common randomness in information theory and cryptography i secret sharing as the first part of a study of problems involving common randomness at distance locations information theoretic models of secret sharing generating a common random key at two terminals without letting an eavesdropper obtain information about this key are considered the concept of key capacity is defined single letter formulas of key capacity are obtained for several models and bounds to key capacity are derived for other models

group gradings on associative algebras abstract let r g g rg be a g graded ring we describe all types of gradings on r if g is torsion free and r is artinian semisimple if r is a matrix algebra over an algebraically closed field f then we give a description of all g gradings on r provided that g is an abelian group in the case of an abelian group g we also classify all finite dimensional graded simple algebras and finite dimensional graded division algebras over an algebraically closed field of characteristic zero

graded identities of matrix algebras and the universal graded algebra in the last decade group gradings and graded identities of finite dimensional central simple algebras have been an active area of research we refer the reader to bahturin et al 6 and 7 there are two basic kinds of group grading elementary and fine it was proved by bahturin and zaicev 7 that any group grading of mn c is given by a certain composition of an elementary grading and a fine grading in this paper we are concerned with fine gradings on mn c and their corresponding graded identities let r be a simple algebra finite dimensional over its center k and g a finite group we say that r is fine graded by g if r g grg is a grading and dimk rg 1 thus any component is either 0 or isomorphic to k as a k vector space it is easy to show that supp r the subset of elements of g for which rg is not 0 is a subgroup of g moreover

polar codes for the two user multiple access channel in this paper polar codes for the m user multiple access channel mac with binary inputs are constructed it is shown that arikan s polarization technique applied individually to each user transforms independent uses of an m user binary input mac into successive uses of extremal macs this transformation has a number of desirable properties 1 the uniform sum rate of the original mac is preserved 2 the extremal macs have uniform rate regions that are not only polymatroids but matroids and thus 3 their uniform sum rate can be reached by each user transmitting either uncoded or fixed bits in this sense they are easy to communicate over a polar code can then be constructed with an encoding and decoding complexity of o n log n where n is the block length a block error probability of o exp n1 2 e and capable of achieving the uniform sum rate of any binary input mac with arbitrary many users applications of this polar code construction to channels with a finite field input alphabet and to the additive white gaussian noise channel are also discussed

secret key agreement by public discussion from common information the problem of generating a shared secret key s by two parties knowing dependent random variables x and y respectively but not sharing a secret key initially is considered an enemy who knows the random variable z jointly distributed with x and y according to some probability distribution p sub xyz can also receive all messages exchanged by the two parties over a public channel the goal of a protocol is that the enemy obtains at most a negligible amount of information about s upper bounds on h s as a function of p sub xyz are presented lower bounds on the rate h s n as n to infinity are derived for the case in which x x sub 1 x sub n y y sub 1 y sub n and z z sub 1 z sub n result from n independent executions of a random experiment generating x sub i y sub i and z sub i for i 1 n it is shown that such a secret key agreement is possible for a scenario in which all three parties receive the output of a binary symmetric source over independent binary symmetric channels even when the enemy s channel is superior to the other two channels

the crease structure of the karush kuhn tucker set in parametric optimization we study optimization problems depending on a parameter vector y in particular we consider the crease structure of the set of pairs x y where x is a karush kuhn tucker point of the problem associated with parameter y the mangasarian fromovitz constraint qualification is assumed to hold we present a lower bound for the fineness of whitney regular stratifications of this provides an approximation of its crease structure

graded embeddings of finite dimensional simple graded algebras abstract let a b be finite dimensional g graded algebras over an algebraically closed field k with char k 0 where g is an abelian group and let id g a be the set of graded identities of a resp id g b we show that if a b are g simple then there is a graded embedding a b if and only if id g b id g a we also give a weaker generalization for the case where a is g semisimple and b is arbitrary

identities and isomorphisms of graded simple algebras abstract let g be an arbitrary abelian group and let a and b be two finite dimensional g graded simple algebras over an algebraically closed field f such that the orders of all finite subgroups of g are invertible in f we prove that a and b are isomorphic if and only if they satisfy the same g graded identities we also describe all isomorphism classes of finite dimensional g graded simple algebras

