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

covert communication over noisy channels a resolvability perspective we consider the situation in which a transmitter attempts to communicate reliably over a discrete memoryless channel while simultaneously ensuring covertness low probability of detection with respect to a warden who observes the signals through another discrete memoryless channel we develop a coding scheme based on the principle of channel resolvability which generalizes and extends prior work in several directions first it shows that irrespective of the quality of the channels it is possible to communicate on the order of sqrt n reliable and covert bits over n channel uses if the transmitter and the receiver share on the order of sqrt n key bits this improves upon earlier results requiring on the order of sqrt n log n key bits second it proves that if the receiver s channel is better than the warden s channel in a sense that we make precise it is possible to communicate on the order of sqrt n reliable and covert bits over n channel uses without a secret key this generalizes earlier results established for binary symmetric channels we also identify the fundamental limits of covert and secret communications in terms of the optimal asymptotic scaling of the message size and key size and we extend the analysis to gaussian channels the main technical problem that we address is how to develop concentration inequalities for low weight sequences the crux of our approach is to define suitably modified typical sets that are amenable to concentration inequalities

scattering for the quasilinear wave equations with null conditions in two dimensions abstract we give a detailed description of the scattering theory of solutions to the quasilinear wave equations with null conditions and small initial data in two dimensions the scattering profile is shown to be the same as the one conjectured by alinhac 1 1995 our proof is based on a framework of fourier transform by deng and pusateri 6 2018 and an expansion of oscillatory integral together with the global well posedness of the solutions by alinhac 2 2001

numerical integration analytic continuation and decomposition one step formulas are developed using the decomposition method to explore relationships with numerical integration analytic continuation and numerical continuation analytic continuation in one or multiple dimensions is considered for decomposition solutions and the effects of chaos

secret key generation for a pairwise independent network model we consider secret key generation for a pairwise independent network model in which every pair of terminals observes correlated sources that are independent of sources observed by all other pairs of terminals the terminals are then allowed to communicate publicly with all such communication being observed by all the terminals the objective is to generate a secret key shared by a given subset of terminals at the largest rate possible with the cooperation of any remaining terminals secrecy is required from an eavesdropper that has access to the public interterminal communication a single letter formula for secret key capacity brings out a natural connection between the problem of secret key generation and a combinatorial problem of maximal packing of steiner trees in an associated multigraph an explicit algorithm is proposed for secret key generation based on a maximal packing of steiner trees in a multigraph the corresponding maximum rate of steiner tree packing is thus a lower bound for the secret key capacity when only two of the terminals or when all the terminals seek to share a secret key the mentioned algorithm achieves secret key capacity in which case the bound is tight

a mathematical model of adomian polynomials the adomian decomposition method is reviewed and a mathematical model of adomian polynomials is introduced to show the convenience of this model for computer programming the mathematica package is used to generate the adomian polynomials for any order

piecewise homotopy methods for nonlinear ordinary differential equations abstract piecewise homotopy perturbation methods are developed for the solution of nonlinear ordinary differential equations these methods are based on the introduction of an artificial or book keeping parameter and the expansion of the solution in a power series of this parameter and provide analytical solutions in open intervals which are smooth everywhere three piecewise adaptive homotopy perturbation methods based on the use of either a fixed number of approximants and a variable step size a variable number of approximants and a fixed step size or a variable number of approximants and a variable step size are presented and applied to eight nonlinear ordinary differential equations it is shown that piecewise adaptive homotopy perturbation methods predict essentially the same solutions as matlab s variable step variable order solvers and variable order transition matrix techniques provided that five term approximations of the decomposition method are applied to both the displacement and the velocity it is also shown that piecewise homotopy perturbation techniques that use three term approximations to both the displacement and the velocity provide essentially the same results as those obtained with a second order accurate time linearization technique when the same step is employed in both schemes

achieving awgn channel capacity with lattice gaussian coding we propose a new coding scheme using only one lattice that achieves the 1 2 log 1 snr capacity of the additive white gaussian noise awgn channel with lattice decoding which is provable for signal to noise ratio snr e at present the scheme applies a discrete gaussian distribution over an awgn good lattice but otherwise does not require a shaping lattice or dither thus it significantly simplifies the default lattice coding scheme of erez and zamir which involves a quantization good lattice as well as an awgn good lattice using the flatness factor we show that the error probability of the proposed scheme under minimum mean square error lattice decoding is almost the same as that of erez and zamir for any rate up to the awgn channel capacity we introduce the notion of good constellations which carry almost the same mutual information as that of continuous gaussian inputs we also address the implementation of gaussian shaping for the proposed lattice gaussian coding scheme

