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Navin Kashyap

Publications and source records attributed to Navin Kashyap.

At least 55 records · Page 3Linked to original sources
arXiv2019

On the PIR capacity of MSR codes

We consider the problem of private information retrieval (PIR) from multiple storage nodes when the underlying database is encoded using regenerating codes, i.e., the database has the ability to recover from individual node failures. We seek optimality in terms of download rates when a user wants to retrieve a message privately (in an information-theoretic sense) from the database. We focus mostly on the more extensively explored Minimum Storage Regenerating (MSR) codes and formally state and prove the PIR capacity of such codes. We also give a simple scheme to achieve this capacity which is based on the similarly motivated construction for MDS codes. Furthermore, we give a PIR scheme for a specific class of Minimum Bandwidth Regenerating (MBR) codes that achieves the same rate as its minimum storage counterpart.

cs.IT↗
arXiv2019

Probabilistic Forwarding of Coded Packets on Networks

We consider a scenario of broadcasting information over a network of nodes connected by noiseless communication links. A source node in the network has $k$ data packets to broadcast, and it suffices that a large fraction of the network nodes receives the broadcast. The source encodes the $k$ data packets into $n \ge k$ coded packets using a maximum distance separable (MDS) code, and transmits them to its one-hop neighbours. Every other node in the network follows a probabilistic forwarding protocol, in which it forwards a previously unreceived packet to all its neighbours with a certain probability $p$. A "near-broadcast" is when the expected fraction of nodes that receive at least $k$ of the $n$ coded packets is close to $1$. The forwarding probability $p$ is chosen so as to minimize the expected total number of transmissions needed for a near-broadcast. In this paper, we analyze the probabilistic forwarding of coded packets on two specific network topologies: binary trees and square grids. For trees, our analysis shows that for fixed $k$, the expected total number of transmissions increases with $n$. On the other hand, on grids, we use ideas from percolation theory to show that a judicious choice of $n$ will significantly reduce the expected total number of transmissions needed for a near-broadcast.

cs.IT↗
arXiv2019

One-Shot Perfect Secret Key Agreement for Finite Linear Sources

We consider a non-asymptotic (one-shot) version of the multiterminal secret key agreement problem on a finite linear source model. In this model, the observation of each terminal is a linear function of an underlying random vector composed of finitely many i.i.d. uniform random variables. Restricting the public discussion to be a linear function of the terminals' observations, we obtain a characterization of the communication complexity (minimum number of symbols of public discussion) of generating a secret key of maximum length. The minimum discussion is achieved by a non-interactive protocol in which each terminal first does a linear processing of its own private observations, following which the terminals all execute a discussion-optimal communication-for-omniscience protocol. The secret key is finally obtained as a linear function of the vector of all observations.

cs.IT↗
arXiv2019

The Effect of Introducing Redundancy in a Probabilistic Forwarding Protocol

This paper is concerned with the problem of broadcasting information from a source node to every node in an ad-hoc network. Flooding, as a broadcast mechanism, involves each node forwarding any packet it receives to all its neighbours. This results in excessive transmissions and thus a high energy expenditure overall. Probabilistic forwarding or gossiping involves each node forwarding a received packet to all its neighbours only with a certain probability $p$. In this paper, we study the effect of introducing redundancy, in the form of coded packets, into a probabilistic forwarding protocol. Specifically, we assume that the source node has $k$ data packets to broadcast, which are encoded into $n \ge k$ coded packets, such that any $k$ of these coded packets are sufficient to recover the original $k$ data packets. Our interest is in determining the minimum forwarding probability $p$ for a "successful broadcast", which we take to be the event that the expected fraction of network nodes that receive at least $k$ of the $n$ coded packets is close to 1. We examine, via simulations and analysis of a number of different network topologies (e.g., trees, grids, random geometric graphs), how this minimum forwarding probability, and correspondingly, the expected total number of packet transmissions varies with the amount of redundancy added. Our simulation results indicate that over network topologies that are highly connected, the introduction of redundancy into the probabilistic forwarding protocol is useful, as it can significantly reduce the expected total number of transmissions needed for a successful broadcast. On the other hand, for trees, our analysis shows that the expected total number of transmissions needed increases with redundancy.

