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Matroid theory is fundamentally connected with index coding and network coding problems. In fact, the reliance of linear index coding and network coding rates on the characteristic of a field has been demonstrated by using the two well-known matroid instances, namely the Fano and non-Fano matroids. This established the insufficiency of linear coding, one of the fundamental theorems in both index coding and network coding. While the Fano matroid is linearly representable only over fields with characteristic two, the non-Fano instance is linearly representable only over fields with odd characteristic. For fields with arbitrary characteristic $p$, the Fano and non-Fano matroids were extended to new classes of matroid instances whose linear representations are dependent on fields with characteristic $p$. However, these matroids have not been well appreciated nor cited in the fields of network coding and index coding. In this paper, we first reintroduce these matroids in a more structured way. Then, we provide a completely independent alternative proof with the main advantage of using only matrix manipulation rather than complex concepts in number theory and matroid theory. In this new proof, it is shown that while the class $p$-Fano matroid instances are linearly representable only over fields with characteristic $p$, the class $p$-non-Fano instances are representable over fields with any characteristic other than characteristic $p$. Finally, following the properties of the class $p$-Fano and $p$-non-Fano matroid instances, we characterize two new classes of index coding instances, respectively, referred to as the class $p$-Fano and $p$-non-Fano index coding, each with a size of $p^2 + 4p + 3$., Comment: arXiv admin note: text overlap with arXiv:2201.10057

This paper explores the implications of guaranteeing privacy by imposing a lower bound on the information density between the private and the public data. We introduce an operationally meaningful privacy measure called pointwise maximal cost (PMC) and demonstrate that imposing an upper bound on PMC is equivalent to enforcing a lower bound on the information density. PMC quantifies the information leakage about a secret to adversaries who aim to minimize non-negative cost functions after observing the outcome of a privacy mechanism. When restricted to finite alphabets, PMC can equivalently be defined as the information leakage to adversaries aiming to minimize the probability of incorrectly guessing randomized functions of the secret. We study the properties of PMC and apply it to standard privacy mechanisms to demonstrate its practical relevance. Through a detailed examination, we connect PMC with other privacy measures that impose upper or lower bounds on the information density. Our results highlight that lower bounding the information density is a more stringent requirement than upper bounding it. Overall, our work significantly bridges the gaps in understanding the relationships between various privacy frameworks and provides insights for selecting a suitable framework for a given application.

Within the machine learning community, reconstruction attacks are a principal attack of concern and have been identified even in federated learning, which was designed with privacy preservation in mind. In federated learning, it has been shown that an adversary with knowledge of the machine learning architecture is able to infer the exact value of a training element given an observation of the weight updates performed during stochastic gradient descent. In response to these threats, the privacy community recommends the use of differential privacy in the stochastic gradient descent algorithm, termed DP-SGD. However, DP has not yet been formally established as an effective countermeasure against reconstruction attacks. In this paper, we formalise the reconstruction threat model using the information-theoretic framework of quantitative information flow. We show that the Bayes' capacity, related to the Sibson mutual information of order infinity, represents a tight upper bound on the leakage of the DP-SGD algorithm to an adversary interested in performing a reconstruction attack. We provide empirical results demonstrating the effectiveness of this measure for comparing mechanisms against reconstruction threats.

Information density and its exponential form, known as lift, play a central role in information privacy leakage measures. $\alpha$-lift is the power-mean of lift, which is tunable between the worst-case measure max-lift ($\alpha=\infty$) and more relaxed versions ($\alpha<\infty$). This paper investigates the optimization problem of the privacy-utility tradeoff (PUT) where $\alpha$-lift and mutual information are privacy and utility measures, respectively. Due to the nonlinear nature of $\alpha$-lift for $\alpha<\infty$, finding the optimal solution is challenging. Therefore, we propose a heuristic algorithm to estimate the optimal utility for each value of $\alpha$, inspired by the optimal solution for $\alpha=\infty$ and the convexity of $\alpha$-lift with respect to the lift, which we prove. The numerical results show the efficacy of the algorithm and indicate the effective range of $\alpha$ and privacy budget $\varepsilon$ with good PUT performance., Comment: This version has developed algorithm representations and updated simulation results

This paper introduces a novel class of PICOD($t$) problems referred to as $g$-group complete-$S$ PICOD($t$) problems. It constructs a multi-stage achievability scheme to generate pliable index codes for group complete PICOD problems when $S = \{s\}$ is a singleton set. Using the maximum acyclic induced subgraph bound, lower bounds on the broadcast rate are derived for singleton $S$, which establishes the optimality of the achievability scheme for a range of values for $t$ and for any $g$ and $s$. For all other values, it is shown that the achievability scheme is optimal among the restricted class of broadcast codes., Comment: Accepted for publication in 2024 IEEE International Symposium on Information Theory

For $\tilde{f}(t) = \exp(\frac{\alpha-1}{\alpha}t)$, this paper proposes a $\tilde{f}$-mean information gain measure. R\'{e}nyi divergence is shown to be the maximum $\tilde{f}$-mean information gain incurred at each elementary event $y$ of channel output $Y$ and Sibson mutual information is the $\tilde{f}$-mean of this $Y$-elementary information gain. Both are proposed as $\alpha$-leakage measures, indicating the most information an adversary can obtain on sensitive data. It is shown that the existing $\alpha$-leakage by Arimoto mutual information can be expressed as $\tilde{f}$-mean measures by a scaled probability. Further, Sibson mutual information is interpreted as the maximum $\tilde{f}$-mean information gain over all estimation decisions applied to channel output., Comment: authorship dispute

Wireless data communications are always facing the risk of eavesdropping and interception. Conventional protection solutions which are based on encryption may not always be practical as is the case for wireless IoT networks or may soon become ineffective against quantum computers. In this regard, Physical Layer Security (PLS) presents a promising approach to secure wireless communications through the exploitation of the physical properties of the wireless channel. Cooperative Friendly Jamming (CFJ) is among the PLS techniques that have received attention in recent years. However, finding an optimal transmit power allocation that results in the highest secrecy is a complex problem that becomes more difficult to address as the size of the wireless network increases. In this paper, we propose an optimization approach to achieve CFJ in large Wi-Fi networks by using a Reinforcement Learning Algorithm. Obtained results show that our optimization approach offers better secrecy results and becomes more effective as the network size and the density of Wi-Fi access points increase.

