This paper studies the problem of simultaneously generating a secret key (SK) and a private key (PK) between Alice and Bob, in a cooperative pairwise-independent network (PIN) with two relays. In the PIN, the pairwise source observed by every pair of terminals is independent of those sources observed by any other pairs. The SK needs to be protected from Eve, while the PK needs to be protected not only from Eve but also from the two relays. Two cooperative SK-PK generation algorithms are proposed: both of them first generate common randomness, based on the well-established pairwise key generation technique and the application of the one-time pad; but then, the two algorithms utilize the XOR operation and a specific random-binning-based SK-PK codebook to generate the expected keys, respectively. The achievable SK-PK rate regions of both the two proposed algorithms are analyzed. Of particular interest is the second algorithm with random-bing based codebook, whose achievable key rate region is demonstrated to be exactly the same as the derived outer bound, a crucial step for establishing the key capacity of this PIN model. Finally, the two proposed SK-PK generation algorithms are extended to a cooperative wireless network, where the correlated source observations are obtained from estimating wireless channels during a training phase.
This paper studies the problem of simultaneously generating a secret key (SK) and a private key (PK) between Alice and Bob, in a cooperative pairwise-independent network (PIN) with two relays. In the PIN, the pairwise source observed by every pair of terminals is independent of those sources observed by any other pairs. The SK needs to be protected from Eve, while the PK needs to be protected not only from Eve but also from the two relays. Two cooperative SK-PK generation algorithms are proposed: both of them first generate common randomness, based on the well-established pairwise key generation technique and the application of the one-time pad; but then, the two algorithms utilize the XOR operation and a specific random-binning-based SK-PK codebook to generate the expected keys, respectively. The achievable SK-PK rate regions of both the two proposed algorithms are analyzed. Of particular interest is the second algorithm with random-bing based codebook, whose achievable key rate region is demonstrated to be exactly the same as the derived outer bound, a crucial step for establishing the key capacity of this PIN model. Finally, the two proposed SK-PK generation algorithms are extended to a cooperative wireless network, where the correlated source observations are obtained from estimating wireless channels during a training phase.
This paper studies the private key generation of a cooperative pairwise-independent network (PIN) with M + 2 terminals, where Alice and Bob wish to generate a private key under the help of the M relays, M ≥ 2; such a private key needs to be protected not only from Eve but also from individual relays simultaneously. In this PIN, the correlated source observed by every pair of terminals is independent of those sources observed by any other pair of terminals. All the terminals can communicate with each other over a public channel which is also observed by Eve noiselessly. The private key capacity of this PIN model is established, whose lower bound is obtained by proposing a novel random binning (RB) based key generation algorithm, and the upper bound is obtained based on the construction of M enhanced source models. The two bounds are shown to be exactly the same. Then, we consider a cooperative wireless network and use the estimates of fading channels to generate private keys. Compared to the existing XOR-based algorithm whose achievable multiplexing gain is ⌊M/2⌋, the proposed RB-based algorithm is demonstrated to achieve the optimal multiplexing gain M - 1.
On the Private Key Capacity of the M -Relay Pairwise Independent Network
We study the problem of private key generation in a cooperative pairwise independent network (PIN), with M + 2 terminals (Alice, Bob, and M relays), M ≥ 2. In the PIN, the correlated source observed by every pair of terminals is independent of the sources observed by any other pairs of terminals. Moreover, all terminals can communicate with each other over a public channel, which is also observed by Eve, noiselessly. The objective is to generate a private key between Alice and Bob with the help of the M relays; such a private key needs to be protected not only from Eve but also from all relays. A single-letter expression for the private key capacity of this PIN model is obtained, where the achievability part is established by proposing a random binning (RB)-based key generation algorithm, and the converse part is established by deriving upper bounds of M enhanced source models. Next, we consider a cooperative wireless network and use the estimates of fading channels to generate private keys. It has been shown that the proposed RB key generation algorithm can achieve a multiplexing gain of M - 1, which is an improvement compared with the existing XOR algorithm, whose achievable multiplexing gain is IM/2J.
