gleason s theorem for rectangular jbw triples a jbw triple b is said to be rectangular if there exists a w algebra a and a pair p q of centrally equivalent elements of the complete orthomodular lattice of projections in a such that b is isomorphic to the jbw triple paq any weak closed injective operator space provides an example of a rectangular jbw triple the principal order ideal of the complete lattice of centrally equivalent pairs of projections in a w algebra a generated by p q forms a complete lattice that is order isomorphic to the complete lattice of weak closed inner ideals in b and to the complete lattice of structural projections on b although not itself in general orthomodular possesses a complementation that allows for definitions of orthogonality centre and central orthogonality to be given a less familiar notion in lattice theory that is well known in the theory of jordan algebras and jordan triple systems is that of rigid collinearity of a pair e2 f2 and e2 f2 of elements of this is defined and characterized in terms of properties of a w algebra a is sometimes thought of as providing a model for a statistical physical system in this case b or equivalently paq may be thought of as providing a model for a fixed sub system of that represented by a therefore may be considered to represent the set consisting of a particular kind of sub system of that represented by paq central orthogonality and rigid collinearity of pairs of elements of may be regarded as representing two different types of disjointness the former classical disjointness and the latter decoherence of the two sub systems it is therefore natural to consider bounded measures m on that are additive on centrally orthogonal and rigidly collinear pairs of elements using results of j d m wright it is shown that provided that neither of the two hereditary sub w algebras pap and qaq of a has a weak closed ideal of type i2 such measures are precisely those that are the restrictions of bounded sesquilinear functionals m on pap qaq with the property that the action of the centroid z b of b commutes with the adjoint operation when b is a complex hilbert space of dimension greater than two this result reduces to gleason s theorem

multipliers and dual operator algebras abstract in a previous paper we showed how the main theorems characterizing operator algebras and operator modules fit neatly into the framework of the noncommutative shilov boundary and more particularly via the left multiplier operator algebra of an operator space as well as giving new characterization theorems the approach of that paper allowed many of the hypotheses of the earlier theorems to be eliminated recent progress of the author with effros and zarikian now enables weak versions of these characterization theorems for example we prove a result analogous to sakai s famous characterization of von neumann algebras as the c algebras with predual namely that the weakly closed unital not necessarily selfadjoint subalgebras of b h for a hilbert space h are exactly the unital operator algebras which possess an operator space predual this removes one of the hypotheses from an earlier characterization due to le merdy we also show that the multiplier operator algebras of dual operator spaces are dual operator algebras using this we refine several known results characterizing dual operator modules

every completely polynomially bounded operator is similar to a contraction it is proved that a bounded operator on a hilbert space is similar to a contraction if and only if it is completely polynomially bounded this gives a partial answer to problem 6 of halmos bull amer math soc 76 1970 877 933 the set of completely bounded maps between c algebras is studied to obtain some structure representation and extension theorems for this class of maps these allow a characterization of the completely bounded representations on a hilbert space of any subalgebra of a c algebra to be obtained the result in the title follows by applying this characterization to the disk algebra

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

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

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

a characterization of operator algebras abstract an operator algebra is a uniformly closed algebra of bounded operators on a hilbert space in this paper we give a characterization of unital operator algebras in terms of their matricial norm structure more precisely if a is an l matricially normed space and also an algebra with a completely contractive multiplication and an identity of norm 1 then there is a completely isometric isomorphism of a onto a unital operator algebra indeed the multiplication on a need not be assumed to be associative for this conclusion to follow examples are given to show that the condition on the identity is necessary it follows from the above that the quotient of an operator algebra by a closed two sided ideal with the natural matricial structure is again an operator algebra up to complete isometric isomorphism

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

generating and counting binary bent sequences two general classes of binary bent sequences bent based and linear based are introduced algorithms that allow easy generation of bent sequences from either class are given based on some simple computation and a computer search the authors conjecture a lower bound on the total number of binary bent sequences of a given order this lower bound is exact for bent sequences of order 16 a list is included from which all such sequences can be derived

