Achievable Scheme Research Articles

Optimizing Distributions for Associated Entropic Vectors via Generative Convolutional Neural Networks.

The complete characterization of the almost-entropic region yields rate regions for network coding problems. However, this characterization is difficult and open. In this paper, we propose a novel algorithm to determine whether an arbitrary vector in the entropy space is entropic or not, by parameterizing and generating probability mass functions by neural networks. Given a target vector, the algorithm minimizes the normalized distance between the target vector and the generated entropic vector by training the neural network. The algorithm reveals the entropic nature of the target vector, and obtains the underlying distribution, accordingly. The proposed algorithm was further implemented with convolutional neural networks, which naturally fit the structure of joint probability mass functions, and accelerate the algorithm with GPUs. Empirical results demonstrate improved normalized distances and convergence performances compared with prior works. We also conducted optimizations of the Ingleton score and Ingleton violation index, where a new lower bound of the Ingleton violation index was obtained. An inner bound of the almost-entropic region with four random variables was constructed with the proposed method, presenting the current best inner bound measured by the volume ratio. The potential of a computer-aided approach to construct achievable schemes for network coding problems using the proposed method is discussed.

Information-theoretic privacy in federated submodel learning

We consider information-theoretic privacy in federated submodel learning, where a global server has multiple submodels. Compared to the privacy considered in the conventional federated submodel learning where secure aggregation is adopted for ensuring privacy, information-theoretic privacy provides the stronger protection on submodel selection by the local machine. We propose an achievable scheme that partially adopts the conventional private information retrieval (PIR) scheme that achieves the minimum amount of download. With respect to computation and communication overhead, we compare the achievable scheme with a naïve approach for federated submodel learning with information-theoretic privacy.

On the Asymptotic Capacity of Information-Theoretic Privacy-Preserving Epidemiological Data Collection.

The paradigm-shifting developments of cryptography and information theory have focused on the privacy of data-sharing systems, such as epidemiological studies, where agencies are collecting far more personal data than they need, causing intrusions on patients' privacy. To study the capability of the data collection while protecting privacy from an information theory perspective, we formulate a new distributed multiparty computation problem called privacy-preserving epidemiological data collection. In our setting, a data collector requires a linear combination of K users' data through a storage system consisting of N servers. Privacy needs to be protected when the users, servers, and data collector do not trust each other. For the users, any data are required to be protected from up to E colluding servers; for the servers, any more information than the desired linear combination cannot be leaked to the data collector; and for the data collector, any single server can not know anything about the coefficients of the linear combination. Our goal is to find the optimal collection rate, which is defined as the ratio of the size of the user's message to the total size of downloads from N servers to the data collector. For achievability, we propose an asymptotic capacity-achieving scheme when E<N-1, by applying the cross-subspace alignment method to our construction; for the converse, we proved an upper bound of the asymptotic rate for all achievable schemes when E<N-1. Additionally, we show that a positive asymptotic capacity is not possible when E≥N-1. The results of the achievability and converse meet when the number of users goes to infinity, yielding the asymptotic capacity. Our work broadens current researches on data privacy in information theory and gives the best achievable asymptotic performance that any epidemiological data collector can obtain.

Secure Distributed Matrix Multiplication Under Arbitrary Collusion Pattern

We study the secure distributed matrix multiplication (SDMM) problem under arbitrary collusion pattern. In the one-sided SDMM problem, where only one matrix of the matrix multiplication needs to be kept secure, we propose an achievable scheme that attains the optimal normalized download cost. The optimal scheme distributes a different number of encoded copies to each server, and the servers that collude more with others are given fewer encoded copies. The converse result is proved using Shearer’s lemma. In the two-sided SDMM problem under arbitrary collusion pattern, where the user would want to keep both matrices of the matrix multiplication secure, we provide an achievable scheme whose key parameters, including the method with which the random matrices are appended, the number of random matrices appended, the number of encoded copies generated, the number of encoded copies distributed to each server, are given by the proposed algorithm. We also demonstrate, via numerical results, the performance of the proposed scheme in terms of normalized upload-download cost trade-off, and show that it is much better than the current known scheme devised for the homogeneous collusion pattern.

