Secret Key Capacity Research Articles

Increasing secret key capacity of OFDM systems: a geometric program approach

SummaryExtracting secret keys from the common randomness of wireless channels has attracted prominent attention recently. Orthogonal frequency‐division multiplexing (OFDM) systems can provide extra randomness in view of the use of multiple subchannels. So far, the secret key capacity of OFDM systems is still an open issue. In this paper, the secret key capacity of OFDM systems based on the subchannel state information is analyzed, and an expression of the secret key capacity is derived under the assumption that the subchannels are independent. To increase the secret key capacity, a power allocation scheme based on geometric program is proposed. Furthermore, an underlying propagation protocol is designed to realize the power allocation scheme. Performance simulations show that the proposed scheme achieves greater secret key capacity in comparison with equal power allocation scheme, especially at low signal‐to‐noise ratio region. Besides, the secret key bits mismatch rate during the secret key generation based on the power allocated subchannels is decreased.

Distillation of secret-key from a class of compound memoryless quantum sources

We consider secret-key distillation from tripartite compound classical-quantum-quantum (cqq) sources with free forward public communication under strong security criterion. We design protocols which are universally reliable and secure in this scenario. These are shown to achieve asymptotically optimal rates as long as a certain regularity condition is fulfilled by the set of its generating density matrices. We derive a multi-letter formula which describes the optimal forward secret-key capacity for all compound cqq sources being regular in this sense. We also determine the forward secret-key distillation capacity for situations where the legitimate sending party has perfect knowledge of his/her marginal state deriving from the source statistics. In this case regularity conditions can be dropped. Our results show that the capacities with and without the mentioned kind of state knowledge are equal as long as the source is generated by a regular set of density matrices. We demonstrate that regularity of cqq sources is not only a technical but also an operational issue. For this reason, we give an example of a source which has zero secret-key distillation capacity without sender knowledge, while achieving positive rates is possible if sender marginal knowledge is provided.

Secret Key Agreement: General Capacity and Second-Order Asymptotics

We revisit the problem of secret key agreement using interactive public communication for two parties and propose a new secret key agreement protocol. The protocol attains the secret key capacity for general observations and attains the second-order asymptotic term in the maximum length of a secret key for independent and identically distributed observations. In contrast to the previously suggested secret key agreement protocols, the proposed protocol uses interactive communication. In fact, the standard one-way communication protocol used prior to this paper fails to attain the asymptotic results above. Our converse proofs rely on a recently established upper bound for secret key lengths. Both our lower and upper bounds are derived in a single-shot setup and the asymptotic results are obtained as corollaries.

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{\'a}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.

Robust secret key extraction from channel secondary random process

AbstractThe vast majority of existing secret key generation protocols exploit the inherent randomness of the wireless channel as a common source of randomness. However, independent noise added at the receivers of the legitimate nodes affects the reciprocity of the channel. In this paper, we propose a new simple technique to generate the secret key that mitigates the effect of noise. Specifically, we exploit the estimated channel to generate a secondary random process (SRP) that is common between the two legitimate nodes. We compare the estimated channel gain and phase to a preset threshold. The moving differences between the locations at which the estimated channel gain and phase exceed the threshold are the realization of our SRP. We study the properties of our generated SRP and derive a closed form expression for the probability mass function of the realizations of our SRP. We simulate an orthogonal frequency division multiplexing system and show that our proposed technique provides a drastic improvement in the key bit mismatch rate between the legitimate nodes when compared with the techniques that exploit the estimated channel gain or phase directly. In addition to that, the secret key generated through our technique is longer than that generated by conventional techniques. Moreover, we compute the conditional probabilities used to estimate the secret key capacity. Copyright © 2016 John Wiley & Sons, Ltd.

Secret-Key Agreement With Public Discussion Subject to an Amplitude Constraint

This paper considers the problem of secret-key agreement with public discussion subject to a peak power constraint $A$ on the channel input. The optimal input distribution is proved to be discrete with finite support. The result is obtained by first transforming the secret-key channel model into an equivalent Gaussian wiretap channel with better noise statistics at the legitimate receiver and then using the fact that the optimal distribution of the Gaussian wiretap channel is discrete. To overcome the computationally heavy search for the optimal discrete distribution, several suboptimal schemes are proposed and shown numerically to perform close to the capacity. Moreover, lower and upper bounds for the secret-key capacity are provided and used to prove that the secret-key capacity converges for asymptotic high values of $A$, to the secret-key capacity with an average power constraint $A^2$. Finally, when the amplitude constraint A is small ($A \to 0$), the secret-key capacity is proved to be asymptotically equal to the capacity of the legitimate user with an amplitude constraint A and no secrecy constraint.