hopf cocycle deformations and invariant theory for a given finite dimensional hopf algebra h we describe the set of all equivalence classes of cocycle deformations of h as an affine variety using methods of geometric invariant theory we show how our results specialize to the universal coefficients theorem in the case of a group algebra and we also give examples from other families of hopf algebras including dual group algebras and bosonizations of nichols algebras in particular we use the methods developed here to classify the cocycle deformations of a dual pointed hopf algebra associated to the symmetric group on three letters we also give an example of a cocycle deformation over a dual group algebra which has only rational invariants but which is not definable over the rational field this differs from the case of group algebras in which every 2 cocycle is equivalent to one which is definable by its invariants

on groups of central type a finite group is said to be of central type if it possesses an irreducible complex character which takes the value zero on all noncentral elements equivalently the degree of this character is the square root of the index of the center in 1964 iwahori and matsumoto 15 conjectured that a group of central type must be solvable the paper of liebler and yellen 16 aims to prove this but there is a gap in their proof as explained below nevertheless they do correctly obtain substantial information about a minimal counterexample to the conjecture the aim of this paper is to fill the gap in the liebler yellen proof using as they do the classification of finite simple groups and to provide further information about which solvable groups can have central type we show for instance that every normal subgroup of the central factor group of a solvable group of central type has the property that its system normalizers have square index it is interesting to contrast this restrictive condition with the result of gagola that every solvable group is embeddable in the central factor group of a group of central type see 6 theorem 1 2 the arguments in 16 show that a minimal counterexample g to the iwahori matsumoto conjecture must have the following structure if k z is any minimal normal subgroup of g z where z z g the center then k is abelian k z is a 2 group s cg k is the maximal solvable normal subgroup of g and any minimal normal subgroup his of g s is the direct product of a number l of copies of gl 3 2 indeed the liebler yellen methods can be used to show that the sylow 2 subgroup of k considered as an h s module is a product o f copies of a gl 3 2 module of dimension 4 this module having a unique minimal submodule which is trivial and an irreducible factor module of dimension 3 the minimal submodules are amalgamated in the product

critical value functions have finite modulus of concavity we consider a smooth finite dimensional parametric optimization problem cal p y with objective function f x y here x and y denote the state variable and the parameter respectively in the case that overline x is a strongly stable karush kuhn tucker point for cal p overline y a neighborhood of overline x contains a unique karush kuhn tucker point x y for cal p y provided that y is sufficiently close to overline y this gives rise to the critical value function y mapsto varphi y f x y y under the additional assumption that the mangasarian fromovitz constraint qualification is satisfied at overline x we show that varphi has finite modulus of concavity that means varphi becomes convex in a neighborhood of overline y by adding to it the function y mapsto alpha 2 cdot y overline y 2 for some alpha 0 moreover we present an explicit upper bound for the alpha to be used the latter bound turns out to be sharp for problem data in general position

sip critical value functions have finite modulus of non convexity we consider semi infinite programming problems rm sip z depending on a finite dimensional parameter z in mathbb r p provided that bar x is a strongly stable stationary point of rm sip bar z there exists a locally unique and continuous stationary point mapping z mapsto x z this defines the local critical value function varphi z f x z z where x mapsto f x z denotes the objective function of rm sip z for a given parameter vector z in mathbb r p we show that varphi is the sum of a convex function and a smooth function in particular this excludes the appearance of negative kinks in the graph of varphi

randomness and dependencies extraction via polarization with applications to slepian wolf coding and secrecy the polarization phenomenon for a single source is extended to a framework with multiple correlated sources it is shown in addition to extracting the randomness of the source the polar transforms take the original arbitrary dependencies to extremal dependencies polar coding schemes for the slepian wolf sw coding problem and for secret key generations are then proposed based on this phenomenon in particular secret keys achieving the secrecy capacity and compression schemes achieving the sw capacity region are obtained with a complexity of o n log n