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

operator space structure of jc triples and tros i we embark upon a systematic investigation of operator space structure of jc triples via a study of the tros ternary rings of operators they generate our approach is to introduce and develop a variety of universal objects including universal tros by which means we are able to describe all possible operator space structures of a jc triple via the concept of reversibility we obtain characterisations of universal tros over a wide range of examples we apply our results to obtain explicit descriptions of operator space structures of cartan factors regardless of dimension

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

characterizations of categories of commutative c subalgebras we aim to characterize the category of injective homomorphisms between commutative c subalgebras of a given c algebra a we reduce this problem to finding a weakly terminal commutative subalgebra of a and solve the latter for various c algebras including all commutative ones and all type i von neumann algebras this addresses a natural generalization of the mackey piron programme which lattices are those of closed subspaces of hilbert space we also discuss the way this categorified generalization differs from the original question

improved lower bound on the number of balanced symmetric functions over gf p the lower bound on the number of n variable balanced symmetric functions over finite fields gf p presented by cusick et al in t w cusick y li p sta nica balanced symmetric functions over gf p ieee trans inform theory 54 3 2008 1304 1307 is improved in this paper an equivalent characterization is also presented for the general case

uniqueness universality and homogeneity of the noncommutative gurarij space we realize the noncommutative gurarij space ng defined by oikhberg as the fraisse limit of the class of finite dimensional 1 exact operator spaces as a consequence we deduce that the noncommutative gurarij space is unique up to completely isometric isomorphism homogeneous and universal among separable 1 exact operator spaces we also prove that ng is the unique separable nuclear operator space with the property that the canonical triple morphism from the universal tro to the triple envelope is an isomorphism we deduce from this fact that ng does not embed completely isometrically into an exact c algebra and it is not completely isometrically isomorphic to a c algebra or to a tro we also provide a canonical construction of ng which shows that the group of surjective complete isometries of ng is universal among polish groups analog results are proved in the commutative setting and more generally for m n spaces in particular we provide a new characterization and canonical construction of the gurarij banach space

state spaces of jb triples an atomic decomposition is proved for banach spaces which satisfy some affine geometric axioms compatible with notions from the quantum mechanical measuring process this is then applied to yield under appropriate assumptions geometric characterizations up to isometry of the unit ball of the dual space of a jb triple and up to complete isometry of one sided ideals in c algebras

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

the kirchberg wassermann operator system is unique in 1998 kirchberg and wassermann constructed a separable nuclear operator system with the property that the canonical unital homomorphism from the maximal c algebra onto the c envelope is injective we show that such an operator system is unique and completely order isomorphic to the operator system associated with the noncommutative poulsen simplex

active lattices determine aw algebras abstract we prove that operator algebras that have enough projections are completely determined by those projections their symmetries and the action of the latter on the former this includes all von neumann algebras and all aw algebras we introduce active lattices which are formed from these three ingredients more generally we prove that the category of aw algebras is equivalent to a full subcategory of active lattices crucial ingredients are an equivalence between the category of piecewise aw algebras and that of piecewise complete boolean algebras and a refinement of the piecewise algebra structure of an aw algebra that enables recovering its total structure

quasi multipliers and algebrizations of an operator space let x be an operator space let be a product on x and let x denote the algebra that one obtains we give necessary and sufficient conditions on the bilinear mapping for the algebra x to have a completely isometric representation as an algebra of operators on some hilbert space in particular we give an elegant geometrical characterization of such products by using the haagerup tensor product our result makes no assumptions about identities or approximate identities our proof is independent of the earlier result of blecher ruan and sinclair d p blecher z j ruan a m sinclair a characterization of operator algebras j funct anal 89 1 1990 188 201 which solved the case when the bilinear mapping has an identity of norm one and our result is used to give a simple direct proof of this earlier result we also develop further the connections between quasi multipliers of operator spaces and their representations on a hilbert space or their embeddings in the second dual and show that the quasi multipliers of operator spaces defined in m kaneda v i paulsen quasi multipliers of operator spaces j funct anal 217 2 2004 347 365 coincide with their c algebraic counterparts

