Source-linked AI summary
The Gaussian Wiretap Channel with a Helping Interferer
Xiaojun Tang, Ruoheng Liu, Predrag Spasojevic, H. Vincent Poor
TL;DR
Wireless broadcast makes transmissions vulnerable to eavesdropping, and the paper studies a Gaussian wiretap channel with a helping interferer. It derives an achievable secrecy rate and a Sato-type upper bound; numerical analysis finds the bound close to the achievable rate for weak symmetric interference and under broader asymmetric-channel conditions.
Problem
Wireless broadcast lets multiple receivers hear transmissions, making confidential communication susceptible to eavesdropping.
Source: §I · p. 1Method
The paper analyzes a Gaussian wiretap channel where an independent helper adds randomization, and derives an achievable secrecy rate and a Sato-type upper bound.
Source: §I · p. 1, §II · p. 2, A. Achievable Secrecy Rate · p. 2Results
Takeaways & Limitations
Interference can benefit secret wireless communication in the studied Gaussian wiretap channel with a helping interferer.
Source: §VI · pp. 4–5
Abstract
from arXiv · showhide
Due to the broadcast nature of the wireless medium, wireless communication is susceptible to adversarial eavesdropping. This paper describes how eavesdropping can potentially be defeated by exploiting the superposition nature of the wireless medium. A Gaussian wire-tap channel with a helping interferer (WTC-HI) is considered in which a transmitter sends confidential messages to its intended receiver in the presence of a passive eavesdropper and with the help of an interferer. The interferer, which does not know the confidential message assists the confidential message transmission by sending a signal that is independent of the transmitted message. An achievable secrecy rate and a Sato-type upper bound on the secrecy capacity are given for the Gaussian WTC-HI. Through numerical analysis, it is found that the upper bound is close to the achievable secrecy rate when the interference is weak for symmetric interference channels, and under more general conditions for asymmetric Gaussian interference channels.
I. INTRODUCTION
The paper studies whether a helping interferer can use wireless superposition to improve secrecy against a passive eavesdropper. It presents an achievable secrecy rate and a Sato-type upper bound, with numerical analysis finding the bound close under specified interference conditions.
I. INTRODUCTION
- A helper can use superposition to improve secrecy against broadcast-enabled eavesdropping by interfering in a communication session compromised by a passive eavesdropper.The paper frames this as using one wireless-medium property to address security issues caused by another.Source: §I · p. 1
- The WTC-HI has a transmitter sending confidential messages to an intended receiver with an interferer’s help, while a passive eavesdropper listens; secrecy is measured by equivocation rate.The paper requires the eavesdropper to remain in total ignorance of the intended message.Source: §I · p. 1
- The helper adds randomization without knowing the message; prior schemes differ because cooperative jamming uses independent Gaussian noise and noise forwarding requires intended-receiver decoding.The paper also notes that earlier work did not describe a computable secrecy-capacity upper bound for the Gaussian WTC-HI.Source: §I · p. 1
- The paper gives an achievable secrecy rate and power-control strategy, plus a Sato-type secrecy-capacity upper bound for the Gaussian WTC-HI.Numerical analysis finds the bound close to the achievable rate for weak symmetric interference and under more general asymmetric-channel conditions.Source: §I · p. 1
II. SYSTEM MODEL
The model has a transmitter, intended receiver, independent helping interferer, and passive but intelligent eavesdropper. Stochastic encoders supply randomness, while secrecy is defined through equivocation and achievable rates through reliable coding under power constraints.
II. SYSTEM MODEL
- The system includes a transmitter, intended receiver, independent interferer, and passive intelligent eavesdropper who knows both codebooks.The transmitter’s confidential message is sent with helper assistance; the channel outputs at the receiver and eavesdropper are specified in the model.Source: §II · p. 2
- The transmitter uses a message-conditioned stochastic encoder, while the helper independently randomizes its signal, adding secrecy randomization without knowing the message.The model imposes average block power constraints on the channel inputs.Source: §II · p. 2
- An (M, n, Pe) code includes both encoders and a deterministic decoder that estimates the confidential message from the intended receiver’s output.The secrecy capacity is defined as the maximal achievable secrecy rate.Source: §II · p. 2
- Secrecy is measured by the eavesdropper’s equivocation rate; the paper notes that this definition is weak but can be strengthened using extractor functions without loss of secrecy rate.Achievability is required for every ε > 0 under sufficiently large blocklengths.Source: §II · p. 2
III. ACHIEVABLE SECRECY RATE
This section considers achievable secrecy rates when transmitter and interferer powers are bounded. The power-control issue is deferred to Subsection III-B.
