This paper appeared in the Proceedings of the 2008 Allerton Conference on Communication, Control, and Computing, September 2008. Theorems 1 and 2 in this paper derived the Equivocation when using Nested Lattice Codes, leading to the "one-bit" result for the secrecy rate
Recent results have shown that lattice codes can be used to construct good channel codes, source codes and physical layer network codes for Gaussian channels. On the other hand, for Gaussian channels with secrecy constraints, efforts to date rely on random codes. In this work, we provide a tool to bridge these two areas so that the secrecy rate can be computed when lattice codes are used. In particular, we address the problem of bounding equivocation rates under nonlinear modulus operation that is present in lattice encoders/decoders. The technique is then demonstrated in two Gaussian channel examples: (1) a Gaussian wiretap channel with a cooperative jammer, and (2) a multi-hop line network from a source to a destination with untrusted intermediate relay nodes from whom the information needs to be kept secret. In both cases, lattice codes are used to facilitate cooperative jamming. In the second case, interestingly, we demonstrate that a non-vanishing positive secrecy rate is achievable regardless of the number of hops.
We consider lattice coding for the Gaussian wiretap channel, where the challenge is to ensure reliable communication between two authorized parties while preventing an eavesdropper from learning the transmitted messages. Recently, a measure called the secrecy function of a lattice coding scheme was proposed as a design criterion to characterize the eavesdropper's probability of correct decision. In this paper, the family of formally unimodular lattices is presented and shown to possess the same secrecy function behavior as unimodular and isodual lattices. Based on Construction A, we provide a …
Polar lattices, which are constructed from polar codes, are provably good for the additive white Gaussian noise (AWGN) channel. In this work, we propose a new polar lattice construction that achieves the secrecy capacity under the strong secrecy criterion over the mod-$Λ$ Gaussian wiretap channel. This construction leads to an AWGN-good lattice and a secrecy-good lattice simultaneously. The design methodology is mainly based on the equivalence in terms of polarization between the $Λ/Λ'$ channel in lattice coding and the equivalent channel derived from the chain rule of mutual information in mu…
Lattice coding for the Gaussian wiretap channel is considered, where the goal is to ensure reliable communication between two authorized parties while preventing an eavesdropper from learning the transmitted messages. Recently, a measure called secrecy gain was proposed as a design criterion to quantify the secrecy-goodness of the applied lattice code. In this paper, the theta series of the so-called formally unimodular lattices obtained by Construction $\text{A}_4$ from codes over $\mathbb{Z}_4$ is derived, and we provide a universal approach to determine their secrecy gains. Initial results …
In this work, an explicit wiretap coding scheme based on polar lattices is proposed to achieve the secrecy capacity of the additive white Gaussian noise (AWGN) wiretap channel. Firstly, polar lattices are used to construct secrecy-good lattices for the mod-$Λ_s$ Gaussian wiretap channel. Then we propose an explicit shaping scheme to remove this mod-$Λ_s$ front end and extend polar lattices to the genuine Gaussian wiretap channel. The shaping technique is based on the lattice Gaussian distribution, which leads to a binary asymmetric channel at each level for the multilevel lattice codes. By emp…
Recent results have shown that structured codes can be used to construct good channel codes, source codes and physical layer network codes for Gaussian channels. For Gaussian channels with secrecy constraints, however, efforts to date rely on random codes. In this work, we advocate that structured codes are useful for providing secrecy, and show how to compute the secrecy rate when structured codes are used. In particular, we solve the problem of bounding equivocation rates with one important class of structured codes, i.e., nested lattice codes. Having established this result, we next demonst…
It is known that given the real sum of two independent uniformly distributed lattice points from the same nested lattice codebook, the eavesdropper can obtain at most 1 bit of information per channel regarding the value of one of the lattice points. In this work, we study the effect of this 1 bit information on the equivocation expressed in three commonly used information theoretic measures, i.e., the Shannon entropy, the Renyi entropy and the min entropy. We then demonstrate its applications in an interference channel with a confidential message. In our previous work, we showed that nested la…
We propose the notion of secrecy gain as a code design criterion for wiretap lattice codes to be used over an additive white Gaussian noise channel. Our analysis relies on the error probabilites of both the legitimate user and the eavesdropper. We focus on geometrical properties of lattices, described by their theta series, to characterize good wiretap codes.
