 k provided a i 2 22 for each 1 i k for such g the bound 3k improves on the previous best upper bound 4 k o 1 our methods include an application of knuth s result on two way rounding and of the existence of spanning regular cyclic caterpillars in the hypercube

tight bounds for chevalley warning ax katz type estimates with improved applications following the work of chevalley and warning ax obtained a bound on the p divisibility of exponential sums involving multi variable polynomials of fixed degree d over a finite field of characteristic p this bound was subsequently improved by katz more recently moreno and moreno and adolphson and sperber derived bounds that in many instances improved upon the ax katz result here we derive a tight bound on the p divisibility of the exponential sums while exact computation of this bound requires the solution of a system of modular equations approximations are provided which in several classes of examples improve on the results of chevalley and warning ax and katz adolphson and sperber and moreno and moreno all of the above results readily translate into bounds on the p divisibility of the number of zeros of multi variable polynomials an important consequence of one of our main results is a method to find classes of examples for which bounds on divisibility of the number of solutions of a system of polynomial equations over finite fields are tight in particular we give classes of examples for which the moreno moreno bound is tight it is important to note that we have also found applications of our results to coding theory computation of the covering radius of certain codes and to waring s problem over finite fields these will be described elsewhere

displacement structure theory and applications in this survey paper we describe how strands of work that are important in two different fields matrix theory and complex function theory have come together in some work on fast computational algorithms for matrices with what we call displacement structure in particular a fast triangularization procedure can be developed for such matrices generalizing in a striking way an algorithm presented by schur 1917 j reine angew math 147 1917 pp 205 232 in a paper on checking when a power series is bounded in the unit disc this factorization algorithm has a surprisingly wide range of significant applications going far beyond numerical linear algebra we mention among others inverse scattering analytic and unconstrained rational interpolation theory digital filter design adaptive filtering and state space least squares estimation

embedding of grids into optimal hypercubes this paper addresses the following graph embedding question given a grid and the smallest hypercube with at least as many nodes as the grid has grid points how can grid points be assigned to hypercube nodes with at most one grid point per node so as to keep grid neighbors as near each other as possible in the hypercube for two dimensional grids a simple embedding strategy is introduced which ensures that grid neighbors are always mapped to hypercube nodes that are within a distance of two edges of each other moreover this embedding algorithm takes time linear to the number of grid points extending and adding further concepts an embedding strategy is formulated for d dimensional grids which ensures that grid neighbors are distanced by o d edges in the hypercube this algorithm takes o d times text the number of grid points time

look ahead methods for block hankel systems in this paper the updating formulas used by three look ahead methods for solving hankel systems are generalized to the square block case each of the original methods was described in the literature using the terminology of different but strongly related fields formal orthogonal polynomials pade approximants structured matrices this paper gives several of these connections generalized to the block case and shows that each viewpoint has its own merits

monomial and quadratic bent functions over the finite fields of odd characteristic considered are p ary bent functions having the form f x tr sub n spl sigma sub i 0 sup s a sub i x sup di a new class of ternary monomial regular bent function with the dillon exponent is discovered the existence of dillon bent functions in the general case is an open problem of deciding whether a certain kloosterman sum can take on the value 1 also described is the general gold like form of a bent function that covers all the previously known monomial quadratic cases the weak regularity of the new as well as of known monomial bent functions is discussed and the first example of a not weakly regular bent function is given finally some criteria for an arbitrary quadratic function to be bent are proven

weil serre type bounds for cyclic codes we give a new method in order to obtain weil serre type bounds on the minimum distance of arbitrary cyclic codes over fpe of length coprime to p where e ges 1 is an arbitrary integer in an earlier paper we obtained weil serre type bounds for such codes only when e 1 or e 2 using lengthy explicit factorizations which seems hopeless to generalize the new method avoids such explicit factorizations and it produces an effective alternative using our method we obtain weil serre type bounds in various cases by examples we show that our bounds perform very well against bose chaudhuri hocquenghem bch bound and they yield the exact minimum distance in some cases

