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Jamming Games in the MIMO Wiretap Channel With an Active Eavesdropper

Amitav Mukherjee, A. Lee Swindlehurst

arXiv:1011.5274v2cs.IT

TL;DR

The transmitter must establish a reliable communication link robust to potential jamming. The paper models the transmitter–eavesdropper interaction as a zero-sum game, derives equilibria across strategic and sequential settings, and finds that parameter changes can shift equilibrium outcomes between pure and mixed strategies.

  • Problem

    The transmitter faces the problem of establishing a reliable communication link to the receiver that is robust to potential jamming.

  • Method

    The paper formulates the multi-antenna transmitter and dual-mode eavesdropper/jammer as a zero-sum game and derives equilibrium policies for pure, mixed, strategic, and extensive-form settings.

  • Results

    Numerical results show that changing a single parameter set while others remain constant can shift the equilibrium from a pure to a mixed Nash equilibrium, or vice versa.

  • Takeaways & Limitations

    Equilibrium behavior depends sensitively on parameter settings, with the same game potentially producing pure or mixed Nash equilibrium outcomes.

Abstract

from arXiv · show

This paper investigates reliable and covert transmission strategies in a multiple-input multiple-output (MIMO) wiretap channel with a transmitter, receiver and an adversarial wiretapper, each equipped with multiple antennas. In a departure from existing work, the wiretapper possesses a novel capability to act either as a passive eavesdropper or as an active jammer, under a half-duplex constraint. The transmitter therefore faces a choice between allocating all of its power for data, or broadcasting artificial interference along with the information signal in an attempt to jam the eavesdropper (assuming its instantaneous channel state is unknown). To examine the resulting trade-offs for the legitimate transmitter and the adversary, we model their interactions as a two-person zero-sum game with the ergodic MIMO secrecy rate as the payoff function. We first examine conditions for the existence of pure-strategy Nash equilibria (NE) and the structure of mixed-strategy NE for the strategic form of the game.We then derive equilibrium strategies for the extensive form of the game where players move sequentially under scenarios of perfect and imperfect information. Finally, numerical simulations are presented to examine the equilibrium outcomes of the various scenarios considered.

I. INTRODUCTION

The paper studies a MIMO wiretap channel with an adversary that can either eavesdrop or jam, forcing the transmitter to balance reliable delivery against confidentiality. It formulates these opposing choices as a zero-sum game and analyzes equilibrium behavior in strategic and sequential settings.

  • Background: The MIMO wiretap channel extends secrecy-capacity analysis to multiple-antenna transmitters, receivers, and eavesdroppers.A non-zero secrecy capacity requires the eavesdropper’s channel to be lower quality than the intended recipient’s.
  • Problem setting: Because the eavesdropper’s instantaneous channel state is unavailable, the transmitter combines secret-message transmission with optional artificial interference.The interference is intended to degrade the eavesdropper’s channel.
  • Problem setting: The adversary can passively eavesdrop or actively jam, creating a trade-off between protecting secrecy and maintaining a reliable legitimate link.The adversary operates under a half-duplex constraint.
  • Game formulation: The interaction is modeled as a two-player zero-sum game whose payoff is the ergodic MIMO secrecy rate.The formulation reflects the players’ mutually opposite interests.
  • Contributions: The paper characterizes pure-strategy Nash equilibria, optimal mixed strategies, and sequential equilibrium outcomes under perfect and imperfect information.It also studies the resulting equilibrium outcomes through numerical simulations.

II. SYSTEM MODEL

The system comprises multi-antenna Alice, Bob, and Eve, with Eve choosing between listening and jamming. Alice can allocate power entirely to data or split it with artificial interference designed to spare Bob while degrading unintended receivers.

  • Network model: The model contains Alice with Na antennas, Bob with Nb antennas, and Eve with Ne antennas.Eve can either eavesdrop on Alice or jam Bob under a half-duplex constraint.
  • Alice’s strategy: Alice either uses all available power for data or splits it between the information vector and artificial interference.The interference is transmitted to jam unintended receivers other than Bob.
  • Power allocation: Eve uses her full available power Pe when jamming, while Alice uses her full power Pa across data and artificial-interference components.The model imposes maximum power constraints on both players.
  • Artificial interference: Artificial interference is precoded to be orthogonal to Bob’s received information signal when Alice knows the Alice–Bob channel.The precoding matrices can be selected from disjoint right singular vectors of the legitimate channel.
  • Single-antenna boundary: With a single transmit antenna, artificial interference cannot be eliminated at Bob, so it degrades both Bob’s and Eve’s SNR and restricts Alice’s strategy.The paper therefore associates the useful interference scheme with the multi-antenna setting.