In this work, we consider achievable secrecy rates for symmetric $K$-user ($K \ge 3$) interference channels with confidential messages. We find that nested lattice codes and layered coding are useful in providing secrecy for these channels. Achievable secrecy rates are derived for very strong interference. In addition, we derive the secure degrees of freedom for a range of channel parameters. As a by-product of our approach, we also demonstrate that nested lattice codes are useful for K-user symmetric interference channels without secrecy constraints in that they yield higher degrees of freedo…
In this paper, we use the hardness of quantization over general lattices as the basis of developing a physical layer secrecy system. Assuming that the channel state observed by the legitimate receiver and the eavesdropper are distinct, this asymmetry is used to develop a cryptosystem that resembles the McEliece cryptosystem, designed to be implemented at the physical layer. We ensure that the legitimate receiver observes a specific lattice over which decoding is known to be possible in polynomial-time. while the eavesdropper observes a lattice over which decoding will prove to have the complex…
We show that for every $\ell>1$, there is a counterexample to the $\ell$-modular secrecy function conjecture by Oggier, Solé and Belfiore. These counterexamples all satisfy the modified conjecture by Ernvall-Hytönen and Sethuraman. Furthermore, we provide a method to prove or disprove the modified conjecture for any given $\ell$-modular lattice rationally equivalent to a suitable amount of copies of $\mathbb{Z}\oplus \sqrt{\ell}\,\mathbb{Z}$ with $\ell \in \{3,5,7,11,23\}$. We also provide a variant of the method for strongly $\ell$-modular lattices when $\ell\in \{6,14,15\}$.
We propose a new scheme of wiretap lattice coding that achieves semantic security and strong secrecy over the Gaussian wiretap channel. The key tool in our security proof is the flatness factor which characterizes the convergence of the conditional output distributions corresponding to different messages and leads to an upper bound on the information leakage. We not only introduce the notion of secrecy-good lattices, but also propose the {flatness factor} as a design criterion of such lattices. Both the modulo-lattice Gaussian channel and the genuine Gaussian channel are considered. In the lat…
Lattice coding over a Gaussian wiretap channel, where an eavesdropper listens to transmissions between a transmitter and a legitimate receiver, is considered. A new lattice invariant called the secrecy gain is used as a code design criterion for wiretap lattice codes since it was shown to characterize the confusion that a chosen lattice can cause at the eavesdropper: the higher the secrecy gain of the lattice, the more confusion. In this paper, a formula for the secrecy gain of unimodular lattices is derived. Secrecy gains of extremal odd unimodular lattices as well as unimodular lattices in d…
We consider the Gaussian wiretap channel, where two legitimate players Alice and Bob communicate over an additive white Gaussian noise (AWGN) channel, while Eve is eavesdropping, also through an AWGN channel. We propose a coding strategy based on lattice coset encoding. We analyze Eve's probability of decoding, from which we define the secrecy gain as a design criterion for wiretap lattice codes, expressed in terms of the lattice theta series, which characterizes Eve's confusion as a function of the channel parameters. The secrecy gain is studied for even unimodular lattices, and an asymptotic…
This paper shows that structured transmission schemes are a good choice for secret communication over interference networks with an eavesdropper. Structured transmission is shown to exploit channel asymmetries and thus perform better than randomly generated codebooks for such channels. For a class of interference channels, we show that an equivocation sumrate that is within two bits of the maximum possible legitimate communication sum-rate is achievable using lattice codes.
