h asymptotic stability by fixed point in neutral nonlinear differential equations with delay abstract we use fixed point theory to obtain h asymptotic stability results about the zero solution of nonlinear differential equations with functional delay neither boundedness of the deviations nor a fixed sign on the coefficients functions are asked an explicit estimation of the solutions is obtained and a necessary and sufficient condition is determined classical results are improved and generalized

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

classification of bent monomials constructions of bent multinomials and upper bounds on the nonlinearity of vectorial functions this paper is composed of two main parts related to the nonlinearity of vectorial functions the first part is devoted to maximally nonlinear n m functions the so called bent vectorial functions which contribute to an optimal resistance to both linear and differential attacks on symmetric cryptosystems they can be used in block ciphers at the cost of additional diffusion compression expansion layers or as building blocks for the construction of substitution boxes s boxes and they are also useful for constructing robust codes and algebraic manipulation detection codes a main issue on bent vectorial functions is to characterize bent monomial functions tr m n lambda x d from mathbb f 2 n to mathbb f 2 m where m is a divisor of n leading to a classification of those bent monomials we also treat the case of functions with multiple trace terms involving general results and explicit constructions furthermore we investigate some open problems raised by pasalic et al and muratovic ribic et al in a series of papers on vectorial functions the second part is devoted to the nonlinearity of n m functions no tight upper bound is known when frac n 2 the covering radius bound is the only known upper bound in this range the sidelnikov chabaud vaudenay bound coincides with it when m n 1 and it has no sense when m finding better bounds is an open problem since the 1990s moreover no bound has been found during the last 23 years which improve upon the covering radius bound for a large part of n m functions we derive such upper bounds for functions which are sufficiently unbalanced or which satisfy some conditions these upper bounds imply some necessary conditions for vectorial functions to have large nonlinearity

three classes of minimal linear codes over the finite fields of odd characteristic minimal linear codes are a special subclass of linear codes and have significant applications in secret sharing and secure two party computation in this paper we focus on constructing minimal linear codes with frac w min w max leq frac p 1 p for any odd prime p based on a generic construction of linear codes where w min and w max denote the minimum and maximum nonzero weights in a code respectively first we present two new infinite families of minimal linear codes with two or three weights by selecting suitable subcode of linear codes which are not minimal second we also present an infinite family of minimal linear codes by employing partial spreads which can be viewed as a generalization of the construction of ding et al in addition we determine the weight distributions of all these minimal linear codes

linear codes from some 2 designs a classical method of constructing a linear code over mathrm gf q with a t design is to use the incidence matrix of the t design as a generator matrix over mathrm gf q of the code this approach has been extensively investigated in the literature in this paper a different method of constructing linear codes using specific classes of 2 designs is studied and linear codes with a few weights are obtained from almost difference sets difference sets and a type of 2 designs associated to semibent functions two families of the codes obtained in this paper are optimal the linear codes presented in this paper have applications in secret sharing and authentication schemes in addition to their applications in consumer electronics communication and data storage systems a coding theory approach to the characterization of highly nonlinear boolean functions is presented

transform domain analysis of des the data encryption standard des can be regarded as a nonlinear feedback shift register nlfsr with input from this point of view the tools for pseudo random sequence analysis are applied to the s boxes in des the properties of the s boxes of des under the fourier transform hadamard transform extended hadamard transform and the avalanche transform are investigated two important results about the s boxes of des are found the first result is that nearly two thirds of the total 32 functions from gf 2 sup 6 to gf 2 which are associated with the eight s boxes of des have the maximal linear span g3 and the other one third have linear span greater than or equal to 57 the second result is that for all s boxes the distances of the s boxes approximated by monomial functions has the same distribution as for the s boxes approximated by linear functions some new criteria for the design of permutation functions for use in block cipher algorithms are discussed

characterization of m sequences of lengths 2 2k 1 and 2 k 1 with three valued cross correlation considered is the distribution of the cross correlation between in sequences of length 22k 1 where m 2k and m sequences of shorter length 2k 1 new pairs of m sequences with three valued cross correlation are found and the complete correlation distribution is determined finally we conjecture that there are no more cases with a three valued cross correlation apart from the ones proven here