B. CSI Model

Alice lacks Eve’s instantaneous CSI and relies only on its statistical distribution, while receiver-side CSI is available for the corresponding legitimate or eavesdropping link. These assumptions lead Eve and Alice to use uniform jamming across their available transmit dimensions.

  • CSI availability: Alice knows Hba for precoding, and Bob and Eve know their respective channel and interference-plus-noise information.Other CSI is treated as non-informative beyond independent CN(0, 1) channel entries.
  • Jamming model: Eve’s lack of Hbe knowledge and half-duplex operation prevent correlated jamming, leading her to distribute jamming uniformly across Ne dimensions.Alice’s artificial interference is likewise uniformly distributed across the Na−d available dimensions.
  • Power allocation: For simplicity and robustness, Alice also distributes information-signal power uniformly rather than performing power loading.This assumption is used to simplify the resulting expressions.

C. Secrecy Rates and Transmit Strategies

The section defines ergodic MIMO secrecy rates for Alice’s transmission choices and Eve’s eavesdropping or jamming actions. Under statistical CSI for Eve, the analysis reduces each player’s relevant options to two strategies.

  • Ergodic secrecy rate averages over channel matrices because Alice and Eve lack the complete instantaneous CSI needed for instantaneous rates.Alice has instantaneous CSI for Bob but only statistical CSI for Eve.
  • Alice chooses between full-power information transmission and artificial interference, while Eve chooses passive eavesdropping or full-power jamming.The resulting actions are denoted X = {F, A} for Alice and Y = {E, J} for Eve.
  • The four action combinations produce secrecy rates R_i,k, with i ∈ {F, A} and k ∈ {E, J}.When Eve jams, her mutual information is zero, so the effective secrecy rates are R_FJ and R_AJ.
  • Artificial interference must use a power fraction and number of dimensions chosen without knowing Eve’s strategy.The selected parameters are held fixed across Eve’s possible actions.
  • When Eve jams, Alice’s code rate may require adjustment based on Eve’s action, assumed observable through feedback from Bob.The paper treats minor coding and decoding adaptations as rapid relative to the ergodic-rate timescale.
  • Artificial interference cannot improve Bob’s mutual information, while Alice’s optimized artificial-interference rate is at least as large as her full-power eavesdropping rate.These properties support the later reduction to two relevant strategies.

III. STRATEGIC WIRETAP GAME

The paper formulates Alice’s and Eve’s interaction as a zero-sum game whose payoff is the achievable MIMO secrecy rate. Although the underlying action spaces are continuous, best-response properties reduce the strategic analysis to two actions per player.

  • Alice maximizes the secrecy-rate payoff while Eve minimizes it in a strictly competitive zero-sum formulation.Alice optimizes transmission parameters, whereas Eve chooses jamming power, including zero power for passive eavesdropping.
  • The continuous strategy spaces contain Alice’s pair (d, ρ) and Eve’s jamming power P_j ∈ [0, P_e].Alice chooses d ∈ {1, …, r} and ρ ∈ [0, 1].
  • Alice’s relevant best responses are full-power transmission or artificial-interference transmission, while Eve’s are zero or full jamming power.These choices correspond to Alice’s F and A and Eve’s E and J actions.
  • Therefore, the game can be analyzed using X = {F, A} and Y = {E, J}.All other pure strategies receive zero probability in an optimal mixed strategy.

A. Pure-strategy Equilibria

The simultaneous strategic game is represented by a 2 × 2 secrecy-rate payoff matrix. Its pure-strategy equilibrium depends on the ordering of the four resulting secrecy rates rather than on one universally optimal action.