III. ACHIEVABLE SECRECY RATE
- The achievable-secrecy-rate analysis assumes transmitter power P1 ≤ P̄1 and interferer power P2 ≤ P̄2.These are the power bounds for the section’s rate analysis.Source: §III · p. 2
- The section studies achievable secrecy rate under bounded transmitter and interferer powers.It explicitly leaves power control for Subsection III-B.Source: §III · p. 2
- Power control is treated separately in Subsection III-B rather than in this section’s stated setup.Source: §III · p. 2
A. Achievable Secrecy Rate
Theorem 1 gives an achievable secrecy rate for the Gaussian WTC-HI as a piecewise function of channel parameters and transmission powers. The scheme uses independent Gaussian codebooks, and the paper states that positive secrecy rate is possible only when a < 1.
A. Achievable Secrecy Rate
- Theorem 1 gives an achievable secrecy rate as a piecewise function of the channel parameters and powers.The stated cases divide according to thresholds involving b, β1, and β2.Source: A. Achievable Secrecy Rate · p. 2
- The achievability scheme uses two independent Gaussian codebooks: the transmitter randomly selects from the message’s bin, while the helper randomly selects from its single-bin codebook.The code parameters (R1, Rs, R2) are chosen to derive the rate in Theorem 1.Source: A. Achievable Secrecy Rate · p. 2
- The paper notes that sufficiently large interference power P2 can benefit secrecy and permit a positive rate, except in a stated case.Source: A. Achievable Secrecy Rate · p. 3
- Positive secrecy rate can be achieved only when a < 1.Source: A. Achievable Secrecy Rate · p. 3
B. Power Control
The section develops power choices that maximize achievable secrecy rate across channel regimes. It balances decoding and canceling interference against limiting the interference reaching the intended receiver.
B. Power Control
- Power control is needed to limit interference at the intended receiver while allowing it to decode and cancel helpful interference before the confidential transmission.The transmitter selects its power and the interferer may need to control its power.Source: B. Power Control · p. 3
- When a≥1, the maximizing power-control scheme depends on the channel regime, as specified in Proposition 1.For b<a−1, the receiver treats interference as noise and the interferer controls its power to avoid excessive interference.Source: B. Power Control · p. 3
- For a>1, positive secrecy is feasible when b>1 or b≤a−1 if helper power is large enough; for b>1, the receiver decodes and cancels interference.In the b>1 regime, the interferer uses its full power and the transmitter chooses its power to enable decoding first.Source: B. Power Control · p. 3
- When a<1, Proposition 2 specifies a maximizing power-control scheme, and positive secrecy is always feasible.For a<b<1, the receiver treats interference as noise; when transmitter power is insufficient, interference hurts the intended receiver more than the eavesdropper.Source: B. Power Control · p. 3
C. Power-unconstrained Secrecy Rate
The section characterizes the achievable secrecy rate when transmitter power is unconstrained. It gives channel-regime-dependent rates and identifies a secrecy-rate correspondence that holds without a transmitter–interferer channel.