new families of binary sequences with low correlation for a positive integer n new families s and u of binary sequences of period 2 sup n 1 with low correlations are proposed where for some positive integer e s is defined for odd n e and u for even n e the family s has four valued correlations and is a generalization of the family of gold like sequences introduced by s boztas and p v kumar see ibid vol 40 p 532 7 1994 the family u which is also a generalization of the sequence family defined by p udaya polyphase and frequency hopping sequences obtained from finite rings ph d dissertation dept elec eng indian inst technol kanpur 1992 has six valued correlations the relationship between gold like sequences and gold sequences is the same as the relationship between the family s and the family constructed from the binary sequences partially contributed by r gold see ibid vol it 14 p 154 6 1968 t kasami see coordinated sci lab univ of illinois urbana champaign tech rep r 285 ad 632574 1966 and welch using a lifting idea no j s and kumar p v ibid vol 35 p 371 9 1989 for the families s and u families of binary sequences with the same correlation distributions and large linear span are also constructed

on certain values of kloosterman sums let k q n a be a kloosterman sum over the finite field f q n of characteristic p in this note so called subfield conjecture is proved if a ne 0 belongs to the proper subfield f q of f q n then k q n a ne 1 this completes recent works on the subfield conjecture by shparlinski and moisio and lisonek the problem is motivated by some applications to bent functions moreover in the course of the proof a large class of translates of dickson polynomials are shown to be irreducible

new constructions of quadratic bent functions in polynomial form new quadratic bent functions in polynomial form are constructed in this paper the constructions give new boolean bent generalized boolean bent and p ary bent functions based on z 4 valued quadratic forms a simple method provides several new constructions of generalized boolean bent functions from these generalized boolean bent functions a method is presented to transform them into boolean bent and semi bent functions moreover many new p ary bent functions can also be obtained by applying similar methods

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

secret sharing schemes from three classes of linear codes secret sharing has been a subject of study for over 20 years and has had a number of real world applications there are several approaches to the construction of secret sharing schemes one of them is based on coding theory in principle every linear code can be used to construct secret sharing schemes but determining the access structure is very hard as this requires the complete characterization of the minimal codewords of the underlying linear code which is a difficult problem in general in this paper a sufficient condition for all nonzero codewords of a linear code to be minimal is derived from exponential sums some linear codes whose covering structure can be determined are constructed and then used to construct secret sharing schemes with nice access structures

several new infinite families of bent functions and their duals bent functions are optimal combinatorial objects since their introduction substantial efforts have been directed toward their study in the last three decades a complete classification of bent functions is elusive and looks hopeless today therefore not only their characterization but also their generation are challenging problems this paper is devoted to the construction of bent functions first we provide several new effective constructions of bent functions self dual bent functions and antiself dual bent functions second we provide seven new infinite families of bent functions by explicitly calculating their dual

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

fixed points stability and exact linearization abstract we study the scalar equation x f t x x x b t g x t l 0 by means of contraction mappings conditions are obtained to ensure that each solution x t x t 0 0 as t the conditions allow f to grow as large as t but not as large as t 2 this is parallel to the classical result of smith quart j math oxford ser 12 2 1961 123 for the linear equation without a delay

the lower bounds on the second order nonlinearity of three classes of boolean functions with high nonlinearity the rth order nonlinearity of boolean functions is an important cryptographic criterion associated with some attacks on stream and block ciphers it is also very useful in coding theory since it is related to the covering radii of reed muller codes this paper tightens the lower bounds of the second order nonlinearity of three classes of boolean functions in the form f x tr x d in n variables where 1 d 2 m 1 3 and n 2m or 2 d 2 m 2 m 1 2 1 n 2m and m is odd or 3 d 2 2 r 2 r 1 1 and n 4r

