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

large sets of disjoint spectra plateaued functions inequivalent to partially linear functions in this paper we give an efficient method for constructing a large set of disjoint spectra functions without linear structures which are not equivalent to partially linear functions this positively answers the open problem how to construct a large set of disjoint spectra functions which are not linearly equivalent to partially linear functions raised by zhang and xiao at the same time this significantly extends a recent result of zhang where a method of specifying four disjoint spectra functions was given it is demonstrated that such sets can be utilized in the design of highly nonlinear resilient functions in the second part based on a generalization of the indirect sum method we give an alternative approach for designing sets of disjoint spectra functions of even larger cardinality than already given ones but these functions then admit linear structures in addition it is shown that using suitable initial functions in the generalized indirect sum method we can specify highly nonlinear resilient boolean functions in odd number of input variables n whose nonlinearity in many cases exceeds the current best known values moreover we design some balanced functions for odd n that also achieve the highest nonlinearity known

a closed form formula for the intersection of two complex matrices under the star order abstract we present an interesting method for computing the intersection a b under the star order for arbitrary pairs of complex m n matrices a and b

a short proof of the xiao massey lemma correlation immune combining functions a short and lucid proof of the xiao massey lemma is given it provides various ways of generalizing the lemma

decrypting a class of stream ciphers using ciphertext only pseudonoise sequences generated by linear feedback shift registers 1 with some nonlinear combining function have been proposed 2 5 for cryptographic applications as running key generators in stream ciphers in this correspondence it will be shown that the number of trials to break these ciphers can be significantly reduced by using correlation methods by comparison of computer simulations and theoretical results based on a statistical model the validity of this analysis is demonstrated rubin 6 has shown that it is computationally feasible to solve a cipher proposed by pless 2 in a known plaintext attack using as few as 15 characters here the number of ciphertext symbols is determined to perform a ciphertext only attack on the pless cipher using the correlation attack our conclusion from the analysis is that the pseudonoise generator s output sequence and the sequences generated by the linear feedback shift registers should be uncorrelated this leads to constraints for the nonlinear combining function to be used

on the existence of 10 2 7 488 resilient functions using a heuristic search combined with some algebraic techniques several examples for 10 variable boolean functions with nonlinearity 488 algebraic degree 7 and resiliency degree 2 were constructed this construction affirmatively answers the open problem about the existence of such functions

on the existence of 9 3 5 240 resilient functions using a heuristic search technique several examples for 9 variable boolean functions with nonlinearity 240 algebraic degree 5 and resiliency degree 3 were constructed this construction affirmatively answers the open problem about the existence of such functions

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

generalized and hypergeneralized projectors abstract generalizations of the class of projectors are considered within the classes of normal and ep matrices some properties are derived and results on the sum difference and product of generalized and hypergeneralized projectors are given

a spectral characterization of correlation immune combining functions it is shown that a boolean combining function f x of n variables is mth order correlation immune if and only if its walsh transform f omega vanishes for all omega with hamming weight between 1 and m inclusive this result is used to extend slightly siegenthaler s ieee trans comput vol c 34 pp 81 85 jan 1985 characterization of the algebraic normal form of correlation immune combining functions

correlation immunity of nonlinear combining functions for cryptographic applications corresp pseudonoise generators for cryptographic applications consisting of several linear feedback shift registers with a nonlinear combining function have been proposed as running key generators in stream ciphers these running key generators eau sometimes be broken by ciphertext only correlation attacks on individual subsequences a new class of combining functions is presented which provides better security against such attacks the security is quantified by the smallest number m 1 of subsequences that must be simultaneously considered in a correlation attack a necessary condition for such m th order correlation immunity is proved a recursive construction is given that permits the construction of an m th order immune combining function for n subsequences for any m and n with 1 leq m finally the trade off between the length of the linear equivalent of the nonlinear generator and the order m of its immunity against correlation attacks is considered

