Achievable Secret Key Rate Research Articles

Jamming-Resilient Frequency Hopping-Aided Secure Communication for Internet-of-Things in the Presence of an Untrusted Relay

In this paper, we propose a light-weight jamming-resistant scheme for the Internet-of-Things (IoT) in 5G networks to ensure high-quality communication in a two-hop cooperative network. In the considered system model, a source communicates with a destination in the presence of an untrusted relay and a powerful multi-antenna adversary jammer. The untrusted relay is an authorized necessary helper who may wiretap the confidential information. Meanwhile, the jammer is an external attacker who tries to damage both the training and transmission phases. Different from traditional frequency hopping spread spectrum (FHSS) techniques that require a pre-determined pattern between communicating nodes, in our scheme, the source and destination enjoy the local observations of the two-hop channels. Then they exploit the measured channel as the source of common randomness to generate shared secret keys. By collecting multiple time slots into a frame, the sequence of channels observed in each frame is utilized to specify the adopted FHSS sequence in the next frame. Based on the derived FHSS sequence from the key generation phase, the source starts to transmit its message supporting by the the destination-assisted cooperative jamming (DACJ) technique which prevents the untrusted relay from discovering the secret message. For the mentioned system model, we present new closed-form expressions for characterizing the achievable secret key rate (SKR) and ergodic secrecy rate (ESR) to highlight the efficiency of our proposed scheme compared to the state-of-the-art. We next determine the optimal power allocation (OPA) between the pilot and data transmission phases that maximizes the ESR performance while escaping from jamming attack. Finally, several numerical examples and discussions are presented to gain engineering insights behind the studied communication scenario.

Secret Key Generation for Minimally Connected Hypergraphical Sources

This paper investigates the secret key generation in the multiterminal source model, where users observing correlated sources discuss interactively under limited rates to agree on a secret key. We focus on a class of sources representable by minimally connected hypergraphs. For such sources, we give a single-letter explicit characterization of the region of achievable secret key rate and public discussion rate tuple. This is the first result that completely characterizes the achievable rate region for a multiterminal source model, which is beyond the PIN model on a tree. We also obtain an explicit formula for the maximum achievable secret key rate, called the constrained secrecy capacity, as a function of the total discussion rate.

Order statistics and random matrix theory of multicarrier continuous‐variable quantum key distribution

SummaryIn a multicarrier continuous‐variable quantum key distribution (CVQKD) protocol, the information is granulated into Gaussian subcarrier CVs and the physical Gaussian link is divided into Gaussian sub‐channels. Here, we propose a combined mathematical framework of order statistics and random matrix theory for multicarrier continuous‐variable quantum key distribution. The analysis covers the study of the distribution of the sub‐channel transmittance coefficients in the presence of a Gaussian noise and the utilization of the moment generation function (MGF) in the error analysis. We reveal the mathematical formalism of sub‐channel selection and formulation of the transmittance coefficients and show a reduced complexity progressive sub‐channel scanning method. We define a framework to evaluate the statistical properties of the information flowing processes in multicarrier CVQKD protocols. Using random matrix theory, we express the achievable secret key rates and study the efficiency of the adaptive multicarrier quadrature division‐multiuser quadrature allocation (AMQD‐MQA) multiple‐access multicarrier CVQKD. The proposed combined framework is particularly convenient for the characterization of the physical processes of experimental multicarrier CVQKD.

Secret key rates of free‐space optical continuous‐variable quantum key distribution

SummaryIn this letter, we derive the maximal achievable secret key rates for continuous‐variable quantum key distribution (CVQKD) over free‐space optical (FSO) quantum channels. We provide a channel decomposition for FSO‐CVQKD quantum channels and study the SNR (signal‐to‐noise ratio) characteristics. The analytical derivations focus particularly on the low‐SNR scenarios. The results are convenient for wireless quantum key distribution and for the quantum Internet.

Parameter estimation of atmospheric continuous-variable quantum key distribution

Atmospheric effects are the chief threats to the quantum properties of propagating quantum signals and may degrade the performance of quantum key distribution seriously. As one of the most important parts of continuous-variable quantum key distribution (CVQKD), a parameter estimation method has not been specially proposed in an atmospheric channel, and usually the security analysis is based on the assumption that the relevant parameters, especially the excess noise, have been previously obtained. Here we propose a parameter estimation method for Gaussian modulated coherent state continuous-variable quantum key distribution over the atmospheric link, and we investigate the impact of the atmospheric channel on the estimated values of the parameters. Based on this method, we study theoretically the effect of link fluctuations on the achievable secret key rate of CVQKD under different practical transmitted conditions. The results show that this method is unified with the physical model and can effectively resist entanglement-distillation attack. The proposed method fills in the blank of parameter estimation for implementation of practical atmospheric CVQKD.

