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

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

a variational principle associated with elliptic boundary value problems abstract we introduce a variational approach which can be applied to a large class of nonlinear elliptical equations assume that r r and r 2 r are differentiable convex functions we concern ourselves with problems of the form 2 u x u x u x x 2 u x 1 u x x where r n is an open bounded domain with smooth boundary the maps 1 2 are boundary operators given by 1 u u u n and 2 u u n u where n is the outward normal to we shall show that solutions to this boundary value problem can coincide with critical points of a functional f defined by f u 2 u u d x u d x 2 u d 1 u d where and are fenchel legendre dual of and respectively we then use these functionals to prove existence of nontrivial solutions to certain boundary value problems this method offers advantages when compared to using the more standard euler lagrange functional in that solutions have greater regularity and nonlinear boundary conditions can be more easily dealt with

on the affine sub families of quadratic nfsrs grain 128 is a hardware oriented stream cipher based on the cascade connection of a 128 bit linear feedback shift register into a 128 bit quadratic nonlinear feedback shift register nfsr its main register is in essence a quadratic nfsr however its affine sub families could not be solved by the previous methods in this paper it is shown that the family of sequences generated by the main register of grain 128 includes no affine sub families except a small one of order three to achieve this goal a new method is proposed for solving affine sub families of general quadratic nfsrs let nfsr f be an nfsr with a quadratic characteristic function f it is proved that the characteristic function of a linear sub family of the nfsr f divides a linear combination of variables appearing in the quadratic terms of f where the division can be seen as the univariate polynomial division over the finite field mathbb f 2 this facilitates picking up a candidate set of linear sub families through univariate polynomial factorization over mathbb f 2 the affine case is an analogy besides a useful new upper bound on the orders of affine sub families of a quadratic nfsr is given

on the global avalanche characteristics between two boolean functions and the higher order nonlinearity the criterion for the global avalanche characteristics gac of cryptographic functions is an important property to measure the correlation between two arbitrary boolean functions we propose two new criteria called the sum of squares indicator and the absolute indicator of the cross correlation between two boolean functions the two indicators generalize the gac criterion based on the properties of the cross correlation function we deduce the rough lower and the rough upper bounds on the two indicators by hamming weights of two boolean functions and generalize some properties between the walsh spectrum and the cross correlation function furthermore we give the tight upper and the tight lower bounds on the two indicators finally we show some relationships between the upper bounds on the two indicators and the higher order nonlinearity

generating modules efficiently algebraic k theory for noncommutative noetherian rings abstract the well known forster swan theorem and the stable version of this result given by eisenbud and evans give a bound on the number of generators of a module over a commutative ring in terms of local data in this paper we generalise these results to arbitrary noetherian rings as an application we prove versions of serre s theorem and the cancellation theorem over such rings

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

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

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

non convex self dual lagrangians new variational principles of symmetric boundary value problems abstract we study the concept and the calculus of non convex self dual nc sd lagrangians and their derived vector fields which are associated to many partial differential equations and evolution systems they indeed provide new representations and formulations for the superposition of convex functions and symmetric operators they yield new variational resolutions for large class of hamiltonian partial differential equations with variety of linear and nonlinear boundary conditions including many of the standard ones this approach seems to offer several useful advantages it associates to a boundary value problem several potential functions which can often be used with relative ease compared to other methods such as the use of euler lagrange functions these potential functions are quite flexible and can be adapted to easily deal with both nonlinear and homogeneous boundary value problems additionally in most cases the solutions generated using this new method have greater regularity than the solutions obtained using the standard euler lagrange function perhaps most remarkable however are the permanence properties of nc sd lagrangians their calculus is relatively manageable and their applications are quite broad

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

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

a perturbation theory near convex hamiltonian systems soit he x p une famille de fonctions sur r n r n dependant du parametre e r d pour e petit on considere des solutions periodiques aux equations de hamilton dx i dt he x p p i dp i dt he x p x i 1 i et on cherche a les relier aux solutions periodiques du systeme non perturbe dx i dt h 0 x p p i dp i dt h 0 x p x i 1 i