We examine the relationship between privacy metrics that utilize information density to measure information leakage between a private and a disclosed random variable. Firstly, we prove that bounding the information density from above or below in turn implies a lower or upper bound on the information density, respectively. Using this result, we establish new relationships between local information privacy, asymmetric local information privacy, pointwise maximal leakage and local differential privacy. We further provide applications of these relations to privacy mechanism design. Furthermore, we provide statements showing the equivalence between a lower bound on information density and risk-averse adversaries. More specifically, we prove an equivalence between a guessing framework and a cost-function framework that result in the desired lower bound on the information density.

This paper proposes an $\alpha$-leakage measure for $\alpha\in[0,\infty)$ by a cross entropy interpretation of R{\'{e}}nyi entropy. While R\'{e}nyi entropy was originally defined as an $f$-mean for $f(t) = \exp((1-\alpha)t)$, we reveal that it is also a $\tilde{f}$-mean cross entropy measure for $\tilde{f}(t) = \exp(\frac{1-\alpha}{\alpha}t)$. Minimizing this R\'{e}nyi cross-entropy gives R\'{e}nyi entropy, by which the prior and posterior uncertainty measures are defined corresponding to the adversary's knowledge gain on sensitive attribute before and after data release, respectively. The $\alpha$-leakage is proposed as the difference between $\tilde{f}$-mean prior and posterior uncertainty measures, which is exactly the Arimoto mutual information. This not only extends the existing $\alpha$-leakage from $\alpha \in [1,\infty)$ to the overall R{\'{e}}nyi order range $\alpha \in [0,\infty)$ in a well-founded way with $\alpha=0$ referring to nonstochastic leakage, but also reveals that the existing maximal leakage is a $\tilde{f}$-mean of an elementary $\alpha$-leakage for all $\alpha \in [0,\infty)$, which generalizes the existing pointwise maximal leakage., Comment: 7 pages; 1 figure

We present the notion of \emph{reasonable utility} for binary mechanisms, which applies to all utility functions in the literature. This notion induces a partial ordering on the performance of all binary differentially private (DP) mechanisms. DP mechanisms that are maximal elements of this ordering are optimal DP mechanisms for every reasonable utility. By looking at differential privacy as a randomized graph coloring, we characterize these optimal DP in terms of their behavior on a certain subset of the boundary datasets we call a boundary hitting set. In the process of establishing our results, we also introduce a useful notion that generalizes DP conditions for binary-valued queries, which we coin as suitable pairs. Suitable pairs abstract away the algebraic roles of $\varepsilon,\delta$ in the DP framework, making the derivations and understanding of our proofs simpler. Additionally, the notion of a suitable pair can potentially capture privacy conditions in frameworks other than DP and may be of independent interest.

We study a new framework for designing differentially private (DP) mechanisms via randomized graph colorings, called rainbow differential privacy. In this framework, datasets are nodes in a graph, and two neighboring datasets are connected by an edge. Each dataset in the graph has a preferential ordering for the possible outputs of the mechanism, and these orderings are called rainbows. Different rainbows partition the graph of connected datasets into different regions. We show that if a DP mechanism at the boundary of such regions is fixed and it behaves identically for all same-rainbow boundary datasets, then a unique optimal $(\epsilon,\delta)$-DP mechanism exists (as long as the boundary condition is valid) and can be expressed in closed-form. Our proof technique is based on an interesting relationship between dominance ordering and DP, which applies to any finite number of colors and for $(\epsilon,\delta)$-DP, improving upon previous results that only apply to at most three colors and for $\epsilon$-DP. We justify the homogeneous boundary condition assumption by giving an example with non-homogeneous boundary condition, for which there exists no optimal DP mechanism., Comment: arXiv admin note: text overlap with arXiv:2202.03974

We propose and study a variant of pliable index coding (PICOD) where receivers have preferences for their unknown messages and give each unknown message a preference ranking. We call this the preferential pliable index-coding (PPICOD) problem and study the Pareto trade-off between the code length and overall satisfaction metric among all receivers. We derive theoretical characteristics of the PPICOD problem in terms of interactions between achievable code length and satisfaction metric. We also conceptually characterise two methods for computation of the Pareto boundary of the set of all achievable code length-satisfaction pairs. As for a coding scheme, we extend the Greedy Cover Algorithm for PICOD by Brahma and Fragouli, 2015, to balance the number of satisfied receivers and average satisfaction metric in each iteration. We present numerical results which show the efficacy of our proposed algorithm in approaching the Pareto boundary, found via brute-force computation., Comment: An extended version of the same-titled paper accepted for presentation at the 2023 IEEE International Symposium on Information Theory (ISIT)

This paper explores analytical connections between the perturbation methodology of the Australian Bureau of Statistics (ABS) and the differential privacy (DP) framework. We consider a single static counting query function and find the analytical form of the perturbation distribution with symmetric support for the ABS perturbation methodology. We then analytically measure the DP parameters, namely the $(\varepsilon, \delta)$ pair, for the ABS perturbation methodology under this setting. The results and insights obtained about the behaviour of $(\varepsilon, \delta)$ with respect to the perturbation support and variance are used to judiciously select the variance of the perturbation distribution to give a good $\delta$ in the DP framework for a given desired $\varepsilon$ and perturbation support. Finally, we propose a simple sampling scheme to implement the perturbation probability matrix in the ABS Cellkey method. The post sampling $(\varepsilon, \delta)$ pair is numerically analysed as a function of the Cellkey size. It is shown that the best results are obtained for a larger Cellkey size, because the $(\varepsilon, \delta)$ pair post-sampling measures remain almost identical when we compare sampling and theoretical results.