adomian decomposition method for solving the volterra integral form of the lane emden equations with initial values and boundary conditions in this paper we use the systematic adomian decomposition method to handle the integral form of the lane emden equations with initial values and boundary conditions the volterra integral form of the lane emden equation overcomes the singular behavior at the origin x 0 we confirm our belief that the adomian decomposition method provides efficient algorithm for analytic approximate solutions of the equation our results are supported by investigating several numerical examples that include initial value problems and boundary value problems as well finally we consider the modified decomposition method of rach adomian and meyers for the volterra integral form

higher order numeric wazwaz el sayed modified adomian decomposition algorithms in this paper we develop new numeric modified adomian decomposition algorithms by using the wazwaz el sayed modified decomposition recursion scheme and investigate their practicality and efficiency for several nonlinear examples we show how we can conveniently generate higher order numeric algorithms at will by this new approach including by using examples 12th order and 20th order numeric algorithms furthermore we show how we can achieve a much larger effective region of convergence using these new discrete solutions we also demonstrate the superior robustness of these numeric modified decomposition algorithms including a 4th order numeric modified decomposition algorithm over the classic 4th order runge kutta algorithm by example the efficiency of our subroutines is guaranteed by the inclusion of the fast algorithms and subroutines as published by duan for generation of the adomian polynomials to high orders

on the public communication needed to achieve sk capacity in the multiterminal source model the focus of this paper is on the public communication required for generating a maximal rate secret key sk within the multiterminal source model of csiszar and narayan building on the prior work of tyagi for the two terminal scenario we derive a lower bound on the communication complexity r text sk defined to be the minimum rate of public communication needed to generate a maximal rate sk it is well known that the minimum rate of communication for omniscience denoted by r text co is an upper bound on r text sk for the class of pairwise independent network pin models defined on uniform hypergraphs we show that a certain type mathcal s condition which is verifiable in polynomial time guarantees that our lower bound on r text sk meets the r text co upper bound thus the pin models satisfying our condition are r text sk maximal indicating that the upper bound r text sk le r text co holds with equality this allows us to explicitly evaluate r text sk for such pin models we also give several examples of pin models that satisfy our type mathcal s condition finally we prove that for an arbitrary multiterminal source model a stricter version of our type mathcal s condition implies that communication from all terminals omnivocality is needed for establishing an sk of maximum rate for three terminal source models the converse is also true omnivocality is needed for generating a maximal rate sk only if the strict type mathcal s condition is satisfied however for the source models with four or more terminals counterexamples exist showing that the converse does not hold in general

large deviation analysis for quantum security via smoothing of renyi entropy of order 2 it is known that the security evaluation can be done by smoothing of renyi entropy of order 2 in the classical and quantum settings when we apply universal 2 hash functions using the smoothing of renyi entropy of order 2 we derive security bounds for l 1 distinguishability and modified mutual information criterion under the classical and quantum setting and have derived these exponential decreasing rates these results are extended to the case when we apply e almost dual universal 2 hash functions furthermore we apply this analysis to the secret key generation with error correction

applications of ldpc codes to the wiretap channel with the advent of quantum key distribution qkd systems perfect i e information theoretic security can now be achieved for distribution of a cryptographic key qkd systems and similar protocols use classical error correcting codes for both error correction for the honest parties to correct errors and privacy amplification to make an eavesdropper fully ignorant from a coding perspective a good model that corresponds to such a setting is the wire tap channel introduced by wyner in 1975 in this correspondence we study fundamental limits and coding methods for wire tap channels we provide an alternative view of the proof for secrecy capacity of wire tap channels and show how capacity achieving codes can be used to achieve the secrecy capacity for any wiretap channel we also consider binary erasure channel and binary symmetric channel special cases for the wiretap channel and propose specific practical codes in some cases our designs achieve the secrecy capacity and in others the codes provide security at rates below secrecy capacity for the special case of a noiseless main channel and binary erasure channel we consider encoder and decoder design for codes achieving secrecy on the wiretap channel we show that it is possible to construct linear time decodable secrecy codes based on low density parity check ldpc codes that achieve secrecy