weak compactness and fixed point property for affine mappings abstract it is shown that a closed convex bounded subset of a banach space is weakly compact if and only if it has the generic fixed point property for continuous affine mappings the class of continuous affine mappings can be replaced by the class of affine mappings which are uniformly lipschitzian with some constant m 1 in the case of c0 the class of affine mappings which are uniformly lipschitzian with some constant m 6 in the case of quasi reflexive james space j and the class of nonexpansive affine mappings in the case of l embedded spaces

on the approximate fixed point property in abstract spaces let x be a hausdorff topological vector space x its topological dual and z a subset of x in this paper we establish some results concerning the x z approximate fixed point property for bounded closed convex subsets c of x three major situations are studied first when z is separable in the strong topology second when x is a metrizable locally convex space and z x and third when x is not necessarily metrizable but admits a metrizable locally convex topology compatible with the duality our approach focuses on establishing the frechet urysohn property for certain sets with regarding the x z topology the support tools include the brouwer s fixed point theorem and an analogous version of the classical rosenthal s l 1 theorem for l 1 sequences in metrizable case the results are novel and generalize previous work obtained by the authors in banach spaces

on semibent boolean functions we show that any boolean function in even dimension equal to the sum of a boolean function g which is constant on each element of a spread and of a boolean function h whose restrictions to these elements are all linear is semibent if and only if g and h are both bent we deduce a large number of infinite classes of semibent functions in explicit bivariate respectively univariate polynomial form

new families of codebooks achieving the levenstein bound in this paper a construction of codebooks based on a set of bent functions satisfying certain conditions is introduced it includes some earlier constructions of codebooks meeting the levenstein bound as special cases with this construction two new families of codebooks achieving the levenstein bound are obtained the codebooks constructed in this paper could have a very small alphabet size

further results on generalized bent functions and their complete characterization this paper contributes to increase our knowledge on generalized bent functions including generalized bent boolean functions and generalized p ary bent functions with odd prime p by bringing new results on their characterization and construction in arbitrary characteristic more specifically we first investigate relations between generalized bent functions and bent functions by the decomposition of generalized bent functions this enables us to completely characterize generalized bent functions and mathbb z p k bent functions by some affine space associated with the generalized bent functions we also present the relationship between generalized bent boolean functions with an odd number of variables and generalized bent boolean functions with an even number of variables based on the well known maiorana mcfarland class of boolean functions we present some infinite classes of generalized bent boolean functions in addition we introduce a class of generalized hyperbent functions that can be seen as generalized dillon s ps functions finally we solve an open problem related to the description of the dual function of a weakly regular generalized bent boolean function with an odd number of variables via the walsh hadamard transform of their component functions and we generalize these results to the case of odd prime

on the weak approximate fixed point property abstract let x be a banach space and c a bounded closed convex subset of x c is said to have the weak approximate fixed point property if for any norm continuous mapping f c c there exists a sequence x n in c such that x n f x n n converges to 0 weakly it is known that every infinite dimensional banach space with the schur property does not have the weak approximate fixed point property in this article we show that every asplund space has the weak approximate fixed point property applications to the asymptotic fixed point theory are given

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 quadratic bent functions in polynomial forms in this correspondence we construct some new quadratic bent functions in polynomial forms by using the theory of quadratic forms over finite fields the results improve some previous work moreover we solve a problem left by yu and gong in 2006

systematic methods of constructing bent functions and 2 rotation symmetric bent functions in this paper we first present two systematic constructions of bent functions by modifying the truth tables of rothaus s bent function and maiorana mcfarland s bent function respectively the number of the newly constructed bent functions by modifying the truth table of rothaus s bent function is also determined the methods of constructing self dual bent functions are then given after the dual functions of these bent functions being determined finally as an application two constructions of 2 rotation symmetric bent functions are presented in this paper