This paper investigates lift, the likelihood ratio between the posterior and prior belief about sensitive features in a dataset. Maximum and minimum lifts over sensitive features quantify the adversary's knowledge gain and should be bounded to protect privacy. We demonstrate that max and min lifts have a distinct range of values and probability of appearance in the dataset, referred to as \emph{lift asymmetry}. We propose asymmetric local information privacy (ALIP) as a compatible privacy notion with lift asymmetry, where different bounds can be applied to min and max lifts. We use ALIP in the watchdog and optimal random response (ORR) mechanisms, the main methods to achieve lift-based privacy. It is shown that ALIP enhances utility in these methods compared to existing local information privacy, which ensures the same (symmetric) bounds on both max and min lifts. We propose subset merging for the watchdog mechanism to improve data utility and subset random response for the ORR to reduce complexity. We then investigate the related lift-based measures, including $\ell_1$-norm, $\chi^2$-privacy criterion, and $\alpha$-lift. We reveal that they can only restrict max-lift, resulting in significant min-lift leakage. To overcome this problem, we propose corresponding lift-inverse measures to restrict the min-lift. We apply these lift-based and lift-inverse measures in the watchdog mechanism. We show that they can be considered as relaxations of ALIP, where a higher utility can be achieved by bounding only average max and min lifts.

We analytically decide whether the broadcast transmission scheme or the unicast transmission scheme achieves the optimal age of information (AoI) performance of a multiuser system where a base station (BS) generates and transmits status updates to multiple user equipments (UEs). In the broadcast transmission scheme, the status update for all UEs is jointly encoded into a packet for transmission, while in the unicast transmission scheme, the status update for each UE is encoded individually and transmitted by following the round robin policy. For both transmission schemes, we examine three packet management strategies, namely the non-preemption strategy, the preemption in buffer strategy, and the preemption in serving strategy. We first derive new closed-form expressions for the average AoI achieved by two transmission schemes with three packet management strategies. Based on them, we compare the AoI performance of two transmission schemes in two systems, namely, the remote control system and the dynamic system. Aided by simulation results, we verify our analysis and investigate the impact of system parameters on the average AoI. For example, the unicast transmission scheme is more appropriate for the system with a large number UEs. Otherwise, the broadcast transmission scheme is more appropriate.

The study of leakage measures for privacy has been a subject of intensive research and is an important aspect of understanding how privacy leaks occur in computer systems. Differential privacy has been a focal point in the privacy community for some years and yet its leakage characteristics are not completely understood. In this paper we bring together two areas of research -- information theory and the g-leakage framework of quantitative information flow (QIF) -- to give an operational interpretation for the epsilon parameter of local differential privacy. We find that epsilon emerges as a capacity measure in both frameworks; via (log)-lift, a popular measure in information theory; and via max-case g-leakage, which we introduce to describe the leakage of any system to Bayesian adversaries modelled using ``worst-case'' assumptions under the QIF framework. Our characterisation resolves an important question of interpretability of epsilon and consolidates a number of disparate results covering the literature of both information theory and quantitative information flow.

The broadcasting nature of the wireless medium makes exposure to eavesdroppers a potential threat. Physical Layer Security (PLS) has been widely recognized as a promising security measure complementary to encryption. It has recently been demonstrated that PLS can be implemented using off-the-shelf equipment by spectrum-programming enhanced Software-Defined Networking (SDN), where a network controller is able to execute intelligent access point (AP) selection algorithms such that PLS can be achieved and secrecy capacity optimized. In this paper, we provide a basic system model for such implementations. We also introduce a novel secrecy capacity optimization algorithm, in which we combine intelligent AP selection with the addition of Friendly Jamming (FJ) by the not-selected AP.

This paper proposes a novel watchdog privatization scheme by generalizing local information privacy (LIP) to enhance data utility. To protect the sensitive features $S$ correlated with some useful data $X$, LIP restricts the lift, the ratio of the posterior belief to the prior on $S$ after and before accessing $X$. For each $x$, both maximum and minimum lift over sensitive features are measures of the privacy risk of publishing this symbol and should be restricted for the privacy-preserving purpose. Previous works enforce the same bound for both max-lift and min-lift. However, empirical observations show that the min-lift is usually much smaller than the max-lift. In this work, we generalize the LIP definition to consider the unequal values of max and min lift, i.e., considering different bounds for max-lift and min-lift. This new definition is applied to the watchdog privacy mechanism. We demonstrate that the utility is enhanced under a given privacy constraint on local differential privacy. At the same time, the resulting max-lift is lower and, therefore, tightly restricts other privacy leakages, e.g., mutual information, maximal leakage, and $\alpha$-leakage.

Information leakage to a guessing adversary in index coding is studied, where some messages in the system are sensitive and others are not. The non-sensitive messages can be used by the server like secret keys to mitigate leakage of the sensitive messages to the adversary. We construct a deterministic linear coding scheme, developed from the rank minimization method based on fitting matrices (Bar-Yossef et al. 2011). The linear scheme leads to a novel upper bound on the optimal information leakage rate, which is proved to be tight over all deterministic scalar linear codes. We also derive a converse result from a graph-theoretic perspective, which holds in general over all deterministic and stochastic coding schemes., Comment: Accepted by IEEE International Symposium on Information Theory (ISIT) 2022

We study the information leakage to a guessing adversary in index coding with a general message distribution. Under both vanishing-error and zero-error decoding assumptions, we develop lower and upper bounds on the optimal leakage rate, which are based on the broadcast rate of the subproblem induced by the set of messages the adversary tries to guess. When the messages are independent and uniformly distributed, the lower and upper bounds match, establishing an equivalence between the two rates., Comment: Published in Proceedings of IEEE Information Theory Workshop (ITW) 2021

We generalize a previous framework for designing utility-optimal differentially private (DP) mechanisms via graphs, where datasets are vertices in the graph and edges represent dataset neighborhood. The boundary set contains datasets where an individual's response changes the binary-valued query compared to its neighbors. Previous work was limited to the homogeneous case where the privacy parameter $\varepsilon$ across all datasets was the same and the mechanism at boundary datasets was identical. In our work, the mechanism can take different distributions at the boundary and the privacy parameter $\varepsilon$ is a function of neighboring datasets, which recovers an earlier definition of personalized DP as special case. The problem is how to extend the mechanism, which is only defined at the boundary set, to other datasets in the graph in a computationally efficient and utility optimal manner. Using the concept of strongest induced DP condition we solve this problem efficiently in polynomial time (in the size of the graph).

