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

disturbance rejection of left invertible systems abstract a new approach is presented for the study of the disturbance rejection of left invertible systems via proportional state feedback reducing the problem to that of solving a linear algebraic system of equations on the basis of this system of equations the necessary and sufficient conditions for the existence of a solution to the disturbance rejection problem are established in a form of simple algebraic criteria thus reducing the computational effort over known results and the general analytical expressions of all solutions of the controller matrices are derived these solutions are parameterized only by an arbitrary matrix

minimal linear codes from characteristic functions minimal linear codes have interesting applications in secret sharing schemes and secure two party computation this paper uses characteristic functions of some subsets of mathbb f q to construct minimal linear codes by properties of characteristic functions we can obtain more minimal binary linear codes from known minimal binary linear codes which generalizes results of ding et al ieee trans inf theory vol 64 no 10 pp 6536 6545 2018 by characteristic functions corresponding to some subspaces of mathbb f q we obtain many minimal linear codes which generalizes results of ieee trans inf theory vol 64 no 10 pp 6536 6545 2018 and ieee trans inf theory vol 65 no 11 pp 7067 7078 2019 finally we use characteristic functions to present a characterization of minimal linear codes from the defining set method and present a class of minimal linear codes

approximation algorithms for np complete problems on planar graphs this paper describes a general technique that can be used to obtain approximation schemes for various np complete problems on planar graphs the strategy depends on decompos ing a planar graph into subgraphs of a form we call k outerplanar for fixed k the problems of interest are solvable optimally in linear time on k outerplanar graphs by dynamic programming for general planar graphs if the problem is a maximization problem such as maximum independent set this technique gives for each k a linear time algorithm that produces a solution whose size is at least k k 1 optimal if the problem is a minimization problem such as minimum vertex cover it gives for each k a linear time algorithm that produces a solution whose size is at most k 1 k optimal taking k c log log nl or k c log nl where n is the number of nodes and c is some constant we get polynomial time approximation algorithms whose solution sizes converge toward optimal as n increases the class of problems for which this approach provides approximation schemes includes maximum independent set maximum tile salvage partition into triangles maximum h matching minimum vertex cover minimum dominat ing set and minimum edge dominating set for these and certain other problems the proof of solvability on k outerplanar graphs also enlarges the class of planar gmphs for which the problems are known to be solvable in polynomial time

on the covering radius of r 1 m in r 3 m we prove that the covering radius of r 1 11 in r 3 11 is 992 and that the covering radius of r 1 13 in r 3 13 is 4032 both not exceeding the quadratic bound

an information theoretical approach to regulation a new approach to the analysis of control systems by means of information theoretical concepts and techniques is presented which is based on a formulation of the regulation problem in terms of the minimization of a cost function this approach is much more oriented towards real control problems than the usual method based on entropy functions using rate distortion theory the requisite variety of the controller and the requisite information from the measurements which are necessary for a certain performance are derived the results are applied to the disturbance decoupling problem in linear systems principal aspects of cause controlled regulation and error controlled regulation are discussed

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

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

investigations on bent and negabent functions via the nega hadamard transform parker considered a new type of discrete fourier transform called nega hadamard transform we prove several results regarding its behavior on combinations of boolean functions and use this theory to derive several results on negabentness that is flat nega spectrum of concatenations and partially symmetric functions we derive the upper bound n 2 for the algebraic degree of a negabent function on n variables further a characterization of bent negabent functions is obtained within a subclass of the maiorana mcfarland set we develop a technique to construct bent negabent boolean functions by using complete mapping polynomials using this technique we demonstrate that for each l 2 there exist bent negabent functions on n 12l variables with algebraic degree n 4 1 3l 1 it is also demonstrated that there exist bent negabent functions on eight variables with algebraic degrees 2 3 and 4 simple proofs of several previously known facts are obtained as immediate consequences of our work

improved methods for approximating node weighted steiner trees and connected dominating sets in this paper we study the steiner tree problem in graphs for the case when vertices as well as edges have weights associated with them a greedy approximation algorithm based on spider decompositions was developed by klein and ravi for this problem this algorithm provides a worst case approximation ratio of 2 ln k where k is the number of terminals however the best known lower bound on the approximation ratio is 1 o 1 ln k assuming that np not subseteq dtime n o log log n by a reduction from set cover