constructing bent functions outside the maiorana mcfarland class using a general form of rothaus in the mid 1960s rothaus proposed the so called most general form of constructing new bent functions by using three initial bent functions whose sum is again bent in this paper we utilize a special case of rothaus construction when two of these three bent functions differ by a suitably chosen characteristic function of an n 2 dimensional subspace this simplification allows us to treat the induced bent conditions more easily also implying the possibility to specify the initial functions in the partial spread class and most notably to identify several instances of the so called non normal bent functions affine inequivalent bent functions within this class are then identified using a suitable selection of initial bent functions within the partial spread class stemming from the complete desarguesian spread it is also shown that when the initial bent functions belong to the class mathcal d then under certain conditions the constructed functions provably do not belong to the completed maiorana mcfarland class we conjecture that our method potentially generates an infinite class of non normal bent functions all tested ten variable functions are non normal but unfortunately they are weakly normal though there are no efficient computational tools for confirming this

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 bent and semi bent quadratic boolean functions the maximum length sequences also called m sequences have received a lot of attention since the late 1960s in terms of linear feedback shift register lfsr synthesis they are usually generated by certain power polynomials over a finite field and in addition are characterized by a low cross correlation and high nonlinearity we say that such a sequence is generated by a semi bent function some new families of such function represented by f x spl sigma sub i 1 sup n 1 2 c sub i tr x 2 sup i 1 n odd and c sub i spl isin f sub 2 have recently 2002 been introduced by khoo et al we first generalize their results to even n we further investigate the conditions on the choice of c sub i for explicit definitions of new infinite families having three and four trace terms also a class of nonpermutation polynomials whose composition with a quadratic function yields again a quadratic semi bent function is specified the treatment of semi bent functions is then presented in a much wider framework we show how bent and semi bent functions are interlinked that is the concatenation of two suitably chosen semi bent functions will yield a bent function and vice versa finally this approach is generalized so that the construction of both bent and semi bent functions of any degree in certain range for any n spl ges 7 is presented n being the number of input variables

univariate niho bent functions from o polynomials in this paper we discover that univariate form of a niho bent function is a sum of functions having the form of a leander kholosha bent function taken with particular coefficients from mathbb f 2 n for every term we know that the niho bent functions are related to o polynomials the power terms in the univariate niho bent function can be derived by working in a first step on each monomial of the corresponding o polynomial separately and in a second step adding them to obtain the global expression this allows knowing the monomials in an o polynomial to obtain the power terms of the polynomial representing corresponding bent function however the coefficients are not calculated explicitly the explicit form is given for the bent functions obtained from quadratic and cubic o polynomials we also calculate the algebraic degree of any bent function in the leander kholosha class

a renorming of some nonseparable banach spaces with the fixed point property abstract we prove that for every banach space which can be embedded in c 0 for instance reflexive spaces or more generally spaces with m basis there exists an equivalent renorming which enjoys the weak fixed point property for non expansive mappings as a consequence we solve a longtime open question in metric fixed point theory every reflexive banach can be renormed to satisfy the fixed point property furthermore this norm can be chosen arbitrarily closed to the original norm

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

constructions of quadratic bent functions in polynomial forms in this correspondence the constructions and enumerations of all bent functions represented by a polynomial form of f x spl sigma sub i 1 sup n 2 1 c sub i tr x sup 1 2 i c sub n 2 tr sub 1 sup n sup 2 x sup 1 2 n 2 c sub i spl isin sub 2 f sub 2 are presented for special cases of n using an iterative approach the construction of bent functions of n variables with degree n 2 is also provided using the constructed quadratic bent functions

systematic constructions of rotation symmetric bent functions 2 rotation symmetric bent functions and bent idempotent functions rotation symmetric bent functions and their generation two rotation symmetric bent functions are two classes of cryptographically significant boolean functions however few constructions have been presented in the literature which either have restriction on integer n or have algebraic degree no more than 4 in this paper for any even integer n ge 4 three classes of bent functions are presented respectively most notably the proposed n variable rotation symmetric bent functions and two rotation symmetric bent functions can have any possible algebraic degree ranging from 2 to n 2 besides we obtain bent idempotent functions with the maximal algebraic degree n 2