We extend a previous framework for designing differentially private (DP) mechanisms via randomized graph colorings that was restricted to binary functions, corresponding to colorings in a graph, to multi-valued functions. As before, datasets are nodes in the graph and any two neighboring datasets are connected by an edge. In our setting, we assume that each dataset has a preferential ordering for the possible outputs of the mechanism, each of which we refer to as a rainbow. Different rainbows partition the graph of datasets into different regions. We show that if the DP mechanism is pre-specified at the boundary of such regions and behaves identically for all same-rainbow boundary datasets, at most one optimal such mechanism can exist and the problem can be solved by means of a morphism to a line graph. We then show closed form expressions for the line graph in the case of ternary functions. Treatment of ternary queries in this paper displays enough richness to be extended to higher-dimensional query spaces with preferential query ordering, but the optimality proof does not seem to follow directly from the ternary proof., Comment: To appear in the 2022 IEEE International Symposium on Information Theory

It has been known that the insufficiency of linear coding in achieving the optimal rate of the general index coding problem is rooted in its rate's dependency on the field size. However, this dependency has been described only through the two well-known matroid instances, namely the Fano and non-Fano matroids, which, in turn, limits its scope only to the fields with characteristic two. In this paper, we extend this scope to demonstrate the reliance of linear index coding rate on fields with characteristic three. By constructing two index coding instances of size 29, we prove that for the first instance, linear coding is optimal only over the fields with characteristic three, and for the second instance, linear coding over any field with characteristic three can never be optimal. Then, a variation of the second instance is designed as the third index coding instance of size 58. For this instance, it is proved that while linear coding over any field with characteristic three cannot be optimal, there exists a nonlinear code over the fields with characteristic three, which achieves its optimal rate. Connecting the first and third index coding instances in two specific ways, called no-way and two-way connections, will lead to two new index coding instances of size 87 and 91, for which linear coding is outperformed by nonlinear codes. Another main contribution of this paper is the reduction of the key constraints on the space of the linear coding for the first and second index coding instances, each of size 29, into a matroid instance with the ground set of size 9, whose linear representability is dependent on the fields with characteristic three. The proofs and discussions provided in this paper through using these two relatively small matroid instances will shed light on the underlying reason causing the linear coding to become insufficient for the general index coding problem.

In this paper, we analyze the impact of different encoding schemes on the age of information (AoI) performance in a point-to-point system, where a source generates packets based on the status updates collected from multiple sensors and transmits the packets to a destination. In this system, we consider two encoding schemes, namely, the joint encoding scheme and the distributed encoding scheme. In the joint encoding scheme, the status updates from all the sensors are jointly encoded into a packet for transmission. In the distributed encoding scheme, the status update from each sensor is encoded individually and the sensors' packets are transmitted following the round robin policy. To ensure the freshness of packets, the zero-wait policy is adopted in both schemes, where a new packet is immediately generated once the source finishes the transmission of the current packet. We derive closed-form expressions for the average AoI achieved by these two encoding schemes and compare their performances. Simulation results show that the distributed encoding scheme is more appropriate for systems with a relatively large number of sensors, compared with the joint encoding scheme.

The Gaussian distribution is widely used in mechanism design for differential privacy (DP). Thanks to its sub-Gaussian tail, it significantly reduces the chance of outliers when responding to queries. However, it can only provide approximate $(\epsilon, \delta(\epsilon))$-DP. In practice, $\delta(\epsilon)$ must be much smaller than the size of the dataset, which may limit the use of the Gaussian mechanism for large datasets with strong privacy requirements. In this paper, we introduce and analyze a new distribution for use in DP that is based on the Gaussian distribution, but has improved privacy performance. The so-called offset-symmetric Gaussian tail (OSGT) distribution is obtained through using the normalized tails of two symmetric Gaussians around zero. Consequently, it can still have sub-Gaussian tail and lend itself to analytical derivations. We analytically derive the variance of the OSGT random variable and the $\delta(\epsilon)$ of the OSGT mechanism. We then numerically show that at the same variance, the OSGT mechanism can offer a lower $\delta(\epsilon)$ than the Gaussian mechanism. We extend the OSGT mechanism to $k$-dimensional queries and derive an easy-to-compute analytical upper bound for its zero-concentrated differential privacy (zCDP) performance. We analytically prove that at the same variance, the same global query sensitivity and for sufficiently large concentration orders $\alpha$, the OSGT mechanism performs better than the Gaussian mechanism in terms of zCDP.

This paper is concerned with enhancing data utility in the privacy watchdog method for attaining information-theoretic privacy. For a specific privacy constraint, the watchdog method filters out the high-risk data symbols through applying a uniform data regulation scheme, e.g., merging all high-risk symbols together. While this method entirely trades the symbols resolution off for privacy, we show that the data utility can be greatly improved by partitioning the high-risk symbols set and individually privatizing each subset. We further propose an agglomerative merging algorithm that finds a suitable partition of high-risk symbols: it starts with a singleton high-risk symbol, which is iteratively fused with others until the resulting subsets are private.~Numerical simulations demonstrate the efficacy of this algorithm in privately achieving higher utilities in the watchdog scheme., Comment: 5 pages, 3 figures

In this work, we study the secure index coding problem where there are security constraints on both legitimate receivers and eavesdroppers. We develop two performance bounds (i.e., converse results) on the symmetric secure capacity. The first one is an extended version of the basic acyclic chain bound (Liu and Sadeghi, 2019) that takes security constraints into account. The second converse result is a novel information-theoretic lower bound on the symmetric secure capacity, which is interesting as all the existing converse results in the literature for secure index coding give upper bounds on the capacity., Comment: A shortened version submitted to IEEE Information Theory Workshop (ITW) 2021