cs.IT↗
arXiv2018

Upper Bounds via Lamination on the Constrained Secrecy Capacity of Hypergraphical Sources

Hypergraphical sources are a natural class of sources for secret key generation, within which different subsets of terminals sharing secrets are allowed to discuss publicly in order to agree upon a global secret key. While their secrecy capacity, i.e., the maximum rate of a secret key that can be agreed upon by the entire set of terminals, is well-understood, what remains open is the maximum rate of a secret key that can be generated when there is a restriction on the overall rate of public discussion allowed. In this work, we obtain a family of explicitly computable upper bounds on the number of bits of secret key that can be generated per bit of public discussion. These upper bounds are derived using a lamination technique based on the submodularity of the entropy function. In particular, a specific instance of these upper bounds, called the edge-partition bound, is shown to be tight for the pairwise independent network model, a special case of the hypergraphical source. The secret key generation scheme achieving this upper bound is the tree-packing protocol of Nitinawarat et al., thereby resolving in the affirmative the discussion rate optimality of the tree packing protocol.

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arXiv2018

Multiterminal Secret Key Agreement at Asymptotically Zero Discussion Rate

In the multiterminal secret key agreement problem, a set of users want to discuss with each other until they share a common secret key independent of their discussion. We want to characterize the maximum secret key rate, called the secrecy capacity, asymptotically when the total discussion rate goes to zero. In the case of only two users, the capacity is equal to the Gács-Körner common information. However, when there are more than two users, the capacity is unknown. It is plausible that a multivariate extension of the Gács-Körner common information is the capacity, however, proving the converse is challenging. We resolved this for the hypergraphical sources and finite linear sources, and provide efficiently computable characterizations. We also give some ideas of extending the techniques to more general source models.

cs.IT↗
arXiv2018

Feedback capacity and coding for the BIBO channel with a no-repeated-ones input constraint

In this paper, a general binary-input binary-output (BIBO) channel is investigated in the presence of feedback and input constraints. The feedback capacity and the optimal input distribution of this setting are calculated for the case of an $(1,\infty)$-RLL input constraint, that is, the input sequence contains no consecutive ones. These results are obtained via explicit solution of a corresponding dynamic programming optimization problem. A simple coding scheme is designed based on the principle of posterior matching, which was introduced by Shayevitz and Feder for memoryless channels. The posterior matching scheme for our input-constrained setting is shown to achieve capacity using two new ideas: \textit{message history}, which captures the memory embedded in the setting, and \textit{message splitting}, which eases the analysis of the scheme. Additionally, in the special case of an S-channel, we give a very simple zero-error coding scheme that is shown to achieve capacity. For the input-constrained BSC, we show using our capacity formula that feedback increases capacity when the cross-over probability is small.

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arXiv2018

On the Optimality of Secret Key Agreement via Omniscience

For the multiterminal secret key agreement problem under a private source model, it is known that the maximum key rate, i.e., the secrecy capacity, can be achieved through communication for omniscience, but the omniscience strategy can be strictly suboptimal in terms of minimizing the public discussion rate. While a single-letter characterization is not known for the minimum discussion rate needed for achieving the secrecy capacity, we derive single-letter lower and upper bounds that yield some simple conditions for omniscience to be discussion-rate optimal. These conditions turn out to be enough to deduce the optimality of omniscience for a large class of sources including the hypergraphical sources. Through conjectures and examples, we explore other source models to which our methods do not easily extend.

cs.IT↗
arXiv2017

Secret Key Agreement under Discussion Rate Constraints

For the multiterminal secret key agreement problem, new single-letter lower bounds are obtained on the public discussion rate required to achieve any given secret key rate below the secrecy capacity. The results apply to general source model without helpers or wiretapper's side information but can be strengthened for hypergraphical sources. In particular, for the pairwise independent network, the results give rise to a complete characterization of the maximum secret key rate achievable under a constraint on the total discussion rate.

