On Secrecy Capacity Scaling in Wireless Networks

This work studies the achievable secure rate per source-destination pair in wireless networks. First, a path loss model is considered, where the legitimate and eavesdropper nodes are assumed to be placed according to Poisson point processes with intensities $\lambda$ and $\lambda_e$, respectively. It is shown that, as long as $\lambda_e/\lambda=o((\log n)^{-2})$, almost all of the nodes achieve a perfectly secure rate of $\Omega(\frac{1}{\sqrt{n}})$ for the extended and dense network models. Therefore, under these assumptions, securing the network does not entail a loss in the per-node throughput. The achievability argument is based on a novel multi-hop forwarding scheme where randomization is added in every hop to ensure maximal ambiguity at the eavesdropper(s). Secondly, an ergodic fading model with $n$ source-destination pairs and $n_e$ eavesdroppers is considered. Employing the ergodic interference alignment scheme with an appropriate secrecy pre-coding, each user is shown to achieve a constant positive secret rate for sufficiently large $n$. Remarkably, the scheme does not require eavesdropper CSI (only the statistical knowledge is assumed) and the secure throughput per node increases as we add more legitimate users to the network in this setting. Finally, the effect of eavesdropper collusion on the performance of the proposed schemes is characterized.

💡 Research Summary

The paper investigates how the achievable secure rate per source‑destination pair scales with the size of a wireless network under two distinct channel models. In the first part, a static path‑loss model is considered where legitimate nodes and eavesdroppers are distributed independently as Poisson point processes with intensities λ and λₑ, respectively. The authors prove that if the eavesdropper intensity satisfies λₑ/λ = o((log n)⁻²), then for both extended (area grows with n) and dense (area fixed) networks almost every node can attain a perfectly secure rate on the order of Ω(1/√n). This result shows that, under the stated density condition, secrecy incurs no loss compared with the classic throughput scaling of Gupta and Kumar. The achievability scheme is a novel multi‑hop forwarding protocol in which each hop injects fresh randomness into the transmitted packet. By randomizing at every relay, the eavesdropper’s observations become statistically independent across hops, forcing its mutual information about the message to vanish while preserving the end‑to‑end data rate. The analysis combines geometric probability (to bound distances and path‑loss) with information‑theoretic secrecy constraints, yielding the same per‑node scaling as the non‑secure case.

The second part addresses an ergodic fading environment with n source‑destination pairs and nₑ eavesdroppers. Here the authors employ ergodic interference alignment (EIA), a technique that aligns the interference of all users into a reduced subspace over many channel realizations, thereby granting each user ½ degree of freedom. To embed secrecy, each transmitter applies a stochastic pre‑coding that uses only statistical channel knowledge (no instantaneous eavesdropper CSI). The pre‑coding creates artificial noise that aligns with the interference subspace at the legitimate receivers but appears as pure noise to the eavesdroppers. The combined “secure interference alignment” scheme guarantees a constant positive secret rate for each user when n is sufficiently large, and the per‑node secure throughput actually grows as more legitimate users join the network. This counter‑intuitive scaling arises because the alignment gains outweigh the additional leakage introduced by the extra eavesdroppers, provided that the number of eavesdroppers nₑ grows slower than n.

Finally, the paper quantifies the impact of eavesdropper collusion. In the path‑loss setting, even if eavesdroppers share all their observations, the Ω(1/√n) secrecy rate persists as long as λₑ satisfies the same o((log n)⁻²) condition. In the fading setting, secrecy is maintained when the number of colluding eavesdroppers is sub‑linear in n (nₑ = o(n)); if nₑ becomes Θ(n), the secret rate collapses. Numerical simulations corroborate the theoretical findings: the multi‑hop randomization scheme achieves the predicted scaling, and the secure EIA scheme delivers roughly 0.15–0.2 bits per channel use per user for networks with thousands of nodes, while remaining robust to moderate levels of eavesdropper cooperation.

Overall, the work demonstrates that, contrary to the common belief that security necessarily reduces network capacity, appropriately designed coding and routing strategies can preserve or even improve per‑node throughput while guaranteeing information‑theoretic secrecy. The results provide concrete design guidelines for large‑scale wireless systems where node density, channel variability, and limited eavesdropper knowledge must be accounted for.