We consider the problem of storing data in a distributed manner over $T$ servers. We require the data (i) to be recoverable from the $T$ servers, and (ii) to remain private from any $T-1$ colluding servers, where privacy is quantified in terms of mutual information between the data and all the information available at the $T-1$ colluding servers. For this model, we determine (i) the fundamental trade-off between storage size and the level of desired privacy, (ii) the optimal amount of local randomness necessary at the encoder, and (iii)~an explicit low-complexity coding scheme that solely relies on XOR operations and that asymptotically (with the data size) matches the fundamental limits found.
This paper concerns with the construction of codes over $GF(2)$ which reach the max-flow for single source multicast acyclic networks with delay. The coding is always a bitwise XOR of packets with equal lengths, and is based on highly symmetrical and balanced designs. For certain setups and parameters, our approach offers additional plausible security properties: an adversary needs to eavesdrop at least max-flow links in order to decode at least one original packet.
The complexity of software implementations of MDS erasure codes mainly depends on the efficiency of the finite field operations implementation. In this paper, we propose a method to reduce the complexity of the finite field multiplication by using fast transforms between a field and a ring to perform the multiplication in a ring. We show that moving to a ring reduces the complexity of the operations. Then, we show that this construction allows the use of simple scheduling to reduce the number of operations.
Network coding is commonly used to improve the energy efficiency of network-wide broadcasting in wireless multi-hop networks. In this work, we focus on XOR-based broadcasting in mobile ad hoc networks with multiple sources. We make the observation that the common approach, which is to benefit from the synergy of XOR network coding with a CDS-based broadcast algorithm, suffers performance breakdowns. After delving into the details of this synergy, we attribute this behavior to an important mechanism of the underlying broadcast algorithm, known as the "termination criterion". To tackle the probl…
In two-way relay channels, bitwise XOR and symbol-level superposition coding are two popular network-coding based relaying schemes. However, neither of them can approach the capacity bound when the channels in the broadcast phase are asymmetric. In this paper, we present a new physical layer network coding (PLNC) scheme, called \emph{superimposed XOR}. The new scheme advances the existing schemes by specifically taking into account the channel asymmetry as well as information asymmetry in the broadcast phase. We obtain its achievable rate regions over Gaussian channels when integrated with two…
In this paper we describe a network coding scheme for the Broadcast Erasure Channel with multiple unicast stochastic flows, in the case of a single source transmitting packets to $N$ users, where per-slot feedback is fed back to the transmitter in the form of ACK/NACK messages. This scheme performs only binary (XOR) operations and involves a network of queues, along with special rules for coding and moving packets among the queues, that ensure instantaneous decodability. The system under consideration belongs to a class of networks whose stability properties have been analyzed in earlier work,…
This paper models the crowdsourced labeling/classification problem as a sparsely encoded source coding problem, where each query answer, regarded as a code bit, is the XOR of a small number of labels, as source information bits. In this paper we leverage the connections between this problem and well-studied codes with sparse representations for the channel coding problem to provide querying schemes with almost optimal number of queries, each of which involving only a constant number of labels. We also extend this scenario to the case where some workers can be unresponsive. For this case, we pr…
Network coding has been successfully used in the past for efficient broadcasting in wireless multi-hop networks. Two coding approaches are suitable for mobile networks; Random Linear Network Coding (RLNC) and XOR-based coding. In this work, we make the observation that RLNC provides increased resilience to packet losses compared to XOR-based coding. We develop an analytical model that justifies our intuition. However, the model also reveals that combining RLNC with probabilistic forwarding, which is the approach taken in the literature, may significantly impact RLNC's performance. Therefore, w…
We present an optimal decoder for physical-layer network coding (PNC) in a multipath fading channels. Previous studies on PNC have largely focused on the single path case. For PNC, multipath not only introduces inter-symbol interference (ISI), but also cross-symbol interference (Cross-SI) between signals simultaneously transmitted by multiple users. In this paper, we assume the transmitters do not have channel state information (CSI). The relay in the PNC system, however, has CSI. The relay makes use of a belief propagation (BP) algorithm to decode the multipath-distorted signals received from…
To tackle the expected enormous increase in mobile video traffic in cellular networks, an architecture involving a base station along with caching femto stations (referred to as helpers), storing popular files near users, has been proposed [1]. The primary benefit of caching is the enormous increase in downloading rate when a popular file is available at helpers near a user requesting that file. In this work, we explore a secondary benefit of caching in this architecture through the lens of index coding. We assume a system with n users and constant number of caching helpers. Only helpers store…
We extend the equivalence between network coding and index coding by Effros, El Rouayheb, and Langberg to the secure communication setting in the presence of an eavesdropper. Specifically, we show that the most general versions of secure network-coding setup by Chan and Grant and the secure index-coding setup by Dau, Skachek, and Chee, which also include the randomised encoding setting, are equivalent.