The -User DM Broadcast Channel With Two Groupcast Messages: Achievable Rate Regions and the Combination Network as a Case Study

A novel class of achievable rate regions is obtained for the general <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> -receiver discrete memoryless broadcast channel over which two groupcast messages are to be transmitted, with each message required by an arbitrary group of receivers. The associated achievability schemes are parameterized by an expansion of the message set which then determines how random coding techniques are employed. These techniques include generalized versions of up-set message-splitting, the generation of possibly multiple auxiliary codebooks for certain compositions of split messages using superposition coding with subset inclusion order, partial interference decoding at all receivers in general, joint unique decoding at receivers that desire both messages, and non-unique or indirect decoding at receivers that desire only one of the two messages. The generality of the proposed class of schemes implies new achievable rate regions for problems previously not considered as well as those that were studied before, with specific members of that class having rate regions that coincide with previously found capacity regions for special classes of broadcast channels with two private or two nested groupcast messages, wherein the group of receivers desiring one message is contained in that desiring the other. Moreover, new capacity results are established for certain partially ordered classes of broadcast channels for a class of two non-nested groupcast messages. To further show the strength of the proposed achievable rate regions we consider the so-called combination network as a test case. When specialized to the combination network, some members of the class of inner bounds are shown, via converse results, to result in the capacity region when the two messages are (a) intended for two distinct sets of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K{-}1$ </tex-math></inline-formula> receivers each and (b) nested, in which one message is intended for one or two (common) receivers and both messages are intended for all other (private) receivers. In the latter two nested messages cases, we hence recover, in a top-down manner, previous results by Bidokhti, Prabhakaran, and Diggavi, obtained therein using lower complexity network coding schemes based on rate-splitting and linear superposition coding but tailored to the combination network, while in the first case we obtain a new capacity result for a non-nested message set, which was hitherto unknown. Furthermore, we show the achievability of rate pairs in two interesting examples of combination networks, with three and four common receivers each. These examples were proposed in the previous literature to show the sub-optimality of the aforementioned rate-splitting and linear superposition coding scheme, and hence to motivate the additional consideration of a pre-encoding technique and a block-Markov linear superposition coding for the combination network, with the latter then lifted to the general broadcast channel. Our results suggest that the proposed framework here for the general broadcast channel when specialized to the combination network is strong enough to incorporate the enhancements afforded by those two latter techniques, thereby implying that perhaps block-Markov superposition coding is not necessary in the general broadcast channel. Moreover, there is a trade-off between the complexity of the coding scheme within the class of schemes we propose when applied to the combination network and that of the determination of the distribution of the auxiliary random variables and the encoding function that achieve the capacity region. This may have interesting implications for the general broadcast channel as well.

On Distributed Computing With Heterogeneous Communication Constraints

We consider a distributed computing framework where the distributed nodes have different communication capabilities, motivated by the heterogeneous networks in data centers and mobile edge computing systems. Following the structure of MapReduce, this framework consists of Map computation phase, Shuffle phase, and Reduce computation phase. The Shuffle phase allows distributed nodes to exchange intermediate values, in the presence of heterogeneous communication bottlenecks for different nodes (heterogeneous communication load constraints). For this setting, we characterize the minimum total computation load and the minimum worst-case computation load in some cases, under the heterogeneous communication load constraints. While the total computation load depends on the sum of the computation loads of all the nodes, the worst-case computation load depends on the computation load of a node with the heaviest job. We show an interesting insight that, for some cases, there is a tradeoff between the minimum total computation load and the minimum worst-case computation load, in the sense that both cannot be achieved at the same time. The achievability schemes are proposed with careful design on the file assignment and the data shuffling. Beyond the cut-set bound, a novel converse is proposed using the proof by contradiction. For the general case, we identify two extreme regimes in which both the scheme with coding and the scheme without coding are optimal, respectively.

