An Overview of Physical Layer Security With Finite-Alphabet Signaling <s> III. GAUSSIAN WIRETAP CHANNELS WITH FINITE-ALPHABET INPUTS <s> Wyner's results for discrete memoryless wire-tap channels are extended to the Gaussian wire-tap channel. It is shown that the secrecy capacity Cs is the difference between the capacities of the main and wire.tap channels. It is further shown that Rd= Cs is the upper boundary of the achievable rate-equivocation region. <s> BIB001 </s> An Overview of Physical Layer Security With Finite-Alphabet Signaling <s> III. GAUSSIAN WIRETAP CHANNELS WITH FINITE-ALPHABET INPUTS <s> Accessible introduction to the theoretical foundations of modern coding theory Including numerousapplications to wireless transmission systems The author is famous in the field of coding and wireless communications for his work in the area of faded channels & communcations. <s> BIB002 </s> An Overview of Physical Layer Security With Finite-Alphabet Signaling <s> III. GAUSSIAN WIRETAP CHANNELS WITH FINITE-ALPHABET INPUTS <s> The secrecy capacity of a discrete memoryless Gaussian Wire-Tap Channel when the input is from a finite complex constellation is studied. It is shown that the secrecy capacity curves of a finite constellation plotted against the SNR, for a fixed noise variance of the eavesdropper's channel has a global maximum at an internal point. This is in contrast to what is known in the case of Gaussian codebook input where the secrecy capacity curve is a bounded, monotonically increasing function of SNR. Secrecy capacity curves for some well known constellations like BPSK, 4-QAM, 16-QAM and 8-PSK are plotted and the SNR at which the maximum occurs is found through simulation. It is conjectured that the secrecy capacity curves for finite constellations have a single maximum. <s> BIB003 </s> An Overview of Physical Layer Security With Finite-Alphabet Signaling <s> III. GAUSSIAN WIRETAP CHANNELS WITH FINITE-ALPHABET INPUTS <s> This paper investigates the secrecy capacity of the Gaussian wiretap channel with M-PAM inputs, by capitalizing on the relationship between mutual information and minimum mean squared error (MMSE). In particular, we establish optimality conditions for both the M-PAM input power and the M-PAM input distribution, which we specialize to the asymptotic low-power and high-power regimes. By using the properties of the MMSE to establish sufficient conditions for the uniqueness of the solution of some of the underlying non-convex optimization problems, we also propose efficient algorithms to compute the optimal solutions. Interestingly, we show that with M-PAM inputs it is sub-optimal to use all the available power for some range of parameters — this is in sharp contrast to standard Gaussian channels. We also extend the results to the parallel Gaussian wiretap channel with M-PAM inputs. We put forth a mercury-waterfilling interpretation of the optimal power allocation procedure for parallel Gaussian wiretap channels which generalizes the conventional mercury-waterfilling interpretation for parallel Gaussian channels, with the mercury level amending the base level to account for both the non-Gaussianess of the input and the secrecy constraint. <s> BIB004 </s> An Overview of Physical Layer Security With Finite-Alphabet Signaling <s> III. GAUSSIAN WIRETAP CHANNELS WITH FINITE-ALPHABET INPUTS <s> In this paper, we investigate the effect of output quantization on the secrecy capacity of the binary-input Gaussian wiretap channel. As a result, a closed-form expression with infinite summation terms of the secrecy capacity of the binary-input Gaussian wiretap channel is derived for the case when both the legitimate receiver and the eavesdropper have unquantized outputs. In particular, computable tight upper and lower bounds on the secrecy capacity are obtained. Theoretically, we prove that when the legitimate receiver has unquantized outputs while the eavesdropper has binary quantized outputs, the secrecy capacity is larger than that when both the legitimate receiver and the eavesdropper have unquantized outputs or both have binary quantized outputs. Further, numerical results show that in the low signal-to-noise ratio (SNR) (of the main channel) region, the secrecy capacity of the binary input Gaussian wiretap channel when both the legitimate receiver and the eavesdropper have unquantized outputs is larger than the capacity when both the legitimate receiver and the eavesdropper have binary quantized outputs; as the SNR increases, the secrecy capacity when both the legitimate receiver and the eavesdropper have binary quantized outputs tends to overtake. <s> BIB005