We show that on an $n=24m+8k$-dimensional even unimodular lattice, if the shortest vector length is $\geq 2m$, then as the number of vectors of length $2m$ decreases, the secrecy gain increases. We will also prove a similar result on general unimodular lattices. Furthermore, assuming the conjecture by Belfiore and Solé, we will calculate the difference between inverses of secrecy gains as the number of vectors varies. Finally, we will show by an example that there exist two lattices in the same dimension with the same shortest vector length and the same kissing number, but different secrecy ga…
We show that the secrecy function conjecture that states that the maximum of the secrecy function of an $l$-modular lattice occurs at $1/\sqrt{l}$ is false, by proving that the 4-modular lattice $C^(4) = \mathbb{Z} \oplus \sqrt{2}\mathbb{Z} \oplus 2\mathbb{Z}$ fails to satisfy this conjecture. We also indicate how the secrecy function must be modified in the $l$-modular case to have a more reasonable chance for it to have a maximum at $1/\sqrt{l}$, and show that the conjecture, modified with this new secrecy function, is true for various odd 2-modular lattices.
We study physical-layer security in wireless ad hoc networks and investigate two types of multi-antenna transmission schemes for providing secrecy enhancements. To establish secure transmission against malicious eavesdroppers, we consider the generation of artificial noise with either sectoring or beamforming. For both approaches, we provide a statistical characterization and tradeoff analysis of the outage performance of the legitimate communication and the eavesdropping links. We then investigate the networkwide secrecy throughput performance of both schemes in terms of the secrecy transmiss…
We consider a Gaussian two-hop network where the source and the destination can communicate only via a relay node who is both an eavesdropper and a Byzantine adversary. Both the source and the destination nodes are allowed to transmit, and the relay receives a superposition of their transmitted signals. We propose a new coding scheme that satisfies two requirements simultaneously: the transmitted message must be kept secret from the relay node, and the destination must be able to detect any Byzantine attack that the relay node might launch reliably and fast. The three main components of the sc…
Non-orthogonal multiple access (NOMA) has been recognized as a promising technique for providing high data rates in 5G systems. This letter is to study physical layer security in a single-input single-output (SISO) NOMA system consisting of a transmitter, multiple legitimate users and an eavesdropper. The aim of this letter is to maximize the secrecy sum rate (SSR) of the NOMA system subject to the users' quality of service (QoS) requirements. We firstly identify the feasible region of the transmit power for satisfying all users' QoS requirements. Then we derive the closed-form expression of a…
A new random geometric graph model, the so-called secrecy graph, is introduced and studied. The graph represents a wireless network and includes only edges over which secure communication in the presence of eavesdroppers is possible. The underlying point process models considered are lattices and Poisson point processes. In the lattice case, analogies to standard bond and site percolation can be exploited to determine percolation thresholds. In the Poisson case, the node degrees are determined and percolation is studied using analytical bounds and simulations. It turns out that a small density…
We consider compound multi-input multi-output (MIMO) wiretap channels where minimal channel state information at the transmitter (CSIT) is assumed. Code construction is given for the special case of isotropic mutual information, which serves as a conservative strategy for general cases. Using the flatness factor for MIMO channels, we propose lattice codes universally achieving the secrecy capacity of compound MIMO wiretap channels up to a constant gap (measured in nats) that is equal to the number of transmit antennas. The proposed approach improves upon existing works on secrecy coding for MI…
The heterogeneous cellular network (HCN) is a promising approach to the deployment of 5G cellular networks. This paper comprehensively studies physical layer security in a multi-tier HCN where base stations (BSs), authorized users and eavesdroppers are all randomly located. We first propose an access threshold based secrecy mobile association policy that associates each user with the BS providing the maximum \emph{truncated average received signal power} beyond a threshold. Under the proposed policy, we investigate the connection probability and secrecy probability of a randomly located user, …
In a recent paper, the authors introduced a lattice invariant called "Secrecy Gain" which measures the confusion experienced by a passive eavesdropper on the Gaussian Wiretap Channel. We study, here, the behavior of this invariant for unimodular lattices by using tools from Modular Forms and show that, for some families of unimodular lattices, indexed by the dimension, the secrecy gain exponentially goes to infinity with the dimension.