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

numerical algorithms and decomposition abstract relationships are explored between conventional one step methods a new one step version resulting from a discretized adomian decomposition runge kutta and analytic continuation

the optimal use of rate limited randomness in broadcast channels with confidential messages in coding schemes for the wire tap channel or for broadcast channels with confidential messages it is well known that the sender needs to use stochastic encoding to avoid information about the transmitted confidential message from being leaked to an eavesdropper in this paper we investigate the tradeoff between the rate of random numbers needed to realize the stochastic encoding and the rates of common private and confidential messages for the direct theorem we use the superposition coding scheme for the wire tap channel recently proposed by chia and el gamal and its strong security is proved the matching converse theorem is also established our result clarifies that a combination of ordinary stochastic encoding and channel prefixing by channel simulation is suboptimal

secret key generation with limited interaction a basic two terminal secret key generation model is considered where the interactive communication rate between the terminals may be limited and in particular may not be enough to achieve the maximum key rate we first prove a multi letter characterization of the key communication rate region where the number of auxiliary random variables depends on the number of rounds of the communication and then provide an equivalent but simpler characterization in terms of concave envelopes in the case of unlimited number of rounds two extreme cases are given special attention first in the regime of very low communication rates the key bits per interaction bit kbib is expressed with a new symmetric strong data processing constant which has a concave envelope characterization analogous to that of the conventional strong data processing constant the symmetric strong data processing constant can be upper bounded by the supremum of the maximal correlation coefficient over a set of distributions which allows us to determine the kbib for binary symmetric sources and conclude in particular that the interactive scheme is not more efficient than the one way scheme at least in the low communication rate regime second a new characterization of the minimum interaction rate needed for achieving the maximum key rate mimk is given and we resolve a conjecture by tyagi regarding the mimk for possibly nonsymmetric binary sources we also propose a new conjecture for binary symmetric sources that the interactive scheme is not more efficient than the one way scheme at any communication rate

solutions of the initial value problem for nonlinear fractional ordinary differential equations by the rach adomian meyers modified decomposition method abstract in this paper we present the generalized adomian rach theorem and the generalized rach adomian meyers modified decomposition method for solving multi order nonlinear fractional ordinary differential equations we consider different classes of initial value problems for nonlinear fractional ordinary differential equations including the case of real valued orders and another case of rational valued orders which are solved by the present method this method can treat any analytic nonlinearity the coefficients of the solution in the form of a generalized power series are determined by a convenient recurrence scheme which does not involve integration operations compared with the classic adomian decomposition method

information theoretic key agreement of multiple terminals part i we study the problem of information theoretically secure secret key agreement under the well known source model and channel model in both of these models multiple terminals wish to create a shared secret key that is secure from a passive eavesdropper the terminals have access to a noiseless public communication channel and an additional resource that depends on the model in the source model the resource is an external source that repeatedly beams correlated randomness to the terminals whereas in the channel model the resource is a secure but noisy discrete memoryless broadcast channel we derive new lower and upper bounds on the secret key capacity under both the source model and the channel model the technique used for deriving our bound for the source model is to find certain properties of functions of joint probability distributions which applied to the joint distribution of the source will imply that they dominate the secret key capacity and then prove the bound by a verification argument a similar technique is used for the channel model finally we also define a problem of communication for omniscience by a neutral observer and establish the equivalence between this new problem and the problem of secret key agreement this generalizes an earlier result of csiszar and narayan

change of multivariate mutual information from local to global we study the change of multivariate mutual information among a set of random variables when some common randomness is added to or removed from a subset of the random variables this is formulated more precisely as two new multiterminal secret key agreement problems that respectively ask how one can increase the secrecy capacity efficiently by adding common randomness to a small subset of users and how one can simplify the source model by removing redundant common randomness that does not contribute to the secrecy capacity characterizations and strongly polynomial time computations are derived for the rates of change maximum usable increment and redundancy these results can be applied to study the communication complexity for secret key agreement

