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Interference Assisted Secret Communication

Xiaojun Tang, Ruoheng Liu, Predrag Spasojevic, H. Vincent Poor

arXiv:0908.2397v1cs.ITcs.CR

TL;DR

Wireless broadcast transmissions are vulnerable to eavesdropping, motivating the study of whether independent interference can assist secrecy. The paper analyzes the WT-HI using achievable coding schemes and power policies, and derives computable outer bounds for discrete memoryless and Gaussian channels. It shows that interference can benefit secrecy, including positive secrecy rates when the source-destination channel is worse than the source-eavesdropper channel.

  • Problem

    The paper addresses how wireless interference can be used to counter eavesdropping in a wire-tap channel with an independent helper that does not know the confidential message.

  • Method

    The paper develops achievable secrecy rates and optimization policies for discrete memoryless and Gaussian WT-HI channels, together with computable upper bounds on Gaussian secrecy capacity.

  • Results

    Interference can increase secrecy and yield a positive secrecy rate even when the source-destination channel is worse than the source-eavesdropper channel.

  • Takeaways & Limitations

    Independent interference can benefit secret wireless communication within the WT-HI setting.

Abstract

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Wireless communication is susceptible to eavesdropping attacks because of its broadcast nature. This paper illustrates how interference can be used to counter eavesdropping and assist secrecy. In particular, a wire-tap channel with a helping interferer (WT-HI) is considered. Here, a transmitter sends a confidential message to its intended receiver in the presence of a passive eavesdropper and with the help of an independent interferer. The interferer, which does not know the confidential message, helps in ensuring the secrecy of the message by sending an independent signal. An achievable secrecy rate and several computable outer bounds on the secrecy capacity of the WT-HI are given for both discrete memoryless and Gaussian channels.

I. INTRODUCTION

The paper studies whether an independent interferer can improve secrecy in wireless wire-tap channels, where broadcast and superposition expose transmissions to eavesdroppers. It develops achievable secrecy rates and computable bounds for discrete memoryless and Gaussian WT-HI models.

  • Problem and model: The wire-tap channel with a helping interferer (WT-HI) adds an independent transmitter that does not know the confidential message.The helper provides additional randomization while the legitimate transmitter and interferer choose schemes to enhance secrecy.
  • Contributions: The paper proposes an achievable secrecy rate for general discrete memoryless WT-HI models by considering interference patterns and optimizing both coding schemes.The legitimate transmitter’s coding scheme is selected based on the interference codebook’s coding rate.
  • Contributions: For Gaussian WT-HI channels, the paper gives an achievable secrecy rate based on Gaussian codebooks and a power policy for optimization.It also provides several computable upper bounds on secrecy capacity, with different bounds preferable under different channel and power conditions.
  • Main findings: The interferer can increase secrecy and enable a positive secrecy rate even when the source-destination channel is worse than the source-eavesdropper channel.A key Gaussian case has the interferer-receiver channel better than the interferer-eavesdropper channel.
  • Main findings: When the interferer-receiver channel is sufficiently good and transmitter powers are unconstrained, the achieved Gaussian secrecy rate equals the rate obtained when a helper secretly receives and retransmits the message.This result does not assume a secret transmitter-interferer channel that would enable relaying.
  • Relation to prior work: The proposed scheme generalizes prior helper strategies, including independent Gaussian noise and noise forwarding, through different interference-codebook rates.Infinite interference-codebook rate yields an unstructured noise special case, while lower rates allow the intended receiver to decode interference first.

II. SYSTEM MODEL

The WT-HI system has a confidential-message transmitter, intended receiver, independent helping interferer, and passive eavesdropper. Its coding model uses stochastic encoders, secrecy binning, and receiver decoding that may be separate or joint.

  • System components: The system consists of transmitter X1, intended receiver Y1, helping interferer X2, and passive eavesdropper Y2.The transmitter sends a confidential message to the intended receiver while the helper sends an independent signal.
  • Assumptions: The helper does not know the confidential message, and the transmitters do not share common randomness.The eavesdropper is assumed to know both transmitters’ codebooks.
  • Channel model: A WT-HI channel is specified by input and output alphabets X1, X2, Y1, Y2 and transition probability p(y1, y2|x1, x2).The transmitter encodes message w1 into a length-n sequence, while the helper generates its output independently at random.
  • Secrecy criterion: The paper defines secrecy through the equivocation rate and defines secrecy capacity as the maximum achievable secrecy rate.The coding requirement includes arbitrarily small error for sufficiently large blocklengths.
  • Encoding: Both transmitter and helper use independent stochastic codebooks, with the transmitter’s codebook partitioned into secrecy bins.Each confidential message corresponds to a bin, and the transmitter randomly selects a codeword within that bin.
  • Decoding: The intended receiver can decode using separate decoding of the transmitter codeword or joint decoding of transmitter and interferer codewords.An error is declared if neither decoding condition yields a unique jointly typical codeword or codeword pair.