In this paper, we investigate the physical layer security over the Alternate Rician Shadowed fading channel, which is a novel channel model for body-centric wireless links and land mobile satellite. We derive exact closed-form expressions for the average secrecy capacity (ASC), secrecy outage probability (SOP), and probability of non-zero secrecy capacity (PNZ) for two cases: (i) m is a positive real number and (ii) m is a positive integer number, where m describes the level of fluctuation of the line-of-sight component. In the first case, SOP is derived in terms of the Meijer's G-function, wh…
This paper derives upper and lower bounds on the secrecy capacity-memory tradeoff of a wiretap erasure broadcast channel (BC) with Kw weak receivers and Ks strong receivers, where weak receivers, respectively strong receivers, have same erasure probabilities and cache sizes. The lower bounds are achieved by schemes that meticulously combine joint cache-channel coding with wiretap coding and key-aided one-time pads. The presented upper bound holds more generally for arbitrary degraded BCs and arbitrary cache sizes. When only weak receivers have cache memories, upper and lower bounds coincide fo…
In this paper, we study the problem of secret communication over a multiple-access channel with a common message. Here, we assume that two transmitters have confidential messages, which must be kept secret from the wiretapper (the second receiver), and both of them have access to a common message which can be decoded by the two receivers. We call this setting as Multiple-Access Wiretap Channel with Common message (MAWC-CM). For this setting, we derive general inner and outer bounds on the secrecy capacity region for the discrete memoryless case and show that these bounds meet each other for a …
We consider a wiretap multiple-input multiple-output multiple-eavesdropper (MIMOME) channel, where agent Alice aims at transmitting a secret message to agent Bob, while leaking no information on it to an eavesdropper agent Eve. We assume that Alice has more antennas than both Bob and Eve, and that she has only statistical knowledge of the channel towards Eve. We focus on the low-noise regime, and assess the secrecy rates that are achievable when the secret message determines the distribution of a multivariate Gaussian mixture model (GMM) from which a realization is generated and transmitted ov…
The secrecy rate region of wiretap interference channels with a multi-antenna passive eavesdropper is studied under receiver energy harvesting constraints. To stay operational in the network, the legitimate receivers demand energy alongside information, which is fulfilled by power transmission and exploiting a power splitting (PS) receiver. By simultaneous wireless information and power transfer (SWIPT), the amount of leakage to the eavesdropper increases, which in turn reduces the secrecy rates. For this setup, lower-bounds for secure communication rate are derived without imposing any limita…
In the first part of the paper, we consider the relation between kissing number and the secrecy gain. We show that on an $n=24m+8k$-dimensional even unimodular lattice, if the shortest vector length is $\geq 2m$, then as the number of vectors of length $2m$ decreases, the secrecy gain increases. We will also prove a similar result on general unimodular lattices. We will also consider the situations with shorter vectors. Furthermore, assuming the conjecture by Belfiore and Solé, we will calculate the difference between inverses of secrecy gains as the number of vectors varies. We will show by a…
In recent years there has been growing interest in study of multi-antenna transmit designs for providing secure communication over the physical layer. This paper considers the scenario of an intended multi-input single-output channel overheard by multiple multi-antenna eavesdroppers. Specifically, we address the transmit covariance optimization for secrecy-rate maximization (SRM) of that scenario. The challenge of this problem is that it is a nonconvex optimization problem. This paper shows that the SRM problem can actually be solved in a convex and tractable fashion, by recasting the SRM prob…
In this paper, we propose novel non-linear precoders for the downlink of a multi-user MIMO system with the existence of multiple eavesdroppers. The proposed non-linear precoders are designed to improve the physical-layer secrecy rate. Specifically, we combine the non-linear successive optimization Tomlinson-Harashima precoding (SO-THP) with generalized matrix inversion (GMI) technique to maximize the physical-layer secrecy rate. For the purpose of comparison, we examine different traditional precoders with the proposed algorithm in terms of secrecy rate as well as BER performance. We also inve…
