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On Secrecy Capacity Scaling in Wireless Networks

O. Ozan Koyluoglu, C. Emre Koksal, Hesham El Gamal

arXiv:0908.0898v2cs.ITcs.CR

TL;DR

The paper asks how secure per-pair wireless rates scale under path loss and ergodic fading, including when eavesdroppers collude. It combines secrecy-zone-based randomized multi-hop forwarding with secrecy precoding and ergodic interference alignment, obtaining secure scaling in the path loss setting and positive secure rates in fading settings, while identifying information and delay assumptions.

  • Problem

    Wireless broadcasting makes transmissions susceptible to eavesdropping, motivating secrecy as a quality-of-service constraint in network design.

  • Method

    The paper uses secrecy zones with independently randomized multi-hop forwarding for path loss networks, and secrecy precoding with ergodic interference alignment for ergodic fading networks.

  • Results

    The proposed schemes achieve secure rate scaling in the path loss model, while the ergodic fading scheme secures each user at any SNR and achieves [1/n]+ secure DoF per orthogonal dimension at high SNR.

  • Takeaways & Limitations

    Ergodic-fading secrecy requires only statistical eavesdropper knowledge, and the proposed schemes remain effective under analyzed eavesdropper-collusion scenarios.

  • Takeaways & Limitations

    The path loss scheme assumes nodes know whether an eavesdropper exists in a secrecy zone, and the work assumes uniform rates while leaving arbitrary traffic patterns for future study.

Abstract

from arXiv · show

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 $λ$ and $λ_e$, respectively. It is shown that, as long as $λ_e/λ=o((\log n)^{-2})$, almost all of the nodes achieve a perfectly secure rate of $Ω(\frac{1}{\sqrt{n}})$ 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.

A. Background

The paper motivates information-theoretic secrecy in wireless networks, where broadcast communication enables eavesdropping and pre-distributed keys are idealistic. It situates the work among wireless scaling-law and physical-layer security research.

  • Earlier wireless-network results established throughput scaling limits and progressively improved multi-hop protocols, including highway-based forwarding.
  • Wireless broadcast communication is susceptible to eavesdropping, motivating secrecy as a network quality-of-service constraint.
  • Cryptographic approaches are broadly divided into public-key methods based on computational limitations and private-key methods based on shared random keys.
  • The paper studies network secrecy capacity scaling against eavesdroppers with infinite computational power without assuming pre-distributed keys.
  • Information-theoretic secrecy developed from Shannon’s noiseless-channel formulation through Wyner’s noisy wiretap channel and later Gaussian extensions.

B. Contributions

The paper develops secrecy-preserving schemes for path-loss and ergodic-fading wireless networks, then analyzes colluding eavesdroppers. Its results preserve path-loss throughput scaling under stated intensity conditions and provide secure degrees of freedom in fading networks.

  • Path loss model: The path-loss scheme combines a secrecy-aware highway backbone with independent randomization at every hop.Edges require a legitimate node in the relevant square and no eavesdropper in a surrounding secrecy zone.
  • Path loss model: Almost all source-destination pairs achieve a secure rate with high probability, so secrecy causes no per-node throughput scaling loss under the stated assumptions.For extended networks λ = 1, while for dense networks λ = n.
  • Ergodic fading model: The ergodic-fading scheme uses ergodic interference alignment with secrecy pre-coding, achieving positive secrecy rates for most relevant fading distributions.
  • Ergodic fading model: η = [1/n]+ secure degrees of freedom per user are achieved at high SNR without eavesdropper CSI, while per-node performance increases with more legitimate users.
  • Eavesdropper collusion: Under collusion, the path-loss model retains the same scaling with nearly the same eavesdropper-intensity requirement, whereas fading performance is affected.When all fading-model eavesdroppers collude, η = [1/n]+ remains achievable.

C. Organization

The paper introduces its network models first, develops the path-loss secrecy scheme next, and then treats ergodic fading and colluding eavesdroppers before concluding.

  • Section II introduces the path-loss and ergodic-fading network models.
  • Section III develops the novel multi-hop secret encoding scheme for the path-loss model.
  • Section IV proposes an ergodic interference-alignment scheme for security applications, while Section V studies colluding eavesdroppers.
  • Section VI gives concluding remarks, with technical lemmas and proofs placed in the Appendix.

II. NETWORK MODELS

The network model distinguishes legitimate nodes from eavesdroppers and represents transmissions, legitimate receptions, and eavesdropper observations over time. Receiver noise is complex Gaussian, with transmit power and SNR specified by the model.

  • Legitimate nodes form L, eavesdroppers form E, and transmitting users at time t form T(t) ⊂ L.
  • A transmitter i sends X_i(t), while legitimate receivers and eavesdroppers observe Y_j(t) and Y_e(t), respectively.
  • Receivers experience zero-mean circularly symmetric complex Gaussian noise with variance N_0.
  • The model imposes an average transmit-power constraint and defines SNR per complex symbol for an i.i.d. CN(0, P) input distribution.

