[Paper Review] Secrecy Capacity of the Gaussian Wire-Tap Channel with Finite Complex Constellation Input

G. D. Raghava, B. Sundar Rajan|arXiv (Cornell University)|Oct 6, 2010

Wireless Communication Security Techniques13 references18 citations

TL;DR

This paper investigates the secrecy capacity of a Gaussian wire-tap channel when the transmitter uses a finite complex constellation (e.g., BPSK, QAM, PSK) instead of Gaussian codebooks. It analytically demonstrates that the constellation-constrained secrecy capacity (CC-SC) exhibits a global maximum at an optimal SNR, in contrast to the monotonically increasing behavior seen with Gaussian inputs, implying that excessive transmit power degrades secrecy performance for finite constellations.

ABSTRACT

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.

Motivation & Objective

• To analyze the secrecy capacity of a Gaussian wire-tap channel under finite complex constellation input constraints.
• To determine how power allocation affects secrecy performance when using practical modulation schemes like BPSK, QAM, and PSK.
• To challenge the conventional assumption that secrecy capacity monotonically increases with SNR by showing it peaks at an optimal SNR for finite constellations.
• To provide a design guideline for practical coding schemes by identifying the optimal operating SNR for maximum secrecy rate.

Proposed method

• Derives the constellation-constrained secrecy capacity (CC-SC) as the difference between the individual constellation-constrained capacities of the main and eavesdropper channels.
• Uses mutual information and differential entropy to compute the CC-SC for discrete, uniformly distributed input constellations.
• Employs simulations to compute the SNR at which the CC-SC reaches its maximum for BPSK, 4-QAM, 16-QAM, and 8-PSK constellations.
• Varying the eavesdropper’s noise variance (σ²₂), the study analyzes how the optimal SNR and maximum secrecy rate evolve across different channel conditions.
• Constructs 3D plots of CC-SC as a function of both legitimate receiver SNR and eavesdropper noise variance to visualize the trade-off.
• Conjectures that the CC-SC curves for finite constellations have a single global maximum based on observed trends in simulation results.

Experimental results

Research questions

• RQ1 Does the secrecy capacity of a Gaussian wire-tap channel with finite constellation input exhibit a maximum at a finite SNR, contrary to the monotonic increase seen with Gaussian inputs?
• RQ2 What is the optimal SNR for maximizing secrecy capacity across common constellations like BPSK, 4-QAM, 16-QAM, and 8-PSK?
• RQ3 How does the eavesdropper’s noise variance affect the location and value of the maximum secrecy capacity?
• RQ4 Can the CC-SC curve be characterized as having a single peak, and what are the implications for practical code design?

Key findings

• The constellation-constrained secrecy capacity (CC-SC) for finite constellations exhibits a global maximum at a finite, non-infinite SNR, unlike the monotonic increase seen with Gaussian codebooks.
• For BPSK, 4-QAM, 16-QAM, and 8-PSK, the maximum secrecy capacity occurs at a specific, finite SNR that depends on the eavesdropper’s noise variance.
• Beyond the optimal SNR, increasing transmit power reduces secrecy capacity, indicating that higher power is counterproductive for finite constellations.
• The SNR at which the maximum occurs increases with the eavesdropper’s noise variance and tends toward infinity as σ²₂ → ∞.
• The maximum achievable secrecy rate increases monotonically with σ²₂ and converges to the SISO Gaussian capacity as σ²₂ → ∞.
• 3D plots confirm that operating at the optimal SNR is critical for maximizing CC-SC, and uncontrolled power scaling degrades performance.