A Resource-Competitive Jamming Defense

Consider a scenario where Alice wishes to send a message $m$ to Bob in a time-slotted wireless network. However, there exists an adversary, Carol, who aims to prevent the transmission of $m$ by jamming the communication channel. There is a per-slot cost of $1$ to send, receive or jam $m$ on the channel, and we are interested in how much Alice and Bob need to spend relative to Carol in order to guarantee communication. Our approach is to design an algorithm in the framework of resource-competitive analysis where the cost to correct network devices (i.e., Alice and Bob) is parameterized by the cost to faulty devices (i.e., Carol). We present an algorithm that guarantees the successful transmission of $m$ and has the following property: if Carol incurs a cost of $T$ to jam, then both Alice and Bob have a cost of $O(T^{\varphi - 1} + 1)=O(T^{.62}+1)$ in expectation, where $\varphi = (1+ \sqrt{5})/2$ is the golden ratio. In other words, it possible for Alice and Bob to communicate while incurring asymptotically less cost than Carol. We generalize to the case where Alice wishes to send $m$ to $n$ receivers, and we achieve a similar result. Our findings hold even if (1) $T$ is unknown to either party; (2) Carol knows the algorithms of both parties, but not their random bits; (3) Carol can jam using knowledge of past actions of both parties; and (4) Carol can jam reactively, so long as there is sufficient network traffic in addition to $m$.

💡 Research Summary

The paper tackles the classic problem of jamming in a time‑slotted wireless network, where a sender Alice wishes to deliver a message m to a receiver Bob while an adversarial jammer Carol attempts to disrupt the communication. Each slot incurs a unit cost for transmitting, receiving, or jamming, and the central question is how much resource (energy, time, or monetary cost) the legitimate parties must expend relative to the jammer’s expenditure in order to guarantee successful delivery.

To answer this, the authors adopt a resource‑competitive analysis framework. Instead of measuring absolute costs or simply bounding the probability of success, they compare the cost incurred by the correct devices (Alice and Bob) directly to the cost incurred by the faulty device (Carol). The goal is to devise a protocol where, if Carol spends T units of cost, Alice and Bob together spend only O(T^{α}) units, with α < 1. This notion of “pay‑less‑than‑the‑jammer” had not been formalized before in the context of wireless jamming.

The system model assumes discrete, equal‑length slots. In any slot, Alice may choose to transmit m, Bob may listen, and Carol may jam; each action costs 1. Carol’s total budget T is unknown to Alice and Bob, and Carol may have full knowledge of their algorithms but not their random bits. Moreover, Carol can observe past actions and even react in real time (reactive jamming), provided there is sufficient background traffic besides m. The authors also extend the model to a multicast setting where Alice must deliver m to n receivers.

The proposed protocol consists of two phases: a search phase and a transmission phase.

Search Phase – In the early slots, Alice and Bob randomly decide, with a modest probability p₁, whether to transmit or listen. This phase serves to probe the channel while keeping the expected cost constant (O(1)). If Carol begins jamming, the protocol quickly transitions to the transmission phase.
Transmission Phase – Here the protocol uses a carefully calibrated exponential schedule. In slot k of this phase, Alice transmits with probability p_k = 2^{-k} (or an equivalent decreasing function), while Bob listens with the same probability. The probabilities are chosen so that the events “Alice transmits and Bob listens” occur with a constant‑order probability in each slot, despite the decreasing transmission rate. Consequently, even if Carol jams every slot, the expected number of slots until a successful transmission is bounded by a function that grows sub‑linearly in T.

Through a rigorous probabilistic analysis employing Markov chains and algebraic inequalities, the authors prove that the expected total cost incurred by Alice and Bob is