Optimal strategies in jamming resistant uncoordinated frequency hopping systems

Uncoordinated frequency hopping (UFH) has recently emerged as an effective mechanism to defend against jamming attacks. Existing research focuses on the optimal design of the hopping pattern, which implicitly assumes that the strategy of the attacker is fixed. In practice, the attacker might adjust its strategy to maximize its damage on the communication system. In this paper, we study the design of optimal hopping pattern (the defense strategy) as long as the optimal jamming pattern (the attack strategy). In particular, we model the dynamic between the legitimate users and the attacker as a zero sum game, and study the property of this game. We show that when the legitimate users and the jammer can access only one channel at any time, the game has a unique Nash equilibrium. In the Nash equilibrium, the legitimate users and Eve will access or jam only a subset of channels that have good channel quality. Furthermore, the better the channel, the larger the probability that Eve will jam the channel and the smaller the probability the legitimate users will access this channel. We further extend the study to multiple access multiple jamming case and characterize the Nash equilibrium. We also give numerical results to illustrate the analytical results derived in this paper.