some remarks on quasiconvexity and rank one convexity we explore some necessary conditions for quasiconvexity in an attempt to show that rank one convexity does not imply quasiconvexity when the target space for deformations is two dimensional an interesting construction is presented showing how rank one directions may fit with each other making the task harder than in higher dimensions

coding theorems for shannon s cipher system with correlated source outputs and common information source coding problems are treated for shannon s 1949 cipher system with correlated source outputs x y several cases are considered based on whether both x and y only x or only y must be transmitted to the receiver whether both x and y only x or only y must be kept secret or whether the security level is measured by sup 1 spl sol sub k h x sup k spl verbar w sup 1 spl sol sub k h y sup k spl verbar w or sup 1 spl sol sub k h x sup k y sup k spl verbar w where w is a cryptogram the admissible region of cryptogram rate and key rate for a given security level is derived for each case furthermore two new kinds of common information of x and y say c sub 1 x y and c sub 2 x y are considered c sub 1 x y is defined as the rate of the attainable minimum core of x sup k y sup k by removing each private information from x sup k y sup k as much as possible while c sub 2 x y is defined as the rate of the attainable maximum core v sub c such that if one loses v sub c then each uncertainty of x sup k and y sup k becomes h v sub c it is proved that c sub 1 x y i x y and c sub 2 x y min spl lcub h x h y spl rcub c sub 1 x y justifies the author s intuitive feeling that the mutual information represents a common information of x and y

growth rates and explosions in sandpiles we study the abelian sandpile growth model where n particles are added at the origin on a stable background configuration in zd any site with at least 2d particles then topples by sending one particle to each neighbor we find that with constant background height h 2d 2 the diameter of the set of sites that topple has order n1 d this was previously known only for h d our proof uses a strong form of the least action principle for sandpiles and a novel method of background modification we can extend this diameter bound to certain backgrounds in which an arbitrarily high fraction of sites have height 2d 1 on the other hand we show that if the background height 2d 2 is augmented by 1 at an arbitrarily small fraction of sites chosen independently at random then adding finitely many particles creates an explosion a sandpile that never stabilizes

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

secret key and private key constructions for simple multiterminal source models we propose an approach for constructing secret and private keys based on the long known slepian wolf code due to wyner for correlated sources connected by a virtual additive noise channel our work is motivated by results of csiszar and narayan which highlight innate connections between secrecy generation by multiple terminals that observe correlated source signals and slepian wolf near lossless data compression explicit procedures for such constructions and their substantiation are provided the performance of low density parity check channel codes in devising a new class of secret keys is examined

on rank one convex functions that are homogeneous of degree one we show that positively 1 homogeneous rank one convex functions are convex at 0 and at matrices of rank one the result is a special case of an abstract convexity result that we establish for positively 1 homogeneous directionally convex functions defined on an open convex cone in a finite dimensional vector space from these results we derive a number of consequences including various generalizations of the ornstein l1 non inequalities most of the results were announced in c r acad sci paris ser i 349 407 409 2011

microstructures with finite surface energy the two well problem we study solutions of the two well problem i e maps which satisfy ueso n a so n b a c in n where a and b are n n matrices with positive determinants this problem arises in the study of microstructure in solid solid phase transitions under the additional hypothesis that the set e where the gradient lies in so n a has finite perimeter we show that u is locally only a function of one variable and that the boundary of e consists of subsets of hyperplanes which extend to and which do not intersect in this may not be the case if the assumption on e is dropped we also discuss applications of this result to magnetostrictive materials

uniform random number generation from markov chains non asymptotic and asymptotic analyses in this paper we derive non asymptotic achievability and converse bounds on the random number generation with without side information our bounds are efficiently computable in the sense that the computational complexity does not depend on the block length we also characterize the asymptotic behaviors of the large deviation regime and the moderate deviation regime by using our bounds which implies that our bounds are asymptotically tight in those regimes we also show the second order rates of those problems and derive single letter forms of the variances characterizing the second order rates furthermore we address the relative entropy rate and the modified mutual information rate for these problems

communication with imperfectly shared randomness communication complexity investigates the amount of communication needed for two or more players to determine some joint function of their private inputs for many interesting functions the communication complexity can be much smaller than basic information theoretic measures associated with the players inputs such as the input length the entropy or even the conditional entropy communication complexity of many functions reduces further when the players share randomness classical works studied the communication complexity of functions when the interacting players share randomness perfectly i e they get identical copies of randomness from a common source this paper considers the variant of this question when the players share randomness imperfectly i e when they get noisy copies of the randomness produced by some common source our main result shows that any function that can be computed by a k bit protocol in the perfect sharing model has a 2 k bit protocol in the setting of imperfectly shared randomness and such an exponential growth is necessary our upper bound relies on ideas from locality sensitive hashing while lower bounds rely on hypercontractivity and a new invariance principle tailored for communication protocols