projective schur algebras of nilpotent type are brauer equivalent to radical algebras let k be a field and g a finite group for any element h g k k denotes the units of k here as a trivial g module we can form the twisted group algebra kg it is isomorphic to the group algebra as a vector space and its multiplication satisfies the rule xu yu xyf u where f g g k is a 2 cocycle representing it is easily checked that the 2 cocycle condition satisfied by f insures the associativity of the multiplication in kg a projective schur algebra over k is by definition a k central simple algebra which is a homomorphic image of a twisted group algebra kg for some finite group g and some h g k equivalently a k central simple algebra a is a projective schur algebra over k if it contains a subgroup of the units of a which is finite modulo k and spans a as k vector space we often use the notation k clearly a projective schur algebra over k determines an element in the brauer group br k of the field k the subgroup in br k generated by and in fact consisting of such elements is called the projective schur group of k and denoted by ps k the notions of projective schur algebra and projective schur group were introduced in 6 by lorenz and opolka this construction can be viewed as the projective analogue of schur algebras and the schur group denoted by s k of the field k see 10 it is natural to restrict attention to special types of groups g our interest here is in k central simple algebras that are homomorphic images of twisted group algebras kg with g nilpotent resp abelian see 6 we call these algebras projective schur algebras of nilpotent type resp abelian type clearly as above these algebras represent classes in br k and we denote by pnil k resp pab k the subgroup they generate in 6 it is shown that if k is a number field and n is the number of roots of unity in k then pab k nbr k where nb denotes the n torsion subgroup of b more generally since a symbol algebra of degree n over k is a homomorphic

on weak equivalences of gradings abstract when one studies the structure e g graded ideals graded subspaces radicals or graded polynomial identities of graded algebras the grading group itself does not play an important role but can be replaced by any other group that realizes the same grading here we come to the notion of weak equivalence of gradings two gradings are weakly equivalent if there exists an isomorphism between the graded algebras that maps each graded component onto a graded component the following question arises naturally when a group grading on a finite dimensional algebra is weakly equivalent to a grading by a finite group it turns out that this question can be reformulated purely group theoretically in terms of the universal group of the grading namely a grading is weakly equivalent to a grading by a finite group if and only if the universal group of the grading is residually finite with respect to a special subset of the grading group the same is true for all the coarsenings of the grading if and only if the universal group of the grading is hereditarily residually finite with respect to the same subset we show that if n 349 then on the full matrix algebra m n f there exists an elementary group grading that is not weakly equivalent to any grading by a finite semi group and if n 3 then any elementary grading on m n f is weakly equivalent to an elementary grading by a finite group

monomial hopf algebras abstract let k be a field of characteristic 0 containing all roots of unity we classified all the hopf structures on monomial k coalgebras or in dual version on monomial k algebras

the center of the generic g crossed product abstract let g be a finite group and let f be a field of characteristic zero in this paper we construct a generic g crossed product over f using generic graded matrices the center of this generic g crossed product denoted by f g is then the invariant field of a suitable g action on a field of rational functions in several indeterminates the main goal of this paper is to study the extensions f g f given that f contains enough roots of unity and determine how close they are to being purely transcendental in particular we show that f g f is a stably rational extension for g c 2 c 2 n where n is odd and for g n 2 m e 1 1 where g c d n 2 m 1 furthermore we prove that if h k are groups of coprime orders then f h k is stably rationally equivalent to the fraction field of f h f k

constructing pointed hopf algebras by ore extensions abstract we present a general construction producing pointed co frobenius hopf algebras and give some classification results for the examples obtained