The secrecy capacity achieving input distribution over a Gaussian MIMO wiretap channel is proved to be Gaussian BIB001 . However, since Gaussian signals take on a continuum of values, their detection complexity is considerably high. In addition, noting that Gaussian signals are unbounded, they have high peak to average power ratios and they cannot be generated in practice. Accordingly, Gaussian signaling is not used in practice, and instead the transmission is carried out with the aid of discrete inputs drawn from standard constellations such as phase shift keying (PSK) or quadrature amplitude modulation (QAM). Hence, it is important to study the implications of discrete signaling in the context of physical layer security. As a first step in studying physical layer security under the finite-alphabet input assumption, the impacts of standard constellations on the achievable secrecy rates of Gaussian wiretap channels are determined in BIB003 and BIB004 . Raghava and Rajan BIB003 evaluate the constellation-constrained secrecy capacity and highlight an important behavioral difference between finite-alphabet (e.g., PSK and QAM) and Gaussian inputs, that is, for a fixed noise variance at Eve, the secrecy rate curves for PSK or QAM plotted against the SNR have global maxima at finite SNR values. Investigation of the secrecy capacity of the pulse amplitude modulation (PAM) inputs over a degraded Gaussian wiretap channel in BIB004 leads to a similar conclusion. In addition, Rodrigues et al. BIB004 demonstrate that when finite constellations are employed, using all the available power for the information bearing signal transmission may not be optimal. The reason behind these observations has its roots in the mutual information expressions. That is, the capacity under an average power constraint over an AWGN channel is given by in bits per complex dimension, and it is an unbounded function of the SNR. On the other hand, the capacity of the constellation-constrained AWGN channel can be calculated as BIB002  where X is a constellation set and |X | denotes its cardinality. Dissimilar to the capacity expression in (9), the mutual information term in (10) converges to log 2 |X | at high SNRs (see Fig. 2 ). This means that by increasing the transmit power, the mutual information terms I(X; Y) and I(X; Z) (calculated using (10)) will reach their saturation value, and the secrecy rate drops to zero. This is also exemplified in Fig. 3 , where the secrecy rates are obtained by evaluating the difference of the mutual information terms corresponding to the main and eavesdropper's channels, under the assumption that the SNR at the eavesdropper (denoted by SNR e ) is lower than the legitimate receiver's SNR (denoted by SNR b ) by 1.5 dB. This also clarifies that using all the available transmit power for information transmission in the high SNR regime is not optimal BIB004 . Furthermore, it can be observed from Fig. 3 that higher secrecy rates are achieved over Gaussian wiretap channels by increasing the modulation orders. This is because by increasing the modulation order, roughly speaking, the distribution of the channel input can be made to resemble the optimal Gaussian signal more closely for a wider range of SNRs. The impact of output quantization on the secrecy rate of binary-input Gaussian wiretap channels is investigated in BIB005 . It is shown through a theoretical analysis that when the legitimate receiver has unquantized inputs and the eavesdropper has (binary) quantized outputs, higher secrecy rates are achievable with respect to the case where both receivers have binary quantized or unquantized outputs. To conclude this section, we note that, while it may be tempting to argue that, as in the case of transmission over an AWGN channel with no security considerations, one can design general transmission schemes using Gaussian codebooks and simply adopt the ideas to the case of inputs from standard constellations (e.g., PSK, QAM, etc.), this approach does not extend in a straightforward manner to the case of physical layer security. The optimal transmit signal design strategies under the Gaussian input assumption lead to considerable losses when applied to inputs drawn from standard constellations. For instance, it is shown in BIB004 that the optimal power allocation strategy with PAM inputs is in sharp contrast to that of Gaussian inputs. This motivates development of new algorithms and ideas for practical constellation-constrained channels to achieve security at the physical layer.