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Interference Alignment for Secrecy

Onur Ozan Koyluoglu, Hesham El Gamal, Lifeng Lai, H. Vincent Poor

arXiv:0810.1187v1cs.IT

TL;DR

The paper asks how secrecy constraints affect secure degrees of freedom in K-user Gaussian interference channels, including a model with unknown eavesdropper CSI. It combines interference alignment with secrecy precoding and shows positive secure DoF for both confidential messages and external-eavesdropper settings. The resulting rates are K−2/(2K−2) per user in the confidential-messages model and K−2/(2K) per user in the external-eavesdropper model under ergodic operation.

  • Problem

    The impact of secrecy constraints on the degrees of freedom of K-user Gaussian interference channels had not been fully characterized, especially with an external eavesdropper lacking CSI at network users.

  • Method

    The paper combines interference alignment with secrecy precoding for confidential-messages and external-eavesdropper models.

  • Results

    K−2/(2K−2) secure DoF per user is achievable for confidential messages, while K−2/(2K) per user is achievable with an external eavesdropper in the ergodic setting.

  • Takeaways & Limitations

    The results show that interference can positively affect the secrecy capacity region of multi-user wireless networks.

Abstract

from arXiv · show

This paper studies the frequency/time selective $K$-user Gaussian interference channel with secrecy constraints. Two distinct models, namely the interference channel with confidential messages and the one with an external eavesdropper, are analyzed. The key difference between the two models is the lack of channel state information (CSI) about the external eavesdropper. Using interference alignment along with secrecy pre-coding, it is shown that each user can achieve non-zero secure Degrees of Freedom (DoF) for both cases. More precisely, the proposed coding scheme achieves $\frac{K-2}{2K-2}$ secure DoF {\em with probability one} per user in the confidential messages model. For the external eavesdropper scenario, on the other hand, it is shown that each user can achieve $\frac{K-2}{2K}$ secure DoF {\em in the ergodic setting}. Remarkably, these results establish the {\em positive impact} of interference on the secrecy capacity region of wireless networks.

I. INTRODUCTION

The paper examines secrecy in frequency/time-selective K-user Gaussian interference channels, where prior wiretap results imply vanishing secure DoF at high SNR. It studies confidential messages and external-eavesdropper models using interference alignment and secrecy precoding.

  • Prior wiretap-channel results imply vanishing secure DoF in the high-SNR regime because secrecy capacity saturates.
  • The secrecy impact of interference alignment in the K-user network had not been fully characterized.Without secrecy constraints, interference alignment achieves 1/2 DoF per orthogonal dimension for each source-destination pair.
  • The paper distinguishes confidential messages, requiring secrecy from non-intended receivers, from an external eavesdropper model.The external-eavesdropper model differs in its treatment of eavesdropper CSI.
  • Interference alignment is combined with secrecy precoding to align unintended signals while assigning each intended signal to an orthogonal subspace.
  • K−2/(2K−2) secure DoF per user is achieved in the confidential-messages model, while 1/2−1/K DoF per user is achieved ergodically with an external eavesdropper.The external-eavesdropper result also indicates a positive impact of interference on wireless-network secrecy.

A. The Confidential Messages Scenario

The confidential-messages model defines a frequency-selective Gaussian interference channel, secret codebooks, decoding, reliability, and equivocation for protecting each message from the other receivers.

  • The channel uses transmitted symbols across frequency slots and time, with additive complex Gaussian noise at each receiver.
  • Channel coefficients are drawn from a continuous distribution, fixed during communication, and known at every network node.
  • Each source has a message that must remain secret from the other K−1 receivers.
  • A secret codebook contains message sets, encoding functions satisfying an average long-term power constraint, and decoding functions producing message estimates.
  • Reliability is measured by decoding error probability, while secrecy is measured by normalized equivocation for message subsets at non-intended receivers.

B. The External Eavesdropper Scenario

The external-eavesdropper model adds an eavesdropper observing all source signals and evaluates secrecy in an ergodic block-fading setting with unavailable eavesdropper CSI at network users.

  • An external eavesdropper observes the signals transmitted by all K sources.
  • Transmission is divided into B fading blocks of n1 symbol times, with n = n1B.
  • The legitimate channel state H is known to all network nodes, whereas the eavesdropper channel state He is known only to the eavesdropper.Network users have only statistical knowledge of the eavesdropper CSI.
  • The secrecy requirement protects each transmitter’s own message from the external eavesdropper using normalized equivocation over message subsets.
  • The symmetric secure DoF with perfect secrecy is defined analogously to the confidential-messages model.

III. THE K-USER GAUSSIAN INTERFERENCE CHANNEL WITH CONFIDENTIAL MESSAGES

The confidential-messages construction combines interference-alignment beamforming with randomized secrecy codebooks. It creates interference-free desired dimensions while mixing unintended messages sufficiently to secure positive DoF.

  • Three-user construction: The three-user construction uses a (2m+1)-symbol extension and beamforming matrices that map each user’s streams into transmitted vectors.
  • Interference alignment: Non-intended signals align in a subspace of dimension F−m_i at each receiver.
  • Interference alignment: Intended streams occupy an orthogonal subspace relative to the aligned interference at each receiver.
  • Three-user intuition: In the three-user intuition, each eavesdropper observes mixed streams in only m dimensions, allowing m/2 secure streams and 1/4 secure DoF in the large-extension limit.
  • Main result: K−2/(2K−2) secure DoF per frequency-time slot is almost surely achievable for each user.The result is established through an appropriately constructed codebook ensemble as blocklength, symbol extension, and power grow.
  • Secrecy precoding: Randomized codebooks partition codewords into message bins, with bin indices carrying secrecy messages and additional indices providing randomization.
  • Secrecy precoding: Each transmitter maps randomized streams through interference-alignment matrices at every symbol time.
  • Achievability argument: The construction selects secrecy and randomization rates inside the decodability region, enabling reliable decoding and asymptotically perfect secrecy for message subsets.