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

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

characters dual pairs and unipotent representations abstract we lift distribution characters of irreducible unitary representations of classical groups from the group to the lie algebra via the cayley transform then a specific class of these characters admits fourier transform supported on the closure of a single nilpotent coadjoint orbit we calculate also the wave front set of the most singular low rank representations

quasi multipliers of operator spaces we use the injective envelope to study quasi multipliers of operator spaces we prove that all representable operator algebra products that an operator space can be endowed with are induced by quasi multipliers we obtain generalizations of the banach stone theorem

the shilov boundary of an operator space and the characterization theorems abstract we study operator spaces operator algebras and operator modules from the point of view of the noncommutative shilov boundary in this attempt to utilize some noncommutative choquet theory we find that hilbert c modules and their properties which we studied earlier in the operator space framework replace certain topological tools we introduce certain multiplier operator algebras and c algebras of an operator space which generalize the algebras of adjointable operators on a c module and the imprimitivity c algebra it also generalizes a classical banach space notion this multiplier algebra plays a key role here as applications of this perspective we unify and strengthen several theorems characterizing operator algebras and modules we also include some general notes on the commutative case of some of the topics we discuss coming in part from joint work with christian le merdy about function modules

diagonalizing matrices over aw algebras abstract every commuting set of normal matrices with entries in an aw algebra can be simultaneously diagonalized to establish this a dimension theory for properly infinite projections in aw algebras is developed as a consequence passing to matrix rings is a functor on the category of aw algebras

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

c operator systems and crossed products abstract the purpose of this paper is to introduce a consistent notion of universal and reduced crossed products by actions and coactions of groups on operator systems and operator spaces in particular we shall put emphasis to reveal the full power of the universal properties of the universal crossed products it turns out to make things consistent one has to perform the constructions in some bigger categories which allow the right framework for studying the universal properties and which are stable under the construction of crossed products even for non discrete groups in the case of operator systems this larger category is what we call a c operator system i e a selfadjoint subspace x of some b h which contains a c algebra a such that a x x x a in the case of operator spaces the larger category is given by what we call c operator bimodules we introduce the notion of crossed products in these categories and show that the classical imai takai and katayama duality theorems for crossed products by group co actions on c algebras extend to our notion of crossed products by c operator systems and c operator bimodules

on injectivity and nuclearity for operator spaces an injective operator space v which is dual as a banach space has the form er 1 e where r is an injective von neumann algebra and where e is a projection in r this is used to show that an operator space v is nuclear if and only if it is locally reflexive and v ast ast is injective it is also shown that any exact operator space is locally reflexive

the unit ball of an injective operator space has an extreme point abstract we define an a w tro as an off diagonal corner of an a w algebra and show that the unit ball of an a w tro has an extreme point in particular the unit ball of an injective operator space has an extreme point which answers a question raised in 8 affirmatively we also show that an a w tro respectively an injective operator space has an ideal decomposition that is it can be decomposed into the direct sum of a left ideal a right ideal and a two sided ideal in an a w algebra respectively an injective c algebra in particular we observe that an a w tro hence an injective operator space has an algebrization which admits a quasi identity

transcending classical invariant theory let sp2 r sp2 sp be the real symplectic group of rank n let sp denote the two fold cover of sp there is a unitary representation constructed by shale sh and weil wa of sp that is of considerable interest in the theory of automorphic forms we denote this representation by co and call it the oscillator representation the purpose of this paper is to establish about c a fact that should be useful for clarifying the structure of spaces of automorphic forms constructed using co we begin by formulating the result let e be a reductive subgroup of sp and let e denote the inverse image of e in sp denote by r e the set of infinitesimal equivalence classes of continuous irreducible admissible representations of e on locally convex spaces let co be the smooth representation of sp associated to co let co be realized on a hilbert space and let the subspace of smooth vectors on which space co is defined be written denote by r9 e co the set of elements of e which are realized as quotients by co e invariant closed subspaces of t g consider a reductive dual pair g g c sp h2 it is not hard to show that g and g commute with one another hence we may regard co i g g as a representation of gx g it is well known f that r gx g s9 g xr9 g the identification associates to p e r g and p e i g the tensor product p x p we note that p x p is not defined as a topological vector space nevertheless the infinitesimal equivalence class of p op is well defined select p x p e g x g suppose that in fact p p e 9 g g c then clearly p e r g c and p e 9 g c hence g co defines the graph of a correspondence between certain subsets of r9 g cv and j g co in fact the situation is quite precise

