On Extracting Common Random Bits From Correlated Sources on Large Alphabets

• Published in: IEEE transactions on information theory 2014-03, Vol.60 (3), p.1630-1637
• Main Authors: Siu On Chan, Mossel, Elchanan, Neeman, Joe
• Format: Article
• Language: English
• Subjects: Algorithms; Applied sciences; Correlation; Correlation analysis; Exact sciences and technology; hypercontractivity; Information theory; Information, signal and communications theory; Joints; Noise; Noise measurement; Probability distribution; Protocols; Randomness extraction; symmetric channels; Telecommunications and information theory; Upper bound
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2014.2301155

Suppose Alice and Bob receive strings X=(X 1 ,...,X n ) and Y=(Y 1 ,...,Y n ) each uniformly random in [s] n , but so that X and Y are correlated. For each symbol i, we have that Y i =X i with probability 1-ε and otherwise Y i is chosen independently and uniformly from [s]. Alice and Bob wish to use their respective strings to extract a uniformly chosen common sequence from [s] k , but without communicating. How well can they do? The trivial strategy of outputting the first k symbols yields an agreement probability of (1-ε+ε/s) k . In a recent work by Bogdanov and Mossel, it was shown that in the binary case where s=2 and k=k(ε) is large enough then it is possible to extract k bits with a better agreement probability rate. In particular, it is possible to achieve agreement probability (kε) -1/2 ·2 -kε/(2(1-ε/2)) using a random construction based on Hamming balls, and this is optimal up to lower order terms. In this paper, we consider the same problem over larger alphabet sizes s and we show that the agreement probability rate changes dramatically as the alphabet grows. In particular, we show no strategy can achieve agreement probability better than (1-ε) k (1+δ(s)) k where δ(s)→ 0 as s→∞. We also show that Hamming ball-based constructions have much lower agreement probability rate than the trivial algorithm as s→∞. Our proofs and results are intimately related to subtle properties of hypercontractive inequalities.