generating random bits from an arbitrary source fundamental limits suppose we are given a random source and want to use it as a random number generator at what rate can we generate fair bits from it we address this question in an information theoretic setting by allowing for some arbitrarily small but nonzero deviation from ideal random bits we prove our results with three different measures of approximation between the ideal and the obtained probability distributions the variational distance the d bar distance and the normalized divergence two different contexts are studied fixed length and variable length random number generation the fixed length results of this paper provide an operational characterization of the inf entropy rate of a source defined in han and verdu see ibid vol 39 no 3 p 752 772 1993 and the variable length results characterize the liminf of the entropy rate thereby establishing a pleasing duality with the fundamental limits of source coding a feature of our results is that we do not restrict ourselves to ergodic or to stationary sources

packing steiner trees a cutting plane algorithm and computational results in this paper we describe a cutting plane algorithm for the steiner tree packing problem we use our algorithm to solve some switchbox routing problems of vlsi design and report on our computational experience this includes a brief discussion of separation algorithms a new lp based primal heuristic and implementation details the paper is based on the polyhedral theory for the steiner tree packing polyhedron developed in our companion paper this issue and meant to turn this theory into an algorithmic tool for the solution of practical problems

information inequalities for joint distributions with interpretations and applications upper and lower bounds are obtained for the joint entropy of a collection of random variables in terms of an arbitrary collection of subset joint entropies these inequalities generalize shannon s chain rule for entropy as well as inequalities of han fujishige and shearer a duality between the upper and lower bounds for joint entropy is developed all of these results are shown to be special cases of general new results for submodular functions thus the inequalities presented constitute a richly structured class of shannon type inequalities the new inequalities are applied to obtain new results in combinatorics such as bounds on the number of independent sets in an arbitrary graph and the number of zero error source channel codes as well as determinantal inequalities in matrix theory a general inequality for relative entropies is also developed finally revealing connections of the results to literature in economics computer science and physics are explored

perfect omniscience perfect secrecy and steiner tree packing we consider perfect secret key generation for a pairwise independent network model in which every pair of terminals share a random binary string with the strings shared by distinct terminal pairs being mutually independent the terminals are then allowed to communicate interactively over a public noiseless channel of unlimited capacity all the terminals as well as an eavesdropper observe this communication the objective is to generate a perfect secret key shared by a given set of terminals at the largest rate possible and concealed from the eavesdropper first we show how the notion of perfect omniscience plays a central role in characterizing perfect secret key capacity second a multigraph representation of the underlying secrecy model leads us to an efficient algorithm for perfect secret key generation based on maximal steiner tree packing this algorithm attains capacity when all the terminals seek to share a key and in general attains at least half the capacity third when a single helper terminal assists the remaining user terminals in generating a perfect secret key we give necessary and sufficient conditions for the optimality of the algorithm also a weak helper is shown to be sufficient for optimality

channel coding rate in the finite blocklength regime this paper investigates the maximal channel coding rate achievable at a given blocklength and error probability for general classes of channels new achievability and converse bounds are given which are tighter than existing bounds for wide ranges of parameters of interest and lead to tight approximations of the maximal achievable rate for blocklengths n as short as 100 it is also shown analytically that the maximal rate achievable with error probability isclosely approximated by c v n q 1 where c is the capacity v is a characteristic of the channel referred to as channel dispersion and q is the complementary gaussian cumulative distribution function

secrecy capacities for multiple terminals we derive single letter characterizations of strong secrecy capacities for models with an arbitrary number of terminals each of which observes a distinct component of a discrete memoryless multiple source with unrestricted and interactive public communication permitted between the terminals a subset of these terminals can serve as helpers for the remaining terminals in generating secrecy according to the extent of an eavesdropper s knowledge three kinds of secrecy capacity are considered secret key sk private key pk and wiretap secret key wsk capacity the characterizations of the sk and pk capacities highlight the innate connections between secrecy generation and multiterminal source coding without secrecy requirements a general upper bound for wsk capacity is derived which is tight in the case when the eavesdropper can wiretap noisy versions of the components of the underlying multiple source provided randomization is permitted at the terminals these secrecy capacities are seen to be achievable with noninteractive communication between the terminals the achievability results are also shown to be universal