B. Achievable Rate

The paper constructs an achievable secrecy rate by combining joint and separate decoding regions for the intended receiver and eavesdropper. The interferer contributes dummy information that can confuse the eavesdropper.

  • Decoding regions: The achievable scheme defines receiver regions for joint decoding, where both codewords are decoded, and separate decoding, where interferer codewords are treated as noise.The intended receiver’s achievable region is the union of these decoding regions.
  • Decoding regions: The eavesdropper is analyzed with analogous multiple-access decoding regions for the channel (X1, X2) → Y2.The scheme compares receiver and eavesdropper decoding constraints in the R1-R2 plane.
  • Achievable rate: Theorem 1 states an achievable secrecy rate for the WT-HI based on the defined joint and separate decoding regions.The rate uses product input distributions p(x1)p(x2)p(y1, y2|x1, x2).
  • Code construction: The transmitter’s code rate is split as R1 = R1,s + R1,d, with R1,d serving as redundancy sacrificed to confuse the eavesdropper.The interferer independently sends dummy information at rate R2.
  • Proof strategy: The proof uses error analysis and equivocation computation; additional binning is introduced for proof simplicity but is equivalent to the described encoding procedure.The proof assumes double binning for C1 and one binning step for C2.

C. Some Special Cases

The paper studies weak, strong, and very strong interference or eavesdropping regimes, showing how an interferer can increase secrecy rates, including when secrecy without interference may be zero. It also gives a Sato-type upper bound and discusses channel prefixing.

  • Weak interference/eavesdropping: In weak interference, the interferer can increase secrecy when Δ1 ≤ Δ2 by generating artificial noise that neither receiver decodes.When Δ1 > Δ2, it instead facilitates transmission by choosing X2 to maximize Δ1.
  • Strong interference/eavesdropping: In strong interference, the channel without the interferer may have zero achievable secrecy rate, whereas the interferer can enable a positive secrecy rate.The intended receiver can first decode the interferer’s codeword and then decode the confidential transmission.
  • Channel prefixing: Channel prefixing replaces X1 and X2 with auxiliary variables V1 and V2 under a factored input distribution, preserving achievability of the secrecy rate.The resulting factorization is p(v1)p(v2)p(x1|v1)p(x2|v2)p(y1, y2|x1, x2).
  • Channel prefixing: The paper does not use channel prefixing because evaluating the resulting scheme is intractable.This limits the treatment to the non-prefixing achievable scheme.
  • Upper bounds: A Sato-type upper bound is provided for a general WT-HI, with secrecy capacity bounded by I(X1, X2; Ỹ1|Ỹ2) for an associated channel.The bound assumes a genie gives the eavesdropper’s signal Ỹ2 to the intended receiver.
  • Upper bounds: The upper bound is tight for degraded WT-HI channels, where the eavesdropper’s signal does not benefit intended-receiver decoding.The eavesdropper output is a degraded version of the combined signal (Ỹ1, Ỹ2).

IV. GAUSSIAN CHANNELS

For Gaussian WT-HI channels, the paper derives an achievable secrecy rate under power constraints and an explicit power policy. The analysis shows that interference power and power control can create positive secrecy rates across several channel regimes.

  • Channel model: The Gaussian WT-HI uses independent unit-variance Gaussian noises and average block power constraints on transmitter and interferer inputs.The channel outputs are specified for the intended receiver and eavesdropper.
  • Achievable secrecy rate: Theorem 3 gives an achievable secrecy rate for fixed transmit powers (P1, P2), with γ(x) ≜ (1/2) log(1 + x).The rate is achieved using the coding scheme developed for the discrete memoryless WT-HI.
  • Interference management: Interference can help secrecy because the interferer may control its power to avoid excessive interference to the intended receiver.The receiver may decode and cancel helpful interference before decoding the primary transmission.
  • Power control: When a > 1, a positive secrecy rate can be achieved when b > 1 or b ≤ a−1 if the interferer’s available power is sufficiently large.For b > 1, the interferer uses full power and the transmitter selects its power to support interference decoding; for b < a−1, interference is treated as noise.
  • Power control: The Gaussian power policy maximizes the secrecy rate in Theorem 3.The policy selects transmitter and interferer powers according to the channel-gain regimes and available power constraints.

2) Power-unconstrained Secrecy Rate:

The Gaussian WT-HI power-unconstrained secrecy rate is obtained through limiting analysis of an explicit power policy, with a piecewise achievable-rate expression. When the interferer-receiver channel is good, interference can provide a gain equivalent to virtually sending the confidential message through the interferer, without assuming a secret transmitter-interferer channel.