riemann roch and abel jacobi theory on a finite graph abstract it is well known that a finite graph can be viewed in many respects as a discrete analogue of a riemann surface in this paper we pursue this analogy further in the context of linear equivalence of divisors in particular we formulate and prove a graph theoretic analogue of the classical riemann roch theorem we also prove several results analogous to classical facts about riemann surfaces concerning the abel jacobi map from a graph to its jacobian as an application of our results we characterize the existence or non existence of a winning strategy for a certain chip firing game played on the vertices of a graph

secret key generation using correlated sources and channels we study the secret key capacity in a joint source channel coding setup the terminals are connected over a discrete memoryless channel and have access to side information modelled as a pair of discrete memoryless source sequences as our main result we establish the upper and lower bounds on the secret key capacity in the lower bound expression the equivocation terms of the source and channel components are functionally additive even though the coding scheme generates a single secret key by jointly taking into account the source and channel equivocations our bounds coincide thus establishing the capacity when the underlying wiretap channel can be decomposed into a set of independent parallel and reversely degraded channels for the case of parallel gaussian channels and jointly gaussian sources we show that gaussian codebooks achieve the secret key capacity in addition when the eavesdropper also observes a correlated side information sequence we establish the secret key capacity when both the source and channel of the eavesdropper are a degraded version of the legitimate receiver we finally also treat the case when a public discussion channel is available propose a separation based coding scheme and establish its optimality when the channel output symbols of the legitimate receiver and eavesdropper are conditionally independent given the input

the lossy common information of correlated sources the two most prevalent notions of common information ci are due to wyner and gacs korner and both the notions can be stated as two different characteristic points in the lossless gray wyner region although the information theoretic characterizations for these two ci quantities can be easily evaluated for random variables with infinite entropy e g continuous random variables their operational significance is applicable only to the lossless framework the primary objective of this paper is to generalize these two ci notions to the lossy gray wyner network which hence extends the theoretical foundation to general sources and distortion measures we begin by deriving a single letter characterization for the lossy generalization of wyner s ci defined as the minimum rate on the shared branch of the gray wyner network maintaining minimum sum transmit rate when the two decoders reconstruct the sources subject to individual distortion constraints to demonstrate its use we compute the ci of bivariate gaussian random variables for the entire regime of distortions we then similarly generalize gacs and korner s definition to the lossy framework the latter half of this paper focuses on studying the tradeoff between the total transmit rate and receive rate in the gray wyner network we show that this tradeoff yields a contour of points on the surface of the gray wyner region which passes through both the wyner and gacs korner operating points and thereby provides a unified framework to understand the different notions of ci we further show that this tradeoff generalizes the two notions of ci to the excess sum transmit rate and receive rate regimes respectively

security analysis of varepsilon almost dual universal 2 hash functions smoothing of min entropy versus smoothing of renyi entropy of order 2 recently varepsilon almost dual universal2 hash functions have been proposed as a new and wider class of hash functions using this class of hash functions several efficient hash functions were proposed this paper evaluates the security performance when we apply this kind of hash functions we evaluate the security in several kinds of setting based on the l 1 distinguishability criterion and the modified mutual information criterion the obtained evaluation is based on smoothing of renyi entropy of order 2 and or min entropy we clarify the difference between these two methods

key capacity for product sources with application to stationary gaussian processes we show that for product sources rate splitting is optimal for secret key agreement using limited one way communication between two terminals this yields an alternative proof of the tensorization property of a strong data processing inequality originally studied by erkip and cover and amended recently by anantharam et al we derive a water filling solution of the communication rate key rate tradeoff for a wide class of discrete memoryless vector gaussian sources which subsumes the case without an eavesdropper moreover we derive an explicit formula for the maximum secret key per bit of communication for all discrete memoryless vector gaussian sources using a tensorization property and a variation on the enhanced channel technique of weingarten et al finally a one shot information spectrum achievability bound for key generation is proved from which we characterize the communication rate key rate tradeoff for stationary gaussian processes

the common randomness capacity of a pair of independent discrete memoryless channels we study the following problem two agents alice and bob are connected to each other by independent discrete memoryless channels they wish to generate common randomness i e agree on a common random variable by communicating interactively over the two channels assuming that alice and bob are allowed access to independent external random sources at rates in bits per step of communication of h sub a and h sub b respectively we show that they can generate common randomness at a rate of max min h sub a h w q i p v min h sub b h v p i q w bits per step by exploiting the noise on the two channels here v is the channel from alice to bob and w is the channel from bob to alice the maximum is over all probability distributions p and q on the input alphabets of v and w respectively we also prove a strong converse which establishes the above rate as the highest attainable in this situation