  • The simultaneous game uses a 2 × 2 payoff matrix and establishes conditions for Nash equilibria.The matrix represents Alice’s and Eve’s actions when neither observes the other’s move.
  • For arbitrary antenna sizes, transmit powers, and channel gains, the game has unique pure-strategy saddle-points or Nash equilibria under specified rate orderings.Only two of the six valid orderings produce a pure equilibrium.
  • If R_AE ≤ R_AJ, the corresponding rate outcome is a Nash equilibrium because neither player can improve its payoff by deviating.The equilibrium condition follows from the rate ordering together with properties P1 and P2.

B. Mixed-strategy Equilibria

When no pure equilibrium exists, the zero-sum game has a unique mixed equilibrium determined by the four secrecy rates. The same single-stage equilibrium strategies also govern repeated play, including finite horizons.

  • Mixed-strategy equilibria: Four of the six valid rate orderings require mixed strategies because no single pure strategy is always optimal.The finite zero-sum game has a saddle point in randomized strategies.
  • Mixed-strategy equilibria: The mixed equilibrium is unique, with probabilities and game value determined by the four payoff entries and D = R_FE + R_AJ − R_FJ − R_AE.The uniqueness follows from the unique solution of the matrix-game optimization.
  • Mixed-strategy equilibria: For N_a = 5, N_b = 3, N_e = 4, d = 2, P_a = P_e = 20 dB, g_1 = 1.1, and g_2 = 0.9, the rates are R_AE = 5.04 > R_FJ = 5.02 > R_AJ = 2.85 > R_FE = 0.These rates yield p* = 0.307, q* = 0.294, and game value v = 3.45.
  • Mixed-strategy equilibria: Alice favors artificial interference because it guarantees a secrecy rate of at least 2.85, while full-power transmission risks a payoff of zero.Eve correspondingly favors jamming because it moves the game value toward R_AJ.
  • Repeated game: Repeated play preserves the single-stage Nash equilibrium strategies for both infinite and finite horizons.The result follows from the zero-sum structure and stagewise minimization of Alice’s payoff.
  • Repeated game: With imperfect observations of the opponent’s actions, Alice and Eve should randomize over X × Y.The same mixed-strategy framework applies to the more involved repeated-game setting.

IV. EXTENSIVE FORM WIRETAP GAME

The paper models sequential MIMO wiretap interactions as an extensive-form game, analyzing equilibrium outcomes when players move under perfect information. Backward induction yields subgame-perfect equilibrium rates and strategies for either player moving first.

  • IV. EXTENSIVE FORM WIRETAP GAME: The extensive-form MIMO wiretap game models one player moving first, an opponent responding, and possible subsequent strategy or code-rate changes.
  • IV. EXTENSIVE FORM WIRETAP GAME: The analysis uses rooted game trees and backward induction to derive equilibria for sequential interactions with perfect and imperfect information.
  • A. Perfect Information: With perfect information, Eve observes Alice’s move and responds at her decision node, producing three-subgame game trees.
  • A. Perfect Information: Information states identify the decision nodes conditioned on knowledge of the opponent’s previous move, while subgames contain a single-node information state and its successors.
  • A. Perfect Information: A subgame-perfect equilibrium is obtained by eliminating irrational subgame choices and applying backward induction to the extensive game tree.
  • A. Perfect Information: When Alice moves first, the unique pure-strategy equilibrium rate is selected from the relevant rate expressions according to their ordering conditions.
  • A. Perfect Information: When Eve moves first, Alice responds with her subgame-optimal actions, and Eve chooses the action associated with the smaller resulting payoff.
  • A. Perfect Information: Both sequential game orders reproduce the corresponding pure-strategy Nash equilibrium whenever the strategic game has such an equilibrium.

B. Imperfect Information

The imperfect-information analysis replaces direct observation with beliefs about the opponent’s move. Sequentially rational responses are computed from posterior probabilities, while players without causal belief knowledge retain minimax or mixed-strategy behavior.