addendum to non bch triple error correcting codes for original paper see van der vlugt ieee trans inform theory vol 42 p 1612 14 1996 one of the subjects in the above mentioned paper is the determination of the weight distribution of the dual code c sup spl perp of the binary cyclic codes c of length n 2 sup m 1 with zeros spl alpha spl alpha sup 2 t 1 1 spl alpha 2 sup t 1 here m 2t 1 and spl alpha generate the multiplicative group of the finite field f sub 2m after the publication of van der vlugt 1996 tor helleseth drew the present author s attention to the fact that kasami s paper 1969 contains the key to an essentially different way of deriving the weight enumerator of c sup spl perp indeed combining theorem 15 ii 1 in kasami with the fact that the minimum distance d sub min 7 see macwilliams and sloane 1983 also yields the weight distribution of c sup spl perp

maiorana mcfarland class degree optimization and algebraic properties in this paper we consider a subclass of the maiorana mcfarland class used in the design of resilient nonlinear boolean functions we show that these functions allow a simple modification so that resilient boolean functions of maximum algebraic degree may be generated instead of suboptimized degree in the original class preserving a high nonlinearity value immanent to the original construction method together with the degree optimization gives in many cases functions with cryptographic properties superior to all previously known construction methods this approach is then used to increase the algebraic degree of functions in the extended maiorana mcfarland mm class nonlinear resilient functions f gf 2 n rarrgf 2 m derived from linear codes we also show that in the boolean case the same subclass seems not to have an optimized algebraic immunity hence not providing a maximum resistance against algebraic attacks a theoretical analysis of the algebraic properties of extended maiorana mcfarland class indicates that this class of functions should be avoided as a filtering function in nonlinear combining generators

generic constructions of five valued spectra boolean functions whereas the design and properties of bent and plateaued functions have been frequently addressed during the past few decades there are only a few design methods of the so called five valued spectra boolean functions whose walsh spectra take the values in 0 pm 2 lambda 1 pm 2 lambda 2 moreover these design methods mainly regard the specification of these functions in their algebraic normal form anf domain in this paper we give a precise characterization of this class of functions in their spectral domain using the concept of a dual of plateaued functions both necessary and sufficient conditions on the walsh support of these functions are given which then connects their design in the spectral domain to a family of the so called totally non overlap disjoint spectra plateaued functions we identify some suitable families of plateaued functions having this property thus providing some generic methods in the spectral domain furthermore we also provide an extensive analysis of their constructions in the anf domain and provide several generic design methods the importance of this class of functions is manifolded where apart from being suitable for some cryptographic applications we emphasize their property of being constituent functions in the so called four bent decomposition

flips from 4 folds with isolated complete intersection singularities we investigate a flipping contraction g x y from a 4 fold x with at worst isolated complete intersection singularities if y has an anti bicanonical divisor bi elephant with only rational singularities then g carries an inductive structure involving a chain of blow ups which we call la torre pendente and in particular the flip exists this naturally contains reid s pagoda as an anticanonical divisor elephant and its proper transforms

on the covering radii of binary reed muller codes in the set of resilient boolean functions let r sub t n be the set of t resilient boolean functions in n variables and let spl rho spl circ t r n be the maximum distance between t resilient functions and the rth order reed muller code rm r n we prove that spl rho spl circ t 2 6 16 for t 0 1 2 and spl rho spl circ 3 2 7 32 from which we derive the lower bound spl rho spl circ t 2 n spl ges 2 sup n 2 with t spl les n 4 using a result from coding theory on the covering radius of n 3 th and n 4 th order reed muller codes we establish exact values of the covering radius of rm n 3 n in the set of 1 resilient boolean functions in n variables when spl lfloor n 2 spl rfloor 1 mod 2 and lower bounds of rm n 4 n in the set of 2 resilient boolean functions in n variables this result leads again to different lower bounds for general dimensions n and r 0 or 3 mod 4

some special cremona transformations in this paper we continue the study c f c k s t 1 2 of special cremona transformations i e those whose base scheme y is smooth and connected crauder and katz have completely classified these when dim y 2 our main results classify them when they are quadro quadric theorem 2 6 and when codim y 2 theorem 3 2 in the first case they are just the maps given by systems of quadrics through severi varieties by a result of zak there are just four of these the veronese surface v c p5 the segre embedding of p2 x p2 in p8 the grassmannian g 2 6 in p14 and the 16dimensional e6 variety in p26 and in the second either y is a quintic elliptic scroll in p4 or y is defined in pi by the vanishing of the n x n minors of

on fano manifolds of large pseudoindex abstract we describe fano manifolds of large pseudoindex that are rationally connected with respect to some numerically independent families of rational curves