Capacity Bounds for One-Bit MIMO Gaussian Channels With Analog Combining

The use of 1-bit analog-to-digital converters (ADCs) is seen as a promising approach to significantly reduce the power consumption and hardware cost of multiple-input multiple-output (MIMO) receivers. However, the nonlinear distortion due to 1-bit quantization fundamentally changes the optimal communication strategy and also imposes a capacity penalty to the system. In this paper, the capacity of a Gaussian MIMO channel in which the antenna outputs are processed by an analog linear combiner and then quantized by a set of zero threshold ADCs is studied. A new capacity upper bound for the zero threshold case is established that is tighter than the bounds available in the literature. In addition, we propose an achievability scheme which configures the analog combiner to create parallel Gaussian channels with phase quantization at the output. Under this class of analog combiners, an algorithm is presented that identifies the analog combiner and input distribution that maximize the achievable rate. Numerical results are provided showing that the rate of the achievability scheme is tight in the low signal-to-noise ratio (SNR) regime. Finally, a new 1-bit MIMO receiver architecture which employs analog temporal and spatial processing is proposed. The proposed receiver attains the capacity in the high SNR regime.

On the Fundamental Limits of Device-to-Device Private Caching Under Uncoded Cache Placement and User Collusion

In the coded caching problem, as originally formulated by Maddah-Ali and Niesen, a server communicates via a noiseless shared broadcast link to multiple users that have local storage capability. In order for a user to decode its demanded file from the coded multicast transmission, the demands of all the users must be globally known, which may violate the privacy of the users. To overcome this privacy problem, Wan and Caire recently proposed several schemes that attain coded multicasting gain while simultaneously guarantee information theoretic privacy of the users’ demands. In Device-to-Device (D2D) networks, the demand privacy problem is further exacerbated by the fact that each user is also a transmitter, which appears to be needing the knowledge of the files demanded by the remaining users in order to form its coded multicast transmission. This paper shows how to solve this seemingly infeasible problem. The main contribution of this paper is the development of new achievable and converse bounds for D2D coded caching that are to within a constant factor of one another when privacy of the users’ demands must be guaranteed even in the presence of colluding users (i.e., when some users share cached contents and demanded file indices). First, a D2D private caching scheme is proposed, whose key feature is the addition of virtual users in the system in order to “hide” the demands of the real users. By comparing the achievable D2D private load with an existing converse bound for the shared-link model without demand privacy constraint, the proposed scheme is shown to be order optimal, except for the very low memory size regime with more files than users. Second, in order to shed light into the open parameter regime, a new achievable scheme and a new converse bound under the constraint of uncoded cache placement (i.e., when each user stores directly a subset of the bits of the library) are developed for the case of two users, and shown to be to within a constant factor of one another for all system parameters. Finally, the two-user converse bound is extended to any number of users by a cut-set type argument. With this new converse bound, the virtual users scheme is shown to be order optimal in all parameter regimes under the constraint of uncoded cache placement and user collusion.

Sum-GDoF of Symmetric Multi-Hop Interference Channel Under Finite Precision CSIT Using Aligned-Images Sum-Set Inequalities