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

on secret sharing systems a secret sharing system permits a secret to be shared among n trustees in such a way that any k of them can recover the secret but any k 1 have complete uncertainty about it a linear coding scheme for secret sharing is exhibited which subsumes the polynomial interpolation method proposed by shamir and can also be viewed as a deterministic version of blakley s probabilistic method bounds on the maximum value of n for a given k and secret size are derived for any system linear or nonlinear the proposed scheme achieves the lower bound which for practical purposes differs insignificantly from the upper bound the scheme may be extended to protect several secrets methods to protect against deliberate tampering by any of the trustees are also presented

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

enumerating linear systems on graphs the divisor theory of graphs views a finite connected graph g as a discrete version of a riemann surface divisors on g are formal integral combinations of the vertices of g and linear equivalence of divisors is determined by the discrete laplacian operator for g as in the case of riemann surfaces we are interested in the complete linear system d of a divisor d the collection of nonnegative divisors linearly equivalent to d unlike the case of riemann surfaces the complete linear system of a divisor on a graph is always finite we compute generating functions encoding the sizes of all complete linear systems on g and interpret our results in terms of polyhedra associated with divisors and in terms of the invariant theory of the dual of the jacobian group of g if g is a cycle graph our results lead to a bijection between complete linear systems and binary necklaces our results also apply to a model in which the laplacian is replaced by an invertible integral m matrix

feedback based collaborative secrecy encoding over binary symmetric channels in this paper we propose a feedback scheme for transmitting secret messages between two legitimate parties over an eavesdropped communication link relative to wyner s traditional encoding scheme our feedback based encoding often yields larger rate equivocation regions and achievable secrecy rates more importantly by exploiting the channel randomness inherent in the feedback channels our scheme achieves a strictly positive secrecy rate even when the eavesdropper s channel is less noisy than the legitimate receiver s channel all channels are modeled as binary and symmetric we demonstrate the versatility of our feedback based encoding method by using it in three different configurations the stand alone configuration the mixed configuration when it combines with wyner s scheme and the reversed configuration depending on the channel conditions significant improvements over wyner s secrecy capacity can be observed in all configurations

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

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

rate distortion theory for secrecy systems secrecy in communication systems is measured herein by the distortion that an adversary incurs the transmitter and receiver share secret key which they use to encrypt communication and ensure distortion at an adversary a model is considered in which an adversary not only intercepts the communication from the transmitter to the receiver but also potentially has side information in particular the adversary may have causal or noncausal access to a signal that is correlated with the source sequence or the receiver s reconstruction sequence the main contribution is the characterization of the optimal tradeoff among communication rate secret key rate distortion at the adversary and distortion at the legitimate receiver it is demonstrated that causal side information at the adversary plays a pivotal role in this tradeoff it is also shown that measures of secrecy based on normalized equivocation are a special case of the framework

secrecy capacity analysis over kappa mu fading channels theory and applications in this paper we consider the transmission of confidential information over a kappa mu fading channel in the presence of an eavesdropper who also experiences kappa mu fading in particular we obtain novel analytical solutions for the probability of strictly positive secrecy capacity spsc and a lower bound of secure outage probability sop l for independent and non identically distributed channel coefficients without parameter constraints we also provide a closed form expression for the probability of spsc when the mu parameter is assumed to take positive integer values monte carlo simulations are performed to verify the derived results the versatility of the kappa mu fading model means that the results presented in this paper can be used to determine the probability of spsc and sop l for a large number of other fading scenarios such as rayleigh rice nakagami n nakagami m one sided gaussian and mixtures of these common fading models in addition due to the duality of the analysis of secrecy capacity and co channel interference cci the results presented here will have immediate applicability in the analysis of outage probability in wireless systems affected by cci and background noise bn to demonstrate the efficacy of the novel formulations proposed here we use the derived equations to provide a useful insight into the probability of spsc and sop l for a range of emerging wireless applications such as cellular device to device peer to peer vehicle to vehicle and body centric communications using data obtained from real channel measurements

