estimation efficiency under privacy constraints we investigate the problem of estimating a random variable y under a privacy constraint dictated by another correlated random variable x when x and y are discrete we express the underlying privacy utility tradeoff in terms of the privacy constrained guessing probability mathcal h p xy varepsilon and the maximum probability mathsf p mathsf c y z of correctly guessing y given an auxiliary random variable z where the maximization is taken over all p z y ensuring that mathsf p mathsf c x z leq varepsilon for a given privacy threshold varepsilon geq 0 we prove that mathcal h p xy cdot is concave and piecewise linear which allows us to derive its expression in closed form for any varepsilon when x and y are binary in the non binary case we derive mathcal h p xy varepsilon in the high utility regime i e for sufficiently large but nontrivial values of varepsilon under the assumption that y and z have the same alphabets we also analyze the privacy constrained guessing probability for two scenarios in which x y and z are binary vectors when x and y are continuous random variables we formulate the corresponding privacy utility tradeoff in terms of mathsf sensr p xy varepsilon the smallest normalized minimum mean squared error mmse incurred in estimating y from a gaussian perturbation z here the minimization is taken over a family of gaussian perturbations z for which the mmse of f x given z is within a factor 1 varepsilon from the variance of f x for any non constant real valued function f we derive tight upper and lower bounds for mathsf sensr when y is gaussian for general absolutely continuous random variables we obtain a tight lower bound for mathsf sensr p xy varepsilon in the high privacy regime i e for small varepsilon

sharp lower bounds on the extractable randomness from non uniform sources extraction of uniform randomness from noisy non uniform sources is an important primitive in many security applications e g pseudo random number generators privacy preserving biometrics and key storage based on physical unclonable functions generic extraction methods exist using universal hash functions there is a trade off between the length of the extracted bit string and the uniformity of the string in the literature there are proven lower bounds on this length as a function of the desired uniformity the best known bound involves a quantity known as smooth min entropy unfortunately there exist at least three definitions of smooth entropy in this paper we compare three of these definitions and we derive improved lower bounds on the extractable randomness we also investigate the use of almost universal hash functions which are slightly worse at extracting randomness than universal hash functions but are preferable in practice because they require far less resources in devices we show that using them has negligible effect on the extractable randomness

sandwiched renyi divergence satisfies data processing inequality sandwiched quantum renyi divergence has been recently defined in the independent works of wilde et al strong converse for the classical capacity of entanglement breaking channels preprint arxiv 1306 1586 2013 and muller lennert et al on quantum renyi entropies a new definition some properties and several conjectures preprint arxiv 1306 3142v1 2013 this new quantum divergence has already found applications in quantum information theory here we further investigate properties of this new quantum divergence in particular we show that sandwiched renyi divergence satisfies the data processing inequality for all values of 1 moreover we prove that holevo information a variant of holevo information defined in terms of sandwiched renyi divergence is super additive our results are based on holder s inequality the riesz thorin theorem and ideas from the theory of complex interpolation we also employ sion s minimax theorem

quantum achievability proof via collision relative entropy in this paper we provide a simple framework for deriving one shot achievable bounds for some problems in quantum information theory our framework is based on the joint convexity of the exponential of the collision relative entropy and is a partial quantum generalization of the technique of yassaee et al from classical information theory based on this framework we derive one shot achievable bounds for the problems of communication over classical quantum channels quantum hypothesis testing and classical data compression with quantum side information we argue that our one shot achievable bounds are strong enough to give the asymptotic achievable rates of these problems even up to the second order

improved bounds on lossless source coding and guessing moments via renyi measures this paper provides upper and lower bounds on the optimal guessing moments of a random variable taking values on a finite set when side information may be available these moments quantify the number of guesses required for correctly identifying the unknown object and similarly to arikan s bounds they are expressed in terms of the arimoto renyi conditional entropy although arikan s bounds are asymptotically tight the improvement of the bounds in this paper is significant in the non asymptotic regime relationships between moments of the optimal guessing function and the map error probability are also established characterizing the exact locus of their attainable values the bounds on optimal guessing moments serve to improve non asymptotic bounds on the cumulant generating function of the codeword lengths for fixed to variable optimal lossless source coding without prefix constraints non asymptotic bounds on the reliability function of discrete memoryless sources are derived as well relying on these techniques lower bounds on the cumulant generating function of the codeword lengths are derived by means of the smooth renyi entropy for source codes that allow decoding errors

