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Rethinking the Secrecy Outage Formulation: A Secure Transmission Design Perspective
Xiangyun Zhou, Matthew R. McKay, Behrouz Maham, Are Hjorungnes
TL;DR
The paper addresses secure wireless transmission without Eve’s instantaneous CSI, where delay constraints make conventional secrecy-outage measures insufficiently direct. It introduces a transmission-conditioned formulation and designs adaptive and non-adaptive on-off schemes. The designs meet security and QoS requirements while trading feedback overhead against throughput and achievable secrecy outage.
Problem
Without Eve’s instantaneous CSI, existing secrecy-outage measures do not directly distinguish perfect-secrecy failures from reliability outages or suspended transmissions.
Method
The paper conditions secrecy outage on actual transmission and optimizes codeword rate, confidential rate, and on-off SNR threshold under security and QoS constraints.
Results
Adaptive rate design provides significant throughput improvement using Bob’s instantaneous CSI, whereas the non-adaptive design requires only 1 bit of feedback; arbitrarily low secrecy outage is generally impossible.
Takeaways & Limitations
The formulation provides a direct security measure for designing transmission schemes that satisfy target security requirements while managing QoS and feedback overhead.
Abstract
from arXiv · showhide
This letter studies information-theoretic security without knowing the eavesdropper's channel fading state. We present an alternative secrecy outage formulation to measure the probability that message transmissions fail to achieve perfect secrecy. Using this formulation, we design two transmission schemes that satisfy the given security requirement while achieving good throughput performance.
I. INTRODUCTION
The paper addresses secure wireless transmission without the eavesdropper’s instantaneous CSI, especially when stringent delays prevent reliable perfect secrecy. It proposes a direct secrecy-outage measure and two throughput-oriented transmission schemes.
- Stringent delay constraints make perfect secrecy unattainable in every transmission, motivating probabilistic outage-based performance measures.
- The existing outage formulation does not directly indicate the system’s security level because outage events need not represent failures of perfect secrecy.
- The proposed formulation accounts for codeword rate and transmission-suspension conditions when measuring the probability that messages are perfectly secure.
- Two schemes guarantee a specified security level while maximizing throughput, requiring either full legitimate-receiver CSI feedback or only 1 bit of feedback.
II. SYSTEM MODEL
The system considers confidential transmission over independent quasi-static Rayleigh fading channels from Alice to Bob in the presence of Eve. Alice knows channel statistics but not Eve’s instantaneous CSI, while Bob and Eve know their own CSI.
- Alice sends confidential messages to Bob over a Rayleigh fading channel with Eve present, using fixed transmit power at its maximum level.
- The Alice–Bob and Alice–Eve channel gains undergo independent quasi-static fading, with receiver noise variances specified at both receivers.
- The instantaneous SNRs at Bob and Eve are determined by transmit power, channel-gain magnitude, and the corresponding receiver noise variance.
- Bob and Eve know their individual CSI perfectly, whereas Alice knows both channel statistics but lacks Eve’s instantaneous CSI.
A. Existing Secrecy Outage Formulation
The existing secrecy-outage formulation declares outage when the secrecy capacity falls below the target secrecy rate. Although it captures reliable-and-secure transmission probability, it conflates reliability, security, and suspended transmissions.
- The existing outage event is O(Rs) := {Cs < Rs}, where Rs > 0 is the target secrecy rate.
- Secrecy capacity is Cs = [Cb − Ce]+, the positive part of Bob’s channel capacity minus Eve’s channel capacity.
- An outage includes either an unreliable transmission that Bob cannot decode or a transmission that leaks information to Eve.
- The formulation does not distinguish reliability failures from security failures, so an outage need not mean that perfect secrecy was not achieved.
- A deliberate suspension when Cb < Rs is counted as outage even though it is not a failure to achieve perfect secrecy.
III. ALTERNATIVE SECRECY OUTAGE FORMULATION
The alternative formulation conditions secrecy outage on actual transmission and uses Wyner coding rates to measure failure of perfect secrecy directly. It incorporates transmission design choices and supports security-oriented scheme design.
- Wyner’s encoder selects the transmitted-codeword rate Rb and confidential-information rate Rs, with Re = Rb − Rs representing the secrecy cost.