a lossy source coding interpretation of wyner s common information wyner s common information was originally defined for a pair of dependent discrete random variables its significance is largely reflected in and also confined to several existing interpretations in various source coding problems this paper attempts to expand its practical significance by providing a new operational interpretation in the context of the gray wyner network it is established that wyner s common information has a new lossy source coding interpretation specifically it is established that under suitable conditions wyner s common information equals to the smallest common message rate when the total rate is arbitrarily close to the rate distortion function with joint decoding for the gray wyner network a surprising observation is that such equality holds independent of the values of distortion constraints as long as the distortions are within some distortion region the new lossy source coding interpretation provides the first meaningful justification for defining wyner s common information for continuous random variables and the result can also be extended to that of multiple variables examples are given for characterizing the rate distortion region for the gray wyner lossy source coding problem and for identifying conditions under which wyner s common information equals that of the smallest common rate as a by product the explicit expression for the common information between a pair of gaussian random variables is obtained

self organized critical state of sandpile automaton models we study a general bak tang wiesenfeld type automaton model of self organized criticality in which the toppling conditions depend on local height but not on its gradient we characterize the critical state and determine its entropy for an arbitrary finite lattice in any dimension the two point correlation function is shown to satisfy a linear equation the spectrum of relaxation times describing the approach to the critical state is also determined exactly

computational complexity of continuous variable quantum key distribution the continuous variable quantum key distribution has been considered to have the potential to provide high secret key rate however in present experimental demonstrations the secret key can be distilled only under very small loss rates here by calculating explicitly the computational complexity with the channel transmission we show that under high loss rate it is hard to distill the secret key in present continuous variable scheme and one of its advantages the potential of providing high secret key rate may therefore be limited

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

the common randomness capacity of a network of discrete memoryless channels we generalize our previous results on generating common randomness at two terminals to a situation where any finite number of agents interconnected by an arbitrary network of independent point to point discrete memoryless channels wish to generate common randomness by interactive communication over the network our main result is an exact characterization of the common randomness capacity of such a network i e the maximum number of bits of randomness that all the agents can agree on per step of communication as a by product we also obtain a purely combinatorial result viz a characterization of the incidence vectors of the spanning arborescences rooted at a specified vertex in a digraph and having exactly one edge exiting the root as precisely the extreme points of a certain unbounded convex polyhedron described by a system of linear inequalities

distillation of secret key and entanglement from quantum states we study and solve the problem of distilling a secret key from quantum states representing correlation between two parties alice and bob and an eavesdropper eve via oneway public discussion we

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

low dimensional quite noisy bound entanglement with a cryptographic key we provide a class of bound entangled states that have a positive distillable secure key rate the smallest state of this kind is 4 4 our class is a generalization of the class presented in horodecki et al 2008 ieee trans inf theory 54 2621 5 it is much wider containing in particular states from the boundary of ppt entangled states all of the states in the previous class were of this kind and also states inside the set of ppt entangled states even approaching the separable states this generalization comes at a price for the wider class a positive key rate requires in general apart from the one way devetak winter protocol used in the previous case also the recurrence preprocessing and thus is effectively a two way protocol we also analyze the amount of noise that can be admixtured to the states of our class without losing the key distillability property which may be crucial for experimental realization the wider class contains key distillable states with higher entropy up to 3 524 as opposed to 2 564 for the previous class

secret key agreement over non coherent block fading channels with public discussion motivated by recent interest in physical layer secret key generation over wireless fading channels we study the non coherent secret key generation capacity of a block fading wireless channel with channel reciprocity and bi directional two way communication we assume a non coherent main channel i e the realization of channel gains on the main channel is not known to any terminal the eavesdropper is assumed to have both perfect channel state information of its own channel and orthogonal observations from the forward and backward channels as our main result we establish new upper and lower bounds on the secret key generation capacity with public discussion which are structurally similar the upper bound can be expressed as a sum of three terms one of the terms arises due to channel reciprocity while the other two terms correspond to communication on the forward and backward channels respectively in the limit of long coherence period the contribution from channel reciprocity vanishes to zero whereas the other terms prevail the lower bound is based on a separation based scheme in each coherence block we use the first symbol for training while the remainder of the coherence block is devoted to source emulation i e to generate correlated sources between the terminals the lower bound expression also consists of three terms and admits an interpretation similar to the upper bound expression for rayleigh fading channels in the high signal to noise ratio snr regime the gap between the upper and lower bounds is shown to vanish inversely with the coherence period numerical results indicate significant performance gains over training only schemes even for moderate values of snr and small coherence periods