bent function sequences in this paper we construct a new family of nonlinear binary signal sets which achieve welch s lower bound on simultaneous cross correlation and autocorrelation magnitudes given a parameter n with n 0 pmod 4 the period of the sequences is 2 n 1 the number of sequences in the set is 2 n 2 and the cross auto correlation function has three values with magnitudes leq 2 n 2 1 the equivalent linear span of the codes is bound above by sum i 1 n 4 left stackrel n i right these new signal sets have the same size and correlation properties as the small set of kasami codes but they have important advantages for use in spread spectrum multiple access communications systems first the sequences are balances which represents only a slight advantage second the sequence generators are easy to randomly initialize into any assigned code and hence can be rapidly hopped from sequence to sequence for code division multiple access operation most importantly the codes are nonlinear in that the order of the linear difference equation satisfied by the sequence can be orders of magnitude larger than the number of memory elements in the generator that produced it this high equivalent linear span assures that the code sequence cannot be readily analyzed by a sophisticated enemy and then used to neutralize the advantages of the spread spectrum processing

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

on the walsh transform of a class of functions from niho exponents in this paper a class of functions from niho exponents with four valued walsh transform is obtained for any prime by a uniform method and the distribution of the walsh transform values is also completely determined in particular this class of functions is proven to be bent for a special case although it is shown that the obtained bent functions are equivalent to the leander kholosha s class of bent functions a direct and much simpler proof for the bentness of this kind of niho functions is provided

on balanced binary sequences with two level autocorrelation functions recently maschietti des codes cryptogr vol 14 p 89 1998 constructed several families of 2 sup m 1 2 sup m 1 1 2 sup m 2 1 cyclic difference sets from monomial hyperovals in this correspondence we consider the binary sequences associated to the difference sets constructed by maschietti we give algebraic proof of the fact that these sequences have two level autocorrelation functions also we discuss the linear spans of these sequences

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

generalized bent criteria for boolean functions i generalizations of the bent property of a boolean function are presented by proposing spectral analysis with respect to a well chosen set of local unitary transforms quadratic boolean functions are related to simple graphs and it is shown that the orbit generated by successive local complementations on a graph can be found within the transform spectra under investigation the flat spectra of a quadratic boolean function are related to modified versions of its associated adjacency matrix

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

perron type stability theorems for neutral equations abstract in this paper we present two perron type asymptotic stability results for a neutral functional differential equation of the form x t sx t px t r d d t q t x t g t x t when the linear part x t sx t px t r is asymptotically stable in particular q and g are allowed to be unbounded functions of t and q need not be differentiable the results are based on krasnoselskii s fixed point theorem it is to be emphasized that unlike perron we obtain only asymptotic stability because of the unboundedness of q and g

optimal codebooks from binary codes meeting the levenshtein bound in this paper a generic construction of codebooks based on binary codes is introduced with this generic construction a few previous constructions of optimal codebooks are extended and a new class of codebooks almost meeting the levenshtein bound is presented exponentially many codebooks meeting or almost meeting the levenshtein bound from binary codes are obtained in this paper the codebooks constructed in this paper have alphabet size 4 as a byproduct three bounds on the parameters of binary codes are derived

vectorial hyperbent trace functions from the mathcal ps _ rm ap class their exact number and specification to identify and specify trace bent functions of the form tr p x where p x in mathbb f 2 n x has been an important research topic lately we characterize a class of vectorial hyper bent functions of the form f x tr k n sum i 0 2 k a ix i 2 k 1 where n 2k in terms of finding an explicit expression for the coefficients a i so that f is vectorial hyperbent these coefficients only depend on the choice of the interpolating polynomial used in the lagrange interpolation of the elements of mathcal u and some prespecified outputs where mathcal u is the cyclic group of 2 n 2 1 th roots of unity in mathbb f 2 n we show that these interpolation polynomials can be chosen in exactly 2 k 1 2 k 1 ways and this is the exact number of vectorial hyperbent functions of the form tr n k sum i 0 2 k a ix i 2 k 1 furthermore a simple optimization method is proposed for selecting the interpolation polynomials that give rise to trace polynomials with a few nonzero coefficients