renyi resolvability and its applications to the wiretap channel the conventional channel resolvability problem refers to the determination of the minimum rate required for an input process so that the output distribution approximates a target distribution in either the total variation distance or the relative entropy in contrast to previous works in this paper we use the normalized or unnormalized renyi divergence with the renyi parameter in 0 2 cup infty to measure the level of approximation we also provide asymptotic expressions for normalized renyi divergence when the renyi parameter is larger than or equal to 1 as well as lower and upper bounds for the case when the same parameter is smaller than 1 we characterize the renyi resolvability which is defined as the minimum rate required to ensure that the renyi divergence vanishes asymptotically the renyi resolvabilities are the same for both the normalized and unnormalized divergence cases in addition when the renyi parameter smaller than 1 consistent with the traditional case where the renyi parameter is equal to 1 the renyi resolvability equals the minimum mutual information over all input distributions that induce the target output distribution when the renyi parameter is larger than 1 the renyi resolvability is in general larger than the mutual information the optimal renyi divergence is proven to vanish at least exponentially fast for both of these two cases as long as the code rate is larger than the renyi resolvability the optimal exponential rate of decay for i i d random codes is also characterized exactly we apply these results to the wiretap channel and completely characterize the optimal tradeoff between the rates of the secret and non secret messages when the leakage measure is given by the unnormalized renyi divergence this tradeoff differs from the conventional setting when the leakage is measured by the traditional mutual information

arimoto renyi conditional entropy and bayesian m ary hypothesis testing this paper gives upper and lower bounds on the minimum error probability of bayesian m ary hypothesis testing in terms of the arimoto renyi conditional entropy of an arbitrary order alpha the improved tightness of these bounds over their specialized versions with the shannon conditional entropy alpha 1 is demonstrated in particular in the case where m is finite we show how to generalize fano s inequality under both the conventional and list decision settings as a counterpart to the generalized fano s inequality allowing m to be infinite a lower bound on the arimoto renyi conditional entropy is derived as a function of the minimum error probability explicit upper and lower bounds on the minimum error probability are obtained as a function of the arimoto renyi conditional entropy for both positive and negative alpha furthermore we give upper bounds on the minimum error probability as functions of the renyi divergence in the setup of discrete memoryless channels we analyze the exponentially vanishing decay of the arimoto renyi conditional entropy of the transmitted codeword given the channel output when averaged over a random coding ensemble

analysis of remaining uncertainties and exponents under various conditional renyi entropies we analyze the asymptotics of the normalized remaining uncertainty of a source when a compressed or hashed version of it and correlated side information is observed for this system commonly known as slepian wolf source coding we establish the optimal minimum rate of compression of the source to ensure that the remaining uncertainties vanish we also study the exponential rate of decay of the remaining uncertainty to zero when the rate is above the optimal rate of compression in this paper we consider various classes of random universal hash functions instead of measuring remaining uncertainties using traditional shannon information measures we do so using two forms of the conditional renyi entropy among other techniques we employ new one shot bounds and the moments of type class enumerator method see merhav for these evaluations we show that these asymptotic results are generalizations of the strong converse exponent and the error exponent of the slepian wolf problem under maximum a posteriori decoding

the ziv zakai renyi bound for joint source channel coding shannon s capacity and rate distortion function combined with the separation principle provide tight bounds for the minimum possible distortion in joint source channel coding these bounds however are usually achievable only in the limit of a large block length in their 1973 paper ziv and zakai introduced a family of alternative capacity and rate distortion functions based on functionals satisfying the data processing inequality which potentially give tighter bounds for systems with a small block length there is a considerable freedom as to how to choose those functionals and the ways of finding the best possible functionals yielding the best bounds for a given source channel combination are not specified we examine recently conjectured high snr asymptotic expressions for the ziv zakai bounds based on the renyi divergence functional we derive nonasymptotic bounds on the ziv zakai renyi rate distortion function and capacity for a broad class of sources and additive noise channels which hold for arbitrary snr and prove the conjectured asymptotic expressions in the limit of a small distortion high snr the results lead to new bounds on the best achievable distortion in finite dimensional joint source channel coding examples are presented where the new bounds achieve significant improvement upon shannon s original bounds