We investigate the problem of secure communications in a Gaussian multi-way relay channel applying the compute-and-forward scheme using nested lattice codes. All nodes employ half-duplex operation and can exchange confidential messages only via an untrusted relay. The relay is assumed to be honest but curious, i.e., an eavesdropper that conforms to the system rules and applies the intended relaying scheme. We start with the general case of the single-input multiple-output (SIMO) L-user multi-way relay channel and provide an achievable secrecy rate region under a weak secrecy criterion. We show…
The many-to-one interference channel has received interest by virtue of embodying the essence of an interference network while being more tractable than the general K-user interference channel. In this paper, we introduce information theoretic secrecy to this model and consider the many-to-one interference channel with confidential messages, in which each receiver, in particular, the one subject to interference, is also one from which the interfering users' messages need to be kept secret from. We derive the achievable secrecy sum rate for this channel using nested lattice codes, as well as an…
With the completion of standardization of fifth-generation (5G) networks, the researchers have begun visioning sixth-generation (6G) networks that are predicted to be human-centric. Hence, similar to 5G networks, besides high data rate, providing secrecy and privacy will be the center of attention by the wireless research community. To support the visions beyond 5G (B5G) and 6G, in this paper we propose a secure radio frequency (RF)-free-space optical (FSO) mixed framework under the attempt of wiretapping by an eavesdropper at the RF hop. We assume the RF links undergo alpha-eta-mu fading wher…
Cooperative relaying is often deployed to enhance the communication reliability (i.e., diversity order) and consequently the end-to-end achievable rate. However, this raises several security concerns when the relays are untrusted since they may have access to the relayed message. In this paper, we study the achievable secrecy diversity order of cooperative networks with untrusted relays. In particular, we consider a network with an N-antenna transmitter (Alice), K single-antenna relays, and a single-antenna destination (Bob). We consider the general scenario where there is no relation between …
Intelligent reflecting surfaces (IRS) can improve the physical layer security (PLS) by providing a controllable wireless environment. In this paper, we propose a novel PLS technique with the help of IRS implemented by an intelligent mirror array for the visible light communication (VLC) system. First, for the IRS aided VLC system containing an access point (AP), a legitimate user and an eavesdropper, the IRS channel gain and a lower bound of the achievable secrecy rate are derived. Further, to enhance the IRS channel gain of the legitimate user while restricting the IRS channel gain of the eav…
The point of this note is to prove that the secrecy function attains its maximum at y=1 on all known extremal even unimodular lattices. This is a special case of a conjecture by Belfiore and Solé. Further, we will give a very simple method to verify or disprove the conjecture on any given unimodular lattice.
We consider a Gaussian multiple access channel with $K$ transmitters, a (intended) receiver and an external eavesdropper. The transmitters wish to reliably communicate with the receiver while concealing their messages from the eavesdropper. This scenario has been investigated in prior works using two different coding techniques; the random i.i.d. Gaussian coding and the signal alignment coding. Although, the latter offers promising results in a very high SNR regime, extending these results to the finite SNR regime is a challenging task. In this paper, we propose a new lattice alignment scheme …
We consider a fading wiretap channel model where the transmitter has only statistical channel state information, and the legitimate receiver and eavesdropper have perfect channel state information. We propose a sequence of non-random lattice codes which achieve strong secrecy and semantic security over ergodic fading channels. The construction is almost universal in the sense that it achieves the same constant gap to secrecy capacity over Gaussian and ergodic fading models.