an efficient characterization of a family of hyperbent functions charpin and gong recently characterized a large class of hyperbent functions defined on fields of order 2n which include the well known monomial functions with the dillon exponent as a special case we give a reformulation of the charpin gong criterion in terms of the number of rational points on certain hyperelliptic curves we present two applications of our result the time needed to check the hyperbentness of a specific function is now polynomial in n and hyperbent functions with subfield coefficients can be constructed

spaces not containing l1 have weak approximate fixed point property abstract a nonempty closed convex bounded subset c of a banach space is said to have the weak approximate fixed point property if for every continuous map f c c there is a sequence x n in c such that x n f x n converge weakly to 0 we prove in particular that c has this property whenever it contains no sequence equivalent to the standard basis of l 1 as a byproduct we obtain a characterization of banach spaces not containing l 1 in terms of the weak topology

enumeration of quadratic functions with prescribed walsh spectrum the walsh transform f over cap of a quadratic function f f p n f p satisfies vertical bar f over cap vertical bar epsilon 0 p n s 2 for an integer 0 s n 1 depending on f in this paper quadratic functions of the form f p f n x tr n sigma k i 0 a i x pt 1 are studied with the restriction that a i is an element of f p 0 i k three methods for enumeration of such functions are presented when the value for s is prescribed this paper extends earlier enumeration results significantly for instance the generating function for the counting function is obtained when n is odd and relatively prime to p or when n 2m for odd m and p 2 the number of bent and semibent functions for various classes of n is also obtained

value distributions of exponential sums from perfect nonlinear functions and their applications in this paper we present a unified way to determine the values and their multiplicities of the exponential sums sigmaxisinf q zetap tr af x bx a bisinfq q pm pges3 for all perfect nonlinear functions f which is a dembowski ostrom polynomial or p 3 f x 3 k 1 2 where k is odd and k m 1 as applications we determine 1 the correlation distribution of the m sequence alambda tr gammalambda lambda 0 1 and the sequence blambda tr f gammalambda lambda 0 1 over fp where gamma is a primitive element of fq and 2 the weight distributions of the linear codes over fp defined by f

fixed points and stability in differential equations with variable delays abstract in this paper we consider a linear scalar differential equation with variable delays and give conditions to ensure that the zero solution is asymptotically stable by means of fixed point theory these conditions do not require the boundedness of delays nor do they ask for a fixed sign on the coefficient functions an asymptotic stability theorem with a necessary and sufficient condition is proved

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

cyclotomic linear codes of order 3 in this correspondence two classes of cyclotomic linear codes over gf q of order 3 are constructed and their weight distributions are determined the two classes are two weight codes and contain optimal codes they are not equivalent to irreducible cyclic codes in general when q 2

on cryptographic properties of the cosets of r 1 m we introduce a new approach for the study of weight distributions of cosets of the reed muller code of order 1 our approach is based on the method introduced by kasami 1968 using pless 1963 identities by interpreting some equations we obtain a necessary condition for a coset to have a high minimum weight most notably we are able to distinguish such cosets which have three weights only we then apply our results to the problem of the nonlinearity of boolean functions we particularly study the links between this criterion and the propagation characteristics of a function

asymptotic integration of a system resulting from the perturbation of an h system z systems were introduced in lo with the intention of obtaining results about stability for a weakly stable system at least weaker than those given by exponential stability under some perturbations in this paper we get asymptotic formulae for these systems which state new results about asymptotic integration for nonlinear systems under general hypotheses moreover the corresponding results for linear systems give new insight about levinson s theorem on asymptotic integration in this way using the h systems we get a uniform treatment for the concept of stability and we verify that several classical theorems of stability take a precise form by the asymptotic formulae in section 2 we recall the definition and some properties of an h system and we state two new theorems about asymptotic behavior of solutions of perturbed h systems in section 3 some general results about asymptotic integration are given these results extend those given in lo in section 4 we apply the technique to the linear case finally in section 5 we show examples and applications and we study the emden fowler equation which originally does not satisfy the general hypothesis but can be transformed into one where they are satisfied 2 h systems let us consider the differential system