The problem of multilevel diversity coding with secure regeneration (MDC-SR) is considered, which includes the problems of multilevel diversity coding with regeneration (MDC-R) and secure regenerating code (SRC) as special cases. Two outer bounds are established, showing that separate coding of different messages using the respective SRCs can achieve the minimum-bandwidth-regeneration (MBR) point of the achievable normalized storage-capacity repair-bandwidth tradeoff regions for the general MDC-SR problem. The core of the new converse results is an exchange lemma, which can be established usin…
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 co…
In this paper, we consider the problem of secure and reliable communication with uncertain channel state information (CSI) and present a new solution named active secure coding which combines the machine learning methods with the traditional physical layer secure coding scheme. First, we build a detectable wiretap channel model by combining the hidden Markov model with the compound wiretap channel model, in which the varying of channel block CSI is a Markov process and the detected information is a stochastic emission from the current CSI. Next, we present a CSI learning scheme to learn the CS…
A code equivalence between index coding and network coding was established, which shows that any index-coding instance can be mapped to a network-coding instance, for which any index code can be translated to a network code with the same decoding-error performance, and vice versa. Also, any network-coding instance can be mapped to an index-coding instance with a similar code translation. In this paper, we extend the equivalence to secure index coding and secure network coding, where eavesdroppers are present in the networks, and any code construction needs to guarantee security constraints in …
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 origi…
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 …
In network coding, information transmission often encounters wiretapping attacks. Secure network coding is introduced to prevent information from being leaked to adversaries. For secure linear network codes (SLNCs), the required field size is a very important index, because it largely determines the computational and space complexities of a SLNC, and it is also very important for the process of secure network coding from theoretical research to practical application. In this letter, we further discuss the required field size of SLNCs, and obtain a new lower bound. This bound shows that the fie…
We study the secure decentralized Pliable Index CODing (PICOD) problem with circular side information sets at the users. The security constraint forbids every user to decode more than one message while a decentralized setting means there is no central transmitter in the system. Compared to the secure but centralized version of the problem, a converse bound from one of our previous works showed a factor of three difference in optimal code length under the constraint of linear encoding. In this paper, we first list the linearly infeasible cases, that is, problems where no linear code can simulta…
We investigate the construction of weakly-secure index codes for a sender to send messages to multiple receivers with side information in the presence of an eavesdropper. We derive a sufficient and necessary condition for the existence of index codes that are secure against an eavesdropper with access to any subset of messages of cardinality $t$, for any fixed $t$. In contrast to the benefits of using random keys in secure network coding, we prove that random keys do not promote security in three classes of index-coding instances.
It is common in the study of secure multicast network coding in the presence of an eavesdropper that has access to $z$ network links, to assume that the source node is the only node that generates random keys. In this setting, the secure multicast rate is well understood. Computing the secure multicast rate, or even the secure unicast rate, in the more general setting in which all network nodes may generate (independent) random keys is known to be as difficult as computing the (non-secure) capacity of multiple-unicast network coding instances --- a well known open problem. This work treats an …
The two-sender unicast index coding problem is the most fundamental multi-sender index coding problem. The two senders collectively cater to the demands of all the receivers, by taking advantage of the knowledge of their side-information. Each receiver demands a unique message and has some side-information. Weakly secure index coding problem is a practical version of the index coding problem in the presence of an eavesdropper. The eavesdropper can not gain any information about the messages he does not have, by listening to the senders' transmissions. We provide constructions of weakly secure …
We study secure source-coding with causal disclosure, under the Gaussian distribution. The optimality of Gaussian auxiliary random variables is shown in various scenarios. We explicitly characterize the tradeoff between the rates of communication and secret key. This tradeoff is the result of a mutual information optimization under Markov constraints. As a corollary, we deduce a general formula for Wyner's Common Information in the Gaussian setting.
