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

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

designing plateaued boolean functions in spectral domain and their classification the design of plateaued functions over gf 2 n also known as 3 valued walsh spectra functions taking the values from the set 0 pm 2 lceil n s 2 rceil has been commonly approached by specifying a suitable algebraic normal form which then induces this particular walsh spectral characterization in this paper we consider the reversed design method which specifies these functions in the spectral domain by specifying a suitable allocation of the nonzero spectral values and their signs we analyze the properties of trivial and nontrivial plateaued functions as affine inequivalent distinct subclasses which are distinguished by their walsh support s f the subset of gf 2 n having the nonzero spectral values in terms of whether it is an affine subspace or not the former class exactly corresponds to partially bent functions and admits linear structures whereas the latter class may contain functions without linear structures a simple sufficient condition on s f which ensures the nonexistence of linear structures is derived and some generic design methods of nontrivial plateaued functions without linear structures are given the extended affine equivalence of plateaued functions is also addressed using the concept of dual of plateaued functions furthermore we solve the problem of specifying disjoint spectra non trivial plateaued functions of maximal cardinality whose concatenation can be used to construct bent functions in a generic manner this approach may lead to new classes of bent functions due to large variety of possibilities to select underlying duals that define these disjoint spectra plateaued functions an additional method of specifying affine inequivalent plateaued functions obtained by applying a nonlinear transform to their input domain is also given

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

some good dilutions of fuzzy compactness abstract the obvious generalisation of compactness to fuzzy topology is known not to be a good extension in the sense of lowen following lowen s approach to defining a good fuzzy compactness definitions are suggested for fuzzy almost nearly and weak compactness which are proved to satisfy goodness fuzzy s closedness and rs compactness are treated to a similar approach

boolean and vectorial plateaued functions and apn functions boolean plateaued functions and vectorial functions with plateaued components play a significant role in cryptography sequences for communications and the related combinatorics and designs our knowledge on them is not at a level corresponding to their importance we introduce new characterizations of plateaued boolean functions we give the characterizations of vectorial functions whose components are all plateaued with possibly different amplitudes that we simply call plateaued by means of the value distributions of their derivatives we characterize similarly those functions whose components are partially bent and autocorrelation functions and of the power moments of their walsh transform this allows us to derive several characterizations of almost perfect nonlinear apn functions in this framework we prove that all the main results known for quadratic apn functions extend to plateaued functions allowing the study of their apn ness to be simplified we show that if additionally the component functions are all unbalanced this study is still simpler the apn ness of such functions depends only on their value distribution this allows proving for instance that any plateaued n n function n even having similar value distribution as the apn power functions is apn and has the same extended walsh spectrum as the apn gold functions as by products we obtain a few other new results for instance any plateaued function in even dimension which is carlet charpin zinoviev ccz equivalent to a gold or kasami apn function is necessarily extended affine ea equivalent to it

thermal operators and cluster topology in the q state potts model we discuss a new class of identities between correlation functions which arise from a local 2 invariance of the partition function of the q state potts model on general graphs or lattices their common feature is to relate the thermal operators of the potts model to some topological properties of the fortuin kasteleyn clusters in particular it turns out that any even correlation function can be expressed in terms of observables which probe the linking properties of these clusters this generalizes a class of analogous relations recently found in the ising model

secondary constructions of highly nonlinear boolean functions and disjoint spectra plateaued functions in this paper we modify a generalized indirect sum construction to construct functions with high nonlinearity by utilizing the modified construction highly nonlinear functions in n m variables can be obtained from known bent functions in n variables and highly nonlinear functions in m variables it is possible to obtain new n 15 variable functions with nonlinearity 2 n 1 5 1 2 n 1 5 1 2 20x2 n 2 and new 12 variable 2 resilient functions with nonlinearity 2000 and algebraic degree 8 which achieve optimal algebraic immunity moreover the modified construction can also be used as an iterative construction of a quadruple of disjoint spectra plateaued functions in addition we present sufficient conditions for a quadruple of disjoint spectra plateaued functions to have no nonzero linear structure

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

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

the status of costas arrays the definition the basic properties and all the currently known systematic constructions for costas arrays are presented as well as a table of the number c n of costas arrays of order n for 2 les n les 26 it is proved that lim supnrarrinfin c n infin and the conjecture liminfnrarrinfin c n 0 is discussed a costas array of order n is known to be equivalent to a permutation 1 2 n for which the difference triangle contains no repeated elements in any row a generalized costas array of order n q 1 or n q 2 is defined as a permutation of the nonzero elements or also excluding 1 of the q element field for which the difference triangle contains no repeated elements in any row two new constructions for these generalized costas arrays are described and illustrated