partial regularity of strong local minimizers of quasiconvex integrals with p q growth defined on vector valued maps u r n 1 here denotes a bounded open set in r n 2 and f r r is a c function satisfying suitable assumptions described below the existence and the partial regularity of minimizers of f are classical issues in the modern calculus of variations and they have been extensively studied especially over the last twenty years specifically in the present paper we will focus on gradient regularity under the basic assumption that f is quasiconvex that is

the commitment capacity of the gaussian channel is infinite we prove that the commitment capacity of the power constrained gaussian channel i e the optimal rate at which this channel can be used for implementing commitment schemes is infinite

on the im possibility of non interactive correlation distillation we study the problem of non interactive correlation distillation nicd suppose that alice and bob each have a string denoted by a a0a1 an 1 and b b0b1 bn 1 respectively furthermore for every k 0 1 n 1 ak bk is drawn independently from a distribution known as the noise model alice and bob wish to distill the correlation non interactively i e they wish to each apply a function to their strings and output one random bit denoted by x and y such that pr x y can be made as close to 1 as possible the problem is for what noise models can they succeed this problem is related to various topics in computer science including information reconciliation and random beacons in fact if nicd is indeed possible for some general class of noise models then some of these topics would in some sense become straightforward corollaries we prove two negative results on nicd for various noise models we prove that for these models it is impossible to distill the correlation to be arbitrarily close to 1 we also give an example where alice and bob can increase their correlation with one bit of communication in this case they need to each output two bits this example which may be of interest on its own demonstrates that even the smallest amount of communication is provably more powerful than no communication

tight exponential analysis of universally composable privacy amplification and its applications motivated by the desirability of universal composability we analyze in terms of l1 distinguishability the task of secret key generation from a joint random variable under this secrecy criterion using the renyi entropy of order 1 s for s 0 1 we derive a new upper bound of eve s distinguishability under the application of the universal2 hash functions it is also shown that this bound gives the tight exponential rate of decrease in the case of independent and identical distributions the result is applied to the wiretap channel model and to secret key generation distillation by public discussion

secure key rate of the bb84 protocol using finite sample bits we improve the non asymptotic key rate shown by scarani and renner by proposing several methods to construct tighter conservative confidence intervals of the phase error rate than one previously shown in addition we show that the accurate channel estimation method non asymptotically increases the key rate over the amplitude damping channel as well as the asymptotic case in the bb84 protocol

robust fuzzy extractors and authenticated key agreement from close secrets consider two parties holding samples from correlated distributions w and w prime respectively where these samples are within distance t of each other in some metric space the parties wish to agree on a close to uniformly distributed secret key r by sending a single message over an insecure channel controlled by an all powerful adversary who may read and modify anything sent over the channel we consider both the keyless case where the parties share no additional secret information and the keyed case where the parties share a long term secret ssr sk ssr ext that they can use to generate a sequence of session keys r j using multiple pairs w j w prime j the former has applications to e g biometric authentication while the latter arises in e g the bounded storage model with errors we show solutions that improve upon previous work in several respects the best prior solution for the keyless case with no errors i e t 0 requires the min entropy of w to exceed 2n 3 where n is the bit length of w our solution applies whenever the min entropy of w exceeds the minimal threshold n 2 and yields a longer key

secret key generation for correlated gaussian sources secret key generation by multiple terminals is considered based on their observations of jointly distributed gaussian signals followed by public communication among themselves exploiting an inherent connection between secrecy generation and lossy data compression two main contributions are made the first is a characterization of strong secret key capacity and entails a converse proof technique that is valid for real valued and not necessarily gaussian as well as finite valued signals the capacity formula acquires a simple form when the terminals observe symmetrically correlated jointly gaussian signals for the latter setup with two terminals considering schemes that involve quantization at one terminal the best rate of an achievable secret key is characterized as a function of quantization rate secret key capacity is attained as the quantization rate tends to infinity structured codes are shown to attain the optimum tradeoff between secret key rate and quantization rate constituting our second main contribution