Aligned-Images Sum-set Inequalities are used in this work to study the Generalized Degrees of Freedom (GDoF) of the symmetric layered multi-hop interference channel under the robust assumption that the channel state information at the transmitters (CSIT) is limited to finite precision. First, the sum-GDoF value is characterized for the <inline-formula> <tex-math notation="LaTeX">$2\times 2\times 2$ </tex-math></inline-formula> setting that is comprised of 2 sources, 2 relays, and 2 destinations. It is shown that the sum-GDoF does not improve even if perfect CSIT is allowed in the first hop, as long as the CSIT in the second hop is limited to finite precision. The sum GDoF characterization is then generalized to the <inline-formula> <tex-math notation="LaTeX">$2\times 2\times \cdots \times 2$ </tex-math></inline-formula> setting that is comprised of <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula> hops. Remarkably, for large <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula>, the sum-GDoF value approaches that of the one-hop broadcast channel that is obtained by full cooperation among the two transmitters of the last hop, with finite precision CSIT. Previous studies of multi-hop interference networks either identified sophisticated GDoF optimal schemes under perfect CSIT, such as aligned interference neutralization and network diagonalization, that are powerful in theory but too fragile to be practical, or studied robust achievable schemes like classical amplify/decode/compress-and-forward without claims of information-theoretic optimality. In contrast, under finite precision CSIT, we show that the benefits of fragile schemes are lost, while a combination of classical random coding schemes that are simpler and much more robust, namely a rate-splitting between decode-and-forward and amplify-and-forward, is shown to be GDoF optimal. As such, this work represents another step towards bridging the gap between theory (optimality) and practice (robustness) with the aid of Aligned-Images Sum-set Inequalities.

On the Optimal Memory-Load Tradeoff of Coded Caching for Location-Based Content

Caching at the wireless edge nodes is a promising way to boost the spatial and spectral efficiency, for the sake of alleviating networks from content-related traffic. Coded caching originally introduced by Maddah-Ali and Niesen significantly speeds up communication efficiency by transmitting multicast messages simultaneously useful to multiple users. Most prior works on coded caching are based on the assumption that each user may request all content in the library. However, in many applications the users are interested only in a limited set of content that depends on their location. For example, assisted self-driving vehicles may access super High-Definition maps of the area through which they are travelling. Motivated by these considerations, this paper formulates the coded caching problem for location-based content with edge cache nodes. The considered problem includes a content server with access to <inline-formula> <tex-math notation="LaTeX">${\mathsf N}$ </tex-math></inline-formula> location-based files (e.g., High-Definition maps), <inline-formula> <tex-math notation="LaTeX">${\mathsf K}$ </tex-math></inline-formula> edge cache nodes located at different regions, and <inline-formula> <tex-math notation="LaTeX">${\mathsf K}$ </tex-math></inline-formula> users (i.e., vehicles) each of which is in the serving region of one cache node and can retrieve the cached content of this cache node with negligible cost. Depending on the location, each user only requests a file from a location-dependent subset of the library. The objective is to minimize the worst-case load (i.e., the worst-case number of broadcasted bits from the content server among all possible demands). For this novel coded caching problem, we propose a highly non-trivial converse bound under uncoded cache placement (i.e., each cache node directly copies some library bits in its cache), which shows that a simple achievable scheme is optimal under uncoded cache placement. In addition, this achievable scheme is also proved to be generally order optimal within a factor of 3. Finally, we extend the coded caching problem for location-based content to the multiaccess coded caching topology originally proposed by Hachem <i>et al.</i>, where each user is connected to <inline-formula> <tex-math notation="LaTeX">${\mathsf L}$ </tex-math></inline-formula> nearest cache nodes. When <inline-formula> <tex-math notation="LaTeX">${\mathsf L}\geq 2$ </tex-math></inline-formula>, we characterize the exact optimality on the worst-case load.

Oblivious Transfer Over Compound Binary Erasure Channels

Oblivious Transfer (OT) is a broadly studied cryptographic primitive which involves two mutually distrustful users who wish to interact with each other in order to transfer messages in an oblivious manner. Noise is an important resource to realize information-theoretically secure cryptographic protocols including oblivious transfer. However, oftentimes these noisy channels may be poorly characterized and the complete channel characteristics might be unknown to both users. Such channels are studied in the literature through the lens of compound channels. We study 1-out-of-2 oblivious transfer over a compound binary erasure channel (compound-BEC) and characterize the oblivious transfer capacity of compound-BEC. Our OT capacity characterization involves a general alphabet compound-DMC converse, subsequently specialized to compound-BEC along with an optimal and computationally-efficient achievability scheme for honest-but-curious users.