α z renyi relative entropies we consider a two parameter family of renyi relative entropies d z that are quantum generalisations of the classical renyi divergence d p q this family includes many known relative entropies or divergences such as the quantum relative entropy the recently defined quantum renyi divergences as well as the quantum renyi relative entropies all its members satisfy the quantum generalizations of renyi s axioms for a divergence we consider the range of the parameters z for which the data processing inequality holds we also investigate a variety of limiting cases for the two parameters obtaining explicit formulas for each one of them

determinants of matrices of the conference type an n by n conference type matrix has 0 s on the main diagonal and 1 s elsewhere we investigate the largest possible determinant of such a matrix the literature is extensive for n even but for n odd the question has not been previously studied we determine the maxima up to n 11 for n 3 5 7 9 11 they are 2 22 394 8760 240786 it will require a new idea or a huge computation to go beyond n 11

entropy of spatial network ensembles we analyze complexity in spatial network ensembles through the lens of graph entropy mathematically we model a spatial network as a soft random geometric graph i e a graph with two sources of randomness namely nodes located randomly in space and links formed independently between pairs of nodes with probability given by a specified function the pair connection function of their mutual distance we consider the general case where randomness arises in node positions as well as pairwise connections i e for a given pair distance the corresponding edge state is a random variable classical random geometric graph and exponential graph models can be recovered in certain limits we derive a simple bound for the entropy of a spatial network ensemble and calculate the conditional entropy of an ensemble given the node location distribution for hard and soft probabilistic pair connection functions under this formalism we derive the connection function that yields maximum entropy under general constraints finally we apply our analytical framework to study two practical examples ad hoc wireless networks and the us flight network through the study of these examples we illustrate that both exhibit properties that are indicative of nearly maximally entropic ensembles

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

on the conditional renyi entropy the renyi entropy of general order unifies the well known shannon entropy with several other entropy notions like the min entropy or collision entropy in contrast to the shannon entropy there seems to be no commonly accepted definition for the conditional renyi entropy several versions have been proposed and used in the literature in this paper we reconsider the definition for the conditional renyi entropy of general order as proposed by arimoto in the seventies we show that this particular notion satisfies several natural properties in particular we show that it satisfies monotonicity under conditioning meaning that conditioning can only reduce the entropy and a weak form of chain rule which implies that the decrease in entropy due to conditioning is bounded by the number of bits one conditions on none of the other suggestions for the conditional renyi entropy satisfies both these properties finally we show a natural interpretation of the conditional renyi entropy in terms of unconditional renyi divergence and we show consistency with a recently proposed notion of conditional renyi entropy in the quantum setting

on quantum renyi entropies a new generalization and some properties the renyi entropies constitute a family of information measures that generalizes the well known shannon entropy inheriting many of its properties they appear in the form of unconditional and conditional entropies relative entropies or mutual information and have found many applications in information theory and beyond various generalizations of renyi entropies to the quantum setting have been proposed most prominently petz s quasi entropies and renner s conditional min max and collision entropy however these quantum extensions are incompatible and thus unsatisfactory we propose a new quantum generalization of the family of renyi entropies that contains the von neumann entropy min entropy collision entropy and the max entropy as special cases thus encompassing most quantum entropies in use today we show several natural properties for this definition including data processing inequalities a duality relation and an entropic uncertainty relation

the maximum principle for viscosity solutions of fully nonlinear second order partial differential equations we prove that viscosity solutions in w1 of the second order fully nonlinear equation f d2u du u 0 are unique when i f is degenerate elliptic and decreasing in u or ii f is uniformly elliptic and nonincreasing in u we do not assume that f is convex the method of proof involves constructing nonlinear approximation operators which map viscosity subsolutions and supersolutions onto viscosity subsolutions and supersolutions respectively this method is completely different from that used in lions 8 9 for second order problems with f convex in d 2 u and from that used by crandall lions 3 and crandall evans lions 2 for fully nonlinear first order problems

the maximum determinant of 1 matrices we give a new proof for the bound on the value of the determinant of a 1 matrix of dimension n 1 mod 4 first given by barba adapting a construction of a e brouwer we give examples to show that the bound is sharp for infinitely many values of n this in turn gives an infinite family of examples which attain the bound given by h ehlich and by m wojtas for the determinant of a 1 matrix of dimension n 2 mod 4 for n 3 mod4 we construct an infinite family of 1 examples which attain slightly more than of the bound given by ehlich elsevier science inc 1997