the conditional common information in classical and quantum secret key distillation in this paper we consider two extensions of the gacs korner common information to three variables the conditional common information cci and the coarse grained conditional common information ccci both quantities are shown to be useful technical tools in the study of classical and quantum resource transformations in particular the ccci is shown to have an operational interpretation as the optimal rate of secret key extraction from an eavesdropped classical source p xyz when alice x and bob y are unable to communicate but share common randomness with the eavesdropper eve z moving to the quantum setting we consider two different ways of generating a tripartite quantum state from classical correlations p xyz 1 coherent encodings sum xyz sqrt p xyz xyz rangle and 2 incoherent encodings sum xyz p xyz xyz rangle langle xyz we study how well can alice and bob extract secret key from these quantum sources using quantum operations compared with the extraction of key from the underlying classical sources p xyz using classical operations while the power of quantum mechanics increases alice and bob s ability to generate shared randomness it also equips eve with a greater arsenal of eavesdropping attacks therefore it is not obvious who gains the greatest advantage for distilling secret key when replacing a classical source with a quantum one we first demonstrate that the classical key rate of p xyz is equivalent to the quantum key rate for an incoherent quantum encoding of the distribution for coherent encodings we next show that the classical and quantum rates are generally incomparable and in fact their difference can be arbitrarily large in either direction finally we introduce a zoo of entangled tripartite states all characterized by the conditional common information of their encoded probability distributions remarkably for these states almost all entanglement measures such as alice and bob s entanglement cost squashed entanglement and relative entropy of entanglement can be sharply bounded or even exactly expressed in terms of the conditional common information in the latter case we thus present a rare instance in which the various entropic entanglement measures of a quantum state can be explicitly calculated

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

secure multiplex coding attaining channel capacity in wiretap channels it is known that a message can be transmitted safely against any wiretapper via a noisy channel without a secret key if the coding rate is less than the so called secrecy capacity cs which is usually smaller than the channel capacity c in order to remove the loss c cs we propose a multiplex coding scheme with plural independent messages in this paper it is shown that the proposed multiplex coding scheme can attain the channel capacity as the total rate of the plural messages and the perfect secrecy for each massage several bounds of achievable multiplex coding rate region are derived for general wiretap channels in the sense of information spectral methods by extending hayashi s proof in which the coding of the channel resolvability is applied to wiretap channels furthermore the exact region for deterministic coding is determined for stationary memoryless full rank wiretap channels

wiretap channel with causal state information a lower bound on the secrecy capacity of the wiretap channel with state information available causally at both the encoder and the decoder is established the lower bound is shown to be strictly larger than that for the noncausal case by liu and chen achievability is proved using block markov coding shannon strategy and key generation from common state information the state sequence available at the end of each block is used to generate a key to enhance the transmission rate of the confidential message in the following block an upper bound on the secrecy capacity when the state is available noncausally at the encoder and the decoder is established and is shown to coincide with the aforementioned lower bound for several classes of wiretap channels with state

on extracting common random bits from correlated sources suppose alice and bob receive strings of unbiased independent but noisy bits from some random source they wish to use their respective strings to extract a common sequence of random bits with high probability but without communicating how many such bits can they extract the trivial strategy of outputting the first k bits yields an agreement probability of 1 e k 2 1 44ke where e is the amount of noise we show that no strategy can achieve agreement probability better than 2 ke 1 e on the other hand we show that when k 10 2 1 e e there exists a strategy which achieves an agreement probability of 0 003 ke 1 2 2 ke 1 e

coding theorems on the threshold scheme for a general source in this paper coding theorems on the t m threshold scheme for a general source are discussed where m means the number of the shares and t means a threshold the t m threshold scheme treated in this paper encrypts n source outputs xn to m shares at once and is required to satisfy the two conditions that 1 xn is reproduced from arbitrary t shares and 2 almost no information of xn is revealed from any t 1 shares it is shown that the t m threshold scheme must satisfy certain inequalities including the limit inferiors in probability one of the inequalities is closely related to the minimum length of the fair random bits needed to a dealer for realizing the t m threshold scheme in addition it is shown that a certain variation of shamir s threshold scheme meets the two conditions the same approach can be taken to the problems of shannon s cipher system with the perfect secrecy and fixed length source coding with vanishing decoding error probability it is shown that the same kind of inequalities which indicate the converse coding theorems hold in both two cases

an example of a quasiconvex function that is not polyconvex in two dimensions we study the different notions of convexity for the function f 2 2 2 det where e 2 2 introduced by dacorogna marcellini we show that f is convex polyconvex quasiconvex rank one convex if and only if 2 3 2 1 1 for some 0 2 3 respectively

linear systems on tropical curves a tropical curve gamma is a metric graph with possibly unbounded edges and tropical rational functions are continuous piecewise linear functions with integer slopes we define the complete linear system d of a divisor d on a tropical curve gamma analogously to the classical counterpart we investigate the structure of d as a cell complex and show that linear systems are quotients of tropical modules finitely generated by vertices of the cell complex using a finite set of generators d defines a map from gamma to a tropical projective space and the image can be modified to a tropical curve of degree equal to mathrm deg d the tropical convex hull of the image realizes the linear system d as a polyhedral complex

a geometric realization of the periodic discrete toda lattice and its tropicalization an explicit formula concerning curve intersections equivalent to the time evolution of the periodic discrete toda lattice pdtl is presented first the time evolution is realized as a point addition on a hyperelliptic curve which is the spectral curve of the pdtl then the point addition is translated into curve intersections next it is shown that the curves which appear in the curve intersections are explicitly given by using the conserved quantities of the pdtl finally the formulation is lifted to the framework of tropical geometry and a tropical geometric realization of the periodic box ball system is constructed via tropical curve intersections

