The benefit of a 1-bit jump-start, and the necessity of stochastic encoding, in jamming channels

TL;DR

This paper explores how a one-bit delay in a malicious jammer's decision process and stochastic encoding at the transmitter can significantly increase communication capacity in adversarial channels.

Contribution

It introduces a novel code construction demonstrating capacity increase with one-bit delay and stochastic encoding, and a new jamming strategy showing deterministic encoding's limitations.

Findings

Capacity increases from 1-2p to 1-p with one-bit delay and stochastic encoding.

Deterministic encoding cannot surpass rate 1-2p in the one-bit-delay setting.

Stochastic encoding is crucial for achieving higher capacity against delayed jamming.

Abstract

We consider the problem of communicating a message $m$ in the presence of a malicious jamming adversary (Calvin), who can erase an arbitrary set of up to $pn$ bits, out of $n$ transmitted bits $(x_1,\ldots,x_n)$. The capacity of such a channel when Calvin is exactly causal, i.e. Calvin's decision of whether or not to erase bit $x_i$ depends on his observations $(x_1,\ldots,x_i)$ was recently characterized to be $1-2p$. In this work we show two (perhaps) surprising phenomena. Firstly, we demonstrate via a novel code construction that if Calvin is delayed by even a single bit, i.e. Calvin's decision of whether or not to erase bit $x_i$ depends only on $(x_1,\ldots,x_{i-1})$ (and is independent of the "current bit" $x_i$) then the capacity increases to $1-p$ when the encoder is allowed to be stochastic. Secondly, we show via a novel jamming strategy for Calvin that, in the single-bit-delay setting, if the encoding is deterministic (i.e. the transmitted codeword is a deterministic function of the message $m$) then no rate asymptotically larger than $1-2p$ is possible with vanishing probability of error, hence stochastic encoding (using private randomness at the encoder) is essential to achieve the capacity of $1-p$ against a one-bit-delayed Calvin.