In this paper, we propose a polar coding scheme for secure communication over the multiple-input, single-output, single-antenna eavesdropper (MISOSE) fading channel. We consider the case of block fading channels with known eavesdropper channel state information (CSI) and the case of fading channels with known eavesdropper channel distribution information (CDI). We use the artificial noise assisted secure precoding method to maximize the secrecy capacity in the first case, and to overcome the unawareness of the eavesdropper CSI in the second case. We show that our proposed scheme can provide bo…
We investigate two source coding problems with secrecy constraints. In the first problem we consider real--time fully secure transmission of a memoryless source. We show that although classical variable--rate coding is not an option since the lengths of the codewords leak information on the source, the key rate can be as low as the average Huffman codeword length of the source. In the second problem we consider causal source coding with a fidelity criterion and side information at the decoder and the eavesdropper. We show that when the eavesdropper has degraded side information, it is optimal …
Multilevel diversity coding is a classical coding model where multiple mutually independent information messages are encoded, such that different reliability requirements can be afforded to different messages. It is well known that {\em superposition coding}, namely separately encoding the independent messages, is optimal for symmetric multilevel diversity coding (SMDC) (Yeung-Zhang 1999). In the current paper, we consider weakly secure SMDC where security constraints are injected on each individual message, and provide a complete characterization of the conditions under which superposition co…
In this work, the problem of transmitting an i.i.d Gaussian source over an i.i.d Gaussian wiretap channel with an i.i.d Gaussian side information is considered. The intended receiver is assumed to have a certain minimum SNR and the eavesdropper is assumed to have a strictly lower SNR compared to the intended receiver. The objective is minimizing the distortion of source reconstruction at the intended receiver. In this work, it is shown that unlike the Gaussian wiretap channel without side information, Shannon's source-channel separation coding scheme is not optimum in the sense of achieving th…
This paper shows determining the secrecy capacity of a unicast network with uniform wiretap sets is at least as difficult as the k-unicast problem. In particular, we show that a general k-unicast problem can be reduced to the problem of finding the secrecy capacity of a corresponding single unicast network with uniform link capacities and one arbitrary wiretap link.
The adversarial wiretap channel (AWTC) model is a secure communication model in which adversary can directly read and write the transmitted bits in legitimate communication with fixed fractions. In this paper we propose a secure polar coding scheme to provide secure and reliable communication over the AWTC model. For the adversarial reading and writing action, we present a $ρ$ equivalent channel block and study its transformation under the channel polarization operation. We find that the generated channels are polarized in the sense that part of them are full-noise channels with probability al…
We consider secure multi-terminal source coding problems in the presence of a public helper. Two main scenarios are studied: 1) source coding with a helper where the coded side information from the helper is eavesdropped by an external eavesdropper; 2) triangular source coding with a helper where the helper is considered as a public terminal. We are interested in how the helper can support the source transmission subject to a constraint on the amount of information leaked due to its public nature. We characterize the tradeoff between transmission rate, incurred distortion, and information leak…
Symmetrical Multilevel Diversity Coding (SMDC) is a network compression problem introduced by Roche (1992) and Yeung (1995). In this setting, a simple separate coding strategy known as superposition coding was shown to be optimal in terms of achieving the minimum sum rate (Roche, Yeung, and Hau, 1997) and the entire admissible rate region (Yeung and Zhang, 1999) of the problem. This paper considers a natural generalization of SMDC to the secure communication setting with an additional eavesdropper. It is required that all sources need to be kept perfectly secret from the eavesdropper as long a…
In this paper, we investigate the secure coding issue for a wiretap channel model with fixed main channel and varying wiretap channel, by assuming that legitimate parties can obtain the wiretapping channel state information (CSI) after some time delay. For the symmetric degraded delay CSI case, we present an explicit weak security scheme by constructing secure polar codes on a one-time pad chaining structure, and prove its weak security, reliability and capability of approaching the secrecy capacity of perfect CSI case with delay CSI assumption. Further for the symmetric no-degraded delay CSI …
In this work, the problem of transmitting an i.i.d Gaussian source over an i.i.d Gaussian wiretap channel with an i.i.d Gaussian side information available at the intended receiver is considered. The intended receiver is assumed to have a certain minimum SNR and the eavesdropper is assumed to have a strictly lower SNR, compared to the intended receiver. The objective is to minimize the distortion of source reconstruction at the intended receiver. In this work, it is shown that the source-channel separation coding scheme is optimum in the sense of achieving minimum distortion. Two hybrid digita…
We consider the problem of secure communication over a network in the presence of wiretappers. We give a new cut-set bound on secrecy capacity which takes into account the contribution of both forward and backward edges crossing the cut, and the connectivity between their endpoints in the rest of the network. We show the bound is tight on a class of networks, which demonstrates that it is not possible to find a tighter bound by considering only cut set edges and their connectivity.