Providing security guarantees for wireless communication is critically important for today's applications. While previous work in this area has concentrated on radio frequency (RF) channels, providing security guarantees for RF channels is inherently difficult because they are prone to rapid variations due small scale fading. Wireless optical communication, on the other hand, is inherently more secure than RF communication due to the intrinsic aspects of the signal propagation in the optical and near-optical frequency range. In this paper, secure communication over wireless optical links is ex…
In this paper, we consider an eavesdropping attack on a multi-hop, UnderWater Acoustic Sensor Network (UWASN) that consists of $M+1$ underwater sensors which report their sensed data via Orthogonal Frequency Division Multiplexing (OFDM) scheme to a sink node on the water surface. Furthermore, due to the presence of a passive malicious node in nearby vicinity, the multi-hop UnderWater Acoustic (UWA) channel between a sensor node and the sink node is prone to eavesdropping attack on each hop. Therefore, the problem at hand is to do (helper/relay) node selection (for data forwarding onto the next…
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 lat…
Despite several works on secrecy coding for fading and MIMO wiretap channels from an error probability perspective, the construction of information-theoretically secure codes over such channels remains an open problem. In this paper, we consider a fading wiretap channel model where the transmitter has only partial statistical channel state information. Our channel model includes static channels, i.i.d. block fading channels, and ergodic stationary fading with fast decay of large deviations for the eavesdropper's channel. We extend the flatness factor criterion from the Gaussian wiretap channel…
In this paper, we consider communication on a two-hop channel, in which a source wants to send information reliably and securely to the destination via a relay. We consider both the untrusted relay case and the external eavesdropper case. In the untrusted relay case, the relay behaves as an eavesdropper and there is a cooperative node which sends a jamming signal to confuse the relay when the it is receiving from the source. We propose two secure transmission schemes using the scaled compute-and-forward technique. One of the schemes is based on a random binning code and the other one is based …
We investigate the $K$-user many-to-one interference channel with confidential messages in which the $K$th user experiences interference from all other $K-1$ users, and is at the same time treated as an eavesdropper to all the messages of these users. We derive achievable rates and an upper bound on the sum rate for this channel and show that the gap between the achievable sum rate and its upper bound is $\log_2(K-1)$ bits per channel use under very strong interference, when the interfering users have equal power constraints and interfering link channel gains. The main contributions of this wo…
In next-generation wireless networks, providing secure transmission and delay guarantees are two critical goals. However, either of them requires a concession on the transmission rate. In this paper, we consider a visible light network consisting of multiple access points and multiple users. Our first objective is to mathematically evaluate the achievable rate under constraints on delay and security. The second objective is to provide a cell formation with customized statistical delay and security guarantees for each user. First, we propose a user-centric design called secure cell formation, i…
This paper studies an untrusted relay channel, in which the destination sends artificial noise simultaneously with the source sending a message to the relay, in order to protect the source's confidential message. The traditional amplify-and-forward (AF) scheme shows poor performance in this situation because of the interference power dilemma: providing better security by using stronger artificial noise will decrease the confidential message power from the relay to the destination. To solve this problem, a modulo-and-forward (MF) operation at the relay with nested lattice encoding at the source…
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 po…
Providing physical-layer security for mobile users in future broadband wireless networks is of both theoretical and practical importance. In this paper, we formulate an analytical framework for resource allocation in a downlink OFDMA-based broadband network with coexistence of secure users (SU) and normal users (NU). The SU's require secure data transmission at the physical layer while the NU's are served with conventional best-effort data traffic. The problem is formulated as joint power and subcarrier allocation with the objective of maximizing average aggregate information rate of all NU's …
Nested codes have been employed in a large number of communication applications as a specific case of superposition codes, for example to implement binning schemes in the presence of noise, in joint network-channel coding, or in physical-layer secrecy. Whereas nested lattice codes have been proposed recently for continuous-input channels, in this paper we focus on the construction of nested linear codes for joint channel-network coding problems based on algebraic protograph LDPC codes. In particular, over the past few years several constructions of codes have been proposed that are based on ra…
We study the multi-receiver wiretap channel with public and confidential messages. In this channel, there is a transmitter that wishes to communicate with two legitimate users in the presence of an external eavesdropper. The transmitter sends a pair of public and confidential messages to each legitimate user. While there are no secrecy constraints on the public messages, confidential messages need to be transmitted in perfect secrecy. We study the discrete memoryless multi-receiver wiretap channel as well as its Gaussian multi-input multi-output (MIMO) instance. First, we consider the degraded…
Under the assumption of perfect channel state information at the transmitters (CSIT), it is known that structured codes offer significant advantages for secure communication in an interference network, e.g., structured jamming signals based on lattice codes may allow a receiver to decode the sum of the jamming signal and the signal being jammed, even though they cannot be separately resolved due to secrecy constraints, subtract the aggregate jammed signal, and then proceed to decode desired codewords at lower power levels. To what extent are such benefits of structured codes fundamentally limi…
Network Oblivious Transfer via Noisy Broadcast Channels
Institute for Communications Technology, Technische Universität Braunschweig（不伦瑞克工业大学通信技术研究所）；Technische Universität München（慕尼黑工业大学）；Munich Quantum Valley (MQV)（慕尼黑量子谷）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）