new constructions of resilient functions with strictly almost optimal nonlinearity via non overlap spectra functions abstract the design of n variable t resilient functions with strictly almost optimal sao nonlinearity 2 n 1 2 n 2 n even appears to be a rather difficult task the known construction methods commonly use a rather large number exactly i t 1 n 2 n 2 i of affine subfunctions in n 2 variables which can induce some algebraic weaknesses making these functions susceptible to certain types of guess and determine cryptanalysis and dynamic cube attacks in this paper the concept of non overlap spectra functions is introduced which essentially generalizes the idea of disjoint spectra functions on different variable spaces two general methods to obtain a large set of non overlap spectra functions are given and a new framework for designing infinite classes of resilient functions with sao nonlinearity is developed based on these unlike previous construction methods our approach employs only a few n 2 variable affine subfunctions in the design resulting in a more favourable algebraic structure it is shown that these new resilient sao functions properly include all the existing classes of resilient sao functions as a subclass moreover it is shown that the new class provides a better resistance against fast algebraic attacks than the known functions with sao nonlinearity and in addition these functions are more robust to guess and determine cryptanalysis and dynamic cube attacks

exploiting product distributions to identify relevant variables of correlation immune functions a boolean function f is correlation immune if each input variable is independent of the output under the uniform distribution on inputs for example the parity function is correlation immune we consider the problem of identifying relevant variables of a correlation immune function in the presence of irrelevant variables we address this problem in two different contexts first we analyze skewing a heuristic method that was developed to improve the ability of greedy decision tree algorithms to identify relevant variables of correlation immune boolean functions given examples drawn from the uniform distribution page and ray 2003 we present theoretical results revealing both the capabilities and limitations of skewing second we explore the problem of identifying relevant variables in the product distribution choice pdc learning model a model in which the learner can choose product distributions and obtain examples from them we prove a lemma establishing a property of boolean functions that may be of independent interest using this lemma we give two new algorithms for finding relevant variables of correlation immune functions in the pdc model

correlation immune functions over finite fields we give a series of constructions of correlation immune function over finite fields we prove that f sub 2 and f sub 3 are the only finite fields f sub q with the property that every n 1 th correlation immune function in n 2 variables over f sub q is linear we also show that by choosing larger finite fields one can alleviate the tradeoff between the length of the linear equivalent and the order of correlation immunity this is useful for the design of various cryptosystems

further constructions of resilient boolean functions with very high nonlinearity one well known method of generating key stream sequences for stream ciphers is to combine the outputs of several linear feedback shift registers lfsr using a combining boolean function here we concentrate on the design of good combining boolean functions we provide resilient boolean functions with currently best known nonlinearity these functions were not known earlier and the issues related to their existence were posed as open questions in the literature some of the functions we construct here achieve the provable upper bound on nonlinearity for resilient boolean functions our technique interlinks mathematical results with classical computer search

constructions of almost optimal resilient boolean functions on large even number of variables in this paper a technique on constructing nonlinear resilient boolean functions is described by using several sets of disjoint spectra functions on a small number of variables an almost optimal resilient function on a large even number of variables can be constructed it is shown that given any m one can construct infinitely many re variable n even m resilient functions with nonlinearity 2n 1 2n 2 a large class of highly nonlinear resilient functions which were not known are obtained then one method to optimize the degree of the constructed functions is proposed last an improved version of the main construction is given

small contractions of symplectic 4 folds we classify small contractions of holomorphically symplectic 4 folds ams msc 14e30 14j35

search for boolean functions with excellent profiles in the rotation symmetric class for the first time boolean functions on 9 variables having nonlinearity 241 are discovered that remained as an open question in literature for almost three decades such functions are found by heuristic search in the space of rotation symmetric boolean functions rsbfs this shows that there exist boolean functions on n odd variables having nonlinearity 2n 1 2n 1 2 if and only if n 7 using similar search technique balanced boolean functions on 9 10 and 11 variables are attained having autocorrelation spectra with maximum absolute value 2lceiln 2rceil on odd number of variables earlier such functions were known for 15 21 variables there was no evidence of such functions at all on even number of variables in certain cases our functions can be affinely transformed to obtain first order resiliency or first order propagation characteristics moreover 10 variable functions having first order resiliency and nonlinearity 492 are presented that had been posed as an open question at crypto 2000 the functions reported in this paper are discovered using a suitably modified steepest descent based iterative heuristic search in the rsbf class along with proper affine transformations it seems elusive to get a construction technique to match such functions