vectorial bent functions from multiple terms trace functions in this paper we provide necessary and sufficient conditions for a function of the form f x trk2k i 1taixri 2k 1 to be bent three equivalent statements all of them providing both the necessary and sufficient conditions are derived in particular one characterization provides an interesting link between the bentness and the evaluation of f on the cyclic group of the 2k 1 th primitive roots of unity in gf 22k more precisely for this group of cardinality 2k 1 given by u u gf 22k u2k 1 1 it is shown that the property of being vectorial bent implies that im f gf 2k 0 if f is evaluated on u that is f u takes all possible values of gf 2k exactly once and the zero value is taken twice when u ranges over u this condition is then reformulated in terms of the evaluation of certain elementary symmetric polynomials related to f which in turn gives some necessary conditions on the coefficients ai for binomial trace functions that can be stated explicitly finally we show that a bent trace monomial of dillon s type trk2k xr 2k 1 is never a vectorial bent function

on some cosets of the first order reed muller code with high minimum weight we study a family of particular cosets of the first order reed muller code r 1 m those generated by special codewords the idempotents thus we obtain new maximal weight distributions of cosets of r 1 7 and 84 distinct almost maximal weight distributions of cosets of r 1 9 that is with minimum weight 240 this leads to crypotographic applications in the context of stream ciphers

the boundedness and stability of neutral delay differential equations with damped stochastic perturbations via fixed point theory in this paper a class of neutral stochastic differential equation is studied by using the fixed point method some conditions are established for ensuring the equation is bounded asymptotically stable and j c stable in mean square then with borel cantelli lemma several sufficient conditions for almost surely asymptotic stability and j c stability are given the conditions do not require the boundedness of delays nor do they ask for the continuousness on the coefficient functions the results of this paper are new and they extend and improve previously known results

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

hamming weights of the duals of cyclic codes with two zeros cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms in this paper let bbb f r be a finite field with r q m suppose that g 1 g 2 in bbb f r are not conjugates over bbb f q operatorname ord g 1 n 1 operatorname ord g 2 n 2 d gcd n 1 n 2 and n n 1n 2 d let bbb f q g 1 bbb f q m 1 bbb f q g 2 bbb f q m 2 and text t i denote the trace function from bbb f q m i to bbb f q for i 1 2 we define a cyclic code mathcal c q m n 1 n 2 c a b a in bbb f q m 1 b in bbb f q m 2 where c a b text t 1 ag 1 0 text t 2 bg 2 0 text t 1 ag 1 1 text t 2 bg 2 1 ldots text t 1 ag 1 n 1 text t 2 bg 2 n 1 we mainly use gauss periods to present the weight distribution of the cyclic code mathcal c q m n 1 n 2 as applications we determine the weight distribution of cyclic code mathcal c q m q m 1 1 q m 2 1 with gcd m 1 m 2 1 in particular it is a three weight cyclic code if gcd q 1 m 1 m 2 1 we also explicitly determine the weight distributions of some classes of cyclic codes including several classes of four weight cyclic codes

on the derivative imbalance and ambiguity of functions in 2007 carlet and ding introduced two parameters denoted by nb f and nb f quantifying respectively the balancedness of general functions f between finite abelian groups and the global balancedness of their derivatives d a f x f x a f x a in g setminus 0 providing an indicator of the nonlinearity of the functions these authors studied the properties and cryptographic significance of these two measures they provided inequalities relating the nonlinearity mathcal nl f to nb f for s box and specifically obtained an upper bound on the nonlinearity that unifies sidelnikov chabaud vaudenay s bound and the covering radius bound at the workshop wcc 2009 and in its postproceedings in 2011 a further study of these parameters was made in particular the first parameter was applied to the functions f l where l is affine providing more nonlinearity parameters in 2010 motivated by the study of costas arrays two parameters called ambiguity and deficiency were introduced by panario et al for permutations over finite abelian groups to measure the injectivity and surjectivity of the derivatives respectively these authors also studied some fundamental properties and cryptographic significance of these two measures further studies followed without comparing the second pair of parameters to the first one in this paper we observe that ambiguity is the same parameter as nb f up to additive and multiplicative constants i e up to rescaling we perform the necessary work of comparison and unification of the results on nb f and on ambiguity which have been obtained in the five papers devoted to these parameters we generalize some known results to any finite abelian groups more importantly we derive many new results on these parameters