representations of completely bounded multilinear operators a deli ion of a completely bounded multilinear operator from one c algebra into another is introduced each completely bounded multilinear operator from a algebra into the algebra of bounded linear operators on a hilbert space is shown to be representable in terms of representations of the c algebra and interlacing operators this result extends wittstock s theorem that decomposes a completely bounded linear operator from a c algebra into an injective c algebra into completely positive linear operators 0 1987 academic press inc stinespring s theorem gives a useful representation for a completely positive linear operator from a c algebra into the algebra bl h for continuous linear operators on a hilbert space h 12 11 using this representation with wittstock s theorem that decomposes a completely bounded linear operator as a finite linear combination of completely positive linear operators 14 lo one obtains a representation of a completely bounded linear operator from a c algebra into bl h our main result is a representation theorem for completely bounded multilinear operators from a c algebra into bl h which generalizes this representation of completely bounded linear operators corollaries give multilinear generalizations of several results known for completely bounded linear operators we shall briefly describe the type of representations to be 151

classification of contractively complemented hilbertian operator spaces abstract we construct some separable infinite dimensional homogeneous hilbertian operator spaces h m r and h m l which generalize the row and column spaces r and c the case m 0 we show that a separable infinite dimensional hilbertian jc triple is completely isometric to one of h m r h m l h m r h m l or the space spanned by creation operators on the full anti symmetric fock space in fact we show that h m l respectively h m r is completely isometric to the space of creation respectively annihilation operators on the m respectively m 1 anti symmetric tensors of the hilbert space together with the finite dimensional case studied in m neal b russo representation of contractively complemented hilbertian operator spaces on the fock space proc amer math soc 134 2006 475 485 this gives a full operator space classification of all rank one jc triples in terms of creation and annihilation operator spaces we use the above structural result for hilbertian jc triples to show that all contractive projections on a c algebra a with infinite dimensional hilbertian range are expansions which we define precisely of normal contractive projections from a onto a hilbertian space which is completely isometric to r c r c or this generalizes the well known result first proved for b h by robertson in a g robertson injective matricial hilbert spaces math proc cambridge philos soc 110 1991 183 190 that all hilbertian operator spaces that are completely contractively complemented in a c algebra are completely isometric to r or c we use the above representation on the fock space to compute various completely bounded banach mazur distances between these spaces or

unipotent representations of complex semisimple groups algebraic group over r or c in this paper we restrict attention to c we generalize arthur s definition slightly or perhaps simply make it more precise all of the resulting representations except for a finite set are then unitarily induced from representations of the same kind on proper parabolic subgroups we call the finite set remaining special unipotent representations a precise definition will be given later definition 1 17 our main result theorem iii of this introduction is a character formula for any special unipotent representation of course such a formula can be deduced from the kazhdan lusztig conjecture cf v3 the advantages of theorem iii are that it is in closed form and that it lends itself to verification of some conjectures of arthur in a so let g be a connected complex semisimple lie group and q its lie algebra choose