on the weight distributions of optimal cosets of the first order reed muller codes we study the weight distributions of cosets of the first order reed muller code r 1 m for odd m whose minimum weight is greater than or equal to the so called quadratic bound some general restrictions on the weight distribution of a coset of r 1 m are obtained by partitioning its words according to their weight divisibility most notably we show that there are exactly five weight distributions for optimal cosets of r 1 7 in r 5 7 and that these distributions are related to the degree of the function generating the coset moreover for any odd m spl ges 9 we exhibit optimal cubic cosets of r 1 m whose weights take on exactly five values

computing a tree having a small vertex cover abstract we consider a new steiner tree problem called vertex cover weighted steiner tree problem this problem defines the weight of a steiner tree as the minimum weight of vertex covers in the tree and seeks a minimum weight steiner tree in a given vertex weighted undirected graph the problem admits an o log n approximation algorithm in general graphs with n vertices and this approximation factor is tight up to a constant because it is np hard to achieve an o log n approximation for the vertex cover weighted steiner tree problem in general graphs even if the given vertex weights are uniform and a spanning tree is required in this paper we present constant factor approximation algorithms for the problem in unit disk graphs and in graphs excluding a fixed minor for the latter graph class our algorithm can be also applied for the steiner tree activation problem

new approximations for minimum weighted dominating sets and minimum weighted connected dominating sets on unit disk graphs given a node weighted graph the minimum weighted dominating set mwds problem is to find a minimum weighted vertex subset such that for any vertex it is contained in this subset or it has a neighbor contained in this set and the minimum weighted connected dominating set mwcds problem is to find a mwds such that the graph induced by this subset is connected in this paper we study these two problems on a unit disk graph a 4 e approximation algorithm for an mwds based on a dynamic programming algorithm for a min weight chromatic disk cover is presented meanwhile we also propose a 1 e approximation algorithm for the connecting part by showing a polynomial time approximation scheme for a node weighted steiner tree problem when the given terminal set is c local and thus obtain a 5 e approximation algorithm for an mwcds

planar formulae and their uses we define the set of planar boolean formulae and then show that the set of true quantified planar formulae is polynomial space complete and that the set of satisfiable planar formulae is np complete using these results we are able to provide simple and nearly uniform proofs of np completeness for planar node cover planar hamiltonian circuit and line geometric connected dominating set and of polynomial space completeness for planar generalized geography the np completeness of planar node cover and planar hamiltonian circuit and line were first proved elsewhere m r garey and d s johnson the rectilinear steiner tree is np complete siam j appl math 32 1977 pp 826 834 and m r garey d s johnson and r e tarjan the planar hamilton circuit problem is np complete siam j comp 5 1976 pp 704 714

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

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

the covering radius of the 2 15 16 reed muller code is at least 16276 two interesting cosets of the first order reed muller code of block length 2 15 are described they provide counterexamples to a conjecture on the covering radius

approximation schemes for covering and packing problems in image processing and vlsi a unified and powerful approach is presented for devising polynomial approximation schemes for many strongly np complete problems such schemes consist of families of approximation algorithms for each desired performance bound on the relative error e o with running time that is polynomial when e is fixed though the polynomiality of these algorithms depends on the degree of approximation e being fixed they cannot be improved owing to a negative result stating that there are no fully polynomial approximation schemes for strongly np complete problems unless np p the unified technique that is introduced here referred to as the shifting strategy is applicable to numerous geometric covering and packing problems the method of using the technique and how it varies with problem parameters are illustrated a similar technique independently devised by b s baker was shown to be applicable for covering and packing problems on planar graphs