the asymptotics of posterior entropy and error probability for bayesian estimation we consider the bayesian parameter estimation problem where the value of a finitary parameter x should be decided on the basis of i i d sample y sup n of size n in this context the amount of missing information on x after observing y sup n may be evaluated by the posterior entropy which is often called the equivocation or the conditional entropy of x given y sup n while it is well known that the minimum possible probability of error in estimating x is achieved by the maximum a posteriori probability map estimator in this work the focus is on the asymptotic relation between the posterior entropy and the map error probability as the sample size n becomes sufficiently large it is shown that if the sample size n is large enough the posterior entropy as well as the map error probability decay with n to zero at the identical exponential rate and that the maximum achievable exponent for this decay is determined by the minimum chernoff information over all the possible pairs of distinct parameter values

some properties of renyi entropy and renyi entropy rate in this paper we define the conditional renyi entropy and show that the so called chain rule holds for the renyi entropy then we introduce a relation for the rate of renyi entropy and use it to derive the rate of the renyi entropy for an irreducible aperiodic markov chain we also show that the bound for the renyi entropy rate is simply the shannon entropy rate

largest j simplices in d cubes some relatives of the hadamard maximum determinant problem abstract this paper studies the computationally difficult problem of finding a largest j dimensional simplex in a given d dimensional cube the case in which j d is of special interest for it is equivalent to the hadamard maximum determinant problem it has been solved for infinitely many values of d but not for d 14 the subcase in which j d 3 mod 4 subsumes the famous problem on the existence of hadamard matrices the known results for the case j d are here summarized and used but the main focus is on fixed small values of j when j 1 the problem is trivial and when j 2 or j 3 it is here solved completely i e for all d beyond that the results are fragmentary but numerous and they lead to several attractive conjectures some other problems involving simplices in cubes are mentioned and the relationship of largest simplices to d optimal weighing designs is discussed

motion of level sets by mean curvature i we continue our investigation of the level set technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature the principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by k brakke we prove that almost every level set of the solution to the mean curvature evolution pde is in fact aunit density varifold moving according to its mean curvature in particular a e level set is endowed with a kind of geometric structure the proof utilizes compensated compactness methods to pass to limits in various geometric expressions

monotonicity of a relative renyi entropy we show that a recent definition of relative renyi entropy is monotone under completely positive trace preserving maps this proves a recent conjecture of muller lennert et al on quantum renyi entropies a new definition some properties j math phys 54 122203 2013 e print arxiv 1306 3142v1 see also e print arxiv 1306 3142

fuzzy extractors how to generate strong keys from biometrics and other noisy data we provide formal definitions and efficient secure techniques for turning noisy information into keys usable for any cryptographic application and in particular reliably and securely authenticating biometric data our techniques apply not just to biometric information but to any keying material that unlike traditional cryptographic keys is 1 not reproducible precisely and 2 not distributed uniformly we propose two primitives a fuzzy extractor reliably extracts nearly uniform randomness r from its input the extraction is error tolerant in the sense that r will be the same even if the input changes as long as it remains reasonably close to the original thus r can be used as a key in a cryptographic application a secure sketch produces public information about its input w that does not reveal w and yet allows exact recovery of w given another value that is close to w thus it can be used to reliably reproduce error prone biometric inputs without incurring the security risk inherent in storing them we define the primitives to be both formally secure and versatile generalizing much prior work in addition we provide nearly optimal constructions of both primitives for various measures of closeness of input data such as hamming distance edit distance and set difference

conditional renyi entropies there is no generally accepted definition of conditional renyi entropy the unconditional renyi entropy depends on a parameter which for the case of min entropy takes the value even for this particular case there are several proposals for the definition of conditional entropy this paper describes three general definitions of conditional renyi entropy that were found or suggested in the literature their properties are studied and their values as a function of are compared the particular case of min entropy is widely used in cryptography as a security parameter this case is studied in some detail

a pseudorandom generator from any one way function pseudorandom generators are fundamental to many theoretical and applied aspects of computing we show how to construct a pseudorandom generator from any one way function since it is easy to construct a one way function from a pseudorandom generator this result shows that there is a pseudorandom generator if and only if there is a one way function

