We consider the fast Rayleigh fading wiretap channel, over which a legitimate transmitter wishes to have secure communication with a legitimate receiver in the presence of an eavesdropper. We consider an average power constraint on the input, and assume that no channel state information (CSI) is available to any user. We show that the input distribution that achieves the secrecy capacity for this wiretap channel is discrete with a finite number of mass points.
We consider the fast Rayleigh fading wiretap channel, over which a legitimate transmitter wishes to have secure communication with a legitimate receiver in the presence of an eavesdropper. We consider an average power constraint on the input, and assume that no channel state information (CSI) is available to any user. We show that the input distribution that achieves the secrecy capacity for this wiretap channel is discrete with a finite number of mass points.
We consider the secure transmission in ergodic fast Rayleigh fading multiple-input single-output single-antenna-eavesdropper (MISOSE) wiretap channels. We assume that the statistics of both the legitimate and eavesdropper channels are the only available channel state information at the transmitter (CSIT). By introducing a new secrecy capacity upper bound, we prove that the secrecy capacity is achieved by the Gaussian input without prefixing. To attain this result, we form another MISOSE channel for upper-bounding by relaxing the equivocation constraint, and tighten the bound by carefully selecting correlations between the legitimate and eavesdropper channel gains. The resulting upper bound is tighter than the others in the literature which are based on modifying the correlation between the noises at the legitimate receiver and eavesdropper. Next, we fully characterize the secrecy capacity by showing that the optimal channel input covariance matrix is a scaled identity matrix. The key to solve such a stochastic optimization problem is by exploiting the completely monotone property of the secrecy capacity. Finally, we prove that with only statistical CSIT of both channels, the capacity will neither scale with signal-to-noise ratio (SNR) nor the number of antenna. Our numerical results also match these observations and further confirm that having the legitimate CSIT (realizations) is very beneficial to increase the secrecy capacity.
We consider the block Rayleigh fading multiple-input multiple-output (MIMO) wiretap channel with no prior channel state information (CSI) available at any of the terminals. The channel gains remain constant in a coherence time of T symbols, and then change to another independent realization. The transmitter, the legitimate receiver and the eavesdropper have nt, nrand neantennas, respectively. We determine the exact secure degrees of freedom (s.d.o.f.) of this system when T ≥ 2 min(nt, nr). We show that, in this case, the s.d.o.f. is exactly (min(nt, nr) − ne)+(T − min(nt, nr))/T. The first term can be interpreted as the eavesdropper with neantennas taking away neantennas from both the transmitter and the legitimate receiver. The second term can be interpreted as a fraction of s.d.o.f. being lost due to the lack of CSI at the legitimate receiver. In particular, the fraction loss, min(nt, nr)/T, can be interpreted as the fraction of channel uses dedicated to training the legitimate receiver for it to learn its own CSI. We prove that this s.d.o.f. can be achieved by employing a constant norm channel input, which can be viewed as a generalization of discrete signalling to multiple dimensions.
The secure transmission of information over wireless systems in the presence of an eavesdropper has attracted much attention. In this paper, we consider the case that full CSI of the legitimate user but only the average channel gain of the eavesdropper is known at the transmitter. In such setting, some information will be inevitably leaked to the eavesdropper from the information-theoretic view and this was not quantitatively evaluated before. A secrecy transmission framework with leakage threshold is thereby proposed. The secrecy capacity satisfying this leakage probability constraints is derived along with an optimal power allocation strategy. For Rayleigh and Nakagami-m fading channels, numerical results indicate that our framework can work well with transmitter knowledge of only average channel gain of the eavesdropper.
We consider the block Rayleigh fading multiple-input multiple-output (MIMO) wiretap channel with no prior channel state information (CSI) available at any of the terminals. The channel gains remain constant within a coherence interval of T symbols, and then change to another independent realization in the next coherence interval. The transmitter, the legitimate receiver, and the eavesdropper have nt, nr, and ne antennas, respectively. We determine the exact secure degrees of freedom (s.d.o.f.) of this system when T ≥ 2min(nt,nr). We show that, in this case, the s.d.o.f. is exactly equal to (min(nt,nr)-ne)+(T -min(nt,nr))/T. The first term in this expression can be interpreted as the eavesdropper with ne antennas taking away ne antennas from both the transmitter and the legitimate receiver. The second term can be interpreted as a fraction of the s.d.o.f. being lost due to the lack of CSI at the legitimate receiver. In particular, the fraction loss, min(nt,nr)/T, can be interpreted as the fraction of channel uses dedicated to training the legitimate receiver for it to learn its own CSI. We prove that this s.d.o.f. can be achieved by employing a constant norm channel input, which can be viewed as a generalization of discrete signalling to multiple dimensions.
In this paper, we consider secure broadcasting over fast fading channels. Assuming imperfect main channel state information (CSI) at the transmitter, we first provide an upper and a lower bounds on the ergodic secrecy capacity when a common message is broadcasted to multiple legitimate receivers in the presence of one eavesdropper. For this case, we show that the secrecy rate is limited by the legitimate receiver having, on average, the worst main channel link. Then, we present an expression for the achievable secrecy sum-rate when each legitimate receiver is interested in an independent message. The special cases of high SNR, perfect and no-main CSI are also analyzed. Numerical results are presented to illustrate the obtained results for the case of independent but not necessarily identically distributed Rayleigh fading channels.
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.