  • The explicit power policy enables limiting analysis that yields an achievable power-unconstrained secrecy rate for the Gaussian WT-HI.The result assumes ab̸ = 0.
  • The achievable power-unconstrained secrecy rate is given piecewise according to the channel parameter b and the thresholds max(1, 1 and min(1, 1.The supplied expression includes branches with 2 log2 b and ab.
  • A gain of (1/2) log2 b over the power-unconstrained rate without interference help is observed when the interferer-receiver channel is good.
  • This gain is equivalent to the power-unconstrained secrecy rate obtained when the confidential message is sent from the interferer to the intended receiver despite the eavesdropper.The paper describes this as if the message were virtually given secretly to the interferer, which would act as a cognitive transmitter.
  • The interpretation does not rely on a secret transmitter-interferer channel that would enable the interferer to relay the transmission.
  • B. Upper Bounds: The Gaussian WT-HI secrecy capacity is upper bounded by the main-channel capacity without a secrecy constraint, with additional Sato-type and other outer bounds described.The Sato-type bound is obtained by specializing the corresponding bound to the Gaussian WT-HI model.

2) A Z-channel upper bound:

The paper develops computable upper bounds and evaluates achievable secrecy rates for Gaussian WT-HI channels. Interference can improve secrecy, including when the source-destination channel is weaker than the source-eavesdropper channel.

  • Z-channel upper bound: The Gaussian WT-HI secrecy capacity is upper bounded by a computable Z-channel bound derived by giving the interference codeword to the intended receiver.This genie-aided model lets the intended receiver cancel interference and become interference-free.
  • Numerical examples: The achievable secrecy rate first decreases with a for a < 1, increases for 1 < a ≤ 3.26, and decreases again for a > 3.26.In the middle range, the intended receiver can decode and cancel interference while the eavesdropper treats it as noise.
  • Numerical examples: Without interference help, a positive secrecy rate requires a < 1; with help, it can be achieved when a < 1 + ¯P2.Larger interference power can improve the secrecy rate.
  • Numerical examples: Each of three upper bounds is best over some ¯P2 range: the Sato-type bound for small ¯P2, the Z-channel bound for larger ¯P2, and the main-channel bound at sufficiently large interference.At large ¯P2, secret signals are hidden in strong eavesdropper interference.
  • Numerical examples: The Sato-type upper bound is close to the achievable secrecy rate when ab is near 1 and is tight in the degraded case ab = 1.The Z-channel bound can be quite loose for some settings, especially when a > 1.
  • Conclusions: The helper sends an independent random codeword that the intended receiver can decode and subtract, while the eavesdropper cannot decode it.This selectively interferes with the eavesdropper and increases secrecy even when the legitimate source channel is weaker.

APPENDIX I PROOF OF THEOREM 1

The appendix proves the general discrete-memoryless WT-HI achievable secrecy rate using random codebooks, implicit double binning, and separate or joint decoding cases.

  • APPENDIX I PROOF OF THEOREM 1: Independent random codebooks are generated for the transmitter and interferer, with the transmitter codebook organized through implicit double binning.Codewords are indexed by confidential-message and randomization indices.
  • APPENDIX I PROOF OF THEOREM 1: The transmitter selects random intra-bin indices for each confidential message, while the interferer selects its codeword index uniformly at random.This randomization supports reliability and secrecy analysis.
  • APPENDIX I PROOF OF THEOREM 1: The intended receiver uses separate or joint decoding, and the message is decoded reliably with arbitrarily small error probability for sufficiently large blocklength.The decoding rule accepts a uniquely jointly typical codeword or codeword pair.
  • APPENDIX I PROOF OF THEOREM 1: The proof analyzes two interferer-rate cases according to whether R2 is below or above I(X2; Y2|X1).These cases correspond to different decoding regions at the eavesdropper.
  • APPENDIX I PROOF OF THEOREM 1: Fano-based bounds show that the eavesdropper’s equivocation satisfies the secrecy constraint in the considered cases.The argument uses rate conditions and bounds on conditional entropy.

B. Case II: R2 > I(X2; Y2|X1)

Case II treats an interferer code rate at least I(X2; Y2|X1), using binning at the helper and joint decoding arguments to establish secrecy.