  • B. Imperfect Information: Imperfect-information games require sequential equilibrium concepts because the second-moving player cannot directly observe the first move.
  • B. Imperfect Information: When Eve cannot determine Alice’s first-stage action, she assigns prior probabilities to Alice’s moves and randomizes over passive listening and jamming.
  • B. Imperfect Information: Eve’s information state can support pure listening, pure jamming, or randomization over both actions.
  • B. Imperfect Information: Probabilistic backward induction produces a sequential equilibrium equivalent to the strategic game’s mixed-strategy Nash equilibrium in the stated no-information case.
  • B. Imperfect Information: A belief vector represents posterior probabilities about the opponent’s move and can be estimated from signal samples sent through public feedback.
  • B. Imperfect Information: Bob’s minimum-probability-of-error detector tests whether Eve is listening or jamming using channel-based observations and assumed priors.
  • B. Imperfect Information: Alice computes posterior beliefs with Bayes’ rule and chooses the action whose expected payoff is optimal given those beliefs.
  • B. Imperfect Information: Without causal knowledge of Eve’s beliefs, Eve retains minimax behavior and Alice uses her maximin strategy when moving first.

V. SIMULATION RESULTS

Simulations show that equilibrium type and payoff depend strongly on transmit powers, antenna ratios, move order, and information quality. The results compare pure and mixed strategic equilibria with sequential outcomes under perfect and imperfect information.

  • V. SIMULATION RESULTS: The simulations evaluate equilibrium secrecy-rate payoffs across channel, user, power, and information configurations using numerically computed results.
  • V. SIMULATION RESULTS: As Alice’s power increases, the pure Nash equilibrium disappears because RAE grows faster than RAJ.
  • V. SIMULATION RESULTS: Under Alice-and-Eve jamming, increased Alice power improves Bob’s rate and reduces Eve’s rate, whereas under Alice jamming it only improves Bob’s rate.
  • V. SIMULATION RESULTS: For equal transmit powers and the stated antenna configuration, the equilibrium changes near Ne/Na = 1 from a pure-strategy saddle point to a mixed-strategy equilibrium.
  • V. SIMULATION RESULTS: In sequential games, Alice moving second is always beneficial, particularly as Eve’s jamming power increases.
  • V. SIMULATION RESULTS: The extensive-form simulations assume transmission parameters are chosen independently of Eve’s actions, despite allowing such adjustment in the general game.
  • V. SIMULATION RESULTS: With Pe = 2Pa, imperfect-information hypothesis testing significantly improves Alice’s payoff over the no-information case when she receives a noisy observation of Eve’s move.
  • V. SIMULATION RESULTS: As Pa increases, Bob’s inference improves because Eve’s power also increases, while Eve’s inference does not improve because the data-to-artificial-noise ratio remains nearly constant.

VI. CONCLUSION

The paper formulates the transmitter–wiretapper interaction as a zero-sum game and analyzes equilibria across strategic and extensive forms. Numerical results show that changing one parameter set can switch the equilibrium between pure and mixed Nash outcomes.

  • The transmitter and dual-mode eavesdropper are modeled as a zero-sum game whose payoff is the ergodic MIMO secrecy rate.
  • The paper derives equilibrium conditions and optimal policies for pure and mixed strategies in the strategic game.
  • It analyzes subgame-perfect and sequential equilibria for extensive-form games with and without perfect information.
  • Changing a single parameter set while holding the others constant can shift the equilibrium from a pure Nash outcome to a mixed one, or the reverse.

Eve

The paper examines strategic and sequential MIMO wiretap games involving Alice and Eve, including mixed strategies, antenna-ratio effects, and perfect-information move orders.

  • Strategic form: The strategic game is represented by a payoff matrix for Alice and Eve.The matrix defines the strategic-form MIMO wiretap game.
  • Mixed strategies: Mixed-strategy analysis varies Alice’s and Eve’s mixing probabilities to evaluate the game value.The example uses Na = 5, Nb = 3, Ne = 4, and d = 2.
  • Extensive form: Perfect-information extensive-form games consider Alice moving first or Eve moving first.The two game trees are labeled Γe,1 and Γe,2, respectively.
  • Strategic form: The strategic game is evaluated for Pe = 4Pa with Na = Ne = 8, Nb = 6, and d = 4.The figure also specifies g1 = 1.2 and g2 = 0.75.
  • Numerical scenarios: Additional evaluations vary the eavesdropper-to-transmitter antenna ratio and examine extensive-form games under specified antenna and power settings.The settings include fixed Pe = Pa = 100 in one case and Na = Nb = Ne = 3 with Pa = 20dB in another.
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