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

trace representation of some generalized cyclotomic sequences of length pq this paper contributes to trace representation of some generalized cyclotomic sequences of length pq p qprime which are defined by ding and helleseth from the relations between these sequences and the legendre sequence we firstly confirm the defining pairs of these sequences of arbitrary order then we obtain their trace representation from which we give their linear complexity using key s method it can be seen that bai et al s conclusion is a special case of our result when the order is two finally an example is given to illustrate the validity of our result

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

sudoku based space filling designs sudoku is played by millions of people across the globe it has simple rules and is very addictive the game board is a nine by nine grid of numbers from one to nine several entries within the grid are provided and the remaining entries must be filled in subject to no row column or three by three subsquare containing duplicate numbers by exploiting these three types of uniformity we propose an approach to constructing a new type of design called a sudoku based space filling design such a design can be divided into groups of subdesigns so that the complete design and each subdesign achieve maximum uniformity in univariate and bivariate margins examples are given illustrating the proposed construction method applications of such designs include computer experiments with qualitative and quantitative factors linking parameters in engineering and crossvalidation copyright 2011 oxford university press

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

multiplicative characters the weil bound and polyphase sequence families with low correlation power residue and sidelnikov sequences are polyphase sequences with low correlation and variable alphabet sizes represented by multiplicative characters in this paper sequence families constructed from the shift and addition of the polyphase sequences are revisited initially 0 1 is assumed for multiplicative characters to represent power residue and sidelnikov sequences in a simple form the weil bound on multiplicative character sums is refined for the assumption where the character sums are equivalent to the correlations of sequences represented by multiplicative characters general constructions of polyphase sequence families that produce some of known families as the special cases are then presented the refined weil bound enables the efficient proofs on the maximum correlation magnitudes of the sequence families from the constructions it is shown that m ary known sequence families with large size can be partitioned into m 1 disjoint subsequence families with smaller maximum correlation magnitudes more generalized constructions are also considered by the addition of multiple cyclic shifts of power residue and sidelnikov sequences

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

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 binary cyclic codes with codewords of weight three and binary sequences with the trinomial property golomb and gong 1997 and 1999 considered binary sequences with the trinomial property in this correspondence we shall show that the sets of those sequences are quite trivially closely connected with binary cyclic codes with codewords of weight three this approach gives us another way to deal with trinomial property problems after disproving one conjecture formulated by golomb and gong we exhibit an infinite class of sequences which do not have the trinomial property corresponding to binary cyclic codes of length 2 sup m 1 with minimum distance exactly four

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

polynomials with linear structure and maiorana mcfarland construction in this paper we study permutation polynomials over the finite fields that have linear structures we present some results on such a permutation which transforms a hyperplane to another hyperplane we fully characterize the bilinear polynomial with linear structure the most important result of this paper is to show the relation between a maiorana mcfarland function with an affine derivative and a polynomial with a linear structure moreover we highlight this result in the context of resilient functions which are based on maiorana mcfarland construction

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

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

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

efficient bit parallel gf 2 m multiplier for a large class of irreducible pentanomials this work studies efficient bit parallel multiplication in gf 2m for irreducible pentanomials based on the so called shifted polynomial bases spbs we derive a closed expression of the reduced spb product for a class of polynomials xm xk s xk s 1 hellip xk 1 1 with ks k1 les m 1 2 then we apply the above formulation to the case of pentanomials the resulting multiplier outperforms or is as efficient as the best proposals in the technical literature but it is suitable for a much larger class of pentanomials than those studied so far unlike previous works this property enables the choice of pentanomials optimizing different field operations for example inversion yet preserving an optimal implementation of field multiplication as discussed and quantitatively proved in the last part of the paper

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

iterations of linear maps over finite fields we study the dynamics of the evolution of ducci sequences and the martin odlyzkowolfram cellular automaton by iterating their respective linear maps on z n a fter a review of an algebraic characterization of cycle lengths we deduce the relationship between the maximal cycle lengths of these two maps from a simple connection between them for n odd we establish a conjugacy relationship that provides a more direct identification of their dynamics we give an alternate geometric proof of the maximal cycle length relationship based on this conjugacy and a symmetry property we show that the cyclic dynamics of both maps in dimension 2n can be deduced from their periodic behavior in dimension n this link is generalized to a larger class of maps with restrictions shared by both maps we obtain a formula for the number of vectors in dimension 2n belonging to a cycle of length q that expresses this number in terms of the analogous values in dimension n