On the Ergodic Secret-Key Agreement Over Spatially Correlated Multiple-Antenna Channels With Public Discussion

We consider secret-key agreement with public discussion over multiple-input multiple-output (MIMO) Rayleigh fast-fading channels under correlated environment. We assume that transmit, legitimate receiver, and eavesdropper antennas are correlated. The legitimate receiver and the eavesdropper are assumed to have perfect channel knowledge while the transmitter has only knowledge of the correlation matrices. First, we derive the expression of the secret-key capacity under the considered setup. We prove that the optimal transmit strategy achieving the secret-key capacity consists in transmitting independent Gaussian signals along the eingenvectors of the transmit correlation matrix. The powers allocated to each channel mode are determined as the solution to a numerical optimization problem. A necessary and sufficient condition for beamforming (i.e., transmitting along the strongest channel mode) to be capacity-achieving is derived. Moreover, we analyze the impact of correlation matrices on the system performance. Finally, we study the system’s performance in the two extreme power regimes. In the high-power regime, we provide closed-form expressions of the gain/loss due to correlation. In the low signal-to-noise ratio (SNR) regime, we investigate the energy efficiency of the system by determining the minimum energy required for sharing a secret-key bit and the wideband slope while highlighting the impact of correlation matrices.

Secret-Key Generation Using Compound Sources and One-Way Public Communication

In the classical secret-key generation model, common randomness is generated by two terminals based on the observation of correlated components of a common source, while keeping it secret from a non-legitimate observer. It is assumed that the statistics of the source are known to all participants. In this paper, the secret-key generation based on a compound source is studied where the realization of the source statistic is unknown. The protocol should guarantee the security and reliability of the generated secret-key, simultaneously for all possible realizations of the compound source. A single-letter lower-bound of the secret-key capacity for a finite compound source is derived as a function of the public communication rate constraint. A multi-letter capacity formula is further computed for a finite compound source for the case in which the public communication is unconstrained. Finally, a single-letter capacity formula is derived for a degraded compound source with an arbitrary (possibly infinite) set of source states and a finite set of marginal states.

Secret Key Agreement for Security in Multi-Emitter Visible Light Communication Systems

This letter examines secret key agreement for physical layer security in multiple-input single-output (MISO) visible light communications (VLC) systems. Upper and lower bounds are presented for the single-input single-output VLC secret-key capacities, followed by the analysis of an MISO VLC secret-key transmission strategy. A two-stage numerical approach based on fractional programming is presented for the construction of a secret-key MISO transmit beamforming vector. Numerical evaluations are then presented to validate the analysis and illustrate the secret-key rates achievable in VLC systems.

Polar Coding for Secret-Key Generation

Practical implementations of secret-key generation are often based on sequential strategies, which handle reliability and secrecy in two successive steps, called reconciliation and privacy amplification. In this paper, we propose an alternative approach based on polar codes that jointly deals with reliability and secrecy. Specifically, we propose secret-key capacity-achieving polar coding schemes for the following models: (i) the degraded binary memoryless source (DBMS) model with rate-unlimited public communication, (ii) the DBMS model with one-way rate-limited public communication, (iii) the 1-to-m broadcast model and (iv) the Markov tree model with uniform marginals. For models (i) and (ii) our coding schemes remain valid for non-degraded sources, although they may not achieve the secret-key capacity. For models (i), (ii) and (iii), our schemes rely on pre-shared secret seed of negligible rate; however, we provide special cases of these models for which no seed is required. Finally, we show an application of our results to secrecy and privacy for biometric systems. We thus provide the first examples of low-complexity secret-key capacity-achieving schemes that are able to handle vector quantization for model (ii), or multiterminal communication for models (iii) and (iv).

Joint optimisation of secret key capacity and sparse channel estimation based on pilot power allocation

Pilot power allocation is investigated under the framework of physical layer secure communications in time-division duplex systems, where the secret keys are generated from the estimates of sparse wireless channels. The joint optimisation of secret key capacity and sparse channel estimation performance based on pilot power allocation is formulated as a convex optimisation problem. Considering the fairness between these two sides, a scaling factor is introduced. Then, a scheme is proposed to fast solve the problem by looking up a table and using existing optimisation solvers. Simulation results show that a proper scaling factor can make a trade-off between the secret key capacity and sparse channel estimation performance.