Robust, Private and Secure Cache-Aided Scalar Linear Function Retrieval From Coded Servers

This work investigates a system where each user aims to retrieve a scalar linear function of the files of a library, which are Maximum Distance Separable coded and stored at multiple distributed servers. The system needs to guarantee <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">robust decoding</i> in the sense that each user must decode its demanded function with signals received from any subset of servers whose cardinality exceeds a threshold. In addition, (a) the content of the library must be kept secure from a wiretapper who obtains all the signals from the servers; (b) any subset of users together can not obtain any information about the demands of the remaining users; and (c) the users’ demands must be kept private against all the servers even if they collude. Achievable schemes are derived by modifying existing Placement Delivery Array (PDA) constructions, originally proposed for single-server single-file retrieval coded caching systems without any privacy or security or robustness constraints. It is shown that the PDAs describing the original Maddah-Ali and Niesen’s coded caching scheme result in a load-memory tradeoff that is optimal to within a constant multiplicative gap, except for the small memory regime when the number of file is smaller than the number of users. As by-products, improved order optimality results are derived for three less restrictive systems in all parameter regimes.

Distributed Linearly Separable Computation

This paper formulates a distributed computation problem, where a master asks <inline-formula> <tex-math notation="LaTeX">${\mathsf N}$ </tex-math></inline-formula> distributed workers to compute a linearly separable function. The task function can be expressed as <inline-formula> <tex-math notation="LaTeX">${\mathsf K}_{\mathrm{ c}}$ </tex-math></inline-formula> linear combinations of <inline-formula> <tex-math notation="LaTeX">${\mathsf K}$ </tex-math></inline-formula> messages, where each message is a function of one dataset. Our objective is to find the optimal tradeoff between the computation cost (number of uncoded datasets assigned to each worker) and the communication cost (number of symbols the master must download), such that from the answers of any <inline-formula> <tex-math notation="LaTeX">${\mathsf N}_{\mathrm{ r}}$ </tex-math></inline-formula> out of <inline-formula> <tex-math notation="LaTeX">${\mathsf N}$ </tex-math></inline-formula> workers the master can recover the task function with high probability, where the coefficients of the <inline-formula> <tex-math notation="LaTeX">${\mathsf K}_{\mathrm{ c}}$ </tex-math></inline-formula> linear combinations are uniformly i.i.d. over some large enough finite field. The formulated problem can be seen as a generalized version of some existing problems, such as distributed gradient coding and distributed linear transform. In this paper, we consider the specific case where the computation cost is minimum, and propose novel achievability schemes and converse bounds for the optimal communication cost. Achievability and converse bounds coincide for some system parameters; when they do not match, we prove that the achievable distributed computing scheme is optimal under the constraint of a widely used &#x2018;cyclic assignment&#x2019; scheme on the datasets. Our results also show that when <inline-formula> <tex-math notation="LaTeX">${\mathsf K}= {\mathsf N}$ </tex-math></inline-formula>, with the same communication cost as the optimal distributed gradient coding scheme proposed by Tandon <i>et al</i>. from which the master recovers one linear combination of <inline-formula> <tex-math notation="LaTeX">${\mathsf K}$ </tex-math></inline-formula> messages, our proposed scheme can let the master recover any additional <inline-formula> <tex-math notation="LaTeX">${\mathsf N}_{\mathrm{ r}}-1$ </tex-math></inline-formula> independent linear combinations of messages with high probability.