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

further result of compressing maps on primitive sequences modulo odd prime powers let be the integer residue ring with odd prime and integer for a sequence over there is an unique adic expansion where each is a sequence over and can be regarded as a sequence over the prime field gf naturally let be a strongly primitive polynomial over and the set of all primitive sequences generated by over suppose that gf gf it is shown that any function in is an injective map from to gf and the derived sequences of different functions are also different that is if and only if and for and these injective functions in can be considered as good candidates for the keys of a stream cipher

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 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 the phase sequence set of slm ofdm scheme for a crest factor reduction the crest factor distribution of orthogonal frequency division multiplexing ofdm symbol sequences is evaluated and it is shown that ofdm symbol sequences with a short period are expected to have a high crest factor the crest factor relationship between two input symbol sequences hamming distance d apart is also derived using these two results two criteria are proposed for a phase sequence set of the selected mapping slm ofdm scheme and suggest the rows of the cyclic hadamard matrix constructed from an m sequence as a near optimal phase sequence set of the slm ofdm scheme

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 the dual of certain ternary weakly regular bent functions in 2006 helleseth and kholosha conjectured and partially proved the existence of a class of ternary weakly regular monomial bent functions and also the expression for the dual bent function up to the sign value the bentness was finally proved later in 2009 using a complicated technique that employs stickelberger s theorem extensively using the previously found results and approaches in this paper a surprisingly short proof for the conjectured expression of the dual is given but without resolving the sign ambiguity furthermore we resolve the sign by finding the trace representation of the dual function

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

basic element theory of noncommutative noetherian rings abstract in this paper we propose a notion of basic element for noncommutative noetherian rings and derive the corresponding basic element theorem as in d eisenbud and e g evans jr generating modules efficiently theorems from algebraic k theory j algebra 27 1973 278 305 as corollaries to this theorem we obtain a number of results of stafford such as his generalizations of serre s splitting off theorem and bass s cancellation theorem

on linear complexity of sequences over gf 2 n in this paper we consider some aspects related to determining the linear complexity of sequences over gf 2n in particular we study the effect of changing the finite field basis on the minimal polynomials and thus on the linear complexity of sequences defined over gf 2n but given in their binary representation let a ai be a sequence over gf 2n then ai can be represented by ai j 0n 1 aij j aij gf 2 where is the root of the irreducible polynomial defining the field consider the sequence b bi whose elements are obtained from the same binary representation of a but assuming a different set of basis say 0 1 n 1 i e bi j 0r 1 aij j we study the relation between the minimal polynomial of a and b

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

on secrecy capacity of minimum storage regenerating codes in this paper we revisit the problem of characterizing the secrecy capacity of minimum storage regenerating msr codes under the passive l 1 l 2 eavesdropper model where the eavesdropper has access to data stored on l 1 nodes and the repair data for an additional l 2 nodes we study it from the information theoretic perspective first some general properties of msr codes as well as a simple and generally applicable upper bound on secrecy capacity are given second a new concept of stable msr codes is introduced where the stable property is shown to be closely linked with secrecy capacity finally a comprehensive and explicit result on secrecy capacity in the linear msr scenario is present which generalizes all related works in the literature and also predicts certain results for some unexplored linear msr codes

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

explicit criteria for construction of plateaued functions plateaued functions are very important cryptographic functions due to their desirable cryptographic characteristics we find explicit criteria for the construction of p ary r plateaued functions with an odd prime p we point out that 0 plateaued functions are bent functions and so plateaued functions generalize the notion of bent functions we first derive an explicit form for the walsh hadamard transform of a p ary r plateaued function we then obtain an upper bound on the degree of p ary r plateaued functions and we classify p ary n 1 plateaued functions in n variables we also obtain explicit criteria for the existence of p ary r plateaued functions accordingly these results lead to improved bounds on the existence of p ary bent functions