a non shannon type conditional inequality of information quantities given n discrete random variables spl omega x sub 1 x sub n associated with any subset spl alpha of 1 2 n there is a joint entropy h x sub spl alpha where x sub spl alpha x sub i i spl isin spl alpha this can be viewed as a function defined on 2 sup 1 2 n taking values in 0 spl infin we call this function the entropy function of spl omega the nonnegativity of the joint entropies implies that this function is nonnegative the nonnegativity of the conditional joint entropies implies that this function is nondecreasing and the nonnegativity of the conditional mutual information implies that this function is two alternative these properties are the so called basic information inequalities of shannon s information measures an entropy function can be viewed as a 2 sup n 1 dimensional vector where the coordinates are indexed by the subsets of the ground set 1 2 n as introduced by yeng see ibid vol 43 no 6 p 1923 34 1997 spl gamma sub n stands for the cone in ir 2 sup n 1 consisting of all vectors which have all these properties let spl gamma sub n be the set of all 2 sup n 1 dimensional vectors which correspond to the entropy functions of some sets of n discrete random variables a fundamental information theoretic problem is whether or not spl gamma sub n spl gamma sub n here spl gamma sub n stands for the closure of the set spl gamma sub n we show that spl gamma sub n is a convex cone spl gamma sub 2 spl gamma sub 2 spl gamma sub 3 spl ne spl gamma sub 3 but spl gamma sub 3 spl gamma sub 3 for four random variables we have discovered a conditional inequality which is not implied by the basic information inequalities of the same set of random variables this lends an evidence to the plausible conjecture that spl gamma sub n spl ne spl gamma sub n for n 3

pauli gradings on lie superalgebras and graded codimension growth abstract we introduce grading on certain finite dimensional simple lie superalgebras of type p t by elementary abelian 2 group this grading gives rise to pauli matrices and is a far generalization of z 2 z 2 grading on lie algebra of 2 2 traceless matrices we use this grading for studying numerical invariants of polynomial identities of lie superalgebras in particular we compute graded pi exponent corresponding to pauli grading

graded polynomial identities of matrices we consider g graded polynomial identities of the p p matrix algebra mp k over a field k of characteristic 0 graded by an arbitrary group g we find relations between the g graded identities of the g graded algebra mp k and the g h graded identities of the tensor product of mp k and the h graded algebra mq k with a fine h grading we also find a basis of the g graded identities of mp k with an elementary grading such that the identity component coincides with the diagonal of mp k

channel polarization a method for constructing capacity achieving codes for symmetric binary input memoryless channels a method is proposed called channel polarization to construct code sequences that achieve the symmetric capacity i w of any given binary input discrete memoryless channel b dmc w the symmetric capacity is the highest rate achievable subject to using the input letters of the channel with equal probability channel polarization refers to the fact that it is possible to synthesize out of n independent copies of a given b dmc w a second set of n binary input channels wn i 1 les i les n such that as n becomes large the fraction of indices i for which i wn i is near 1 approaches i w and the fraction for which i wn i is near 0 approaches 1 i w the polarized channels wn i are well conditioned for channel coding one need only send data at rate 1 through those with capacity near 1 and at rate 0 through the remaining codes constructed on the basis of this idea are called polar codes the paper proves that given any b dmc w with i w 0 and any target rate r i w there exists a sequence of polar codes cfrn nges1 such that cfrn has block length n 2n rate ges r and probability of block error under successive cancellation decoding bounded as pe n r les o n 1 4 independently of the code rate this performance is achievable by encoders and decoders with complexity o n logn for each

polynomial identities and noncommutative versal torsors abstract to any cleft hopf galois object i e any algebra h obtained from a hopf algebra h by twisting its multiplication with a two cocycle we attach two universal algebras a h and u h the algebra a h is obtained by twisting the multiplication of h with the most general two cocycle formally cohomologous to the cocycle takes values in the field of rational functions on h by construction a h is a cleft h galois extension of a big commutative algebra b h any form of h can be obtained from a h by a specialization of b h and vice versa if the algebra h is simple then a h is an azumaya algebra with center b h the algebra u h is constructed using a general theory of polynomial identities that we set up for arbitrary comodule algebras it is the universal comodule algebra in which all comodule algebra identities of h are satisfied we construct an embedding of u h into a h this embedding maps the center z h of u h into b h when the algebra h is simple in this case under an additional assumption a h b h z h u h thus turning a h into a central localization of u h we completely work out these constructions in the case of the four dimensional sweedler algebra