Role of Shared Key for Secure Communication Over 2-User Gaussian Z-Interference Channel

In this paper, the role of secret key with finite rate is studied to enhance the secrecy performance of the system when users are operating in interference limited scenarios. To address this problem, a 2-user Gaussian Z-interference channel with secrecy constraint at the receiver is considered. The paper proposes novel achievable schemes, where the schemes differ from each other based on how the key has been used in the encoding process. The first achievable scheme uses a combination of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">key rate splitting</i> , <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">one-time pad</i> , <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">stochastic encoding</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">superposition coding</i> . In this scheme, one part of the key is used for one-time pad and the remaining part of the key is used for stochastic encoding. The encoding is performed such that the receiver experiencing interference can decode some part of the interference without violating the secrecy constraint. As a special case of the derived result, one can obtain the secrecy rate region when the key is completely used for one-time pad or part of the stochastic encoding. The second scheme uses the shared key to encrypt the message using one-time pad and in contrast to the previous case no interference is decoded at the receiver. The paper also derives outer bound on the sum rate and secrecy rate. The main novelty of deriving outer bound lies in the selection of side information provided to the receiver and using the secrecy constraint. The derived outer bounds are found to be tight for certain channel conditions and rate of the key. The scaling behaviour of key rate is also explored for different schemes using the notion of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">secure generalized degrees of freedom</i> . The optimality of different schemes are characterized for some specific cases. The developed results show the importance of key rate splitting in enhancing the secrecy performance of the system.

Encoding Individual Source Sequences for the Wiretap Channel.

We consider the problem of encoding a deterministic source sequence (i.e., individual sequence) for the degraded wiretap channel by means of an encoder and decoder that can both be implemented as finite-state machines. Our first main result is a necessary condition for both reliable and secure transmission in terms of the given source sequence, the bandwidth expansion factor, the secrecy capacity, the number of states of the encoder and the number of states of the decoder. Equivalently, this necessary condition can be presented as a converse bound (i.e., a lower bound) on the smallest achievable bandwidth expansion factor. The bound is asymptotically achievable by Lempel–Ziv compression followed by good channel coding for the wiretap channel. Given that the lower bound is saturated, we also derive a lower bound on the minimum necessary rate of purely random bits needed for local randomness at the encoder in order to meet the security constraint. This bound too is achieved by the same achievability scheme. Finally, we extend the main results to the case where the legitimate decoder has access to a side information sequence, which is another individual sequence that may be related to the source sequence, and a noisy version of the side information sequence leaks to the wiretapper.

Coordinated Multi Point Transmission and Reception for Mixed-Delay Traffic

This paper analyzes the multiplexing gains (MG) for simultaneous transmission of delay-sensitive and delay-tolerant data over interference networks. In the considered model, only delay-tolerant data can profit from coordinated multipoint (CoMP) transmission or reception techniques, because delay-sensitive data has to be transmitted without further delay. Transmission of delay-tolerant data is also subject to a delay constraint, which is however less stringent than the one on delay-sensitive data. Different coding schemes are proposed, and the corresponding MG pairs for delay-sensitive and delay-tolerant data characterized for Wyner’s linear symmetric network and for Wyner’s two-dimensional hexagonal network with and without sectorization. Information-theoretic converses are established for these models. For Wyners linear symmetric network the bounds match whenever the cooperation rates are sufficiently large or the delay-sensitive MG is small or moderate. These results show that on Wyner’s symmetric linear network and for sufficiently large cooperation rates, the largest MG for delay-sensitive data can be achieved without penalizing the maximum sum-MG of both delay-sensitive and delay-tolerant data. Our achievable schemes show that a similar conclusion holds for Wyner’s hexagonal network only for the model with sectorization. In the model without sectorization, a penalty in sum-MG is incurred whenever one insists on a positive delay-sensitive MG.

The Capacity of Private Information Retrieval Under Arbitrary Collusion Patterns for Replicated Databases

We study the private information retrieval (PIR) problem under arbitrary collusion patterns for replicated databases. We find a general characterization of the PIR capacity, which is the same as the capacity of the original PIR problem with the number of databases N replaced by a number S <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*</sup> . S <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*</sup> is the optimal solution to a linear programming problem that is a function of the incidence matrix of the collusion pattern. Hence, the essence of any collusion pattern can be distilled into one number S <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*</sup> . In the proposed achievable scheme, databases are non-uniformly queried according to the optimal solution of a linear programming problem based on the collusion pattern. It can be seen that the databases who collude more with others are queried less. In the converse proof, Shearer's lemma is applied, in place of Han's inequality, to a linear combination of inequalities, where each inequality corresponds to one colluding set in the collusion pattern. The weights of the linear combination come from the optimal solution of another linear programming problem based on the collusion pattern. Finally, by noting the interesting fact that the two seemingly different linear programming problems, one used in the achievability proof and the other used in the converse proof, are in fact dual problems, we characterize the capacity of the PIR problem under arbitrary collusion patterns.