C. Power-unconstrained Secrecy Rate
- The power-unconstrained secrecy rate is the secrecy rate when transmitter power is unconstrained and depends only on channel conditions.The section obtains the Gaussian WTC-HI result through limiting analysis.Source: C. Power-unconstrained Secrecy Rate · p. 3
- Weak interference yields a gain of (1/2) log2(1/b), while strong interference yields a power-unconstrained secrecy rate of (1/2) log2 b under the stated channel conditions.Weak means b<a−1 for a≥1 or b<1 for a<1; strong means b>1 for a≥1 or b>a−1 for a<1.Source: C. Power-unconstrained Secrecy Rate · p. 3
- The strong-interference rate matches the power-unconstrained secrecy rate for sending the confidential message from the interferer to the intended receiver, despite no assumed transmitter–interferer channel.The stated setup therefore does not rely on that channel to relay the transmission.Source: C. Power-unconstrained Secrecy Rate · p. 3
IV. SATO-TYPE UPPER BOUND
The section gives a Sato-type upper bound on WTC-HI secrecy capacity and evaluates it for the Gaussian channel. The bound uses a genie-aided construction and can be computed by optimizing a function of transmit powers and noise covariance.
IV. SATO-TYPE UPPER BOUND
- The secrecy capacity depends on the two conditional output marginals, not on further structure in their joint distribution.This permits replacing the original outputs with arbitrarily correlated outputs having the same marginals.Source: §IV · pp. 3–4
- The Sato-type quantity R_u upper-bounds WTC-HI secrecy capacity through I(X1, X2; Ỹ1|Ỹ2).The construction assumes the intended receiver receives the eavesdropper’s signal as side information.Source: §IV · p. 4
- For the Gaussian channel, the bound is expressed as f(P1, P2, ρ), with ρ the covariance of the correlated unit-variance noises.The evaluation uses Gaussian inputs to calculate the conditional mutual information.Source: §IV · p. 4
- Because f increases with both transmit powers and is convex in ρ, its maximum uses the power limits and the minimizing ρ⋆.The resulting bound is f(P̄1, P̄2, ρ⋆(P̄1, P̄2)).Source: §IV · p. 4
V. NUMERICAL EXAMPLES
Numerical examples compare the achievable secrecy rate with the modified Sato-type upper bound for symmetric and general Gaussian WTC-HI channels. The bound is close in weak-interference conditions, but can become loose as symmetric interference grows.
Symmetric channel
- For symmetric channels, the upper bound is good when a ≤ 1, but becomes quite loose when a > 1 and large.The looseness is attributed to giving the intended receiver too much information in the genie-aided bound.Source: §V · p. 4
- For the symmetric example, Rs decreases for a < 1, rises for 1 < a ≤ 1.73, then decreases for a > 1.73.In the middle range, the intended receiver can decode and cancel interference while the eavesdropper treats it as noise.Source: §V · p. 4
General channel
- For general parameter settings, the upper bound is good when ab ≤ 1; at ab = 1, the degraded case, it is always tight.The examples use P̄1 = P̄2 = 2.Source: §V · p. 4
VI. CONCLUSIONS
The paper studies how independent interference from a helper can exploit wireless superposition to alleviate eavesdropping. It provides an achievable secrecy rate and a Sato-type upper bound, and concludes that interference can benefit secret wireless communication.
VI. CONCLUSIONS
- The paper studies a Gaussian wiretap channel where a helper injects interference independent of the confidential message.The stated motivation is to use wireless superposition to alleviate eavesdropping associated with broadcasting.Source: §VI · p. 4
- The analysis provides an achievable secrecy rate and a Sato-type upper bound on secrecy capacity..Source: §VI · p. 4
- The results show that interference can benefit secret wireless communication.The conclusion contrasts this with Gaussian problems that do not involve secrecy, where interference seldom offers an advantage.Source: §VI · pp. 4–5
APPENDIX
The appendix proves the Sato-type bound by applying the secrecy requirement and Fano’s inequality, then evaluating the resulting conditional mutual information for the Gaussian channel.
APPENDIX
- The proof applies Fano’s inequality and bounds message information using the conditional mutual information between the outputs and channel inputs.It uses the Markov relation from the message through the inputs to the outputs.Source: APPENDIX · p. 5
- Since secrecy capacity depends only on marginal output distributions, the proof replaces the outputs with the correlated pair Ỹ1, Ỹ2.The replacement preserves the required output marginals.Source: APPENDIX · p. 5
- For the Gaussian evaluation, the conditional mutual information is evaluated using Gaussian variables and the maximum-entropy theorem.The proof expresses it through joint-output entropies and conditional entropies.Source: APPENDIX · p. 5