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

approximations for steiner trees with minimum number of steiner points abstract given n terminals in the euclidean plane and a positive constant find a steiner tree interconnecting all terminals with the minimum number of steiner points such that the euclidean length of each edge is no more than the given positive constant this problem is np hard with applications in vlsi design wdm optical networks and wireless communications in this paper we show that a the steiner ratio is 1 4 that is the minimum spanning tree yields a polynomial time approximation with performance ratio exactly 4 b there exists a polynomial time approximation with performance ratio 3 and c there exists a polynomial time approximation scheme under certain conditions

finding minimum weight connected dominating set in stochastic graph based on learning automata finding the minimum weight connected dominating set mcds in an arbitrary graph is an np hard problem and several heuristics and approximation methods have been proposed to solve it forwarding the messages along the virtual backbone induced by the connected dominating set cds significantly reduces the routing overhead as well as the power consumption by reducing the routing nodes to the backbone nodes this paper first defines the stochastic mcds problem where the probability distribution function pdf of the random weight associated with the graph vertices is unknown then it presents several learning automata based algorithms algorithms 1 6 to solve the stochastic mcds problem taking advantage of learning automata the proposed algorithms significantly reduce the number of samples that must be taken from the graph to construct the mcds it is proved that by the proper choice of the learning rate the probability of finding the mcds is close enough to unity the standard sampling method ssm is the baseline with which we compare the performance of the proposed algorithms experimental results show that algorithm 6 significantly outperforms the ssm and the other proposed algorithms in terms of the sampling rate

best affine and quadratic approximations of particular classes of boolean functions in this paper we consider the problem of computing best low order approximations of boolean functions we focus on the best quadratic approximations of a subclass of cubic functions with arbitrary number of variables and we provide formulas for their efficient calculation our methodology is developed upon properties of the best affine approximations of quadratic functions for which formulas for their direct computation not by means of the walsh hadamard transform are given we determine the cubic functions in the above subclass that achieve the maximum second order nonlinearity thus yielding a lower bound for the covering radius of the second order reed muller code ssr rm 2 n in ssr rm 3 n simple extensions of these results to some special cases of higher degree functions are seen to hold furthermore a preliminary analysis of well known constructions for bent functions in terms of their second order nonlinearity is performed that indicates potential weaknesses if construction parameters are not properly chosen

symmetric boolean functions we present an extensive study of symmetric boolean functions especially of their cryptographic properties our main result establishes the link between the periodicity of the simplified value vector of a symmetric boolean function and its degree besides the reduction of the amount of memory required for representing a symmetric function this property has some consequences from a cryptographic point of view for instance it leads to a new general bound on the order of resiliency of symmetric functions which improves siegenthaler s bound the propagation characteristics of these functions are also addressed and the algebraic normal forms of all their derivatives are given we finally detail the characteristics of the symmetric functions of degree at most 7 for any number of variables most notably we determine all balanced symmetric functions of degree less than or equal to 7

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

transfer function conditions for c a b pairs state space conditions for the solution of the output feedback disturbance rejection problem have been given in a recent paper 1 in terms of c a b pairs this paper develops constructive and simple solvability conditions for this problem in terms of the rational matrix equation g s x s n s h s

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

efficient sub 5 approximations for minimum dominating sets in unit disk graphs abstract a unit disk graph is the intersection graph of n congruent disks in the plane dominating sets in unit disk graphs are widely studied due to their applicability in wireless ad hoc networks because the minimum dominating set problem for unit disk graphs is np hard numerous approximation algorithms have been proposed in the literature including some ptass however since the proposal of a linear time 5 approximation algorithm in 1995 the lack of efficient algorithms attaining better approximation factors has aroused attention we introduce an o n m algorithm that takes the usual adjacency representation of the graph as input and outputs a 44 9 approximation this approximation factor is also attained by a second algorithm which takes the geometric representation of the graph as input and runs in o n log n time regardless of the number of edges additionally we propose a 43 9 approximation which can be obtained in o n 2 m time given only the graph s adjacency representation it is noteworthy that the dominating sets obtained by our algorithms are also independent sets

a generalisation of dillon s apn permutation with the best known differential and nonlinear properties for all fields of size 2 4k 2 the existence of almost perfect nonlinear apn permutations operating on an even number of variables was a long standing open problem until an example with six variables was exhibited by dillon et al in 2009 however it is still unknown whether this example can be generalized to any even number of inputs in a recent work perrin et al described an infinite family of permutations named butterflies operating on 4k 2 variables and with differential uniformity at most 4 which contains the dillon apn permutation in this paper we generalize this family and we completely solve the two open problems raised by perrin et al indeed we prove that all functions in this larger family have the best known nonlinearity we also show that this family does not contain any apn permutation besides the dillon permutation implying that all other functions have differential uniformity exactly four