cs.IT↗
arXiv2016

Improving the Performance of Nested Lattice Codes Using Concatenation

A fundamental problem in coding theory is the design of an efficient coding scheme that achieves the capacity of the additive white Gaussian (AWGN) channel. The main objective of this short note is to point out that by concatenating a capacity-achieving nested lattice code with a suitable high-rate linear code over an appropriate finite field, we can achieve the capacity of the AWGN channel with polynomial encoding and decoding complexity. Specifically, we show that using inner Construction-A lattice codes and outer Reed-Solomon codes, we can obtain capacity-achieving codes whose encoding and decoding complexities grow as $O(N^2)$, while the probability of error decays exponentially in $N$, where $N$ denotes the blocklength. Replacing the outer Reed-Solomon code by an expander code helps us further reduce the decoding complexity to $O(N\log^2N)$. This also gives us a recipe for converting a high-complexity nested lattice code for a Gaussian channel to a low-complexity concatenated code without any loss in the asymptotic rate. As examples, we describe polynomial-time coding schemes for the wiretap channel, and the compute-and-forward scheme for computing integer linear combinations of messages.

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arXiv2016

Some "Goodness" Properties of LDA Lattices

We study some structural properties of Construction-A lattices obtained from low density parity check (LDPC) codes over prime fields. Such lattices are called low density Construction-A (LDA) lattices, and permit low-complexity belief propagation decoding for transmission over Gaussian channels. It has been shown that LDA lattices achieve the capacity of the power constrained additive white Gaussian noise (AWGN) channel with closest lattice-point decoding, and simulations suggested that they perform well under belief propagation decoding. We continue this line of work, and prove that these lattices are good for packing and mean squared error (MSE) quantization, and that their duals are good for packing. With this, we can conclude that codes constructed using nested LDA lattices can achieve the capacity of the power constrained AWGN channel, the capacity of the dirty paper channel, the rates guaranteed by the compute-and-forward protocol, and the best known rates for bidirectional relaying with perfect secrecy.

cs.IT↗
arXiv2016

Bounds on the Communication Rate Needed to Achieve SK Capacity in the Hypergraphical Source Model

In the multiterminal source model of Csisz$\text{á}$r and Narayan, the communication complexity, $R_{\text{SK}}$, for secret key (SK) generation is the minimum rate of communication required to achieve SK capacity. An obvious upper bound to $R_{\text{SK}}$ is given by $R_{\text{CO}}$, which is the minimum rate of communication required for \emph{omniscience}. In this paper we derive a better upper bound to $R_{\text{SK}}$ for the hypergraphical source model, which is a special instance of the multiterminal source model. The upper bound is based on the idea of fractional removal of hyperedges. It is further shown that this upper bound can be computed in polynomial time. We conjecture that our upper bound is tight. For the special case of a graphical source model, we also give an explicit lower bound on $R_{\text{SK}}$. This bound, however, is not tight, as demonstrated by a counterexample.

cs.IT↗
arXiv2016

A Lattice Coding Scheme for Secret Key Generation from Gaussian Markov Tree Sources

In this article, we study the problem of secret key generation in the multiterminal source model, where the terminals have access to correlated Gaussian sources. We assume that the sources form a Markov chain on a tree. We give a nested lattice-based key generation scheme whose computational complexity is polynomial in the number, N , of independent and identically distributed samples observed by each source. We also compute the achievable secret key rate and give a class of examples where our scheme is optimal in the fine quantization limit. However, we also give examples that show that our scheme is not always optimal in the limit of fine quantization.

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arXiv2015

On the Public Communication Needed to Achieve SK Capacity in the Multiterminal Source Model