- Bob decodes correctly when Cb > Rb, while perfect secrecy fails when Eve’s capacity exceeds the redundancy rate, Ce > Re.
- The new secrecy-outage probability is conditioned on a message actually being transmitted rather than counting suspended transmissions.
- The formulation incorporates codeword rate and transmission conditions, enabling explicit security characterization under different legitimate-CSI availability assumptions.
IV. SECURE TRANSMISSION DESIGN
The design maximizes throughput while enforcing minimum security and QoS requirements. It uses the transmission probability as a QoS measure and considers on-off transmission with either full or limited Bob-CSI knowledge.
- Throughput is defined as η = ptxRs, where ptx is the probability of transmission and also represents a QoS measure.
- The design constrains security and transmission probability through minimum requirements ǫ and σ, respectively.The controllable parameters are Rb, Rs, and the transmission condition.
- On-off transmission occurs when Bob’s SNR γb exceeds a threshold µ, subject to µ ≥ 2^Rs − 1.
- Two transmission schemes are designed using either full or limited knowledge of Bob’s instantaneous CSI at Alice.
A. Adaptive Encoder with On-Off Transmission
The adaptive design uses Bob’s instantaneous SNR to select the transmitted-codeword rate while controlling on-off transmission through an optimized threshold. Its threshold exposes a security–QoS trade-off, and the constant confidential rate is a stated scope boundary.
- A. Adaptive Encoder with On-Off Transmission: The adaptive encoder sets Rb arbitrarily close to Bob’s instantaneous capacity Cb using feedback of γb, avoiding decoding errors at Bob.
- A. Adaptive Encoder with On-Off Transmission: A larger on-off threshold µ improves security but compromises QoS.
- A. Adaptive Encoder with On-Off Transmission: The optimization maximizes throughput subject to pso(µ, Rs) ≤ ǫ, ptx(µ) ≥ σ, µ ≥ 2^Rs − 1, and Rs > 0.
- A. Adaptive Encoder with On-Off Transmission: Because pso decreases with µ, the throughput-maximizing threshold is the smallest feasible value satisfying the security constraint.
- A. Adaptive Encoder with On-Off Transmission: The design keeps Rs constant over time rather than adaptively changing it with instantaneous SNR because the adaptive optimization is difficult to obtain in closed form.
B. Non-Adaptive Encoder with On-Off Transmission
The non-adaptive design keeps the codeword rate constant and requires only one feedback bit for on-off transmission. Its optimization minimizes the threshold and selects the largest secure confidential-information rate for each feasible codeword rate.
- B. Non-Adaptive Encoder with On-Off Transmission: The non-adaptive encoder keeps Rb constant over time and uses only 1 bit of feedback to enable on-off transmission.
- B. Non-Adaptive Encoder with On-Off Transmission: Its secrecy outage probability is independent of the on-off threshold µ and equals P(Ce > Rb − Rs).
- B. Non-Adaptive Encoder with On-Off Transmission: For a fixed Rb, minimizing µ maximizes the transmission probability because secrecy outage does not depend on µ.
- B. Non-Adaptive Encoder with On-Off Transmission: The codeword rate must satisfy Rb ≤ log2(1 + γ̄b ln σ^-1) for the QoS constraint to be feasible.
- B. Non-Adaptive Encoder with On-Off Transmission: For each feasible Rb, the throughput-maximizing confidential rate is Rs = Rb − κ, with κ = log2(1 + γ̄e ln ǫ^-1).
V. NUMERICAL RESULTS AND DISCUSSION
The numerical results show that the existing and new secrecy outage formulations can differ significantly, so the existing formulation cannot directly measure security level. Adaptive codeword-rate selection improves throughput, while the non-adaptive design reduces feedback overhead.
- The existing and new formulations predict significantly different secrecy outage probabilities, so the existing formulation cannot directly measure the security level.
- The throughput comparison fixes the QoS constraint at σ = 0.5 and evaluates adaptive and non-adaptive encoder designs across security constraints.
- Arbitrarily low secrecy outage probability is generally impossible, consistent with the lower bounds derived in Section IV.
- Adaptive codeword-rate changes based on Bob’s instantaneous CSI provide a significant throughput improvement.
- The non-adaptive encoder requires only 1 bit of feedback, minimizing feedback overhead.