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

ambiguity and deficiency of permutations over finite fields with linearized difference map the concepts of ambiguity and deficiency for a bijection on a finite abelian group were recently introduced in this paper we present some further fundamental results on the ambiguity and deficiency of functions in particular we note that they are invariant under the well known carlet charpin zinoviev equivalence we obtain upper and lower bounds on the ambiguity and deficiency of differentially k uniform functions and we give a lower bound on the nonlinearity of functions that achieve the lower bound of ambiguity and deficiency in addition we provide an explicit formula in terms of the ranks of matrices on the ambiguity and deficiency of a dembowski ostrom do polynomial and using this technique we find exact values for known cases of do permutations with few terms we also derive exact values for the ambiguities and deficiencies of do permutations obtained from trace functions the key relationship between the above polynomials is that they all have linearized difference map

new families of almost perfect nonlinear power mappings a power mapping f x x sup d over gf p sup n is said to be differentially k uniform if k is the maximum number of solutions x spl isin gf p sup n of f x a f x b where a b spl isin gf p sup n and a spl ne 0 a 2 uniform mapping is called almost perfect nonlinear apn we construct several new infinite families of nonbinary apn power mappings

almost perfect nonlinear power functions in odd characteristic in this paper two new families of functions in odd characteristic are constructed and they are proved to be almost perfect nonlinear apn functions some of the open cases which were introduced by helleseth are answered by these new apn functions

paracompactness in l fuzzy topological spaces abstract in this paper we introduce a good definition of paracompactness in l fuzzy topological spaces where l is a fuzzy lattice we define paracompactness on general l fuzzy subsets and study its properties

a new family of ternary almost perfect nonlinear mappings a mapping f x from gf pn to gf pn is differentially k uniform if k is the maximum number of solutions x isin gf pn of f x a f x b where a b isin gf pn and a ne 0 a 2 uniform mapping is called almost perfect nonlinear apn this correspondence describes new families of ternary apn mappings over gf 3n n 3 odd of the form f x uxd xd 2 where d1 3n 1 2 1 and d2 3n 2

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

fuzzy perfect maps and fuzzy paracompactness abstract in this paper we prove that s paracompactness s paracompactness fuzzy paracompactness and fuzzy paracompactness are invariants and inverse invariants of various types of fuzzy perfect maps

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

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

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

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

on the nonlinearity of exponential welch costas functions we study the nonlinearity of the exponential welch costas functions using the fourier transform on z m these functions have been proposed for use in nonbinary cryptosystems high nonlinearity is required to ensure resistance to linear cryptanalysis we prove some properties of the nonlinearity of these functions and we suggest a plausible connection of the nonlinearity to the class number of a quadratic field

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

generalized plateaued functions and admissible plateaued functions plateaued functions are very important crypto graphic functions due to their various desirable cryptographic characteristics we point out that plateaued functions are more general than bent functions that is functions with maximum nonlinearity some boolean plateaued functions have large nonlinearity which provides protection against fast correlation attacks when they are used as combiners or filters in stream ciphers and contributes when they are the component functions of the substitution boxes in block ciphers to protection against linear cryptanalysis p ary plateaued functions have attracted recently some attention in the literature and many activities on generalized p ary functions have been carried out this paper increases our knowledge on plateaued functions in the general context of generalized p ary functions we first introduce two new versions of plateaued functions which we shall call generalized plateaued functions and admissible plateaued functions the generalized plateaued functions extend the standard notion of plateaued p ary functions to those whose outputs are in the ring mathbb z p k next we study the generalized plateaued functions and use admissible plateaued functions to characterize the generalized plateaued functions by means of their components finally we provide for the first time two constructions of generalized plateaued functions in particular we generalize a known secondary construction of binary generalized bent functions and derive constructions of binary generalized plateaued functions with different amplitudes

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

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

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

paracompactness and full normality in ditopological texture spaces in this paper the authors lay the foundation for a theory of dicovers of ditopological texture spaces and use this to define notions of paracompactness and full normality some applications to fuzzy topology are also mentioned

on fuzzy topological spaces abstract in this paper a few separation properties and some aspects of subspace fuzzy topology have been studied where both the crisp and the fuzzy elements have been taken into consideration since the conventional definition of compactness is not quite meaningful in hausdorff fuzzy spaces as introduced by us a new more natural definition of proper compactness is given and a few properties resulting from this are established

a new attack on the filter generator the filter generator is an important building block in many stream ciphers the generator consists of a linear feedback shift register of length n that generates an m sequence of period 2n 1 filtered through a boolean function of degree d that combines bits from the shift register and creates an output bit zt at any time t the previous best attacks aimed at reconstructing the initial state from an observed keystream have essentially reduced the problem to solving a nonlinear system of d sigmai 1 d n i equations in n unknowns using techniques based on linear algebra this attack needs about d bits of keystream and the system can be solved in complexity o domega where omega can be taken to be strassen s reduction exponent omega log2 7 ap2 807 this paper describes a new algorithm that recovers the initial state of most filter generators after observing o d keystream bits with complexity o d n 2 apo d after a pre computation with complexity o d log2d 3