Secret-Key Capacity Regions for Multiple Enrollments With an SRAM-PUF

We introduce the multiple enrollment scheme for SRAM-physical unclonable functions (PUFs). During each enrollment, the binary power-on values of the SRAM are observed, and a corresponding key and helper data are generated. Each key can later be reconstructed from an additional observation and the helper data. The helper data do not reveal information about the keys to an attacker. It is our goal to use the additional enrollments to consecutively increase the entropy of the generated key material. We analyze two alternative settings. First, we present a regular setting, where each additional key is independent of all previous keys. Second, we introduce a key-replacement setting, where instead of an additional independent key, a new key (of increased length) is generated that replaces the old key. We characterize the capacity regions for both the settings. We show that the total achievable secret-key rate is equal to the mutual information between all enrollment observations and a single (reconstruction) observation. We derive our results based on a statistical model for the SRAM-PUF that has been proposed in the literature. This model implies a permutation symmetry property of the SRAM-PUF which plays a key role in our proofs.

Atmospheric effects on continuous-variable quantum key distribution

Compared to fiber continuous-variable quantum key distribution (CVQKD), atmospheric link offers the possibility of a broader geographical coverage and more flexible transmission. However, there are many negative features of the atmospheric channel that will reduce the achievable secret key rate, such as beam extinction and a variety of turbulence effects. Here we show how these factors affect performance of CVQKD, by considering our newly derived key rate formulas for fading channels, which involves detection imperfections, thus form a transmission model for CVQKD. This model can help evaluate the feasibility of experiment scheme in practical applications. We found that performance deterioration of horizontal link within the boundary layer is primarily caused by transmittance fluctuations (including beam wandering, broadening, deformation, and scintillation), while transmittance change due to pulse broadening under weak turbulence is negligible. Besides, we also found that communication interruptions can also cause a perceptible key rate reduction when the transmission distance is longer, while phase excess noise due to arrival time fluctuations requires new compensation techniques to reduce it to a negligible level. Furthermore, it is found that performing homodyne detection enables longer transmission distances, whereas heterodyne allows higher achievable key rate over short distances.

Secret-Key Generation and Convexity of the Rate Region Using Infinite Compound Sources

In secret-key generation using a compound source, the actual statistics of the source are unknown to the participants. It is assumed rather that the actual source belongs to a set (compound set) which is known to the participants. The secret-key generation protocol should guarantee in this case reliability and security of the generated secret-key simultaneously for all elements of the compound set. In this paper, secret-key generation based on a three-party compound source is studied in which an eavesdropper’s side information is also taken into account and strong secrecy is guaranteed. At the same time, the public communication rate constraint between the legitimate users is part of the secret-key generation protocol. In this setting, the achievable secret-key rates for finite compound sources are first reformulated as a region of secret-key rate versus communication rate constraint pairs. It is shown that this region is in general convex, even if the compound set is infinite. Based on this, the secret-key capacity results are extended to be valid for arbitrary (possibly infinite) compound sources with a finite set of marginals. In this case, the secret-key capacity is completely characterized as a function of the forward communication rate parameter between the legitimate users.

Cooperative Group Secret Key Generation Based on Secure Network Coding

Most of the existing works on group secret key (GSK) generation are based on pairwise key which wasting the total time and power of the group users. We present a cooperative GSK generation algorithm via star topology with one central node and one reference node. After channel estimation, secure network coding technique is employed by the central node to help all the group users exploit the randomness of reference channel with nothing about that leaking to the eavesdropper. Then all the group users agree on a secret key from the correlative observations of the reference channel in one information reconciliation phase. In addition, the expression of achievable secret key rate is derived. Theoretical and Monte Carlo simulation results validate the performance of our proposed algorithm.

Quantization of high dimensional Gaussian vector using permutation modulation with application to information reconciliation in continuous variable QKD

This paper is focused on the problem of Information Reconciliation (IR) for continuous variable Quantum Key Distribution (QKD). The main problem is quantization and assignment of labels to the samples of the Gaussian variables observed at Alice and Bob. Trouble is that most of the samples, assuming that the Gaussian variable is zero mean which is de-facto the case, tend to have small magnitudes and are easily disturbed by noise. Transmission over longer and longer distances increases the losses corresponding to a lower effective Signal-to-Noise Ratio (SNR) exasperating the problem. Quantization over higher dimensions is advantageous since it allows for fractional bit per sample accuracy which may be needed at very low SNR conditions whereby the achievable secret key rate is significantly less than one bit per sample. In this paper, we propose to use Permutation Modulation (PM) for quantization of Gaussian vectors potentially containing thousands of samples. PM is applied to the magnitudes of the Gaussian samples and we explore the dependence of the sign error probability on the magnitude of the samples. At very low SNR, we may transmit the entire label of the PM code from Bob to Alice in Reverse Reconciliation (RR) over public channel. The side information extracted from this label can then be used by Alice to characterize the sign error probability of her individual samples. Forward Error Correction (FEC) coding can be used by Bob on each subset of samples with similar sign error probability to aid Alice in error correction. This can be done for different subsets of samples with similar sign error probabilities leading to an Unequal Error Protection (UEP) coding paradigm.

