[Paper Review] A characterization of the capacity of online (causal) binary channels

Zitan Chen, Sidharth Jaggi+1 more|arXiv (Cornell University)|Dec 19, 2014

Wireless Communication Security Techniques8 references3 citations

TL;DR

This paper provides a tight characterization of the capacity of binary online (causal) channels under both bit-flip and erasure corruption models, where the adversary decides to corrupt each bit based only on previously transmitted bits. The authors propose a novel coding scheme that achieves the previously conjectured upper bounds, proving the capacity is $ C^{flip}_p = \min_{\bar{p} \in [0,p]} \left[ (1 - 4(p - \bar{p})) \left(1 - H\left(\frac{\bar{p}}{1 - 4(p - \bar{p})}\right)\right) \right] $ for $ p \leq 1/4 $, and $ C^{erase}_p = 1 - 2p $ for $ p \leq 1/2 $, with zero capacity beyond these thresholds.

ABSTRACT

In the binary online (or "causal") channel coding model, a sender wishes to communicate a message to a receiver by transmitting a codeword $\mathbf{x} =(x_1,\ldots,x_n) \in \{0,1\}^n$ bit by bit via a channel limited to at most $pn$ corruptions. The channel is "online" in the sense that at the $i$th step of communication the channel decides whether to corrupt the $i$th bit or not based on its view so far, i.e., its decision depends only on the transmitted bits $(x_1,\ldots,x_i)$. This is in contrast to the classical adversarial channel in which the error is chosen by a channel that has a full knowledge on the sent codeword $\mathbf{x}$. In this work we study the capacity of binary online channels for two corruption models: the {\em bit-flip} model in which the channel may flip at most $pn$ of the bits of the transmitted codeword, and the {\em erasure} model in which the channel may erase at most $pn$ bits of the transmitted codeword. Specifically, for both error models we give a full characterization of the capacity as a function of $p$. The online channel (in both the bit-flip and erasure case) has seen a number of recent studies which present both upper and lower bounds on its capacity. In this work, we present and analyze a coding scheme that improves on the previously suggested lower bounds and matches the previously suggested upper bounds thus implying a tight characterization.

Motivation & Objective

• To resolve the open problem of characterizing the exact capacity of binary causal (online) channels under adversarial bit-flip and erasure models.
• To close the gap between previously known upper and lower bounds on the capacity of causal channels by constructing a coding scheme that achieves the upper bounds.
• To analyze the robustness of the capacity under limited lookahead by the adversary, showing minimal degradation even with small lookahead.
• To provide a rigorous achievability proof for the capacity expressions using a structured coding scheme with list decoding and secret sharing.

Proposed method

• Propose a novel coding scheme that partitions the codeword into left and right mega sub-codewords, using a two-phase decoding process.
• Use a secret-sharing mechanism with random secrets to construct the right mega sub-codeword, ensuring resilience against adversarial choices.
• Apply list decoding in the right sub-codeword, where the list size is bounded and the correct message is recovered with high probability.
• Define a consistency check at a designated decoding point $ t^* $, ensuring the correct message is selected with high probability over random secrets.
• Use probabilistic analysis to bound the failure probability, showing decoding succeeds with probability at least $ 1 - 2^{-n\theta^3/32} $.
• Prove that even with $ n\epsilon $-lookahead, the capacity is only marginally reduced, showing robustness to limited adversary foresight.

Experimental results

Research questions

• RQ1 What is the exact capacity of the binary causal bit-flip channel as a function of the error fraction $ p $?
• RQ2 What is the exact capacity of the binary causal erasure channel as a function of the erasure fraction $ p $?
• RQ3 Can a coding scheme be constructed that achieves the known upper bounds on the capacity of causal channels?
• RQ4 How does the capacity of causal channels change if the adversary has limited lookahead (e.g., $ n\epsilon $)?

Key findings

• The capacity of the binary causal bit-flip channel is $ C^{flip}_p = \min_{\bar{p} \in [0,p]} \left[ (1 - 4(p - \bar{p})) \left(1 - H\left(\frac{\bar{p}}{1 - 4(p - \bar{p})}\right)\right) \right] $ for $ p \leq 1/4 $, and zero for $ p \geq 1/4 $.
• The capacity of the binary causal erasure channel is $ C^{erase}_p = 1 - 2p $ for $ p \leq 1/2 $, and zero for $ p \geq 1/2 $.
• The proposed coding scheme achieves the known upper bounds, thereby proving the capacity expressions are tight.
• The scheme ensures decoding success with probability at least $ 1 - 2^{-n\theta^3/32} $, which approaches 1 as $ n $ increases.
• Even with $ n\epsilon $-lookahead, the capacity is only reduced by a continuous function $ f(\epsilon) $, showing robustness to limited adversary foresight.
• The analysis confirms that the capacity remains unchanged under small lookahead, implying the causal model is stable under mild deviations from strict causality.