Abstract We study the problem of private key generation in a cooperative pairwise independent network (PIN), with M + 2 terminals (Alice, Bob, and M relays), M ≥ 2. In the PIN, the correlated source observed by every pair of terminals is independent of the sources observed by any other pairs of terminals. Moreover, all terminals can communicate with each other over a public channel, which is also observed by Eve, noiselessly. The objective is to generate a private key between Alice and Bob with the help of the M relays; such a private key needs to be protected not only from Eve but also from all relays. A single-letter expression for the private key capacity of this PIN model is obtained, where the achievability part is established by proposing a random binning (RB)-based key generation algorithm, and the converse part is established by deriving upper bounds of M enhanced source models. Next, we consider a cooperative wireless network and use the estimates of fading channels to generate private keys. It has been shown that the proposed RB key generation algorithm can achieve a multiplexing gain of M - 1, which is an improvement compared with the existing XOR algorithm, whose achievable multiplexing gain is IM/2J.
This paper investigates the cooperative jamming based key generation scheme in a cooperative wireless network, where Alice and Bob aim to generate a secret key (SK) that is secret from Eve and a private key (PK) that is secret from both the relay and Eve, with the assistance of a relay. The correlation between legitimate channels and eavesdropping channels is considered. The SK and PK rates are evaluated based on minimum mean square error (MMSE) and zero forcing (ZF) estimation methods, respectively. The analytical expressions are further simplified with asymptotic forms in the high signal-to-noise ratio (SNR) regime. We demonstrate that the MMSE method is optimal to estimate channels, and the ZF method is asymptotically optimal in the high SNR regime. Moreover, we prove that the double-hop key generation strategy cannot achieve a higher rate than the single-hop strategy. Finally, numerical results validate the analysis and demonstrate that the cooperative jamming based key generation scheme significantly outperforms the traditional pairwise key generation scheme.
This paper investigates the performance of key generation between two nodes assisted by a relay in the presence of correlated eavesdropping channels. A cooperative jamming scheme is utilized to impose superimposed channel measurements on the relay and eavesdropper. Both lower and upper bounds on key capacities for both secret key (SK) and private key (PK) generation are evaluated, where the lower bounds are derived by using minimum mean square error and zero forcing methods for channel estimation, and the upper bounds are derived by formulating several enhanced discrete memoryless source (DMS) models. The analytical expressions are further simplified in the high signal-to-noise ratio (SNR) regime. We discover that one of the two legitimate channels should specialize in playing a role of jamming the relay or eavesdropper. We also demonstrate that the derived lower and upper bounds are tight when the eavesdropping channels are lowly or highly correlated. When the eavesdropping channels are uncorrelated, the SK and PK capacities can be determined since the corresponding upper and lower bounds are equal. Moreover, at high SNRs, a constant gap exists between the SK/PK upper and lower bounds as the correlation coefficient becomes one.
This paper explores physical layer group key generation in wireless relay networks with a star topology. In this setup, the relay node plays the role of either a trusted or untrusted central node, while one legitimate node (Alice) acts as the reference node. The channel between the relay and Alice serves as the reference channel. To enhance security during the channel measurement stage, a cooperative jamming-based scheme is proposed in this paper. This scheme allows the relay to obtain superimposed channel observations from both the reference channel and other relay channels. Then, a public discussion is utilized to enable all nodes to obtain estimates of the reference channel. Subsequently, the legitimate nodes can agree on a secret key (SK) that remains secret from the eavesdropper (Eve), or a private key (PK) that needs to be secret from both the relay and Eve. This paper also derives the lower and upper bounds of the SK/PK capacity. Notably, it demonstrates that there exists only a small constant difference between the SK/PK upper and lower bounds in the high signal-to-noise ratio (SNR) regime. Simulation results confirm the effectiveness of the proposed scheme for ensuring security and efficiency of group key generation.