The Sender-Excited Secret Key Agreement Model: Capacity, Reliability, and Secrecy Exponents

We consider the secret key generation problem when sources are randomly excited by the sender and there is a noiseless public discussion channel. Our setting is thus similar to recent works on channels with action-dependent states where the channel state may be influenced by some of the parties involved. We derive single-letter expressions for the secret key capacity through a type of source emulation analysis. We also derive lower bounds on the achievable reliability and secrecy exponents, i.e., the exponential rates of decay of the probability of decoding error and of the information leakage. These exponents allow us to determine a set of strongly-achievable secret key rates. For degraded eavesdroppers the maximum strongly-achievable rate equals the secret key capacity; our exponents can also be specialized to previously known results. In deriving our strong achievability results we introduce a coding scheme that combines wiretap coding (to excite the channel) and key extraction (to distill keys from residual randomness). The secret key capacity is naturally seen to be a combination of both source- and channel-type randomness. Through examples we illustrate a fundamental interplay between the portion of the secret key rate due to each type of randomness. We also illustrate inherent tradeoffs between the achievable reliability and secrecy exponents. Our new scheme also naturally accommodates rate limits on the public discussion. We show that under rate constraints we are able to achieve larger rates than those that can be attained through a pure source emulation strategy.

Fundamental rate-loss tradeoff for optical quantum key distribution.

Since 1984, various optical quantum key distribution (QKD) protocols have been proposed and examined. In all of them, the rate of secret key generation decays exponentially with distance. A natural and fundamental question is then whether there are yet-to-be discovered optical QKD protocols (without quantum repeaters) that could circumvent this rate-distance tradeoff. This paper provides a major step towards answering this question. Here we show that the secret key agreement capacity of a lossy and noisy optical channel assisted by unlimited two-way public classical communication is limited by an upper bound that is solely a function of the channel loss, regardless of how much optical power the protocol may use. Our result has major implications for understanding the secret key agreement capacity of optical channels-a long-standing open problem in optical quantum information theory-and strongly suggests a real need for quantum repeaters to perform QKD at high rates over long distances.

Separation of Reliability and Secrecy in Rate-Limited Secret-Key Generation

For a discrete or a continuous source model, we study the problem of secret-key generation with one round of rate-limited public communication between two legitimate users. Although we do not provide new bounds on the wiretap secret-key (WSK) capacity for the discrete source model, we use an alternative achievability scheme that may be useful for practical applications. As a side result, we conveniently extend known bounds to the case of a continuous source model. Specifically, we consider a sequential key-generation strategy, that implements a rate-limited reconciliation step to handle reliability, followed by a privacy amplification step performed with extractors to handle secrecy. We prove that such a sequential strategy achieves the best known bounds for the rate-limited WSK capacity (under the assumption of degraded sources in the case of two-way communication). However, we show that, unlike the case of rate-unlimited public communication, achieving the reconciliation capacity in a sequential strategy does not necessarily lead to achieving the best known bounds for the WSK capacity. Consequently, reliability and secrecy can be treated successively but not independently, thereby exhibiting a limitation of sequential strategies for rate-limited public communication. Nevertheless, we provide scenarios for which reliability and secrecy can be treated successively and independently, such as the two-way rate-limited SK capacity, the one-way rate-limited WSK capacity for degraded binary symmetric sources, and the one-way rate-limited WSK capacity for Gaussian degraded sources.

Strong Secrecy From Channel Resolvability

We analyze physical-layer security based on the premise that the coding mechanism for secrecy over noisy channels is tied to the notion of channel resolvability. Instead of considering capacity-based constructions, which associate to each message a subcode that operates just below the capacity of the eavesdropper's channel, we consider channel-resolvability-based constructions, which associate to each message a subcode that operates just above the resolvability of the eavesdropper's channel. Building upon the work of Csiszár and Hayashi, we provide further evidence that channel resolvability is a powerful and versatile coding mechanism for secrecy by developing results that hold for strong secrecy metrics and arbitrary channels. Specifically, we show that at least for symmetric wiretap channels, random capacity-based constructions fail to achieve the strong secrecy capacity, while channel-resolvability-based constructions achieve it. We then leverage channel resolvability to establish the secrecy-capacity region of arbitrary broadcast channels with confidential messages and a cost constraint for strong secrecy metrics. Finally, we specialize our results to study the secrecy capacity of wireless channels with perfect channel state information (CSI), mixed channels, and compound channels with receiver CSI, as well as the secret-key capacity of source models for secret-key agreement. By tying secrecy to channel resolvability, we obtain achievable rates for strong secrecy metrics with simple proofs.