We present a framework for designing differentially private (DP) mechanisms for binary functions via a graph representation of datasets. Datasets are nodes in the graph and any two neighboring datasets are connected by an edge. The true binary function we want to approximate assigns a value (or true color) to a dataset. Randomized DP mechanisms are then equivalent to randomized colorings of the graph. A key notion we use is that of the boundary of the graph. Any two neighboring datasets assigned a different true color belong to the boundary. Under this framework, we show that fixing the mechanism behavior at the boundary induces a unique optimal mechanism. Moreover, if the mechanism is to have a homogeneous behavior at the boundary, we present a closed expression for the optimal mechanism, which is obtained by means of a \emph{pullback} operation on the optimal mechanism of a line graph. For balanced mechanisms, not favoring one binary value over another, the optimal $(\epsilon,\delta)$-DP mechanism takes a particularly simple form, depending only on the minimum distance to the boundary, on $\epsilon$, and on $\delta$., Comment: Submitted to IEEE ISIT 2021

We study the information leakage to a guessing adversary in zero-error source coding. The source coding problem is defined by a confusion graph capturing the distinguishability between source symbols. The information leakage is measured by the ratio of the adversary's successful guessing probability after and before eavesdropping the codeword, maximized over all possible source distributions. Such measurement under the basic adversarial model where the adversary makes a single guess and allows no distortion between its estimator and the true sequence is known as the maximum min-entropy leakage or the maximal leakage in the literature. We develop a single-letter characterization of the optimal normalized leakage under the basic adversarial model, together with an optimum-achieving scalar stochastic mapping scheme. An interesting observation is that the optimal normalized leakage is equal to the optimal compression rate with fixed-length source codes, both of which can be simultaneously achieved by some deterministic coding schemes. We then extend the leakage measurement to generalized adversarial models where the adversary makes multiple guesses and allows certain level of distortion, for which we derive single-letter lower and upper bounds., Comment: A shortened version has been submitted to ISIT 2021

Insufficiency of linear coding for the network coding problem was first proved by providing an instance which is solvable only by nonlinear network coding (Dougherty et al., 2005).Based on the work of Effros, et al., 2015, this specific network coding instance can be modeled as a groupcast index coding (GIC)instance with 74 messages and 80 users (where a message can be requested by multiple users). This proves the insufficiency of linear coding for the GIC problem. Using the systematic approach proposed by Maleki et al., 2014, the aforementioned GIC instance can be cast into a unicast index coding (UIC) instance with more than 200 users, each wanting a unique message. This confirms the necessity of nonlinear coding for the UIC problem, but only for achieving the entire capacity region. Nevertheless, the question of whether nonlinear coding is required to achieve the symmetric capacity (broadcast rate) of the UIC problem remained open. In this paper, we settle this question and prove the insufficiency of linear coding, by directly building a UIC instance with only 36users for which there exists a nonlinear index code outperforming the optimal linear code in terms of the broadcast rate.

This paper proposes an $\alpha$-lift measure for data privacy and determines the optimal privatization scheme that minimizes the $\alpha$-lift in the watchdog method. To release data $X$ that is correlated with sensitive information $S$, the ratio $l(s,x) = \frac{p(s|x)}{p(s)} $ denotes the `lift' of the posterior belief on $S$ and quantifies data privacy. The $\alpha$-lift is proposed as the $L_\alpha$-norm of the lift: $\ell_{\alpha}(x) = \| (\cdot,x) \|_{\alpha} = (E[l(S,x)^\alpha])^{1/\alpha}$. This is a tunable measure: When $\alpha < \infty$, each lift is weighted by its likelihood of appearing in the dataset (w.r.t. the marginal probability $p(s)$); For $\alpha = \infty$, $\alpha$-lift reduces to the existing maximum lift. To generate the sanitized data $Y$, we adopt the privacy watchdog method using $\alpha$-lift: Obtain $\mathcal{X}_{\epsilon}$ containing all $x$'s such that $\ell_{\alpha}(x) > e^{\epsilon}$; Apply the randomization $r(y|x)$ to all $x \in \mathcal{X}_{\epsilon}$, while all other $x \in \mathcal{X} \setminus \mathcal{X}_{\epsilon}$ are published directly. For the resulting $\alpha$-lift $\ell_{\alpha}(y)$, it is shown that the Sibson mutual information $I_{\alpha}^{S}(S;Y)$ is proportional to $E[ \ell_{\alpha}(y)]$. We further define a stronger measure $\bar{I}_{\alpha}^{S}(S;Y)$ using the worst-case $\alpha$-lift: $\max_{y} \ell_{\alpha}(y)$. We prove that the optimal randomization $r^*(y|x)$ that minimizes both $I_{\alpha}^{S}(S;Y)$ and $\bar{I}_{\alpha}^{S}(S;Y)$ is $X$-invariant, i.e., $r^*(y|x) = R(y), \forall x\in \mathcal{X}_{\epsilon}$ for any probability distribution $R$ over $y \in \mathcal{X}_{\epsilon}$. Numerical experiments show that $\alpha$-lift can provide flexibility in the privacy-utility tradeoff.

In this paper, we propose a new scalar linear coding scheme for the index coding problem called update-based maximum column distance (UMCD) coding scheme. The central idea in each transmission is to code messages such that one of the receivers with the minimum size of side information is instantaneously eliminated from unsatisfied receivers. One main contribution of the paper is to prove that the other satisfied receivers can be identified after each transmission, using a polynomial-time algorithm solving the well-known maximum cardinality matching problem in graph theory. This leads to determining the total number of transmissions without knowing the coding coefficients. Once this number and what messages to transmit in each round are found, we then propose a method to determine all coding coefficients from a sufficiently large finite field. We provide concrete instances where the proposed UMCD coding scheme has a better broadcast performance compared to the most efficient existing coding schemes, including the recursive scheme (Arbabjolfaei and Kim, 2014) and the interlinked-cycle cover (ICC) scheme (Thapa et al., 2017). We prove that the proposed UMCD coding scheme performs at least as well as the MDS coding scheme in terms of broadcast rate. By characterizing two classes of index coding instances, we show that the gap between the broadcast rates of the recursive and ICC schemes and the UMCD scheme grows linearly with the number of messages. Then, we extend the UMCD coding scheme to its vector version by applying it as a basic coding block to solve the subinstances.