new higher order numerical one step methods based on the adomian and the modified decomposition methods abstract we develop new higher order numerical one step methods and apply them to several examples to investigate approximate discrete solutions of nonlinear differential equations these new algorithms are derived from the adomian decomposition method adm and the rach adomian meyers modified decomposition method mdm to present an alternative to such classic schemes as the explicit runge kutta methods for engineering models which require high accuracy with low computational costs for repetitive simulations in contrast to a one size fits all philosophy this new approach incorporates the notion of analytic continuation which extends the region of convergence without resort to other techniques that are also used to accelerate the rate of convergence such as the diagonal pade approximants or the iterated shanks transforms hence global solutions instead of only local solutions are directly realized albeit in a discretized representation we observe that one of the difficulties in applying explicit runge kutta one step methods is that there is no general procedure to generate higher order numeric methods it becomes a time consuming tedious endeavor to generate higher order explicit runge kutta formulas because it is constrained by the traditional picard formalism as used to represent nonlinear differential equations the adm and the mdm rely instead upon adomian s representation and the adomian polynomials to permit a straightforward universal procedure to generate higher order numeric methods at will such as a 12th order or 24th order one step method if need be another key advantage is that we can easily estimate the maximum step size prior to computing data sets representing the discretized solution because we can approximate the radius of convergence from the solution approximants unlike the runge kutta approach with its intrinsic linearization between computed data points we propose new variable step size variable order and variable step size variable order algorithms for automatic step size control to increase the computational efficiency and reduce the computational costs even further for critical engineering models

strong secrecy from channel resolvability we analyze physical layer security based on the premise that the coding mechanism for secrecy over noisy channels is tied to the notion of channel resolvability instead of considering capacity based constructions which associate to each message a subcode that operates just below the capacity of the eavesdropper s channel we consider channel resolvability based constructions which associate to each message a subcode that operates just above the resolvability of the eavesdropper s channel building upon the work of csiszar and hayashi we provide further evidence that channel resolvability is a powerful and versatile coding mechanism for secrecy by developing results that hold for strong secrecy metrics and arbitrary channels specifically we show that at least for symmetric wiretap channels random capacity based constructions fail to achieve the strong secrecy capacity while channel resolvability based constructions achieve it we then leverage channel resolvability to establish the secrecy capacity region of arbitrary broadcast channels with confidential messages and a cost constraint for strong secrecy metrics finally we specialize our results to study the secrecy capacity of wireless channels with perfect channel state information csi mixed channels and compound channels with receiver csi as well as the secret key capacity of source models for secret key agreement by tying secrecy to channel resolvability we obtain achievable rates for strong secrecy metrics with simple proofs

equivocations exponents and second order coding rates under various renyi information measures we evaluate the asymptotics of equivocations their exponents as well as their second order coding rates under various renyi information measures specifically we consider the effect of applying a hash function on a source and we quantify the level of non uniformity and dependence of the compressed source from another correlated source when the number of copies of the sources is large unlike previous works that use shannon information measures to quantify randomness information or uniformity we define our security measures in terms of a more general class of information measures the renyi information measures and their gallager type counterparts a special case of these renyi information measure is the class of shannon information measures we prove tight asymptotic results for the security measures and their exponential rates of decay we also prove bounds on the second order asymptotics and show that these bounds match when the magnitudes of the second order coding rates are large we do so by establishing new classes non asymptotic bounds on the equivocation and evaluating these bounds using various probabilistic limit theorems asymptotically

amortized communication complexity in this work we study the direct sum problem with respect to communication complexity consider a relation f defined over 0 1 n times 0 1 n can the communication complexity of simultaneously computing f on cal l instances x 1 y 1 ldots x cal l y cal l be smaller than the communication complexity of computing f on the cal l instances separately let the amortized communication complexity of f be the communication complexity of simultaneously computing f on cal l instances divided by cal l we study the properties of the amortized communication complexity we show that the amortized communication complexity of a relation can be smaller than its communication complexity more precisely we present a partial function whose deterministic communication complexity is theta log n and its amortized deterministic communication complexity is o 1 similarly for randomized protocols we present a function whose randomized communication complexity is theta log n and its amortized randomized communication complexity is o 1 we also give a general lower bound on the amortized communication complexity of any function f in terms of its communication complexity c f for every function f the amortized communication complexity of f is omega left sqrt c f log n right