optimal approximate fixed point results in locally convex spaces abstract let c be a convex subset of a locally convex space we provide optimal approximate fixed point results for sequentially continuous maps f c c first we prove that if f c is totally bounded then it has an approximate fixed point net next it is shown that if c is bounded but not totally bounded then there is a uniformly continuous map f c c without approximate fixed point nets we also exhibit an example of a sequentially continuous map defined on a compact convex set with no approximate fixed point sequence in contrast it is observed that every affine not necessarily continuous self mapping of a bounded convex subset of a topological vector space has an approximate fixed point sequence moreover we construct an affine sequentially continuous map from a compact convex set into itself without fixed points

stability results for neutral stochastic functional differential equations via fixed point methods in this paper we prove some results on the mean square asymptotic stability of a class of neutral stochastic differential systems with variable delays by using a contraction mapping principle name

linear codes from perfect nonlinear mappings and their secret sharing schemes in this paper error correcting codes from perfect nonlinear mappings are constructed and then employed to construct secret sharing schemes the error correcting codes obtained in this paper are very good in general and many of them are optimal or almost optimal the secret sharing schemes obtained in this paper have two types of access structures the first type is democratic in the sense that every participant is involved in the same number of minimal access sets in the second type of access structures there are a few dictators who are in every minimal access set while each of the remaining participants is in the same number of minimal access sets

approximate fixed point theorems in banach spaces with applications in game theory in this paper some new approximate fixed point theorems for multifunctions in banach spaces are presented and a method is developed indicating how to use approximate fixed point theorems in proving the existence of approximate nash equilibria for non cooperative games

generic construction of bent functions and bent idempotents with any possible algebraic degrees as a class of optimal combinatorial objects bent functions have important applications in cryptography sequence design and coding theory bent idempotents are a subclass of bent functions and of great interest since they can be stored in less space and allow faster computation of the walsh hadamard transform the objective of this paper is to present a generic construction of bent functions from known ones it includes the previous constructions of bent functions by mesnager and xu et al as special cases and produces new bent functions which cannot be produced by earlier ones in particular it also generates infinite families of bent idempotents over f 2 2m of any algebraic degree between 2 and m this together with a recent construction by su and tang gives a positive answer to an open problem on bent idempotents proposed by carlet in addition an infinite family of anti self dual bent functions is obtained in which the sum of any three distinct functions is again an anti self dual bent function in this family this solves an open problem recently proposed by mesnager

further results on niho bent functions this paper consists of two main contributions first the niho bent function consisting of 2r exponents discovered by leander and kholosha is studied the dual of the function is found and it is shown that this new bent function is not of the niho type second all known univariate representations of niho bent functions are analyzed for their relation to the completed maiorana mcfarland class m in particular it is proven that two families do not belong to the completed class m the latter result gives a positive answer to an open problem whether the class h of bent functions introduced by dillon in his thesis of 1974 differs from the completed class m

a class of binomial bent functions over the finite fields of odd characteristic this paper studies a class of binomial functions over the finite fields of odd characteristic and characterizes their bentness in terms of the kloosterman sums numerical results show that the proposed class contains bent functions that are affinely inequivalent to all known monomial and binomial ones

several new classes of bent functions from dillon exponents several new classes of binary and p ary regular bent functions are obtained in this paper the bentness of all these functions is determined by some exponential sums over finite fields most of which have close relations with the well known kloosterman sums

bent functions with 2 r niho exponents in this note a new primary construction of bent functions consisting of a linear combination of 2r niho exponents is presented this construction generalizes one of the constructions proven by h dobbertin g leander a canteaut c carlet p felke and p gaborit