We study the problem of privacy preservation in data sharing, where $S$ is a sensitive variable to be protected and $X$ is a non-sensitive useful variable correlated with $S$. Variable $X$ is randomized into variable $Y$, which will be shared or released according to $p_{Y|X}(y|x)$. We measure privacy leakage by \emph{information privacy} (also known as \emph{log-lift} in the literature), which guarantees mutual information privacy and differential privacy (DP). Let $\Xepsc \subseteq \X$ contain elements n the alphabet of $X$ for which the absolute value of log-lift (abs-log-lift for short) is greater than a desired threshold $\eps$. When elements $x\in \Xepsc$ are randomized into $y\in \Y$, we derive the best upper bound on the abs-log-lift across the resultant pairs $(s,y)$. We then prove that this bound is achievable via an \emph{$X$-invariant} randomization $p(y|x) = R(y)$ for $x,y\in\Xepsc$. However, the utility measured by the mutual information $I(X;Y)$ is severely damaged in imposing a strict upper bound $\eps$ on the abs-log-lift. To remedy this and inspired by the probabilistic ($\eps$, $\delta$)-DP, we propose a relaxed ($\eps$, $\delta$)-log-lift framework. To achieve this relaxation, we introduce a greedy algorithm which exempts some elements in $\Xepsc$ from randomization, as long as their abs-log-lift is bounded by $\eps$ with probability $1-\delta$. Numerical results demonstrate efficacy of this algorithm in achieving a better privacy-utility tradeoff.

We study the relationship between randomized low influence functions and differentially private mechanisms. Our main aim is to formally determine whether differentially private mechanisms are low influence and whether low influence randomized functions can be differentially private. We show that differential privacy does not necessarily imply low influence in a formal sense. However, low influence implies approximate differential privacy. These results hold for both independent and non-independent randomized mechanisms, where an important instance of the former is the widely-used additive noise techniques in the differential privacy literature. Our study also reveals the interesting dynamics between utility, low influence, and independence of a differentially private mechanism. As the name of this paper suggests, we show that any two such features are simultaneously possible. However, in order to have a differentially private mechanism that has both utility and low influence, even under a very mild utility condition, one has to employ non-independent mechanisms.

In this paper, we consider the problem of responding to a count query (or any other integer-valued queries) evaluated on a dataset containing sensitive attributes. To protect the privacy of individuals in the dataset, a standard practice is to add continuous noise to the true count. We design a differentially-private mechanism which adds integer-valued noise allowing the released output to remain integer. As a trade-off between utility and privacy, we derive privacy parameters $\eps$ and $\delta$ in terms of the the probability of releasing an erroneous count under the assumption that the true count is no smaller than half the support size of the noise. We then numerically demonstrate that our mechanism provides higher privacy guarantee compared to the discrete Gaussian mechanism that is recently proposed in the literature.

The groupcast index coding (GIC) problem is a generalization of the index coding problem, where one packet can be demanded by multiple users. In this paper, we propose a new coding scheme called independent user partition multicast (IUPM) for the GIC problem. The novelty of this scheme compared to the user partition multicast (UPM) (Shanmugam \textit{et al.}, 2015) is in removing redundancies in the UPM solution by eliminating the linearly dependent coded packets. We also prove that the UPM scheme subsumes the packet partition multicast (PPM) scheme (Tehrani \textit{et al.}, 2012). Hence, the IUPM scheme is a generalization of both PPM and UPM schemes. Furthermore, inspired by jointly considering users and packets, we modify the approximation partition multicast (CAPM) scheme (Unal and Wagner, 2016) to achieve a new polynomial-time algorithm for solving the general GIC problem. We characterize a class of GIC problems with $\frac{k(k-1)}{2}$ packets, for any integer $k\geq 2$, for which the IUPM scheme is optimal. We also prove that for this class, the broadcast rate of the proposed new heuristic algorithm is $k$, while the broadcast rate of the CAPM scheme is $\mathcal{O}(k^2)$.

Index coding is concerned with efficient broadcast of a set of messages to receivers in the presence of receiver side information. In this paper, we study the secure index coding problem with security constraints on the receivers themselves. That is, for each receiver there is a single legitimate message it needs to decode and a prohibited message list, none of which should be decoded by that receiver. To this end, our contributions are threefold. We first introduce a secure linear coding scheme, which is an extended version of the fractional local partial clique covering scheme that was originally devised for non-secure index coding. We then develop two information-theoretic bounds on the performance of any valid secure index code, namely secure polymatroidal outer bound (on the capacity region) and secure maximum acyclic induced subgraph lower bound (on the broadcast rate). The structure of these bounds leads us to further develop two necessary conditions for a given index coding problem to be securely feasible (i.e., to have nonzero rates)., Comment: 6 pages; a shorted version submitted to the International Symposium on Information Theory and Its Applications (ISITA) 2020

This paper studies the tradeoff in privacy and utility in a single-trial multi-terminal guessing (estimation) framework using a system model that is inspired by index coding. There are $n$ independent discrete sources at a data curator. There are $m$ legitimate users and one adversary, each with some side information about the sources. The data curator broadcasts a distorted function of sources to legitimate users, which is also overheard by the adversary. In terms of utility, each legitimate user wishes to perfectly reconstruct some of the unknown sources and attain a certain gain in the estimation correctness for the remaining unknown sources. In terms of privacy, the data curator wishes to minimize the maximal leakage: the worst-case guessing gain of the adversary in estimating any target function of its unknown sources after receiving the broadcast data. Given the system settings, we derive fundamental performance lower bounds on the maximal leakage to the adversary, which are inspired by the notion of confusion graph and performance bounds for the index coding problem. We also detail a greedy privacy enhancing mechanism, which is inspired by the agglomerative clustering algorithms in the information bottleneck and privacy funnel problems., Comment: 6 pages; accepted by IEEE International Symposium on Information Theory (ISIT) 2020