Secret Key Generation from Sparse Wireless Channels: Ergodic Capacity and Secrecy Outage

This paper investigates generation of a secret key from a reciprocal wireless channel. In particular we consider wireless channels that exhibit sparse structure in the wideband regime and study the impact of sparsity on the secret key capacity. We explore this problem in two steps. First, we study key generation from a state-dependent discrete memoryless multiple source. The state of the source captures the effect of channel sparsity. Secondly, we consider a wireless channel model that captures channel sparsity and correlation between the legitimate users' channel and the eavesdropper's channel. Such dependency can significantly reduce the secret key capacity. According to system delay requirements, two performance measures are considered: (i) ergodic secret key capacity and (ii) outage probability. We show that in the wideband regime when a white sounding sequence is adopted, a sparser channel can achieve a higher ergodic secret key rate than a richer channel can. For outage performance, we show that if the users generate secret keys at a fraction of the ergodic capacity, the outage probability will decay exponentially in signal bandwidth. Moreover, a larger exponent is achieved by a richer channel.

How Many Queries Will Resolve Common Randomness?

A set of m terminals, observing correlated signals, communicate interactively to generate common randomness for a given subset of them. Knowing only the communication, how many direct queries of the value of the common randomness will resolve it? A general upper bound, valid for arbitrary signal alphabets, is developed for the number of such queries by using a query strategy that applies to all common randomness and associated communication. When the underlying signals are independent and identically distributed repetitions of m correlated random variables, the number of queries can be exponential in signal length. For this case, the mentioned upper bound is tight and leads to a single-letter formula for the largest query exponent, which coincides with the secret key capacity of a corresponding multiterminal source model. In fact, the upper bound constitutes a strong converse for the optimum query exponent, and implies also a new strong converse for secret key capacity. A key tool, estimating the size of a large probability set in terms of Rényi entropy, is interpreted separately, too, as a lossless block coding result for general sources. As a particularization, it yields the classic result for a discrete memoryless source.

Common Information and Secret Key Capacity

We study the generation of a secret key of maximum rate by a pair of terminals observing correlated sources and with the means to communicate over a noiseless public com- munication channel. Our main result establishes a structural equivalence between the generation of a maximum rate secret key and the generation of a common randomness that renders the observations of the two terminals conditionally independent. The minimum rate of such common randomness, termed interactive common information, is related to Wyner's notion of common information, and serves to characterize the minimum rate of interactive public communication required to generate an optimum rate secret key. This characterization yields a single-letter expression for the aforementioned communication rate when the number of rounds of interaction are bounded. An application of our results shows that interaction does not reduce this rate for binary symmetric sources. Further, we provide an example for which interaction does reduce the minimum rate of communication. Also, certain invariance properties of common information quantities are established that may be of independent interest.

Exploiting Partial Channel State Information for Secrecy over Wireless Channels

In this paper, we investigate the effect of partial channel state information on the achievable secure communication rates and secret-key generation rates over ergodic fading channels. In particular, we establish the strong secret-key capacity as well as lower bounds for the strong secrecy capacity of ergodic and block-ergodic fading channels with partial Channel State Information at the Transmitter(CSIT). Our analysis sheds light on the usefulness of CSIT to harness the benefits of fading for secrecy and allows us to quantify the penalty incurred by the lack of full CSIT. In particular, we numerically illustrate situations in which little CSIT is required to recover most of the benefits of fading and in which the legitimate terminals have an incentive to precisely characterize their channel.

Message transmission and key establishment: General equality for weak and strong capacities

Secure communication using noisy resources has been first studied in the contexts of secure message transmission (SMT) by Wyner as well as Csiszár-and-Körner, and secret key establishment (SKE) by Ahlswede-and-Csiszár as well as Maurer. The work defines secrecy (resp. secret-key (SK)) capacity as the highest achievable rate of secure transmission (resp. key establishment). Focusing on SKE, Maurer and Wolf later noticed that the secrecy requirement in the SKE study was weak as it required only the “ratio” between the adversary's information and the key length to be negligible. They suggested a stronger definition of the SK capacity by requiring absolute information leakage to be negligible. They provided an interesting proof for the equality of weak and strong SK capacities in the above scenarios (setups).Followup work has since studied SKE in various setups considering the weak SK capacity without discussing whether the results also hold for the strong definition. In this paper, we pose the question whether the equality of weak and strong SK capacities can be derived in general for all setups with discrete memoryless resources. We first extract the requirements that have been implicit in Maurer-and-Wolf's proof and then investigate whether these required conditions can be removed. Our results show that weak and strong SK capacities are equal for any discrete memoryless communication setup. We also extend this study to message transmission and prove the equality of weak and strong secrecy capacities in any discrete memoryless setup.