thermal operators in ising percolation we discuss a new cluster representation for the internal energy and the specific heat of the d dimensional ising model obtained by studying the percolation mapping of an ising model with an arbitrary set of antiferromagnetic links such a representation relates the thermal operators to the topological properties of the fortuin kasteleyn fk clusters of ising percolation and is a powerful tool to get new exact relations on the topological structure of fk clusters of the ising model defined on an arbitrary graph

clusters and droplets in the q state potts model a potts correlated polychromatic percolation is studied the clusters are made of sites corresponding to a given value of the q state potts variables connected by bonds being active with probability pb to treat this problem an s state potts hamiltonian diluted with q state potts variables instead of lattice gas variables is introduced to which the the migdal kadanoff renormalisation group is applied it is found for a particular choice of pb 1 e k where k is the potts coupling constant divided by the boltzmann factor that these clusters called droplets diverge at the potts critical point with potts exponents

two new measures for permutations ambiguity and deficiency we introduce the concepts of weighted ambiguity and deficiency for a mapping between two finite abelian groups of the same size then we study the optimum lower bounds of these measures for permutations of an abelian group a construction of permutations by modifying some permutation functions over finite fields is given their ambiguity and deficiency is investigated most of these functions are apn permutations we show that when they are not optimal the mobius function in the multiplicative group of bbfq is closer to being optimal in ambiguity than the inverse function in the additive group of bbfq we note that the inverse function over bbf28 is used in aes finally we conclude that a twisted permutation polynomial of a finite field is again closer to being optimal in ambiguity than the apn function employed in the safer cryptosystem

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

results of the enumeration of costas arrays of order 27 this correspondence presents the results of the enumeration of costas arrays of order 27 all arrays found except for one are accounted for by the golomb and welch construction methods

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

on the goodness of some types of fuzzy paracompactness abstract we introduce in fuzzy topological spaces definitions of paracompactness almost paracompactness and nearly paracompactness all of which turn out to be good extensions of their classical topological counterparts fuzzy semi paracompact para s closed weakly compact and rs paracompact spaces are treated to a similar approach

interleavers for turbo codes using permutation polynomials over integer rings a class of deterministic interleavers for turbo codes tcs based on permutation polynomials over spl zopf sub n is introduced the main characteristic of this class of interleavers is that they can be algebraically designed to fit a given component code moreover since the interleaver can be generated by a few simple computations storage of the interleaver tables can be avoided by using the permutation polynomial based interleavers the design of the interleavers reduces to the selection of the coefficients of the polynomials it is observed that the performance of the tcs using these permutation polynomial based interleavers is usually dominated by a subset of input weight 2m error events the minimum distance and its multiplicity or the first few spectrum lines of this subset are used as design criterion to select good permutation polynomials a simple method to enumerate these error events for small m is presented searches for good interleavers are performed the decoding performance of these interleavers is close to s random interleavers for long frame sizes for short frame sizes the new interleavers outperform s random interleavers

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

characterizations of the differential uniformity of vectorial functions by the walsh transform for every positive integers n and m and every even positive integer delta we derive inequalities satisfied by the walsh transforms of all vectorial n m functions and prove that the case of equality characterizes differential delta uniformity this provides a generalization to all differentially delta uniform functions of the characterization of almost perfect nonlinear apn functions due to chabaud and vaudenay by means of the fourth moment of the walsh transform such generalization has been missing since the introduction of the notion of differential uniformity by nyberg in 1994 and since chabaud vaudenay s result in the same year moreover for each even delta geq 2 we find several in fact an infinity of such characterizations in particular when delta 2 and delta 4 we have that for any n n function resp any n n 1 function the arithmetic mean of w f 2 u 1 v 1 w f 2 u 2 v 2 w f 2 u 1 u 2 v 1 v 2 when u 1 u 2 range independently over mathbb f 2 n and v 1 v 2 are nonzero and distinct and range independently over mathbb f 2 m is at least 2 3n and that f is apn resp is differentially 4 uniform if and only if this arithmetic mean equals 2 3n which is the value we would get with a bent function if such function could exist these inequalities give more knowledge on the walsh spectrum of n m functions we deduce in particular a property of the walsh support of highly nonlinear functions we also consider the completely open question of knowing if the nonlinearity of apn functions is necessarily non weak as it is the case for known apn functions we prove new lower bounds which cover all power apn functions and hence a large part of known apn functions which explain why their nonlinearities are not bad and we discuss the question of the nonlinearity of apn quadratic functions since almost all other known apn functions are quadratic

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