amorphous surface growth via a level set approach abstract we construct an equation for amorphous surface growth due to some deposition process within the level set approach and derive a unique viscosity solution to this equation from a source particles deposit on the initial hypersurface 0 in any direction and diffuse on the hypersurface for each time t we define thereby a hypersurface t

tunable measures for information leakage and applications to privacy utility tradeoffs we introduce a tunable measure for information leakage called maximal alpha leakage this measure quantifies the maximal gain of an adversary in inferring any potentially random function of a dataset from a release of the data the inferential capability of the adversary is in turn quantified by a class of adversarial loss functions that we introduce as alpha loss alpha in 1 infty cup infty the choice of alpha determines the specific adversarial action and ranges from refining a belief about any function of the data for alpha 1 to guessing the most likely value for alpha infty while refining the alpha text th moment of the belief for alpha in between maximal alpha leakage then quantifies the adversarial gain under alpha loss over all possible functions of the data in particular for the extremal values of alpha 1 and alpha infty maximal alpha leakage simplifies to mutual information and maximal leakage respectively for alpha in 1 infty this measure is shown to be the arimoto channel capacity of order alpha we show that maximal alpha leakage satisfies data processing inequalities and a sub additivity property thereby allowing for a weak composition result building upon these properties we use maximal alpha leakage as the privacy measure and study the problem of data publishing with privacy guarantees wherein the utility of the released data is ensured via a hard distortion constraint unlike average distortion hard distortion provides a deterministic guarantee of fidelity we show that under a hard distortion constraint for alpha 1 the optimal mechanism is independent of alpha and therefore the resulting optimal tradeoff is the same for all values of alpha 1 finally the tunability of maximal alpha leakage as a privacy measure is also illustrated for binary data with average hamming distortion as the utility measure

operational interpretation of renyi information measures via composite hypothesis testing against product and markov distributions we revisit the problem of asymmetric binary hypothesis testing against a composite alternative hypothesis we introduce a general framework to treat such problems when the alternative hypothesis adheres to certain axioms in this case we find the threshold rate the optimal error and strong converse exponents at large deviations from the threshold and the second order asymptotics at small deviations from the threshold we apply our results to find the operational interpretations of various renyi information measures in case the alternative hypothesis is comprised of bipartite product distributions we find that the optimal error and strong converse exponents are determined by the variations of renyi mutual information in case the alternative hypothesis consists of tripartite distributions satisfying the markov property we find that the optimal exponents are determined by the variations of renyi conditional mutual information in either case the relevant notion of renyi mutual information depends on the precise choice of the alternative hypothesis as such this paper also strengthens the view that different definitions of renyi mutual information conditional entropy and conditional mutual information are adequate depending on the context in which the measures are used

strong converse for the classical capacity of entanglement breaking and hadamard channels via a sandwiched renyi relative entropy a strong converse theorem for the classical capacity of a quantum channel states that the probability of correctly decoding a classical message converges exponentially fast to zero in the limit of many channel uses if the rate of communication exceeds the classical capacity of the channel along with a corresponding achievability statement for rates below the capacity such a strong converse theorem enhances our understanding of the capacity as a very sharp dividing line between achievable and unachievable rates of communication here we show that such a strong converse theorem holds for the classical capacity of all entanglement breaking channels and all hadamard channels the complementary channels of the former these results follow by bounding the success probability in terms of a sandwiched renyi relative entropy by showing that this quantity is subadditive for all entanglement breaking and hadamard channels and by relating this quantity to the holevo capacity prior results regarding strong converse theorems for particular covariant channels emerge as a special case of our results

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

on renyi entropy power inequalities this paper gives improved renyi entropy power inequalities r epis consider a sum s n sum k 1 n x k of n independent continuous random vectors taking values on mathbb r d and let alpha in 1 infty an r epi provides a lower bound on the order alpha renyi entropy power of s n that up to a multiplicative constant which may depend in general on n alpha d is equal to the sum of the order alpha renyi entropy powers of the n random vectors x k k 1 n for alpha 1 the r epi coincides with the well known entropy power inequality by shannon the first improved r epi is obtained by tightening the recent r epi by bobkov and chistyakov which relies on the sharpened young s inequality a further improvement of the r epi also relies on convex optimization and results on rank one modification of a real valued diagonal matrix