The Capacity Region of Distributed Multi-User Secret Sharing

In this paper, we study the problem of distributed multi-user secret sharing, including a trusted master node, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N\in \mathbb {N}$ </tex-math></inline-formula> storage nodes, and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> users, where each user has access to the contents of a subset of storage nodes. Each user has an independent secret message with certain rate, defined as the size of the message normalized by the size of a storage node. Having access to the secret messages, the trusted master node places encoded shares in the storage nodes, such that (i) each user can recover its own message from the content of the storage nodes that it has access to, (ii) each user cannot gain any information about the message of any other user. We characterize the capacity region of the distributed multi-user secret sharing, defined as the set of all achievable rate tuples, subject to the correctness and privacy constraints. In the achievable scheme, for each user, the master node forms a polynomial with the degree equal to the number of its accessible storage nodes minus one, where the value of this polynomial at certain points are stored as the encoded shares. The message of that user is embedded in some of the coefficients of the polynomial. The remaining coefficients are determined such that the content of each storage node serves as the encoded shares for all users that have access to that storage node.

Achievable Error Exponents of One-Way and Two-Way AWGN Channels

Achievable error exponents for the one-way with noisy feedback and two-way AWGN channels are derived for the transmission of a finite number of messages M under almost sure (AS) and expected block (EXP) transmit power constraints. In the one-way setting under noisy AWGN feedback, under an AS power constraint, known linear and non-linear passive schemes are modified to incorporate AS constraints in the feedback link as well. In addition, a new active feedback scheme is presented in which the receiver feeds back the most likely pair of codewords, and the transmitter re-transmits which of these two was originally sent. This active feedback scheme outperforms one of the passive feedback schemes for all channel parameters; the linear scheme outperforms the others for low feedback noise variance. Under the EXP constraint, a known achievable error exponent for the transmission of two messages is generalized to any arbitrary but finite number of messages M through the use of simplex codes and erasure decoding. In the two-way AWGN setting, each user has its own message to send in addition to (possibly) aiding in the transmission of feedback for the opposite direction. Two-way error exponent regions are defined and achievable error exponent regions are derived for the first time under both AS and EXP power constraints. For the presented achievability schemes, feedback or interaction leads to error exponent gains in one direction, possibly at the expense of a decrease in the error exponents attained in the other direction. The relationship between M and n supported by our achievable strategies is explored.

On Asymptotic Analysis of Energy-Distortion Tradeoff for Low-Delay Transmission Over Gaussian Channels

Asymptotic energy-distortion performance of zero- and low-delay transmission of Gaussian sources over energy-limited Gaussian channels is studied. A lower bound for the leading term in the negative logarithm of the distortion, termed the energy-distortion exponent, is derived through an achievable scheme based on high-resolution quantization coupled with orthogonal signaling. The higher-order term in the negative logarithm of the distortion, termed the energy-distortion dispersion, is optimized while keeping the leading term, the energy-distortion exponent, at its optimal (respectively, the best known) value for the zero-delay (respectively, low-delay) regime. In contrast with the decaying dispersion previously reported in the literature, the proposed coding scheme achieves a constant dispersion. When the scheme is optimized, this constant can be improved with respect to its naïve value, i.e., that achieved by optimizing purely the source coding performance instead of the end-to-end distortion. Lastly, a tradeoff of achievable energy-distortion exponents is derived for broadcast scenarios by extending the point-to-point scheme to include a successive refinement source coder coupled with two rounds of orthogonal signaling. A simple parametric computation algorithm is derived for the tradeoff.