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 local lifting property for operator spaces we study the local lifting property for operator spaces this is a natural non commutative analogue of the banach space local lifting property but is very different from the local lifting property studied in c algebra theory we show that an operator space has the local lifting property if and only if it is an l 1 space these operator space are completely isomorphic to the operator subspaces of the operator preduals of von neumann algebras and thus locally reflexive moreover we show that an operator space v has the local lifting property if and only if its operator space dual v is injective

la formule des traces stable pour le groupe metaplectique les termes elliptiques soient f un corps de nombres et tilde g le revetement metaplectique de weil de g mathbb a f mathrm sp 2n mathbb a f nous stabilisons les termes elliptiques semi simples de la partie specifique de la formule des trace pour tilde g a l aide du transfert des fonctions test et du lemme fondamental etablis dans un article anterieur nous obtenons ainsi une expression en termes des integrales orbitales stables le long de certaines classes f elliptiques dans mathrm so 2n 1 times mathrm so 2n 1 ce que nous appelons equi singulieres pour les paires n n avec n n n une formule simple des coefficients est aussi obtenue

injectivity of operator spaces we study the structure of injective operator spaces and the existence and uniqueness of the injective envelopes of operator spaces we give an easy example of an injective operator space which is not completely isometric to any c algebra this answers a question of wittstock 23 furthermore we show that an operator space e is injective if and only if there exists an injective c algebra a and two projections p and q in a such that e is completely isometric to paq

type decomposition for von neumann algebra embeddings this paper establishes a type decomposition theory for embeddings of a von neumann algebra a into a von neumann algebra m by studying projections in a that are semifinite resp abelian relative to m an application of this theory is a complete characterization of the subalgebras a of m that have an n equipartition of the identity i e i k 1npk with projections pk a which are equivalent in m from this follows r v kadison s result amer j math 106 1984 1451 1468 that masas of m always have equipartitions and hence that normal operator valued matrices are diagonalizable a second application is an answer to a question by g k pedersen and e stormer indiana univ math j 23 1973 121 129 on the characterization of g finite projections in discrete crossed products

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

local properties of ternary rings of operators and their linking c algebras abstract we show that some local properties such as nuclearity exactness and local reflexivity of ternary rings of operators tros are closely related to the local properties of their linking c algebras we also show some equivalent conditions for nuclear tros and show that haagerup s decomposition property for completely bounded maps and pisier s norm can be naturally generalized to tros

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

contractive projections and operator spaces parallel to the study of finite dimensional banach spaces there is a growing interest in the corresponding local theory of operator spaces we define a family of hilbertian operator spaces h k n 1 k n generalizing the row and column hilbert spaces r n and c n and we show that an atomic subspace x c b h that is the range of a contractive projection on b h is isometrically completely contractive to an sum of the h k n and cartan factors of types 1 to 4 in particular for finite dimensional x this answers a question posed by oikhberg and rosenthal explicit in the proof is a classification up to complete isometry of atomic w closed jw triples without an infinite dimensional rank 1 w closed ideal

on the second order nonlinearities of some bent functions the rth order nonlinearity of boolean functions plays a central role against several known attacks on stream and block ciphers it plays also an important role in coding theory since its maximum equals the covering radius of the rth order reed muller code but it is difficult to calculate and even to bound in this paper we show lower bounds on the second order nonlinearity of two subclasses of well known bent functions we first improve a known lower bound on the second order nonlinearity of the simplest partial spread bent functions whose nonlinearity profile has been bounded by the second author this improvement allows obtaining a better bound for the whole profile we subsequently give a lower bound on the second order nonlinearity of some infinite class of maiorana mcfarland m m bent functions which generalizes a result by gangopadhyay et al

improving the lower bound on the higher order nonlinearity of boolean functions with prescribed algebraic immunity the recent algebraic attacks have received a lot of attention in cryptographic literature the algebraic immunity of a boolean function quantifies its resistance to the standard algebraic attacks of the pseudorandom generators using it as a nonlinear filtering or combining function very few results have been found concerning its relation with the other cryptographic parameters or with the rth order nonlinearity as recalled by carlet at crypto 06 many papers have illustrated the importance of the r th order nonlinearity profile which includes the first order nonlinearity the role of this parameter relatively to the currently known attacks has been also shown for block ciphers recently two lower bounds involving the algebraic immunity on the rth order nonlinearity have been shown by carlet none of them improves upon the other one in all situations in this paper we prove a new lower bound on the rth order nonlinearity profile of boolean functions given their algebraic immunity that improves significantly upon one of these lower bounds for all orders and upon the other one for low orders