Decoy-State Reference-Frame-Independent Measurement-Device-Independent Quantum Key Distribution With Biased Bases

Reference-frame-independent measurement-device-independent quantum key distribution (RFI-MDI-QKD) can eschew the alignment of reference frames in practical systems and defeat all potential detector side channel attacks. Here, we propose the decoy-state RFI-MDI-QKD protocol with biased bases. In this protocol, two legitimate parties Alice and Bob prepare signal states in $Z$ , $X$ , and $Y$ bases and decoy states in $X$ and $Y$ bases, which avoids the futility in $Z$ basis for decoy states and simplifies the operation of existing systems. Considering the security against coherent attacks with statistical fluctuations, we investigate the performance of the decoy-state RFI-MDI-QKD protocol with biased bases in the environment of unknown and slowly drifting reference frames and make comparisons with the original decoy-state RFI-MDI-QKD protocol under the same conditions. Simulation results show that the proposed protocol can increase the achievable secret key rate and transmission distance obviously compared with the original protocol, which is very promising in real-life QKD systems.

Multi-partite entanglement can speed up quantum key distribution in networks

The laws of quantum mechanics allow for the distribution of a secret random key between two parties. Here we analyse the security of a protocol for establishing a common secret key between N parties (i.e. a conference key), using resource states with genuine N-partite entanglement. We compare this protocol to conference key distribution via bipartite entanglement, regarding the required resources, achievable secret key rates and threshold qubit error rates. Furthermore we discuss quantum networks with bottlenecks for which our multipartite entanglement-based protocol can benefit from network coding, while the bipartite protocol cannot. It is shown how this advantage leads to a higher secret key rate.

Multi-Party Secret Key Agreement Over State-Dependent Wireless Broadcast Channels

We consider a group of $m$ trusted and authenticated nodes that aim to create a shared secret key $K$ over a wireless channel in the presence of an eavesdropper Eve. We assume that there exists a state-dependent wireless broadcast channel from one of the honest nodes to the rest of them including Eve. All of the trusted nodes can also discuss over a cost-free, noiseless and unlimited rate public channel which is also overheard by Eve. For this setup, we develop an information-theoretically secure secret key agreement protocol. We show the optimality of this protocol for “linear deterministic” wireless broadcast channels. This model generalizes the packet erasure model studied in the literature for wireless broadcast channels. Here, the main idea is to convert a deterministic channel into multiple independent erasure channels by using superposition coding. For “state-dependent Gaussian” wireless broadcast channels, by using insights from the deterministic problem, we propose an achievability scheme based on a multi-layer wiretap code. By using the wiretap code, we can mimic the phenomenon of converting the wireless channel into multiple independent erasure channels. Then, finding the best achievable secret key generation rate leads to solving a non-convex power allocation problem over these channels (layers). We show that using a dynamic programming algorithm, one can obtain the best power allocation for this problem. Moreover, we prove the optimality of the proposed achievability scheme for the regime of high-SNR and large-dynamic range over the channel states in the (generalized) degrees of freedom sense.

Improved key-rate bounds for practical decoy-state quantum-key-distribution systems

The decoy-state scheme is the most widely implemented quantum key distribution protocol in practice. In order to account for the finite-size key effects on the achievable secret key generation rate, a rigorous statistical fluctuation analysis is required. Originally, a heuristic Gaussian-approximation technique was used for this purpose, which, despite of its analytical convenience, was not sufficiently rigorous. The fluctuation analysis has recently been made rigorous by using the Chernoff bound. There is a considerable gap, however, between the key rate bounds obtained from these new techniques and that obtained from the Gaussian assumption. Here, we develop a tighter bound for the decoy-state method, which yields a smaller failure probability. This improvement results in a higher key rate and increases the maximum distance over which secure key exchange is possible. By optimizing the system parameters, our simulation results show that our new method almost closes the gap between the two previously proposed techniques and achieves a similar performance to that of conventional Gaussian approximations.