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

on the relaxation in bv ω rm of quasi convex integrals abstract given a quasi convex function f with linear growth we find the integral representation in bv r m of the functional f arising from the relaxation of f u f u dx u c 1 r m in the l loc 1 r m topology

quasiconvex functions with subquadratic growth we establish the existence of quasiconvex functions which are not convex and which have subquadratic grow that infinity

rank one convexity does not imply quasiconvexity we consider variational integrals defined for sufficiently regular functions u r m here is a bounded open subset of r n du x denotes the gradient matrix of u at x and f is a continuous function on the space of all real m n matrices m m n one of the important problems in the calculus of variations is to characterise the functions f for which the integral i is lower semicontinuous in this connection the following notions were introduced see 3 9 10

an operational approach to information leakage given two random variables x and y an operational approach is undertaken to quantify the leakage of information from x to y the resulting measure mathcal l left x to y right is called maximal leakage and is defined as the multiplicative increase upon observing y of the probability of correctly guessing a randomized function of x maximized over all such randomized functions a closed form expression for mathcal l left x to y right is given for discrete x and y and it is subsequently generalized to handle a large class of random variables the resulting properties are shown to be consistent with an axiomatic view of a leakage measure and the definition is shown to be robust to variations in the setup moreover a variant of the shannon cipher system is studied in which performance of an encryption scheme is measured using maximal leakage a single letter characterization of the optimal limit of normalized maximal leakage is derived and asymptotically optimal encryption schemes are demonstrated furthermore the sample complexity of estimating maximal leakage from data is characterized up to subpolynomial factors finally the guessing framework used to define maximal leakage is used to give operational interpretations of commonly used leakage measures such as shannon capacity maximal correlation and local differential privacy

secret key agreement over unauthenticated public channels part ii the simulatability condition for pt i see ibid vol 49 no 4 p 822 31 2003 in the first part we showed that when two parties willing to generate a secret key but connected only by a completely insecure communication channel have access to independent repetitions of some random experiment then the possibility of secret key agreement depends on a certain property called simulatability of the probability distribution modeling the parties initial knowledge more generally the simulatability condition is important in the context of identification and authentication among parties sharing some correlated but not necessarily identical partially secret keys unfortunately this condition is a priori not very useful since it is not clear how to decide efficiently whether it is satisfied or not for a given distribution p sub xyz we introduce a new formalism based on a mechanical model for representing the involved quantities that allows for dealing with discrete joint distributions of random variables and their manipulations by noisy channels we show that this representation leads to a simple and efficient characterization of the possibility of secret key agreement secure against active adversaries

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

a tropical proof of the brill noether theorem we produce brill noether general graphs in every genus confirming a conjecture of baker and giving a new proof of the brill noether theorem due to griffiths and harris over any algebraically closed field

communication in key distribution schemes a g b key distribution scheme allows conferences of g users to generate secret keys such that disjoint coalitions of b users cannot gain any information on the generated key in the information theoretic sense we study the relationships between communication and space efficiency of key distribution schemes we prove that communication does not help in the context of unrestricted schemes on the other hand we show that for restricted schemes which are secure only when used by a limited number of conferences communication can substantially improve the space efficiency we also present lower bounds on the space efficiency of restricted schemes

assisted common information with an application to secure two party sampling an important subclass of secure multiparty computation is secure sampling two parties output samples of a pair of jointly distributed random variables such that neither party learns more about the other party s output than what its own output reveals the parties make use of a setup correlated random variables with a different distribution as well as unlimited noiseless communication an upperbound on the rate of producing samples of a desired distribution from a given setup is presented the region of tension developed in this paper measures how well the dependence between a pair of random variables can be resolved by a piece of common information the bounds on rate are a consequence of a monotonicity property a protocol between two parties can only lower the tension between their views connections are drawn between the region of tension and the notion of common information a generalization of the gacs korner common information called the assisted common information which takes into account almost common information ignored by gacs korner common information is defined the region of tension is shown to be related to the rate regions of both the assisted common information and the gray wyner systems and a fortiori wyner s common information

tropical theta functions and riemann roch inequality for tropical abelian surfaces we show that the space of theta functions on tropical tori is identified with a convex polyhedron we also show a riemann roch inequality for tropical abelian surfaces by calculating the self intersection numbers of divisors