9 variable boolean functions with nonlinearity 242 in the generalized rotation symmetric class we give a new lower bound to the covering radius of the first order reed muller code rm 1 n where n 9 11 13 equivalently we present the n variable boolean functions for n 9 11 13 with maximum nonlinearity found till now in 2006 9 variable boolean functions having nonlinearity 241 which is strictly greater than the bent concatenation bound of 240 have been discovered in the class of rotation symmetric boolean functions rsbfs by kavut maitra and yucel to improve this nonlinearity result we have firstly defined some subsets of the n variable boolean functions as the generalized classes of k rsbfs and k dsbfs k dihedral symmetric boolean functions where k is a positive integer dividing n secondly utilizing a steepest descent like iterative heuristic search algorithm we have found 9 variable boolean functions with nonlinearity 242 within the classes of both 3 rsbfs and 3 dsbfs thirdly motivated by the fact that rsbfs are invariant under a special permutation of the input vector we have classified all possible permutations up to the linear equivalence of boolean functions that are invariant under those permutations

high meets low construction of strictly almost optimal resilient boolean functions via fragmentary walsh spectra this paper considers the construction of resilient boolean functions on an odd number of variables with strictly almost optimal sao nonlinearity through introducing the fragmentary walsh transform a construction technique called high meets low is proposed the detailed design procedures of a 39 variable 3 resilient boolean function with sao nonlinearity 2 38 2 19 2 16 2 14 are given it is shown that the nonlinearity of an n variable t resilient boolean function can reach 2 n 1 2 n 1 2 5 cdot 2 n 11 2 or 2 n 1 2 n 1 2 2 n 7 2 which are the largest known values for the corresponding n and t values finally by constructing a 29 variable balanced boolean function with sao nonlinearity 2 28 2 14 2 10 2 9 we show an alternative method to realize the high meets low construction technique

on some characterizations of the star partial ordering for matrices and rank subtractivity abstract the main result is that drazin s star partial ordering a b holds if and only if a b and b a b a where a b is defined by a a a b and aa ba and where a b denotes rank subtractivity here a and b are m n complex matrices and the superscript denotes the moore penrose inverse several other characterizations of a b are given with particular emphasis on what extra condition must be added in order that rank subtractivity becomes the stronger star order a key tool is a new canonical form for rank subtractivity connections with simultaneous singular value decompositions schur complements and idempotent matrices are also mentioned

lattice properties of the order for complex matrices abstract it is shown that the set c m n of complex m n matrices forms a lower semilattice under the partial ordering a b defined by a a a b aa ba where a denotes the conjugate transpose of a as a special case of a result for division rings it is further shown that over any field f for m n 2 and any proper involution of f2 2 the corresponding intersections a b all exist

linear codes in generalized construction of resilient functions with very high nonlinearity we provide a new generalized construction method for highly nonlinear t resilient functions f f sub 2 sup n spl rarr f sub 2 sup m the construction is based on the use of linear error correcting codes together with highly nonlinear multiple output functions given a linear u m t 1 code we show that it is possible to construct n variable m output t resilient functions with very high nonlinearity for n u the method provides the currently best known nonlinearity results for most of the cases

some new three valued crosscorrelation functions for binary m sequences in 1972 niho gave various conjectures about the crosscorrelation between a binary maximum length linear shift register sequence and a decimation of that sequence by an integer d we prove that the crosscorrelation function for two new values of d takes on precisely three values and thereby confirm two of niho s conjectures

spectral methods for cross correlations of geometric sequences families of sequences with low pairwise shifted cross correlations are desirable for applications such as code division multiple access cdma communications often such sequences must have additional properties for specific applications several ad hoc constructions of such families exist in the literature but there are few systematic approaches to such sequence design we introduce a general method of constructing new families of sequences with bounded pairwise shifted cross correlations from old families of such sequences the bounds are obtained in terms of the maximum cross correlation in the old family and the walsh transform of certain functions

cryptographically resilient functions this correspondence studies resilient functions which have applications in fault tolerant distributed computing quantum cryptographic key distribution and random sequence generation for stream ciphers we present a number of new methods for synthesizing resilient functions an interesting aspect of these methods is that they are applicable both to linear and nonlinear resilient functions our second major contribution is to show that every linear resilient function can be transformed into a large number of nonlinear resilient functions with the same parameters as a result we obtain resilient functions that are highly nonlinear and have a high algebraic degree

