Interactive Coding with Small Memory and Improved Rate

TL;DR

This work resolves the problem of robust interactive coding under adversarial noise when the communicating parties have small memory. Building on the rewind-if paradigm of H14, the authors introduce a novel small-space protocol that preserves near-optimal rate while using space $O_\ obreak(\log s \cdot \log |\Pi|)$. A key innovation is the combination of a traditional potential-function analysis with a global accounting for sneaky attacks, ensuring progress except for attacks that are themselves bounded in number. The resulting compiler produces a robust protocol $\Pi'$ that tolerates an $\varepsilon$-fraction of adversarial corruptions with rate $R \ge 1 - O\big(\sqrt{\varepsilon \log\log(1/\varepsilon)}\big)$, and runs in time $T' \le T \cdot \mathrm{poly}( |\Pi|/\varepsilon )$, while maintaining memory usage that matches the best known space-bounded results. These advances close a long-standing gap between high-rate interactive coding and memory efficiency, with practical implications for client-server settings and circuit resilience applications.

Abstract

In this work, we study two-party interactive coding for adversarial noise, when both parties have limited memory. We show how to convert any adaptive protocol $Π$ into a protocol $Π'$ that is robust to an $ε$-fraction of adversarial corruptions, not too much longer than $Π$, and which uses small space. More precisely, if $Π$ requires space $\log(s)$ and has $|Π|$ rounds of communication, then $Π'$ requires $O_ε(\log s \log |Π|)$ memory, and has $$|Π'| = |Π|\cdot\left( 1 + O\left( \sqrt{ ε\log \log 1/ε} \right)\right)$$ rounds of communication. The above matches the best known communication rate, even for protocols with no space restrictions.

Figures (2)

• Figure 1: Steps in an attack that the adversary might make when Alice and Bob use only $\mathrm{MP1}$ and $\mathrm{MP2}$ as in H14, exploiting Alice and Bob's limited memory. The red solid line represents Alice's simulated path; the blue dashed line represents Bob's simulated path. The ovals represent the meeting points that Alice and Bob remember. In steps 2 and 3, the adversary invests an amount of corruptions that is commensurate with the amount that Alice and Bob rewind, which is good. But in steps 4 and 5, the adversary invests few corruptions, and can make Alice and Bob rewind very far back, which is not good.
• Figure 2: Steps in a sneaky attack. The red solid line represents Alice's simulated path; the blue dashed line represents Bob's simulated path. The ovals represent the meeting points that Alice and Bob remember. In Steps 2 and 3, the adversary invests an amount of corruptions that is commensurate with the amount that Alice has to rewind, which is good. But in Step 4, the adversary only has to invest a few corruptions (enough to make Bob think that he and Alice are out of sync and make a short rewind); and then in Step 5, Alice and Bob may rewind a lot.

Theorems & Definitions (155)

• Theorem 1
• Remark 1: Speaking order in Theorem \ref{['thm:main']}
• Remark 2: Where are the inputs?
• Remark 3: What about speaking order?
• Remark 4
• Remark 5
• Definition 1: MP set, scale-$j$ MPs
• Definition 2: Scale-$j$ transition candidates
• Definition 4
• Lemma 2: HR18