Entanglement distribution over 150 km in wavelength division multiplexed channels for quantum cryptography

AbstractGranting information privacy is of crucial importance in our society, notably in fiber communication networks. Quantum cryptography provides a unique means to establish, at remote locations, identical strings of genuine random bits, with a level of secrecy unattainable using classical resources. However, several constraints, such as non‐optimized photon number statistics and resources, detectors' noise, and optical losses, currently limit the performances in terms of both achievable secret key rates and distances. Here, these issues are addressed using an approach that combines both fundamental and off‐the‐shelves technological resources. High‐quality bipartite photonic entanglement is distributed over a 150 km fiber link, exploiting a wavelength demultiplexing strategy implemented at the end‐user locations. It is shown how coincidence rates scale linearly with the number of employed telecommunication channels, with values outperforming previous realizations by almost one order of magnitude. Thanks to its potential of scalability and compliance with device‐independent strategies, this system is ready for real quantum applications, notably entanglement‐based quantum cryptography. image

Secret Key Generation Via Localization and Mobility

We consider secret key generation by a pair of mobile nodes utilizing observations of their relative locations in the presence of a mobile eavesdropper. In our proposed algorithm, the legitimate node pair makes noisy observations of the relative locations of each other. Based on these observations, the nodes generate secret key bits via information reconciliation, data compression, and privacy amplification. We characterize a theoretically achievable secret key bit rate in terms of the observation noise variance at the legitimate nodes and the eavesdropper and show that the performance of our algorithm is comparable to the theoretical bounds. We also test our algorithm in a vehicular setting based on observations made using wireless beacon exchange between the legitimate nodes. To achieve this, we used TelosB wireless radios mounted on the sides of the vehicles on local roads and freeways. Note that our approach relies solely on distance reciprocity, and thus, it is not restricted to the use of wireless radios and can be used with other localization systems (e.g., infrared and ultrasound systems) as well. Overall, this study proves, via both information theoretic and practical analysis, that localization information provides a significant additional resource for secret key generation in mobile networks.

Uncertainty relation for mutual information

We postulate the existence of a universal uncertainty relation between the quantum and classical mutual informations between pairs of quantum systems. Specifically, we propose that the sum of the classical mutual information, determined by two mutually unbiased pairs of observables, never exceeds the quantum mutual information. We call this the complementary-quantum correlation (CQC) relation and prove its validity for pure states, for states with one maximally mixed subsystem, and for all states when one measurement is minimally disturbing. We provide results of a Monte Carlo simulation suggesting that the CQC relation is generally valid. Importantly, we also show that the CQC relation represents an improvement to an entropic uncertainty principle in the presence of a quantum memory, and that it can be used to verify an achievable secret key rate in the quantum one-time pad cryptographic protocol.

Secret Key Agreement by LLR Thresholding and Syndrome Feedback over AWGN Channel

In this paper we propose novel advantage distillation and reconciliation approaches for secret key agreement (SKA) over additive white Gaussian noise (AWGN) (possibly fading) channels, using binary phase shift keying (BPSK) constellations. For advantage distillation Bob selects bits having an absolute log-likelihood ratio (LLR) larger than some threshold. For reconciliation, Alice transmits the syndrome of the selected bits with respect to a globally pre-shared forward error correcting (FEC) code. We show that the syndrome transmission does not diminish the advantage of Bob over Eve on the key knowledge, thus being an efficient solution. For the proposed scheme we derive both a lower bound on the achievable secret key rate with ideal wiretap coding, and the effective throughput achieved by using practical codes.

Long-distance quantum key distribution with imperfect devices

Quantum key distribution over probabilistic quantum repeaters is addressed. We compare, under practical assumptions, two such schemes in terms of their secure key generation rates per quantum memory. The two schemes under investigation are the one proposed by Duan et al. in [Nat. 414, 413 (2001)] and that of Sangouard et al. in [Phys. Rev. A 76, 050301 (2007)]. We consider various sources of imperfection in both protocols, such as nonzero double-photon probabilities at the sources, dark counts in detectors, and inefficiencies in the channel, photodetectors and memories. We also consider memory decay and dephasing processes in our analysis. For the latter system, we determine the maximum value of the double-photon probability beyond which secretkey distillation is not possible. We also find the crossover distance beyond which the repeater schemes outperform the non-repeater ones. We finally compare the two protocols in terms of their achievable secret key generation rates at their optimal settings

Min-entropy and quantum key distribution: Nonzero key rates for “small” numbers of signals

We calculate an achievable secret key rate for quantum key distribution with a finite number of signals, by evaluating the min-entropy explicitly. The min-entropy can be expressed in terms of the guessing probability, which we calculate for d-dimensional systems. We compare these key rates to previous approaches using the von Neumann entropy and find non-zero key rates for a smaller number of signals. Furthermore, we improve the secret key rates by modifying the parameter estimation step. Both improvements taken together lead to non-zero key rates for only 10^4-10^5 signals. An interesting conclusion can also be drawn from the additivity of the min-entropy and its relation to the guessing probability: for a set of symmetric tensor product states the optimal minimum-error discrimination (MED) measurement is the optimal MED measurement on each subsystem.