wyner s common information under renyi divergence measures we study a generalized version of wyner s common information problem also coined the distributed source simulation problem the original common information problem consists in understanding the minimum rate of the common input to independent processors to generate an approximation of a joint distribution when the distance measure used to quantify the discrepancy between the synthesized and target distributions is the normalized relative entropy our generalization involves changing the distance measure to the unnormalized and normalized renyi divergences of order alpha 1 s in 0 2 we show that the minimum rate needed to ensure the renyi divergences between the distribution induced by a code and the target distribution vanishes remains the same as the one in wyner s setting except when the order alpha 1 s 0 this implies that wyner s common information is rather robust to the choice of distance measure employed as a byproduct of the proofs used to the establish the above results the exponential strong converse for the common information problem under the total variation distance measure is established

new recurrence algorithms for the nonclassic adomian polynomials in this article we present new algorithms for the nonclassic adomian polynomials which are valuable for solving a wide range of nonlinear functional equations by the adomian decomposition method and introduce their symbolic implementation in mathematica beginning with rach s new definition of the adomian polynomials we derive the explicit expression for each class of the adomian polynomials e g a email protected k 1 mf k u 0 z m k for the class ii iii and iv adomian polynomials where the z m k are called the reduced polynomials these expressions provide a basis for developing improved algorithmic approaches by introducing the index vectors the recurrence algorithms for the reduced polynomials are suitably deduced which naturally lead to new recurrence algorithms for the class ii and class iii adomian polynomials mathematica programs generating these classes of adomian polynomials are subsequently presented computation shows that for computer generation of the class iii adomian polynomials the new algorithm reduces the running times compared with the definitional formula we also consider the number of summands of these classes of adomian polynomials and obtain the corresponding formulas finally we demonstrate the versatility of the four classes of adomian polynomials with several examples which include the nonlinearity of the form f t u explicitly depending on the argument t

space time resonances and the null condition for first order systems of wave equations in this manuscript we prove global existence and linear asymptotic behavior of small solutions to nonlinear wave equations we assume that the quadratic part of the nonlinearity satisfies a nonresonance condition that is a generalization of the null condition given by klainerman 2013 wiley periodicals inc

a general formula for channel capacity a formula for the capacity of arbitrary single user channels without feedback not necessarily information stable stationary etc is proved capacity is shown to equal the supremum over all input processes of the input output inf information rate defined as the liminf in probability of the normalized information density the key to this result is a new converse approach based on a simple new lower bound on the error probability of m ary hypothesis tests among equiprobable hypotheses a necessary and sufficient condition for the validity of the strong converse is given as well as general expressions for spl epsiv capacity

cognitive interference channels with confidential messages under randomness constraint the cognitive interference channel with confidential messages cicc proposed by liang et al is investigated when the security is considered in coding systems it is well known that the sender needs to use a stochastic encoding to avoid the information about the transmitted confidential message to be leaked to an eavesdropper for the cicc the tradeoff between the rate of the random number to realize the stochastic encoding and the communication rates is investigated and the optimal tradeoff is completely characterized

exact random coding secrecy exponents for the wiretap channel we analyze the exact exponential decay rate of the expected amount of information leaked to the wiretapper in wyner s wiretap channel setting using wiretap channel codes constructed from both i i d and constant composition random codes our analysis for those sampled from i i d random coding ensemble shows that the previously known achievable secrecy exponent using this ensemble is indeed the exact exponent for an average code in the ensemble furthermore our analysis on wiretap channel codes constructed from the ensemble of constant composition random codes leads to an exponent which in addition to being the exact exponent for an average code is larger than the achievable secrecy exponent that has been established so far in the literature for this ensemble which in turn was known to be smaller than that achievable by wiretap channel codes sampled from i i d random coding ensemble we show examples where the exact secrecy exponent for the wiretap channel codes constructed from random constant composition codes is larger than that of those constructed from i i d random codes and examples where the exact secrecy exponent for the wiretap channel codes constructed from i i d random codes is larger than that of those constructed from constant composition random codes we hence conclude that unlike the error correction problem there is no general ordering between the two random coding ensembles in terms of their secrecy exponent

strong functional representation lemma and applications to coding theorems this paper shows that for any random variables x and y it is possible to represent y as a function of x z such that z is independent of x and i x z y le log i x y 1 4 bits we use this strong functional representation lemma sfrl to establish a bound on the rate needed for one shot exact channel simulation for general discrete or continuous random variables strengthening the results by harsha et al and braverman and garg and to establish new and simple achievability results for one shot variable length lossy source coding multiple description coding and gray wyner system we also show that the sfrl can be used to reduce the channel with state noncausally known at the encoder to a point to point channel which provides a simple achievability proof of the gelfand pinsker theorem