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

coding theorems for the shannon cipher system with a guessing wiretapper and correlated source outputs the security level of the shannon cipher system is traditionally measured by equivocation where is a secret plaintext with length and is its cryptogram but merhav and arikan have considered another security criterion which is measured by the number of guesses needed for a wiretapper to uncover from merhav has also considered the third security criterion which measured by the probability of correct guess of a wiretapper on the other hand in the case of the traditional security criterion yamamoto has treated a coding problem for correlated source outputs and such that only is secret against wiretappers and only must be transmitted to a legitimate receiver in this correspondence coding theorems are proved for the case that yamamoto s coding problem is applied to merhav arikan s security criterion or merhav s security criterion

coding theorems for a 2 2 threshold scheme with detectability of impersonation attacks coding theorems on a 2 2 threshold scheme with an opponent are discussed in an asymptotic setup where the opponent tries to impersonate one of the two participants a situation is considered where n secrets s n from a memoryless source is blockwisely encoded to two shares and the two shares are decoded to s n with permitting negligible decoding error we introduce correlation level of the two shares and characterize the minimum attainable rates of the shares and a uniform random number for realizing a 2 2 threshold scheme that is secure against the impersonation attack by the opponent it is shown that if the correlation level between the two shares equals to ell geq 0 the minimum attainable rates coincide with h s ell where h s denotes the entropy of the source and the maximum attainable exponent of the success probability of the impersonation attack equals to ell it is also shown that a simple scheme using an ordinary 2 2 threshold scheme attains all the bounds as well

a riemann roch theorem in tropical geometry recently baker and norine have proven a riemann roch theorem for finite graphs we extend their results to metric graphs and thus establish a riemann roch theorem for divisors on abstract tropical curves

reconciliation of a quantum distributed gaussian key two parties alice and bob wish to distill a binary secret key out of a list of correlated variables that they share after running a quantum key distribution qkd protocol based on continuous spectrum quantum carriers we present a novel construction that allows the legitimate parties to get equal bit strings out of correlated variables by using a classical channel with as little leaked information as possible this opens the way to securely correcting nonbinary key elements in particular the construction is refined to the case of gaussian variables as it applies directly to recent continuous variable protocols for qkd

secret sharing with public reconstruction all known constructions of information theoretic t out of n secret sharing schemes require secure private communication channels among the parties for the reconstruction of the secret we investigate the cost of performing the reconstruction over public communication channels a naive implementation of this task distributes 2n 2 one times pads to each party this results in shares whose size is 2n 1 times the secret size we present three implementations of such schemes that are substantially more efficient a scheme enabling multiple reconstructions of the secret by different subsets of parties with factor o n t increase in the shares size a one time scheme enabling a single reconstruction of the secret with o log n t increase in the shares size a one time scheme enabling a single reconstruction by a set of size exactly t with factor o 1 increase in the shares size we prove that the first implementation is optimal up to constant factors by showing a tight spl omega n t lower bound for the increase in the shares size

relaxation of quasiconvex functional in bv ω ℝp for integrands f x u u in this paper it is shown that if p x u is a quasiconvex function with linear growth then the relaxed functional in bv p of u to int limits omega f x u x nabla u x dx with respect to the l1 topology has an integral representation of the form begin gathered mathfrak f u int limits omega f x u x nabla u x dx int limits sigma u k x u x u x v x dh n 1 x hfill int limits omega f infty x u x dc u hfill end gathered where du u dx u u v dhn 1l u c u the proof relies on a blow up argument introduced by fonseca muller in the case where u w1 1 and on a recent result by alberti showing that the cantor part c u is rank one valued

on lossless coding with coded side information this paper considers the problem first introduced by ahlswede and korner in 1975 of lossless source coding with coded side information specifically let x and y be two random variables such that x is desired losslessly at the decoder while y serves as side information the random variables are encoded independently and both descriptions are used by the decoder to reconstruct x ahlswede and korner describe the achievable rate region in terms of an auxiliary random variable this paper gives a partial solution for an optimal auxiliary random variable thereby describing part of the rate region explicitly in terms of the distribution of x and y

common randomness and distributed control a counterexample when agents collaborate to perform a control task it is of interest to characterize the set of joint probability distributions they can achieve on their joint action space when they are passively provided with external common randomness we give a simple counterexample to a natural conjecture about this class of joint distributions