  • B. Case II: R2 > I(X2; Y2|X1): The helper partitions its interference codebook into bins and independently randomizes both the bin and intra-bin indices.This construction adapts the helper codebook to the high-rate case.
  • B. Case II: R2 > I(X2; Y2|X1): The equivocation bound includes the mutual-information term I(X1, X2; Y2) and the conditional entropy of the residual indices.The resulting bound is used to verify secrecy for Case II.
  • B. Case II: R2 > I(X2; Y2|X1): The eavesdropper’s relevant decoding analysis jointly considers the transmitter’s and helper’s residual indices under side information.The rate pair is selected to make this decoding probability arbitrarily small.
  • B. Case II: R2 > I(X2; Y2|X1): The secrecy constraint is satisfied for Case II as blocklength grows and the error terms vanish.The proof explicitly takes ǫ → 0 as n → ∞.
  • B. Case II: R2 > I(X2; Y2|X1): The WT-HI secrecy capacity depends only on the marginal distributions of the intended and eavesdropper channels.The joint distribution of the two outputs conditioned on the inputs does not affect the capacity.
  • B. Case II: R2 > I(X2; Y2|X1): For Gaussian inputs, the achievable secrecy rate is optimized through coding parameters and power policies across channel regimes.When a ≥ 1 + P2, the secrecy rate is zero; when a ≤ 1 + P2, the intended receiver can decode and cancel interference in one regime.

APPENDIX IV

The appendix characterizes power optimization for the Gaussian achievable secrecy rate by partitioning channel-parameter regimes and comparing active rate functions and boundaries.

  • APPENDIX IV: Only one of three secrecy-rate functions is active for each channel-parameter and power configuration.All three functions are bounded over the allowed power region.
  • APPENDIX IV: A global maximum exists either at the maximum of one active function or at an intersection of two functions.The optimization checks gradients and boundary points.
  • APPENDIX IV: When a > 1 and b > 1, no positive secrecy rate is obtained for ¯P2 ≤ a − 1, so the selected powers are (P1, P2) = (0, 0).For ¯P2 > a − 1, the power choice depends on the relevant stationary point and power constraint.
  • APPENDIX IV: For several parameter regimes, secrecy is optimized by using full available powers, while other regimes select an interior interference power or set P2 = 0.The choice follows the signs of partial derivatives and comparisons between candidate rates.
  • APPENDIX IV: The optimization may require limiting P2 to a stationary point when increasing interference changes the active secrecy-rate function’s derivative from positive to nonpositive.This prevents blindly increasing helper power beyond the maximizing point.

APPENDIX V

The appendix derives Gaussian-channel secrecy-rate expressions and evaluates an upper bound by optimizing transmit powers and noise correlation. It also gives power policies whose achievable secrecy rate approaches 1 in the stated limiting regimes.

  • Achievable secrecy rate: Rs = 1 after taking the limit with respect to ¯P2 when b > max(1, a−1).The corresponding power policy uses (P1, P2) = (P ∗, …).
  • Achievable secrecy rate: Rs = 1 after taking the limit with respect to ¯P1 when b < min(1, a−1).The power policy uses P2 = P ∗ 2, with P ∗ 2 given by (25).
  • Achievable secrecy rate: For other cases, the power policy sets P2 = 0 and the achievable secrecy rate remains Rs = 1.
  • Sato-type upper bound: I(X1, X2; ˜Y1| ˜Y2) depends on transmit powers P1 and P2 and noise covariance ρ, and is denoted through f(P1, P2, ρ).The conditional mutual information is evaluated using the entropy expressions described in the appendix.
  • Sato-type upper bound: The Sato-type upper bound is evaluated at maximum powers and the minimizing correlation, yielding f( ¯P1, ¯P2, ρ∗( ¯P1, ¯P2)).For fixed powers, f is convex in ρ and its minimum occurs at ρ⋆ given by (33).

APPENDIX VII

The appendix derives an outer bound from the secrecy requirement and Fano’s inequality, then bounds information leakage using Gaussian entropy arguments. Combining the intermediate inequalities produces the stated upper bound.

  • Outer-bound derivation: The derivation begins from the secrecy requirement in (46) and Fano’s inequality in (47).
  • Outer-bound derivation: The leakage term I(W1; V n 2 |Y n 2 ) is bounded through mutual-information expansions and entropy differences.The steps introduce I(W1, Xn 1 ; V n 2 |Y n 2 ) and related conditional terms.
  • Gaussian entropy bound: Assuming Xn 1 and Xn 2 are independent, the proof applies the subset sum entropy power inequality to bound the resulting entropy terms.The inputs are further specialized to i.i.d. Gaussian variables when evaluating the bound.
  • Gaussian entropy bound: The entropy-based bound is increasing in t1 and t2, and maximum-entropy reasoning is used before combining the intermediate inequalities.
  • Final upper bound: The resulting upper bound includes the term 2 log 2(1 + a ¯P1)(1 + ¯P2) / (2 + a ¯P1 + ¯P2).The appendix states that combining (61), (62), and (67) yields the upper bound in (34).
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