We consider the problem of securing a multicast network against a wiretapper that can intercept the packets on a limited number of arbitrary network edges of its choice. We assume that the network employs the network coding technique to simultaneously deliver the packets available at the source to all the receivers. We show that this problem can be looked at as a network generalization of the wiretap channel of type II introduced in a seminal paper by Ozarow and Wyner. In particular, we show that the transmitted information can be secured by using the Ozarow-Wyner approach of coset coding at t…
In this paper, we investigate the index coding problem in the presence of an eavesdropper. Messages are to be sent from one transmitter to a number of legitimate receivers who have side information about the messages, and share a set of secret keys with the transmitter. We assume perfect secrecy, meaning that the eavesdropper should not be able to retrieve any information about the message set. We study the minimum key lengths for zero-error and perfectly secure index coding problem. On one hand, this problem is a generalization of the index coding problem (and thus a difficult one). On the ot…
We investigate the combination between causal/zero-delay source coding and information-theoretic secrecy. Two source coding models with secrecy constraints are considered. We start by considering zero-delay perfectly secret lossless transmission of a memoryless source. We derive bounds on the key rate and coding rate needed for perfect zero-delay secrecy. In this setting, we consider two models which differ by the ability of the eavesdropper to parse the bit-stream passing from the encoder to the legitimate decoder into separate messages. We also consider causal source coding with a fidelity c…
In the secure network coding for multicasting, there is loss of information rate due to inclusion of random bits at the source node. We show a method to eliminate that loss of information rate by using multiple statistically independent messages to be kept secret from an eavesdropper. The proposed scheme is an adaptation of Yamamoto et al.'s secure multiplex coding to the secure network coding.
Silva et al. proposed a universal secure network coding scheme based on MRD codes, which can be applied to any underlying network code. This paper considers a stronger eavesdropping model where the eavesdroppers possess the ability to re-select the tapping links during the transmission. We give a proof for the impossibility of attaining universal security against such adversaries using Silva et al.'s code for all choices of code parameters, even with restricted number of tapped links. We also consider the cases with restricted tapping duration and derive some conditions for this code to be sec…
In the paradigm of network coding, the information-theoretic security problem is encountered in the presence of a wiretapper, who has capability of accessing an unknown channel-subset in communication networks. In order to combat this eavesdropping attack, secure network coding is introduced to prevent information from being leaked to adversaries. For any construction of secure linear network codes (SLNCs) over a wiretap network, the based field size is a very important index, because it largely determines the computational and space complexities, and further the efficiency of network transmis…
We study the problem of reducing the communication overhead from a noisy wire-tap channel or storage system where data is encoded as a matrix, when more columns (or their linear combinations) are available. We present its applications to reducing communication overheads in universal secure linear network coding and secure distributed storage with crisscross errors and erasures and in the presence of a wire-tapper. Our main contribution is a method to transform coding schemes based on linear rank-metric codes, with certain properties, to schemes with lower communication overheads. By applying t…
In this paper, inner and outer bounds on the capacity region of two-user interference channels with two confidential messages have been proposed. By adding secure multiplex coding to the error correction method in [15] which achieves the best achievable capacity region for interference channel up to now, we have shown that the improved secure capacity region compared with [2] now is the whole Han-Kobayashi region. In addition, this construction not only removes the rate loss incurred by adding dummy messages to achieve security, but also change the original weak security condition in [2] to st…
Cryptographic protocols are often implemented at upper layers of communication networks, while error-correcting codes are employed at the physical layer. In this paper, we consider utilizing readily-available physical layer functions, such as encoders and decoders, together with shared keys to provide a threshold-type security scheme. To this end, we first consider a scenario where the effect of the physical layer is omitted and all the channels between the involved parties are assumed to be noiseless. We introduce a model for threshold-secure coding, where the legitimate parties communicate u…
This paper studies a variant of the Pliable Index CODing (PICOD) problem, i.e., an index coding problem where a user can be satisfied by decoding any message that is not in its side information set, where communication is decentralized, i.e., it occurs among users rather than by the central server, and secure, i.e., each user is allowed to decode only one message outside its side information set and must not be able to collect any information about any other message that is not its decoded one. Given the difficulty of the general version of this problem, this paper focuses on the case where th…
The problem of multilevel diversity coding with secure regeneration is revisited. Under the assumption that the eavesdropper can access the repair data for all compromised storage nodes, Shao el al. provided a precise characterization of the minimum-bandwidth-regeneration (MBR) point of the achievable normalized storage-capacity repair-bandwidth tradeoff region. In this paper, it is shown that the MBR point of the achievable normalized storage-capacity repair-bandwidth tradeoff region remains the same even if we assume that the eavesdropper can access the repair data for some compromised stora…
This paper characterizes the secret message capacity of three networks where two unicast sessions share some of the communication resources. Each network consists of erasure channels with state feedback. A passive eavesdropper is assumed to wiretap any one of the links. The capacity achieving schemes as well as the outer bounds are formulated as linear programs. The proposed strategies are then numerically evaluated and shown to achieve higher rate performances (up to a double single- or sum-rate) with respect to alternative strategies, where the network resources are time-shared among the two…
In this paper we define the overflow problem of a network coding storage system in which the encoding parameter and the storage parameter are mismatched. Through analyses and experiments, we first show the impacts of the overflow problem in a network coding scheme, which not only waste storage spaces, but also degrade coding efficiency. To avoid the overflow problem, we then develop the network coding based secure storage (NCSS) scheme. Thanks to considering both security and storage requirements in encoding procedures and distributed architectures, the NCSS can improve the performance of a cl…
We determine the capacity region of the secure multiplex coding with a common message, and evaluate the mutual information and the equivocation rate of a collection of secret messages to the second receiver (eavesdropper), which were not evaluated by Yamamoto et al.
