Defeating Jamming With the Power of Silence A Game-Theoretic Analysis
The timing channel is a logical communication channel in which information is encoded in the timing between events. Recently, the use of the timing channel has been proposed as a countermeasure to reactive jamming attacks performed by an energy- constrained malicious node. In fact, while a jammer is able to disrupt the information contained in the attacked packets, timing information cannot be jammed, and therefore, timing channels can be exploited to deliver information to the receiver even on a jammed channel. Since the nodes under attack and the jammer have conflicting interests, their interactions can be modeled by means of game theory. Accordingly, in this paper, a game-theoretic model of the interactions between nodes exploiting the timing channel to achieve resilience to jamming attacks and a jammer is derived and analyzed. More specifically, the Nash equilibrium is studied in terms of existence, uniqueness, and convergence under best response dynamics. Furthermore, the case in which the communication nodes set their strategy and the jammer reacts accordingly is modeled and analyzed as a Stackelberg game, by considering both perfect and imperfect knowledge of the jammer’s utility function. Extensive numerical results are presented, showing the impact of network parameters on the system performance.
A timing channel is a communication channel which exploits silence intervals between consecutive transmissions to encode information. Recently, use of timing channels has been proposed in the wireless domain to support low rate, energy efficient communications as well as covert and resilient communications Timing channels are more although not totally immune from reactive jamming attacks. In fact, the interfering signal begins its disturbing action against the communication only after identifying an ongoing transmission, and thus after the timing information has been decoded by the receiver.
Timing channel-based communication scheme has been proposed to counteract jamming by establishing a low rate physical layer on top of the traditional physical/link layers using detection and timing of failed packet receptions at the receiver.
The energy cost of jamming the timing channel and the resulting trade-offs have been analyzed. The interactions between the jammer and the node whose transmissions are under attack, which we call target node.
Specifically, assume that the target node wants to maximize the amount of information that can be transmitted per unit of time by means of the timing channel, whereas, the jammer wants to minimize such amount of information while reducing the energy expenditure.
The target node and the jammer have conflicting interests; we develop a game theoretical framework that models their interactions. We investigate both the case in which these two adversaries play their strategies.
The situation when the target node (the leader) anticipates the actions of the jammer (the follower). To this purpose, we study both the Nash Equilibria (NEs) and Stackelberg Equilibria (SEs) of our proposed games.
Recently, use of timing channels has been proposed in the wireless domain to support low rate, energy efficient communications as well as covert and resilient communications. In existing system methodologies to detect jamming attacks are illustrated; it is also shown that it is possible to identify which kind of jamming attack is ongoing by looking at the signal strength and other relevant network parameters, such as bit and packet errors. Several solutions against reactive jamming have been proposed that exploit different techniques, such as frequency hopping, power control and UN jammed bits.
- Continuous jamming is very costly in terms of energy consumption for the jammer
- Existing solutions usually rely on users’ cooperation and coordination, which might not be guaranteed in a jammed environment. In fact, the reactive jammer can totally disrupt each transmitted packet and, consequently, no information can be decoded and then used to this purpose.
Our proposed system implementation focus on the resilience of timing channels to jamming attacks. In general, these attacks can completely disrupt communications when the jammer continuously emits a high power disturbing signal, i.e., when continuous jamming is performed.
Analyze the interactions between the jammer and the node whose transmissions are under attack, which we call target node. Specifically, we assume that the target node wants to maximize the amount of information that can be transmitted per unit of time by means of the timing channel, whereas, the jammer wants to minimize such amount of information while reducing the energy expenditure.
As the target node and the jammer have conflicting interests, we develop a game theoretical framework that models their interactions. We investigate both the case in which these two adversaries play their strategies simultaneously and the situation when the target node (the leader) anticipates the actions of the jammer (the follower). To this purpose, we study both the Nash Equilibria (NEs) and Stackelberg Equilibria (SEs) of our proposed games.
- System model the interactions between a jammer and a target node as a jamming game
- We prove the existence, uniqueness and convergence to the Nash equilibrium (NE) under best response dynamics
- We prove the existence and uniqueness of the equilibrium of the Stackelberg game where the target node plays as a leader and the jammer reacts consequently
- We investigate in this latter Stackelberg scenario the impact on the achievable performance of imperfect knowledge of the jammer’s utility function;
- We conduct an extensive numerical analysis which shows that our proposed models well capture the main factors behind the utilization of timing channels, thus representing a promising framework for the design and understanding of such systems.
NASH Equilibrium Analysis:
The Nash Equilibrium points (NEs), in which both players achieve their highest utility given the strategy profile of the opponent. In the following we also provide proofs of the existence, uniqueness and convergence to the Nash Equilibrium under best response dynamics.
Existence of the Nash Equilibrium:
It is well known that the intersection points between bT(y) and bJ(x)are the NEs of the game. Therefore, to demonstrate the existence of at least one NE, it suffices to prove that bT(y) and bJ(x) have one or more intersection points. In other words, it is sufficient to find one or more pairs.
Uniqueness of the Nash Equilibrium:
After proving the NE existence in Theorem, let us prove the uniqueness of the NE, that is, there is only one strategy profile such that no player has incentive to deviate unilaterally.
Convergence to the Nash Equilibrium:
Analyze the convergence of the game to the NE when players follow Best Response Dynamics (BRD). In BRD the game starts from any initial point(x(0),y(0))∈Sand, at each successive step, each player plays its strategy by following its best response function.
The game allows the leader to achieve a utility which is atleast equal to the utility achieved in the ordinary game at the NE, if we assume perfect knowledge, that is, the target node is completely aware of the utility function of the jammer and its parameters, and thus it is able to evaluate bJ(x). Whereas, if some parameters in the utility function of the jammer are unknown at the target node
Our system implementation proposed a game-theoretic model of the interactions between a jammer and a communication node that exploits a timing channel to improve resilience to jamming attacks. Structural properties of the utility functions of the two players have been analyzed and exploited to prove the existence and uniqueness of the Nash Equilibrium. The convergence of the game to the Nash Equilibrium has been studied and proved by analyzing the best response dynamics. Furthermore, as the reactive jammer is assumed to start transmitting its interference signal only after detecting activity of the node under attack, a Stackelberg game has been properly investigated, and proofs on the existence and uniqueness of the Stackelberg Equilibrium has been provided.