a channel under simultaneous jamming and eavesdropping attack correlated random coding capacities under strong secrecy criteria we give a complete characterization of the correlated random coding secrecy capacity of arbitrarily varying wiretap channels avwcs we apply two alternative strong secrecy criteria which both lead to the same multi letter formula the difference of these criteria lies in the treatment of correlated randomness they coincide in the case of uncorrelated codes on the basis of the derived formula we show that the correlated random coding secrecy capacity is continuous as a function of the avwc in contrast to the discontinuous uncorrelated coding secrecy capacity in the proof of the secrecy capacity formula for correlated random codes we apply an auxiliary channel which is compound from the sender to the intended receiver and arbitrarily varying from the sender to the eavesdropper

key generation algorithms for pairwise independent networks based on graphical models we consider two secret key generation problems under a pairwise independent network model and propose low complexity key generation schemes in a framework that connects our problems to network flow problems in graphs our schemes have two components 1 local key generation and 2 global key propagation in the local key generation we use point to point source coding with side information to establish pairwise keys from which we construct a graph with the capacity of each edge being the key rate of the corresponding point to point local key in the global key propagation depending on the particular problem secret keys are delivered to users in the network using various network flow algorithms in particular in the first problem in which one is required to generate a group key for a group of users in the network we propose a network coding based global key propagation approach this approach has a low complexity and has a better performance than the existing approach in the second problem in which one is required to generate multiple keys simultaneously for different pairs of users we propose a multicommodity flow based global key propagation approach we show that the proposed approach is optimal for the case of generating two keys for the general case of generating more than two keys we show that the sum rate of the proposed scheme is larger than an upper bound characterized in this paper divided by a constant

exponential separation of information and communication for boolean functions we show an exponential gap between communication complexity and information complexity by giving an explicit example of a partial boolean function with information complexity o k and distributional communication complexity 2k this shows that a communication protocol cannot always be compressed to its internal information by a result of braverman 2015 our gap is the largest possible by a result of braverman and rao 2014 our example shows a gap between communication complexity and amortized communication complexity implying that a tight direct sum result for distributional communication complexity cannot hold answering a long standing open problem another conceptual contribution of our work is the relative discrepancy method a new rectangle based method for proving communication complexity lower bounds for boolean functions powerful enough to separate information complexity and communication complexity

second order asymptotics in fixed length source coding and intrinsic randomness there is a difference between the optimal rates of fixed length source coding and intrinsic randomness when we care about the second order asymptotics we prove this difference for general information sources and then investigate independent and identically distributed i i d random variables and markovian variables as examples the difference is demonstrated through an investigation of the second order asymptotic behavior of the rates a universal fixed length source code attaining the second order optimal rate is also proposed the difference between the rates of fixed length source coding and intrinsic randomness proves that the outputs of fixed length source codes are not uniformly distributed

general nonasymptotic and asymptotic formulas in channel resolvability and identification capacity and their application to the wiretap channel several nonasymptotic formulas are established in channel resolvability and identification capacity and they are applied to the wiretap channel by using these formulas the epsi capacities of the above three problems are considered in the most general setting where no structural assumptions such as the stationary memoryless property are made on a channel as a result we solve an open problem proposed by han and verduacute moreover we obtain lower bounds of the exponents of error probability and the wiretapper s information in the wiretap channel

on the secrecy capacity of the wiretap channel with imperfect main channel estimation we study the secrecy capacity of fast fading channels under imperfect main channel between the transmitter and the legitimate receiver estimation at the transmitter lower and upper bounds on the ergodic secrecy capacity are derived for a class of independent identically distributed i i d fading channels the achievable rate follows from a standard wiretap code in which a simple on off power control is employed along with a gaussian input the upper bound is obtained using an appropriate correlation scheme of the main and eavesdropper channels and is the best known upper bound so far the upper and lower bounds coincide with recently derived ones in case of perfect main csi furthermore the upper bound is tight in case of no main csi where the secrecy capacity is equal to zero asymptotic analysis at high and low signal to noise ratio snr is also given at high snr we show that the capacity is bounded by providing upper and lower bounds that depend on the channel estimation error at low snr however we prove that the secrecy capacity is asymptotically equal to the capacity of the main channel as if there were no secrecy constraint numerical results are provided for i i d rayleigh fading channels