A. Static Path Loss Model with Stochastic Node Distribution

The path-loss model uses stochastic legitimate and eavesdropper placement, defines secrecy over all hops, and accounts for information accumulation across repeated transmissions of one message.

  • Model and secrecy definition: α > 2 governs path-loss decay as d^-α between nodes.The distance between nodes i and j is denoted d_ij.
  • Model and secrecy definition: λ = 1 models legitimate-node intensity in extended networks, while λ = n models dense networks.The extended region has area n; the dense region has unit area.
  • Model and secrecy definition: A secure rate requires arbitrarily small decoding error and information leakage rate for every eavesdropper as N →∞.The leakage condition concerns the message transmission over its entire path.
  • Model and secrecy definition: H hops require joint evaluation of the eavesdropper’s observations from all hops carrying the same message.The relevant observation at hop h is the length-N vector Y_e(h).
  • Model and secrecy definition: NH observations can accumulate substantial information about W_s,d even when the per-channel-use leakage is made arbitrarily small.This creates a distinction between small leakage rate and total accumulated information.

B. Ergodic Fading Model

The ergodic fading model considers n source-destination pairs under specified legitimate-user and eavesdropper fading assumptions, with secrecy defined against each eavesdropper. Legitimate users know H, while eavesdroppers know both H and He.

  • The fading processes are i.i.d. across time and ergodic, with H denoting legitimate-user channels and He denoting eavesdropper channels.
  • Legitimate-user channel gains are independent across pairs and symmetric around zero.
  • Each eavesdropper's channels from all transmitters are independently drawn from the same distribution, making node locations irrelevant in this model.
  • The model contains n source-destination pairs, each transmitting a secret message to its distinct receiver.
  • Secrecy requires vanishing information leakage I(Wi; Ye, H, He) ≤ ǫ for every eavesdropper as the blocklength becomes sufficiently large.

III. THE PATH LOSS MODEL

The path loss achievability strategy combines secrecy zones, randomized multi-hop forwarding, and percolation-based highway construction. A TDMA schedule organizes transmissions between nearby grid squares while preserving secure per-hop communication.

  • The path loss model studies extended networks with α > 2 and stochastic node placement modeled by Poisson point processes.
  • Independent randomization is added at every hop so an eavesdropper listening across all hops cannot recover the forwarded information.
  • Secrecy zones are used to guarantee secure communication over a single hop.
  • Percolation theory establishes enough horizontal and vertical highways for network-wide forwarding.
  • The network is partitioned into constant-size squares, scheduled through a TDMA frame, with active transmitters communicating to receivers within at most d squares.
  • Lemma 1 characterizes a simultaneously achievable secure rate for each active transmitter-receiver pair in the single-hop scenario.

SNRT R

The paper develops a secrecy-preserving multi-hop highway scheme in which each hop is secured against a boundary eavesdropper, yielding secure highway paths and per-node rate scaling under stated network conditions.

  • Multi-Hop Secrecy: Secrecy on each hop is sufficient to protect a multi-hop path from eavesdroppers observing all transmissions outside the transmitters’ secrecy zones.The scheme adds independent randomization at every hop, while secrecy codes account for the highest possible eavesdropper SNR.
  • Secure Highways: Secure highways exist with high probability, each serving O(√n) nodes while connecting sources and destinations within O(log n) distance.Percolation arguments establish sufficient horizontal and vertical secure highways under λe → 0 and suitable construction constants.
  • Rate Scaling: Each highway node can transmit to its next hop at a constant secure rate, so highway sharing yields per-node secure throughput of Ω(1/√n).The access phase also provides almost all source and destination nodes with secure rates of Ω(1/√n).
  • Extended Networks: If λ = 1 and λe/λ = o((log n)^−2) in an extended network, almost all nodes achieve secure rate Ω(1/√n).The result applies when the expected eavesdropper count is ne = o(n(log n)^−2).
  • Scaling Consequences: Eavesdropper collusion does not change the path-loss scheme’s scaling under its stated assumptions, while the extended-network result matches the optimal unsecured throughput scaling.The construction treats interference as noise at legitimate receivers.
  • Dense Networks: The same Ω(1/√n) secure-rate scaling is achievable for a (1−ε) fraction of nodes in the dense-network path-loss model under the stated eavesdropper-intensity condition.The network uses four time-division phases: access to highways, horizontal transport, vertical transport, and delivery to destinations.

IV. THE ERGODIC FADING MODEL

The ergodic fading scheme combines ergodic interference alignment with secrecy precoding and randomized codebooks to provide secure communication without eavesdropper CSI. Legitimate receivers cancel interference while eavesdroppers retain it, yielding positive secure rates and secure degrees of freedom under broad fading conditions.