IV. THE K-USER GAUSSIAN INTERFERENCE CHANNEL WITH AN EXTERNAL EAVESDROPPER

The external-eavesdropper model achieves positive secure DoF without eavesdropper CSI by combining interference alignment, secrecy coding, and channel ergodicity. With CSI unavailable, each user achieves 1/K secure DoF per frequency-time slot in the ergodic setting.

  • Coding scheme: The scheme combines interference alignment with secrecy pre-coding and uses randomly permuted user orderings across fading blocks to ensure statistical symmetry.Each block uses a permutation of the users to generate the corresponding alignment matrices.
  • Main result: (K−1)/(2K) secure DoF per frequency-time slot is almost surely achievable for each user when eavesdropper CSI is available.The CSI-assisted result is obtained by treating the eavesdropper as a virtual transmitter in a K+1-user alignment network.
  • Main result: 1/K secure DoF per frequency-time slot is achievable for each user in the ergodic setting without eavesdropper CSI.This is the paper's general external-eavesdropper result.
  • Coding scheme: The equivalent channel matrices are identically distributed across users, receivers, and fading blocks, enabling symmetric rate and secrecy analysis.This symmetry permits a common mutual-information expression for all users.
  • Scope and implication: The no-CSI result depends strongly on ergodicity, whereas the CSI-available result holds almost surely for all channel realizations.The paper identifies this dependence on ergodicity as the cost of lacking eavesdropper CSI.
  • Scope and implication: Interference can improve secrecy: unlike the point-to-point external-eavesdropper channel with zero DoF, the network provides non-zero secure DoF for K ≥ 2.Interference alignment separates intended signals from interference at legitimate receivers while packing interference into a low-dimensional subspace that impairs eavesdropper discrimination.

V. CONCLUSIONS

The paper shows that interference alignment with secrecy precoding yields non-zero secure DoF for K-user Gaussian interference channels under two secrecy models. The results distinguish confidential messages from an external eavesdropper with unknown CSI and identify interference as beneficial to secrecy capacity.

  • Interference alignment combined with secrecy precoding enables each network user to achieve non-zero secure DoF.The conclusion states this result for the K-user Gaussian interference channel with secrecy constraints.
  • The results distinguish confidential messages from the external-eavesdropper model with unknown CSI.The paper explicitly contrasts the two network models and the eavesdropper-CSI condition.
  • The findings indicate that interference can increase secrecy capacity in multi-user wireless networks.

APPENDIX I

This appendix proves that sufficiently strong aggregate interference at a receiver provides the stated secrecy level for every subset of unintended users. The proof uses entropy bounds and conditioning arguments.

  • For receiver i, if the aggregate interference level ΔK−i,i is sufficiently close to one, every subset S of other users achieves secrecy level at least d−ε.The lemma states that for any ε>0 and d∈[0,1], an appropriate ε* guarantees ΔS,i≥d−ε whenever ΔK−i,i≥1−ε*.
  • The proof bounds the eavesdropper’s observation through the conditional entropy of all non-intended messages.It assumes a received observation Yi and lower-bounds H(WK−i|Yi) using the aggregate secrecy condition.
  • The entropy inequalities rely on conditioning not increasing entropy and on the chosen ε* relating total and subset message entropies.

APPENDIX II

This appendix establishes that the interference-alignment gain matrices have full column rank with probability one. The argument follows from the alignment construction and the continuous distribution of channel gains.

  • The gain matrix Hi,k V̄k has rank mk with probability one for every transmitter k and receiver i.Because its dimension is F×mk, the result means the matrix has full column rank.
  • The desired-link matrix has rank mk by the interference-alignment construction, and the precoding vectors have linearly independent columns.
  • Continuous channel gains make the diagonal row operations nonzero with probability one, preserving the rank of the precoding matrix.Thus rank(Hi,k V̄k)=rank(V̄k)=mk.

APPENDIX III

This appendix develops mutual-information inequalities used in the secrecy analysis. Conditioning on additional independent user signals bounds the eavesdropper’s information, while random user ordering yields symmetric expected terms.

  • For disjoint user sets M and L, I(X̃M; Ȳe|H,Ḣe) is no greater than I(X̃M;Ȳe|X̃L,H,Ḣe).
  • The inequality follows from expanding mutual information into entropies, using conditioning’s entropy property, and applying message independence.
  • The complementary subset Sc is handled analogously through a chain-rule expansion over its users.
  • Repeating the same conditional term across the users in S yields the factor S in the resulting bound.
  • The same symmetry arises because user ordering is randomized independently at each fading block.

APPENDIX V

Appendix V establishes a rate assignment under which each user’s randomization message is decodable at the eavesdropper. The argument uses mutual-information inequalities for subsets of users and their complements.

  • Randomization-rate assignment: Each user can set its randomization rate to 1/K E[I(˜X_K; Ȳ_e|H, Ĥ_e)].The supplied lemma states this rate assignment explicitly.
  • Decodability condition: The proof reduces the decodability condition to mutual-information inequalities involving a subset S and its complement S^c.The displayed relations compare |S| and K−|S| weighted conditional mutual informations.
  • Decodability conclusion: With this rate assignment, the randomization messages are decodable at the eavesdropper.The passage states that each randomization message, given the secrecy-message bin indices, is decodable.
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