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

improving the lower bound on the maximum nonlinearity of 1 resilient boolean functions and designing functions satisfying all cryptographic criteria in this article we improve the lower bound on the maximum nonlinearity of 1 resilient boolean functions for n even by proposing a method of constructing this class of functions attaining the best nonlinearity currently known thus for the first time at least for small values of n the upper bound on nonlinearity can be reached in a deterministic manner in difference to some heuristic search methods proposed previously the nonlinearity of these functions is extremely close to the maximum nonlinearity attained by bent functions and it might be the case that this is the highest possible nonlinearity of 1 resilient functions apart from this theoretical contribution it turns out that the cryptographic properties of these functions are overall good apart from their moderate resistance to fast algebraic attacks faa this weakness is repaired by a suitable modification of the original functions giving a class of balanced functions with almost optimal resistance to faa whose nonlinearity is better than the nonlinearity of other methods

resilient functions over finite fields resilient functions play an important role in the art of information security in this correspondence we discuss the existence construction and enumeration of resilient functions over finite fields we show that for each finite field gf q with q 3 we can easily construct a large number of q n 1 n 1 resilient functions most of which include mixing terms we give a general structure for q m 1 m 1 resilient functions and present an example which is not of this general structure we prove that q m 2 m 2 resilient functions exist for any m such that 1 2 we prove that q m t m t resilient functions exist for any m t such that 1 3 by making some simple generalizations of former results we also provide some new methods for constructing resilient functions

constructions of bent negabent functions and their relation to the completed maiorana mcfarland class the problem of constructing bent negabent functions that do not belong to the completed maiorana mcfarland class emerges implicitly through a series of construction methods proposed recently these approaches manage to optimize the algebraic degree of bent negabent functions but all of the constructed bent negabent functions belong to the completed maiorana mcfarland class in this paper we use the indirect sum construction proposed by carlet in 2004 for constructing the bent negabent functions that are not provably contained in this class which is the first significant attempt in this direction to achieve this we first provide a class of bent functions with certain desirable properties that does not belong to this class and demonstrate the existence of the class members then embedding these functions in the framework of the indirect sum construction we are able to specify sufficient conditions for bent negabent functions not being contained in the completed maiorana mcfarland class

generalized maiorana mcfarland construction of resilient boolean functions with high nonlinearity and good algebraic properties a new framework concerning the construction of small order resilient boolean functions whose nonlinearity is strictly greater than 2 n 1 2 n 2 is given first a generalized maiorana mcfarland construction technique is described which extends the current approaches by combining the usage of affine and nonlinear functions in a controllable manner it is shown that for any given m this technique can be used to construct a large class of n variable n both even and odd m resilient degree optimized boolean functions with currently best known nonlinearity this class may also provide functions with excellent algebraic properties measured through the resistance to fast algebraic attacks if the number of n 2 variable affine subfunctions used in the construction is relatively low due to a potentially low hardware implementation cost along with overall good cryptographic properties this class of functions is an attractive candidate for the use in certain stream cipher schemes

constructions of resilient s boxes with strictly almost optimal nonlinearity through disjoint linear codes in this paper a novel approach of finding disjoint linear codes is presented the cardinality of a set of u m t 1 disjoint linear codes largely exceeds all the previous best known methods used for the same purpose using such sets of disjoint linear codes not necessarily of the same length we have been able to provide a construction technique of t resilient s boxes f f2n 2m n even with strictly almost optimal nonlinearity this is the first time that the bound 2n 1 2n 2 has been exceeded by multiple output resilient functions actually the nonlinearity of our functions is in many cases equal to the best known nonlinearity of balanced boolean functions a large class of previously unknown cryptographic resilient s boxes is obtained and several improvements of the original approach are proposed some other relevant cryptographic properties are also briefly discussed it is shown that these functions may reach siegenthaler s bound n t 1 and can be either of optimal algebraic immunity or of slightly suboptimal algebraic immunity which was confirmed by simulations