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

cubic boolean functions with highest resiliency we classify those cubic m variable boolean functions which are m 4 resilient we prove that there are four types of such functions depending on the structure of the support of their walsh spectra we are able to determine for each type the walsh spectrum and then the nonlinearity of the corresponding functions we also give the dimension of their linear space this dimension equals m k where k 3 for the first type k 4 for the second type k 5 for the third type and 5 spl les k spl les 9 for the fourth type

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

minimum connected dominating sets and maximal independent sets in unit disk graphs in ad hoc wireless networks a connected dominating set can be used as a virtual backbone to improve the performance many constructions for approximating the minimum connected dominating set are based on the construction of a maximal independent set the relation between the size mis g of a maximum independent set and the size cds g of a minimum connected dominating set in the same graph g plays an important role in establishing the performance ratio of those approximation algorithms previously it is known that mis g 4 cds g 1 for all unit disk graphs g in this paper we improve it by showing mis g 3 8 cds g 1 2

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

almost disturbance decoupling with internal stability the almost disturbance decoupling problem with internal stability is solved in terms of the geometric concepts of linear systems theory the solution gives necessary and sufficient conditions for the existence of a dynamic output feedback controller such that in the closed loop system the disturbances are quenched say in the h sup infinity sense up to any degree of accuracy while maintaining a stable system matrix

reed muller codes achieve capacity on erasure channels we introduce a new approach to proving that a sequence of deterministic linear codes achieves capacity on an erasure channel under maximum a posteriori decoding rather than relying on the precise structure of the codes our method exploits code symmetry in particular the technique applies to any sequence of linear codes where the blocklengths are strictly increasing the code rates converge and the permutation group of each code is doubly transitive in other words we show that symmetry alone implies near optimal performance an important consequence of this result is that a sequence of reed muller codes with increasing blocklength and converging rate achieves capacity this possibility has been suggested previously in the literature but it has only been proven for cases where the limiting code rate is 0 or 1 moreover these results extend naturally to all affine invariant codes and thus to extended primitive narrow sense bch codes this also resolves in the affirmative the existence question for capacity achieving sequences of binary cyclic codes the primary tools used in the proof are the sharp threshold property for symmetric monotone boolean functions and the area theorem for extrinsic information transfer functions

bent functions of maximal degree in this paper a technique for constructing p ary bent functions from plateaued functions is presented this generalizes earlier techniques of constructing bent from near bent functions the fourier spectrum of quadratic monomials is analyzed and examples of quadratic functions with highest possible absolute values in their fourier spectrum are given applying the construction of bent functions to the latter class of functions yields bent functions attaining upper bounds for the algebraic degree when p 3 5 until now no construction of bent functions attaining these bounds was known

transfer function conditions for output feedback disturbance rejection a recent paper 1 has derived state space conditions under which the disturbance transfer function in a linear multivariable system can be zeroed by a dynamic compensator forced by a prescribed set of measurements the present note derives necessary conditions for this problem in terms of the orders of the open loop control disturbance and measurement transfer functions these necessary conditions are shown to be generically sufficient for solvability moreover they provide additional insight into the geometric solvability conditions are simple to check and extend the corresponding results obtained for the state feedback case 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

on cryptographic propagation criteria for boolean functions we determine the functions on gf 2 n which satisfy the propagation criterion of degree n 2 pc n 2 we study subsequently the propagation criterion of degree and order k and its extended version epc we determine those boolean functions on gf 2 n which satisfy pc of order k n 2 we show that none of them satisfies epc of the same order we finally give a general construction of nonquadratic functions satisfying epc of order k this construction uses the existence of nonlinear systematic codes with good minimum distances and dual distances e g kerdock codes and preparata codes