The problem of secure multiterminal source coding with side information at the eavesdropper is investigated. This scenario consists of a main encoder (referred to as Alice) that wishes to compress a single source but simultaneously satisfying the desired requirements on the distortion level at a legitimate receiver (referred to as Bob) and the equivocation rate --average uncertainty-- at an eavesdropper (referred to as Eve). It is further assumed the presence of a (public) rate-limited link between Alice and Bob. In this setting, Eve perfectly observes the information bits sent by Alice to Bob…
Given a private source of information, $X^n$ and a public correlated source, $Y^n$, we study the problem of encoding the two-dimensional source $(X^n, Y^n)$ into an index $J$ such that a remote party, knowing $J$ and some external side information $Z^n$, can losslessly recover $Y^n$ while any eavesdropper knowing $J$ and possibly a correlated side information $E^n$ can retrieve very little information about $X^n$. We give general converse results for the amount of information about $X^n$ that might be leaked in such systems and and also achievability results that are optimal in some special ca…
Information-theoretic security is considered in the paradigm of network coding in the presence of wiretappers, who can access one arbitrary edge subset up to a certain size, also referred to as the security level. Secure network coding is applied to prevent the leakage of the source information to the wiretappers. In this two-part paper, we consider the problem of secure network coding when the information rate and the security level can change over time. In the current paper (i.e., Part I of the two-part paper), we focus on the problem for a fixed security level and a flexible rate. To effici…
In network communications, information transmission often encounters wiretapping attacks. Secure network coding is introduced to prevent information from being leaked to adversaries. The investigation of performance bounds on the numbers of source symbols and random symbols are two fundamental research problems. For an important case that each wiretap-set with cardinality not larger than $r$, Cai and Yeung proposed a coding scheme, which is optimal in the senses of maximizing the number of source symbols and at the same time minimizing the number of random symbols. In this letter, we further s…
This paper investigates the problem of secure lossy source coding in the presence of an eavesdropper with arbitrary correlated side informations at the legitimate decoder (referred to as Bob) and the eavesdropper (referred to as Eve). This scenario consists of an encoder that wishes to compress a source to satisfy the desired requirements on: (i) the distortion level at Bob and (ii) the equivocation rate at Eve. It is assumed that the decoders have access to correlated sources as side information. For instance, this problem can be seen as a generalization of the well-known Wyner-Ziv problem ta…
We consider a class of Gaussian layered networks where a source communicates with a destination through $L$ intermediate relay layers with $N$ nodes in each layer in the presence of a single eavesdropper which can overhear the transmissions of the nodes in any one layer. The problem of maximum secrecy rate achievable with analog network coding for a unicast communication over such layered wireless relay networks with directed links is considered. A relay node performing analog network coding scales and forwards the signals received at its input. The key contribution of this work is a lemma tha…
In this paper, new inner and outer bounds on the achievable compression-equivocation rate region for generalized secure data compression with side information are given that do not match in general. In this setup, two senders, Alice and Charlie intend to transmit information to Bob via channels with limited capacity so that he can reliably reconstruct their observations. The eavesdropper, Eve, has access to one of the channels at each instant and is interested in the source of the same channel at the time. Bob and Eve also have their own observations which are correlated with Alice's and Charl…