global solutions of quasilinear wave equations we show global small data existence for a class of quasilinear wave equations related to einstein s equations in harmonic coordinates these equations do not satisfy the classical null condition and the asymptotic behavior of solutions is not free but the light cones bend at infinity

global solutions of nonlinear hyperbolic equations for small initial data on considere des systemes quasilineaires d equations hyperboliques d ordre 2 qui sont des deformations non lineaires de l equation d onde on etudie l existence globale et le comportement asymptotique des solutions du probleme de cauchy quand la donnee initiale est suffisamment petite

near field and far field approximations by the adomian and asymptotic decomposition methods the adomian decomposition method and the asymptotic decomposition method give the near field approximate solution and far field approximate solution respectively for linear and nonlinear differential equations the pade approximants give solution continuation of series solutions but the continuation is usually effective only on some finite domain and it can not always give the asymptotic behavior as the independent variables approach infinity we investigate the global approximate solution by matching the near field approximation derived from the adomian decomposition method with the far field approximation derived from the asymptotic decomposition method for linear and nonlinear differential equations for several examples we find that there exists an overlap between the near field approximation and the far field approximation so we can match them to obtain a global approximate solution for other nonlinear examples where the series solution from the adomian decomposition method has a finite convergent domain we can match the pade approximant of the near field approximation with the far field approximation to obtain a global approximate solution representing the true entire solution over an infinite domain

new ideas for decomposing nonlinearities in differential equations abstract in this paper we consider the decomposition for the nonlinearity in a differential equation for the solution by decomposition by analyzing and transforming the taylor expansion of the nonlinearity about the initial solution component the decomposition of the nonlinearity is converted to the partitions of the solution sets for a class of diophantine equations this conversion simplifies the discussion and presents a new idea for decompositions we enumerate five types of partitions and their corresponding decomposition polynomials each of the last four types contains infinitely many kinds of decomposition polynomials in the form of finite sums in types 2 3 and 4 there is a parameter q and each value of q corresponds to a class of decomposition polynomials in type 5 each positive integer sequence c j satisfying 1 c 1 c 2 and j c j for j 2 3 corresponds to a class of decomposition polynomials four classes of the adomian polynomials r rach a new definition of the adomian polynomials kybernetes 37 2008 910 955 are derived as particular cases

exponential decreasing rate of leaked information in universal random privacy amplification we derive a new upper bound for eve s information in secret key generation from a common random number without communication this bound improves on bennett s bound based on the renyi entropy of order 2 because the bound obtained here uses the renyi entropy of order 1 s for s 0 1 this bound is applied to a wire tap channel then we derive an exponential upper bound for eve s information our exponent is compared with hayashi s exponent for the additive case the bound obtained here is better the result is applied to secret key agreement by public discussion

a formula for the capacity of the general gel fand pinsker channel we consider the gel fand pinsker problem in which the channel and state are general i e possibly non stationary non memoryless and non ergodic using the information spectrum method and a non trivial modification of the piggyback coding lemma by wyner we prove that the capacity can be expressed as an optimization over the difference of a spectral inf and a spectral sup mutual information rate we consider various specializations including the case where the channel and the state are memoryless but not necessarily stationary

degraded broadcast channel with secrecy outside a bounded range the k receiver degraded broadcast channel with secrecy outside a bounded range is studied in which a transmitter sends k messages to k receivers and the channel quality gradually degrades from receiver k to receiver 1 each receiver k is required to decode message w 1 ldots w k for 1 leq k leq k and to be kept ignorant of w k 2 ldots w k for k 1 ldots k 2 thus each message w k is kept secure from receivers with at least two level worse channel quality i e receivers 1 ldots k 2 the secrecy capacity region is fully characterized the achievable scheme designates one superposition layer to each message with binning employed for each layer joint embedded coding and binning are employed to protect all upper layer messages from lower layer receivers furthermore the scheme allows adjacent layers to share rates so that part of the rate of each message can be shared with its immediate upper layer message to enlarge the rate region more importantly an induction approach is developed to perform fourier motzkin elimination of 2k variables from the order of k 2 bounds to obtain a close form achievable rate region an outer bound is developed that matches the achievable rate region whose proof involves recursive construction of the rate bounds and exploits the intuition gained from the achievable scheme