  • Ergodic interference alignment: Ergodic interference alignment cancels interference for legitimate users but leaves it at eavesdroppers, whose effective channel becomes a SIMO-MAC.This channel asymmetry enables secure transmission at each user for any SNR, depending on the fading processes.
  • Coding construction: Typical-sequence quantization pairs each channel type with its complement, allowing coding over matched fading blocks with negligible rate loss.The construction uses strong typicality to allocate channel uses and repeats each block’s symbols during its complement block.
  • Secrecy precoding: Random binning adds a randomization index to each transmitted message, confusing eavesdroppers while preserving the intended message rate.Each transmitter randomly selects a codeword within the message bin before dividing it across paired fading blocks.
  • Achievable secure rate: A positive constant secure rate is achievable for each user at any fixed SNR when n is sufficiently large under non-degenerate fading.For Rayleigh fading, the relevant interference term diminishes as the number of legitimate users grows.
  • Secure degrees of freedom: The proposed scheme achieves a secure DoF per user for any non-degenerate fading model while requiring only statistical eavesdropper CSI.Its pre-log matches the prior interference-alignment result, but the construction incurs coding delay at least exponential in the number of users.

V. EAVESDROPPER COLLUSION

The paper analyzes colluding eavesdroppers in both channel models. Collusion preserves the path-loss scaling under a slightly modified intensity condition, while the fading scheme retains secure DoF but suffers performance degradation and requires broader eavesdropper information in the path-loss construction.

  • Path-loss model: Colluding eavesdroppers do not change the path-loss model’s scaling result under nearly the same eavesdropper-intensity requirement.The proposed multi-hop scheme continues to secure almost all nodes at the established scaling rate.
  • Path-loss model: The colluding path-loss construction requires legitimate nodes to know whether an eavesdropper lies within a first-layer zone of area (2dfl1 + 1)2c2.Here, fl1 = δ′ log(n), with δ′ arbitrarily small.
  • Path-loss model: The optimal path-loss scaling law is achieved even when eavesdroppers collude under the stated assumptions.This comes with more eavesdropper-related information than the non-colluding construction requires.
  • Ergodic fading model: For ergodic fading, collusion decreases achievable performance by adding independent eavesdropper observations to the received vector.The colluding observations are represented by a joint vector indexed by the selected collusion set.
  • Ergodic fading model: When all eavesdroppers collude, each user still achieves secure DoFs for non-degenerate fading distributions.The result follows by evaluating the achievable rate with the full eavesdropper set as the collusion set.

VI. CONCLUSION

The paper establishes secure throughput scaling for path-loss and ergodic-fading wireless networks, including scenarios with colluding eavesdroppers. It also identifies information about eavesdropper locations and coding delay as important scope considerations.

  • Path-loss model: Secure multi-hop forwarding uses secrecy zones and independent randomization at every hop to protect messages from eavesdroppers.The path-loss analysis assumes nodes can determine whether eavesdroppers occupy relevant secrecy zones.
  • Ergodic fading model: In ergodic fading, secrecy pre-coding combined with ergodic interference alignment secures each user at any SNR under the underlying fading distributions.At high SNR, the scheme achieves a positive secure degrees-of-freedom result per orthogonal dimension.
  • Ergodic fading model: The ergodic-fading scheme requires only statistical eavesdropper knowledge, but its gain comes with large coding delays.No eavesdropper channel-state information is required at legitimate users.
  • Eavesdropper collusion: With colluding eavesdroppers, the path-loss model retains the same per-node throughput scaling under nearly the same intensity requirement, while the fading model achieves secure DoF even under full collusion.The full-collusion fading result is reported as [1/n]+.
  • Future directions: The paper leaves the throughput–eavesdropper-intensity trade-off and cooperation-based security enhancements for future work.Hierarchical cooperation is identified as a possible direction in extended networks.

APPENDIX C PROOF OF THEOREM 10

The appendix extends secrecy-zone and secure-rate arguments to colluding eavesdroppers. Multi-level zones bound the colluding eavesdropper SNR while preserving secure highway and access-rate scaling.

  • Colluding eavesdropper construction: Multi-level secrecy zones are designed with areas and distances chosen to obtain a working bound on the colluding eavesdroppers’ SNR.The zone parameters may differ for highway forwarding and highway access and may depend on n.
  • Secure rate per hop: A secure rate per hop is achievable with high probability when the first secrecy zone contains no eavesdroppers and the zone parameters satisfy the stated conditions.This generalizes the single-hop secrecy-zone result to colluding eavesdroppers.
  • Highway rate: If λ_e=O((log n)^−2), each node on a constructed highway can transmit to its next hop at a constant secure rate.The result further relates highway service capacity to the number of nodes served by each highway.
  • Highway access: Almost all source and destination nodes can access highways with secure rate Ω(1/√n) under the colluding-eavesdropper construction.The access result is obtained with high probability under the appendix’s secrecy-zone choices.
  • End-to-end secrecy: Per-hop security suffices for multi-hop secrecy even when colluding eavesdroppers listen to every hop.Combining this property with the secure-highway percolation result completes the scaling argument.
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