In Part I, we studied the communication for omniscience (CO) problem and proposed a parametric (PAR) algorithm to determine the minimum sum-rate at which a set of users indexed by a finite set $V$ attain omniscience. The omniscience in CO refers to the status that each user in $V$ recovers the observations of a multiple random source. It is called the global omniscience in this paper in contrast to the study of the successive omniscience (SO), where the local omniscience is attained subsequently in user subsets. By inputting a lower bound on the minimum sum-rate for CO, we apply the PAR algorithm to search a complimentary subset $X_* \subsetneq V$ such that if the local omniscience in $X_*$ is reached first, the global omniscience whereafter can still be attained with the minimum sum-rate. We further utilize the outputs of the PAR algorithm to outline a multi-stage SO approach that is characterized by $K \leq |V| - 1$ complimentary subsets $X_*^{(k)}, \forall k \in \{1,\dotsc,K\}$ forming a nesting sequence $X_*^{(1)} \subsetneq \dotsc \subsetneq X_*^{(K)} = V$. Starting from stage $k = 1$, the local omniscience in $X_*^{(k)}$ is attained at each stage $k$ until the final global omniscience in $X_*^{(K)} = V$. A $|X_*{(k)}|$-dimensional local omniscience achievable rate vector is also derived for each stage $k$ designating individual users transmitting rates. The sum-rate of this rate vector in the last stage $K$ coincides with the minimized sum-rate for the global omniscience., Comment: 12 pages, 2 figures

Communication for omniscience (CO) refers to the problem where the users in a finite set $V$ observe a discrete multiple random source and want to exchange data over broadcast channels to reach omniscience, the state where everyone recovers the entire source. This paper studies how to improve the computational complexity for the problem of minimizing the sum-rate for attaining omniscience in $V$. While the existing algorithms rely on the submodular function minimization (SFM) techniques and complete in $O(|V|^2 \cdot \text{SFM}(|V|)$ time, we prove the strict strong map property of the nesting SFM problem. We propose a parametric (PAR) algorithm that utilizes the parametric SFM techniques and reduce the the complexity to $O(|V| \cdot \text{SFM}(|V|)$. The output of the PAR algorithm is in fact the segmented Dilworth truncation of the residual entropy for all minimum sum-rate estimates $\alpha$, which characterizes the principal sequence of partitions (PSP) and solves some related problems: It not only determines the secret capacity, a dual problem to CO, and the network strength of a graph, but also outlines the hierarchical solution to a combinatorial clustering problem. \end{abstract}, Comment: 16 pages, 3 figures

This paper studies pliable index coding, in which a sender broadcasts information to multiple receivers through a shared broadcast medium, and the receivers each have some message a priori and want any message they do not have. An approach, based on receivers that are absent from the problem, was previously proposed to find lower bounds on the optimal broadcast rate. In this paper, we introduce new techniques to obtained better lower bounds, and derive the optimal broadcast rates for new classes of the problems, including all problems with up to four absent receivers., Comment: An extended version of the same-titled paper submitted to a conference

For a group of users in $V$ where everyone observes a component of a discrete multiple random source, the process that users exchange data so as to reach omniscience, the state where everyone recovers the entire source, is called communication for omniscience (CO). We first consider how to improve the existing complexity $O(|V|^2 \cdot \text{SFM}(|V|)$ of minimizing the sum of communication rates in CO, where $\text{SFM}(|V|)$ denotes the complexity of minimizing a submodular function. We reveal some structured property in an existing coordinate saturation algorithm: the resulting rate vector and the corresponding partition of $V$ are segmented in $\alpha$, the estimation of the minimum sum-rate. A parametric (PAR) algorithm is then proposed where, instead of a particular $\alpha$, we search the critical points that fully determine the segmented variables for all $\alpha$ so that they converge to the solution to the minimum sum-rate problem and the overall complexity reduces to $O(|V| \cdot \text{SFM}(|V|))$. For the successive omniscience (SO), we consider how to attain local omniscience in some complimentary user subset so that the overall sum-rate for the global omniscience still remains minimum. While the existing algorithm only determines a complimentary user subset in $O(|V| \cdot \text{SFM}(|V|))$ time, we show that, if a lower bound on the minimum sum-rate is applied to the segmented variables in the PAR algorithm, not only a complimentary subset, but also an optimal rate vector for attaining the local omniscience in it are returned in $O(|V| \cdot \text{SFM}(|V|))$ time., Comment: 8 pages, 3 figures

This paper studies how to attain fairness in communication for omniscience, where a set of users exchange their observations of a discrete multiple random source to attain omniscience---the state that all users recover the entire source. The optimal rate region containing all source coding rate vectors that achieve the omniscience with the minimum sum rate is shown to coincide with the core (the solution set) of a coalitional game. Two game-theoretic fairness solutions are studied: the Shapley value and the egalitarian solution. It is shown that the Shapley value assigns each user the source coding rate measured by his/her remaining information of the multiple source given the common randomness that is shared by all users, while the egalitarian solution simply distributes the rates as evenly as possible in the core. To avoid the exponentially growing complexity of obtaining the Shapley value, a polynomial-time approximation method is proposed by utilizing the fact that the Shapley value is the mean value over all extreme points in the core. In addition, a steepest descent algorithm is proposed which converges in polynomial time to the fractional egalitarian solution in the core that can be implemented by network coding schemes. Finally, it is shown that the game can be decomposed into subgames so that both the Shapley value and the egalitarian solution can be obtained within each subgame in a distributed manner with reduced complexity., Comment: 12 pages, 5 figures

In this paper, we study the information-theoretic converse for the index coding problem. We generalize the definition for the alignment chain, introduced by Maleki et al., to capture more flexible relations among interfering messages at each receiver. Based on this, we derive improved converse results for the single-server index coding problem. Compared to the maximum acyclic induced subgraph (MAIS) bound, the new bounds are always as tight and can strictly outperform the MAIS bound. They can also be useful for large problems, where the generally tighter polymatroidal bound is computationally impractical. We then extend these new bounds to the multi-server index coding problem. We also present a separate, but related result where we identify a smaller single-server index coding instance, compared to those identified in the literature, for which non-Shannon-type inequalities are necessary to give a tighter converse., Comment: A shorter version has been accepted by the 2019 IEEE International Symposium on Information Theory (ISIT)

