On extracting common random bits from correlated sources

TL;DR

This paper investigates the limits and possibilities of extracting common random bits from correlated noisy sources without communication, establishing bounds on agreement probability and proposing strategies for improvement.

Contribution

It provides new theoretical bounds on agreement probability and introduces strategies that approach these bounds for extracting common bits from noisy correlated sources.

Findings

No strategy can surpass the agreement probability of $2^{-k ext{eps}/(1- ext{eps})}$.

A strategy exists achieving an agreement probability close to the theoretical upper bound when $k$ is sufficiently large.

The trivial approach's agreement probability is $(1- ext{eps})^k$, which is significantly lower than the optimal bounds.

Abstract

Suppose Alice and Bob receive strings of unbiased independent but noisy bits from some random source. They wish to use their respective strings to extract a common sequence of random bits with high probability but without communicating. How many such bits can they extract? The trivial strategy of outputting the first $k$ bits yields an agreement probability of $(1 - \eps)^k < 2^{-1.44k\eps}$, where $\eps$ is the amount of noise. We show that no strategy can achieve agreement probability better than $2^{-k\eps/(1 - \eps)}$. On the other hand, we show that when $k \geq 10 + 2 (1 - \eps) / \eps$, there exists a strategy which achieves an agreement probability of $0.1 (k\eps)^{-1/2} \cdot 2^{-k\eps/(1 - \eps)}$.