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

restriction terms and nonlinearity of boolean functions nonlinear characteristics of boolean functions is one of the important issues both in the design and cryptanalysis of private key ciphers or encryption algorithms this paper studies nonlinear properties of functions from three different but closely related perspectives maximal odd weighting subspaces restrictions to cosets and hypergraphs all associated with a function main contributions of this work include 1 by using a duality property of a function we have obtained several results that are related to lower bounds on nonlinearity as well as on the number of terms of the function 2 we show that the restriction of a function on a coset has a significant impact on cryptographic properties of the function 3 we identify relationships between the nonlinearity of a function and the distribution of terms in the algebraic normal form of the function 4 we prove that cycles of odd length in the terms as well as quadratic terms in the algebraic normal form of a function play an important role in determining the nonlinearity of the function we hope that these results contribute to the study of new cryptanalytic attacks on ciphers and more importantly of counter measures against such attacks

on correctable errors of binary linear codes the error correction capability of binary linear codes with minimum distance decoding in particular the number of correctable uncorrectable errors is investigated for general linear codes and the first order reed muller codes for linear codes a lower bound on the number of uncorrectable errors is derived the bound for uncorrectable errors with a weight of half the minimum distance asymptotically coincides with the corresponding upper bound for reed muller codes and random linear codes for the first order reed muller codes the number of correctable uncorrectable errors with a weight of half the minimum distance plus one is determined this result is equivalent to deriving the number of boolean functions of m variables with nonlinearity 2 m 2 1 the monotone error structure and its related notions larger half and trial set which were introduced by helleseth klove and levenshtein are mainly used to derive the results

output feedback disturbance decoupling graph interpretation for structured systems abstract in this paper we consider first the output feedback disturbance decoupling problem for linear systems we derive a set of new necessary or sufficient conditions for the solvability of this problem these conditions are expressed in terms of system structural invariants such as infinite zero orders and essential orders then we consider the output feedback disturbance decoupling problem for structured linear systems necessary or sufficient conditions for the generic solvability of this problem are provided they are expressed in terms of input output path length comparisons on the system associated graph

on the degree nonlinearity algebraic thickness and nonnormality of boolean functions with developments on symmetric functions the two main criteria evaluating from cryptographic viewpoint the complexity of boolean functions are the nonlinearity and the algebraic degree two other criteria can also be considered the algebraic thickness and the nonnormality simple proofs are given that asymptotically almost all boolean functions have high algebraic thicknesses and are deeply nonnormal as well as they have high algebraic degrees and high nonlinearities we also study in detail the relationship between nonnormality and nonlinearity we derive simple proofs of known results on symmetric boolean functions and we prove several new and more general results on a class containing all symmetric functions

binary m sequences with three valued crosscorrelation a proof of welch s conjecture we prove the long standing conjecture of welch stating that for odd n 2m 1 the power function x sup d with d 2 sup m 3 is maximally nonlinear on gf 2 sup n or in other terms that the crosscorrelation function between a binary maximum length linear shift register sequence of degree n and a decimation of that sequence by 2 sup m 3 takes on precisely the three values 1 1 spl plusmn 2 sup m 1

a 5 epsilon porson approximation algorithm for minimum weighted dominating set in unit disk graph we study the minimum weight dominating set problem in weighted unit disk graph and give a polynomial time algorithm with approximation ratio 5 epsilon porson improving the previous best result of 6 epsilon porson in yaochun huang xiaofeng gao zhao zhang weili wu a better constant factor approximation for weighted dominating set in unit disk graph j comb optim issn 1382 6905 2008 1573 2886 print online combining the common technique used in the above mentioned reference we can compute a minimum weight connected dominating set with approximation ratio 9 epsilon porson beating the previous best result of 10 epsilon porson in the same work

frequency domain conditions for the solvability of the multivariable disturbance decoupling problem given a plant description in terms of its transfer matrices necessary and sufficient conditions are derived in the frequency domain for the solvability of the multivariable disturbance decoupling problem with independent controlled and measured outputs

a three valued walsh transform from decimations of helleseth gong sequences the walsh transform of two level autocorrelation sequences has played an important role in the construction of the set of sequences in which any two sequences are orthogonal for p ary sequences there are only two basic classes of two level autocorrelation sequences with no subfield structures for an arbitrary odd prime p one is the class of p ary m sequences and the other is the class of p ary helleseth gong sequences in this paper the walsh transform of a subclass of p ary helleseth gong sequences and the walsh transform of their particular decimations are completely determined and are shown to be three valued

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