optimized halftoning using dot diffusion and methods for inverse halftoning unlike the error diffusion method the dot diffusion method for digital halftoning has the advantage of pixel level parallelism however the image quality offered by error diffusion is still regarded as superior to most of the other known methods we show how the dot diffusion method can be improved by optimization of the so called class matrix by taking the human visual characteristics into account we show that such optimization consistently results in images comparable to error diffusion without sacrificing the pixel level parallelism adaptive dot diffusion is also introduced and then a mathematical description of dot diffusion is derived furthermore inverse halftoning of dot diffused images is discussed and two methods are proposed the first one uses projection onto convex sets pocs and the second one uses wavelets of these methods the wavelet method does not make use of the knowledge of the class matrix embedded multiresolution dot diffusion is also discussed which is useful for rendering at different resolutions and transmitting images progressively

construction of n variable n equiv 2 bmod 4 balanced boolean functions with maximum absolute value in autocorrelation spectra 2 frac n 2 in this paper we consider the maximum absolute value delta f in the autocorrelation spectrum not considering the zero point of a function f in an even number of variables n bent functions possess the highest nonlinearity with delta f 0 the long standing open question for two decades in this area is to obtain a theoretical construction of balanced functions with delta f so far there are only a few examples of such functions for n 10 14 but no general construction technique is known in this paper we mathematically construct an infinite class of balanced boolean functions on n variables having absolute indicator strictly lesser than delta n 2 n 2 2 n 6 4 nonlinearity strictly greater than rho n 2 n 1 2 n 2 2 n 2 3 5 cdot 2 n 2 4 and algebraic degree n 1 where n equiv 2 pmod 4 and n geq 46 while the bound n geq 46 is required for proving the generic result our construction starts from n 18 and we could obtain balanced functions with delta f and nonlinearity 2 n 1 2 n 2 for n 18 22 and 26

a lookahead algorithm for the solution of block toeplitz systems abstract we give a weakly stable algorithm to solve a block toeplitz system of linear equations if the lookahead steps taken to compute the parameters of the inversion formula for the block toeplitz matrix are small compared to the order n of the matrix the algorithm requires o n2 floating point operations the parameters of the inversion formula are interpreted and computed in a recursive way as solutions of certain interpolation problems given the formal laurent series based on the data of the block toeplitz matrix

a duality theorem for the weight distribution of some cyclic codes a close relationship between the weight enumerators of two cyclic codes is presented the weights of the codewords are expressed in terms of kloosterman sums the weil carlitz uchiyama estimate for kloosterman sums is used to obtain bounds for the minimum distance of the codes a relation with negacyclic codes is discussed

computations with gohberg semencul type formulas for toeplitz matrices abstract the inverse of a toeplitz matrix t n can be represented in different ways by gohberg semencul type formulas as the sum of products of upper and lower triangular toeplitz matrices if we have given such a formula we can solve every equation t n b in o n log n steps there are three main questions arising with such representations of t n 1 1 which special linear equations can be solved in order to get generating vectors for gohberg semencul type formulas 2 which algorithm we want to apply to solve these questions and 3 which special gohberg semencul type formula we want to use for evaluating t n 1 b in this paper we present an elementary approach to derive all gohberg semencul type formulas then we introduce representations of t n 1 with special properties in particular we prove that there exists a gohberg semencul type formula such that the generating vectors are pairwise orthogonal finally based on the previous results we give new fast and stable algorithms for solving linear toeplitz systems that can be used to compute generating vectors for gohberg semencul type formulas in the case of a breakdown or near breakdown in the k th step the new method introduces a perturbation in the k th entry t k of t n such that the perturbed submatrix t k is well conditioned by using the sherman morrison woodbury formula it is possible to recover the original problem as soon as the corresponding submatrix t k r is again nonsingular

are 2 weight projective cyclic codes irreducible we prove the following result on a 2 weight projective cyclic code c over fq if q 2 then c is irreducible if qne2 then either c is irreducible or c is the direct sum of two 1 weight irreducible cyclic codes

a decomposition of toeplitz matrices and optimal circulant preconditioning abstract a factorization of the given toeplitz matrix t obtained in this work is used to derive an optimal circulant preconditioner c in the sense of minimizing the frobenius norm of i ct by use of the fast fourier transform the computation of the new preconditioner can be performed in o n log n operations