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

ein operatorwertiger hahn banach satz abstract we generalize arveson s extension theorem for completely positive mappings 1 to a hahn banach principle for matricial sublinear functionals with values in an injective c algebra or an ideal in b h we characterize injective w algebras by a matricial order condition we illustrate the matricial hahn banach principle by three applications 1 let a b b be unital c algebras b a subalgebra of a and b b injective if a b is a completely bounded self adjoint b bihomomorphism then it can be expressed as the difference of two completely positive b bihomomorphism 2 let m be a w algebra containing 1 h on a hilbert space h if m is finite and hyperfinite there exists an invariant expectation mapping p of b h onto m p is an extension of the center trace 3 combes 7 proved that a lower semicontinuous scalar weight on a c algebra is the upper envelope of bounded positive functionals we generalize this result to unbounded completely positive mappings with values in an injective w algebra

diagonalizing matrices over c x a complete characterization of those compact hausdorff spaces is given such that for every n each normal element in the algebra c x mn of continuous functions from x to mn can be continuously diagonalized the conditions are that x be a sub stonean space with dim x 2 and carries no nontrivial g bundles over any closed subset for g a symmetric group or the circle group in particular diagonalization is assured on every totally disconnected sub stonean space but also on connected spaces of the form y y where y is a simply connected noncompact graph

improving the upper bounds on the covering radii of binary reed muller codes by deriving bounds on character sums of boolean functions and by using the characterizations due to kasami of those elements of the reed muller codes whose hamming weights are smaller than twice and a half the minimum distance we derive an improved upper bound on the covering radius of the reed muller code of order 2 and we deduce improved upper bounds on the covering radii of the reed muller codes of higher orders

algebraic immunity for cryptographically significant boolean functions analysis and construction recently algebraic attacks have received a lot of attention in the cryptographic literature it has been observed that a boolean function f used as a cryptographic primitive and interpreted as a multivariate polynomial over f sub 2 should not have low degree multiples obtained by multiplication with low degree nonzero functions in this paper we show that a boolean function having low nonlinearity is also weak against algebraic attacks and we extend this result to higher order nonlinearities next we present enumeration results on linearly independent annihilators we also study certain classes of highly nonlinear resilient boolean functions for their algebraic immunity we identify that functions having low degree subfunctions are weak in terms of algebraic immunity and we analyze some existing constructions from this viewpoint further we present a construction method to generate boolean functions on n variables with highest possible algebraic immunity spl lceil n 2 spl rceil this construction first presented at the 2005 workshop on fast software encryption fse 2005 has been the first one producing such functions these functions are obtained through a doubly indexed recursive relation we calculate their hamming weights and deduce their nonlinearities we show that they have very high algebraic degrees we express them as the sums of two functions which can be obtained from simple symmetric functions by a transformation which can be implemented with an algorithm whose complexity is linear in the number of variables we deduce a very fast way of computing the output to these functions given their input

kazhdan lusztig algorithms for nonlinear groups and applications to kazhdan patterson lifting we establish an algorithm to compute characters of irreducible harish chandra modules for a large class of nonalgebraic lie groups roughly speaking the class of groups consists of those obtained as nonlinear double covers of linear groups in harish chandra s class we then apply this theory to study a particular group the universal cover of the real general linear group and discover a symmetry of the character computations encoded in a character multiplicity duality using this duality theory we reinterpret a kind of representation theoretic shimura correspondence for the general linear group geometrically and find that it is dual to an analogous lifting for indefinite unitary groups it seems likely that this example is illustrative of a general framework for studying similar correspondences

subspaces of c algebras abstract basic properties of matricially normed spaces are considered and a simple matrix norm characterization of the subspaces of c algebras is given the latter result is used to study the haagerup tensor products and quotients of such subspaces