For a communication where agent Alice aims at transmitting to agent Bob a message which must be kept secret to a third agent Eve, we consider terminals equipped with multiple antennas, obtaining the multiple-input multiple-output multiple-output-eavesdropper (MIMOME) channel. In this scenario, we assume that Alice knows the instantaneous channel conditions of the Alice-Bob channel before transmission, while knowing only the statistics of the Alice-Eve channel. For Rayleigh fading channels there is a non-zero probability that Eve gets a better channel than Bob, thus obtaining some information on the secret message (secrecy outage probability). We derive a closed form expression of the secrecy outage probability and then we consider the problem of allocating the power among the space streams, in order to maximize the total rate of the secret message under a constraint on the secrecy outage probability.
For a communication where agent Alice aims at transmitting to agent Bob a message which must be kept secret to a third agent Eve, we consider terminals equipped with multiple antennas, obtaining the multiple-input multiple-output multiple-output-eavesdropper (MIMOME) channel. In this scenario, we assume that Alice knows the instantaneous channel conditions of the Alice-Bob channel before transmission, while knowing only the statistics of the Alice-Eve channel. For Rayleigh fading channels there is a non-zero probability that Eve gets a better channel than Bob, thus obtaining some information on the secret message (secrecy outage probability). We derive a closed form expression of the secrecy outage probability and then we consider the problem of allocating the power among the space streams, in order to maximize the total rate of the secret message under a constraint on the secrecy outage probability.
A transmission between two agents, Alice and Bob, over a set of parallel sub-channels is overheard by a third agent Eve, through a second set of parallel sub-channels. All subchannels are flat with random and independent gains and additive white Gaussian noise (AWGN). Alice splits the total amount of available power among the sub-channels, with the purpose of maximizing the communication rate to Bob, under reliability and secrecy constraints. To this end, two schemes are considered. In one case the secret message is encoded with a single wiretap code and then split among the sub-channels. In the latter case the secret message is first split into a number of sub-messages, each separately encoded and transmitted on a different sub-channel. The achievable secrecy rates under a constraint on the secrecy outage probability (SOP) are derived and closed form expressions for Rayleigh fading sub-channels are obtained. In order to limit the complexity of resources optimization (power and rates) we also consider suboptimal solutions based on the selection of active sub-channels over which power is split either equally or according to a waterfilling algorithm with respect to the Alice-Bob channel.
We consider a system where an agent (Alice) aims at transmitting a message to a second agent (Bob) over a set of parallel channels, while keeping it secret from a third agent (Eve) by using physical layer security techniques. We assume that Alice perfectly knows the set of channels with respect to Bob, but she has only a statistical knowledge of the channels with respect to Eve. We derive bounds on the achievable outage secrecy rates, by considering coding either within each channel or across all parallel channels. Transmit power is adapted to the channel conditions, with a constraint on the average power over the whole transmission. We also focus on the maximum cumulative outage secrecy rate that can be achieved. Moreover, in order to assess the performance in a real life scenario, we consider the use of practical error correcting codes. We extend the definitions of security gap and equivocation rate, previously applied to the single additive white Gaussian noise channel, to Rayleigh distributed parallel channels, on the basis of the error rate targets and the outage probability. Bounds on these metrics are also derived, taking into account the statistics of the parallel channels. Numerical results are provided, that confirm the feasibility of the considered physical layer security techniques.
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 over the channel. In particular, we show that if Eve has fewer antennas than Bob, secret transmission is always possible at low-noise. Moreover, we show that in the low-noise limit the secrecy capacity of our scheme coincides with its unconstrained capacity, by providing a class of covariance matrices that allow to attain such limit without the need of wiretap coding.
In this paper, we consider secure transmissions in ergodic Rayleigh fast-faded multiple-input multiple-output multiple-antenna-eavesdropper (MIMOME) wiretap channels with only statistical channel state information at the transmitter (CSIT). When the antennas at the legitimate receiver are more than (or equal to) those at the eavesdropper, we prove the first MIMOME secrecy capacity with partial CSIT by establishing a new secrecy capacity upper-bound. The key step is to form an MIMOME degraded channel by dividing the legitimate receiver's channel matrix into two submatrices, and setting one of the submatrices to be the same as the eavesdropper's channel matrix. Next, under the total power constraint over all transmit antennas, we analytically solve the channel-input covariance matrix optimization problem to fully characterize the MIMOME secrecy capacity. Typically, the MIMOME optimization problems are non-concave. However, thanks to the proposed degraded channel, we can transform the stochastic MIMOME optimization problem to be a Schur-concave one and then find its solution. Besides total power constraint, we also investigate the secrecy capacity when the transmitter is subject to the practical per-antenna power constraint. The corresponding optimization problem is even more difficult since it is not Schur-concave. Under the two power constraints considered, the corresponding MIMOME secrecy capacities can both scale with the signal-to-noise ratios (SNR) when the difference between numbers of antennas at legitimate receiver and eavesdropper are large enough. However, when the legitimate receiver and eavesdropper have a single antenna each, such SNR scalings do not exist for both cases.
In this paper, we consider secure transmissions in ergodic Rayleigh fast-faded multiple-input multiple-output multiple-antenna-eavesdropper (MIMOME) wiretap channels with only statistical channel state information at the transmitter (CSIT). When the antennas at the legitimate receiver are more than (or equal to) those at the eavesdropper, we prove the first MIMOME secrecy capacity with partial CSIT by establishing a new secrecy capacity upper-bound. The key step is to form an MIMOME degraded channel by dividing the legitimate receiver's channel matrix into two submatrices, and setting one of the submatrices to be the same as the eavesdropper's channel matrix. Next, under the total power constraint over all transmit antennas, we analytically solve the channel-input covariance matrix optimization problem to fully characterize the MIMOME secrecy capacity. Typically, the MIMOME optimization problems are non-concave. However, thanks to the proposed degraded channel, we can transform the stochastic MIMOME optimization problem to be a Schur-concave one and then find its solution. Besides total power constraint, we also investigate the secrecy capacity when the transmitter is subject to the practical per-antenna power constraint. The corresponding optimization problem is even more difficult since it is not Schur-concave. Under the two power constraints considered, the corresponding MIMOME secrecy capacities can both scale with the signal-to-noise ratios (SNR) when the difference between numbers of antennas at legitimate receiver and eavesdropper are large enough. However, when the legitimate receiver and eavesdropper have a single antenna each, such SNR scalings do not exist for both cases.
This paper studies the secrecy performance of multiple-input multiple-output (MIMO) wiretap channels, also termed as multiple-input multiple-output multiple-eavesdropper (MIMOME) channels, under transmit antenna selection (TAS) and BPSK/QPSK modulations. In the main channel between the transmitter and the legitimate receiver, a single transmit antenna is selected to maximize the instantaneous Signal to Noise Ratio (SNR) at the receiver. At the receiver and the eavesdropper, selection combining (SC) is utilized. Additionally, suppose that the transmitted message is modulated by BPSK/QPSK modes. We first derive the closed-form approximated expression for the ergodic secrecy rate under Rayleigh fading, assuming that the channel state information of the eavesdropper (CSIE) is available at the transmitter. Next, analytical formulas for the approximated and asymptotic secrecy outage probability are also developed when CSIE is unavailable (NCSIE). Simulation results are provided to demonstrate the approximation precision of the derived results above. Furthermore, the asymptotic results reveal that the secrecy diversity order degrades into 0 due to the finite-alphabet inputs, which is totally different from that driven by the Gaussian inputs.