The focus of this paper is on the public communication required for generating a maximal-rate secret key (SK) within the multiterminal source model of Csisz{á}r and Narayan. Building on the prior work of Tyagi for the two-terminal scenario, we derive a lower bound on the communication complexity, $R_{\text{SK}}$, defined to be the minimum rate of public communication needed to generate a maximal-rate SK. It is well known that the minimum rate of communication for omniscience, denoted by $R_{\text{CO}}$, is an upper bound on $R_{\text{SK}}$. For the class of pairwise independent network (PIN) models defined on uniform hypergraphs, we show that a certain "Type $\mathcal{S}$" condition, which is verifiable in polynomial time, guarantees that our lower bound on $R_{\text{SK}}$ meets the $R_{\text{CO}}$ upper bound. Thus, PIN models satisfying our condition are $R_{\text{SK}}$-maximal, meaning that the upper bound $R_{\text{SK}} \le R_{\text{CO}}$ holds with equality. This allows us to explicitly evaluate $R_{\text{SK}}$ for such PIN models. We also give several examples of PIN models that satisfy our Type $\mathcal S$ condition. Finally, we prove that for an arbitrary multiterminal source model, a stricter version of our Type $\mathcal S$ condition implies that communication from \emph{all} terminals ("omnivocality") is needed for establishing a SK of maximum rate. For three-terminal source models, the converse is also true: omnivocality is needed for generating a maximal-rate SK only if the strict Type $\mathcal S$ condition is satisfied. Counterexamples exist that show that the converse is not true in general for source models with four or more terminals.

cs.IT↗
arXiv2015

Nested Lattice Codes for Secure Bidirectional Relaying with Asymmetric Channel Gains

The basic problem of secure bidirectional relaying involves two users who want to exchange messages via an intermediate "honest-but-curious" relay node. There is no direct link between the users, all communication must take place via the relay node. The links between the user nodes and the relay are wireless links with Gaussian noise. It is required that the users' messages be kept secure from the relay. In prior work, we proposed coding schemes based on nested lattices for this problem, assuming that the channel gains from the two user nodes to the relay are identical. We also analyzed the power-rate tradeoff for secure and reliable message exchange using our coding schemes. In this paper, we extend our prior work to the case when the channel gains are not necessarily identical, and are known to the relay node but perhaps not to the users. We show that using our scheme, perfect secrecy can be obtained only for certain values of the channel gains, and analyze the power-rate tradeoff in these cases. We also make similar observations for our strongly-secure scheme.

cs.IT↗
arXiv2015

Phase Transitions for the Uniform Distribution in the PML Problem and its Bethe Approximation

The pattern maximum likelihood (PML) estimate, introduced by Orlitsky et al., is an estimate of the multiset of probabilities in an unknown probability distribution $\mathbf{p}$, the estimate being obtained from $n$ i.i.d. samples drawn from $\mathbf{p}$. The PML estimate involves solving a difficult optimization problem over the set of all probability mass functions (pmfs) of finite support. In this paper, we describe an interesting phase transition phenomenon in the PML estimate: at a certain sharp threshold, the uniform distribution goes from being a local maximum to being a local minimum for the optimization problem in the estimate. We go on to consider the question of whether a similar phase transition phenomenon also exists in the Bethe approximation of the PML estimate, the latter being an approximation method with origins in statistical physics. We show that the answer to this question is a qualified "Yes". Our analysis involves the computation of the mean and variance of the $(i,j)$th entry, $a_{i,j}$, in a random $k \times k$ non-negative integer matrix $A$ with row and column sums all equal to $M$, drawn according to a distribution that assigns to $A$ a probability proportional to $\prod_{i,j} \frac{(M-a_{i,j})!}{a_{i,j}!}$.

cs.DM↗
arXiv2015

The communication complexity of achieving SK capacity in a class of PIN models

The communication complexity of achieving secret key (SK) capacity in the multiterminal source model of Csisz$á$r and Narayan is the minimum rate of public communication required to generate a maximal-rate SK. It is well known that the minimum rate of communication for omniscience, denoted by $R_{\text{CO}}$, is an upper bound on the communication complexity, denoted by $R_{\text{SK}}$. A source model for which this upper bound is tight is called $R_{\text{SK}}$-maximal. In this paper, we establish a sufficient condition for $R_{\text{SK}}$-maximality within the class of pairwise independent network (PIN) models defined on hypergraphs. This allows us to compute $R_{\text{SK}}$ exactly within the class of PIN models satisfying this condition. On the other hand, we also provide a counterexample that shows that our condition does not in general guarantee $R_{\text{SK}}$-maximality for sources beyond PIN models.

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