For the privacy funnel (PF) problem, we propose an efficient iterative agglomerative clustering algorithm based on the minimization of the difference of submodular functions (IAC-MDSF). For a data curator that wants to share the data $X$ correlated with the sensitive information $S$, the PF problem is to generate the sanitized data $\hat{X}$ that maintains a specified utility/fidelity threshold on $I(X; \hat{X})$ while minimizing the privacy leakage $I(S; \hat{X})$. Our IAC-MDSF algorithm starts with the original alphabet $\hat{\mathcal{X}} := \mathcal{X}$ and iteratively merges the elements in the current alphabet $\hat{\mathcal{X}}$ that minimizes the Lagrangian function $ I(S;\hat{X}) - \lambda I(X;\hat{X}) $. We prove that the best merge in each iteration of IAC-MDSF can be searched efficiently over all subsets of $\hat{\mathcal{X}}$ by the existing MDSF algorithms. We show that the IAC-MDSF algorithm also applies to the information bottleneck (IB), a dual problem to PF. By varying the value of the Lagrangian multiplier $\lambda$, we obtain the experimental results on a heart disease data set in terms of the Pareto frontier: $ I(S;\hat{X})$ vs. $- I(X;\hat{X})$. We show that our IAC-MDSF algorithm outperforms the existing iterative pairwise merge approaches for both PF and IB and is computationally much less complex., Comment: 6 pages, 4 figures

We study the index coding problem in the presence of an eavesdropper, where the aim is to communicate without allowing the eavesdropper to learn any single message aside from the messages it may already know as side information. We establish an outer bound on the underlying secure capacity region of the index coding problem, which includes polymatroidal and security constraints, as well as the set of additional decoding constraints for legitimate receivers. We then propose a secure variant of the composite coding scheme, which yields an inner bound on the secure capacity region of the index coding problem. For the achievability of secure composite coding, a secret key with vanishingly small rate may be needed to ensure that each legitimate receiver who wants the same message as the eavesdropper, knows at least two more messages than the eavesdropper. For all securely feasible index coding problems with four or fewer messages, our numerical results establish the secure index coding capacity region.

This paper proposes a novel splitting (SPLIT) algorithm to achieve fairness in the multiterminal lossless data compression problem. It finds the egalitarian solution in the Slepian-Wolf region and completes in strongly polynomial time. We show that the SPLIT algorithm adaptively updates the source coding rates to the optimal solution, while recursively splitting the terminal set, enabling parallel and distributed computation. The result of an experiment demonstrates a significant reduction in computation time by the parallel implementation when the number of terminals becomes large. The achieved egalitarian solution is also shown to be superior to the Shapley value in distributed networks, e.g., wireless sensor networks, in that it best balances the nodes' energy consumption and is far less computationally complex to obtain., Comment: 5 pages, 4 figures

We consider the problem of how to determine a fair source coding rate allocation method for the lossless data compression problem in multiterminal networks, e.g, the wireless sensor network where there are a large number of sources to be encoded. We model this problem by a game-theoretic approach and present a decomposition method for obtaining the Shapley value, a fair source coding rate vector in the Slepian-Wolf achievable region. We formulate a coalitional game model where the entropy function quantifies the cost incurred due to the source coding rates in each coalition. In the typical case for which the game is decomposable, we show that the Shapley value can be obtained separately for each subgame. The complexity of this decomposition method is determined by the maximum size of subgames, which is strictly smaller than the total number of sources and contributes to a considerable reduction in computational complexity. Experiments demonstrate large complexity reduction when the number of sources becomes large., Comment: 10 pages, 10 figures

Let a cluster (network) of sensors be connected by the communication links, each link having a capacity upper bound. Each sensor observes a discrete random variable in private and one sensor serves as a cluster header or sink. Here, we formulate the problem of how to let the sensors encode their observations such that the direction of compressed data is a feasible flow towards the sink. We demonstrate that this problem can be solved by an existing maximum independent flow (MIF) algorithm in polynomial time. Further, we reveal that this algorithm in fact determines an optimal solution by recursively pushing the remaining randomness in the sources via unsaturated communication links towards the sink. We then show that the MIF algorithm can be implemented in a distributed manner. For those networks with integral communication capacities, we propose an integral MIF algorithm which completes much faster than MIF. Finally, we point out that the nature of the data compression problem in a sensor cluster is to seek the maximum independent information flow in the intersection of two submodular polyhedra, which can be further utilized to improve the MIF algorithm in the future., Comment: 5 pages, 4 figures

We apply linear network coding (LNC) to broadcast a block of data packets from one sender to a set of receivers via lossy wireless channels, assuming each receiver already possesses a subset of these packets and wants the rest. We aim to characterize the average packet decoding delay (APDD), which reflects how soon each individual data packet can be decoded by each receiver on average, and to minimize it while achieving optimal throughput. To this end, we first derive closed-form lower bounds on the expected APDD of all LNC techniques under random packet erasures. We then prove that these bounds are NP-hard to achieve and, thus, that APDD minimization is an NP-hard problem. We then study the performance of some existing LNC techniques, including random linear network coding (RLNC) and instantly decodable network coding (IDNC). We proved that all throughput-optimal LNC techniques can approximate the minimum expected APDD with a ratio between 4/3 and 2. In particular, the ratio of RLNC is exactly 2. We then prove that all IDNC techniques are only heuristics in terms of throughput optimization and {cannot guarantee an APDD approximation ratio for at least a subset of the receivers}. Finally, we propose hyper-graphic linear network coding (HLNC), a novel throughput-optimal and APDD-approximating LNC technique based on a hypergraph model of receivers' packet reception state. We implement it under different availability of receiver feedback, and numerically compare its performance with RLNC and a heuristic general IDNC technique. The results show that the APDD performance of HLNC is better under all tested system settings, even if receiver feedback is only collected intermittently.