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High entropy random selection protocols. (Extended abstract). (English) Zbl 1171.68504

Charikar, Moses (ed.) et al., Approximation, randomization, and combinatorial optimization. Algorithms and techniques. 10th international workshop, APPROX 2007, and 11th international workshop, RANDOM 2007, Princeton, NJ, USA, August 20–22, 2007. Proceedings. Berlin: Springer (ISBN 978-3-540-74207-4/pbk). Lecture Notes in Computer Science 4627, 366-379 (2007).
Summary: We study the two party problem of randomly selecting a string among all the strings of length \(n\). We want the protocol to have the property that the output distribution has high entropy, even when one of the two parties is dishonest and deviates from the protocol. We develop protocols that achieve high, close to \(n\), entropy.
In the literature the randomness guarantee is usually expressed as being close to the uniform distribution or in terms of resiliency. The notion of entropy is not directly comparable to that of resiliency, but we establish a connection between the two that allows us to compare our protocols with the existing ones.
We construct an explicit protocol that yields entropy \(n - O(1)\) and has \(4\log ^{*} n\) rounds, improving over the protocol of O. Goldreich et al. [“Fault-tolerant computation in the full information model”, SIAM J. Comput. 27, No. 2, 506–544 (1998; Zbl 0912.68037)] that also achieves this entropy but needs \(O(n)\) rounds. Both these protocols need \(O(n ^{2})\) bits of communication.
Next we reduce the communication in our protocols. We show the existence, non-explicitly, of a protocol that has 6 rounds, \(2n + 8\log n\) bits of communication and yields entropy \(n - O(\log n)\) and min-entropy \(n/2 - O(\log n)\). Our protocol achieves the same entropy bound as the recent, also non-explicit, protocol of R. Gradwohl et al. [“Random selection with an adversarial majority”, Lect. Notes Comput. Sci. 4117, 409–426 (2006; Zbl 1161.68568)], however achieves much higher min-entropy: \(n/2 - O(\log n)\) versus \(O(\log n)\).
Finally we exhibit very simple explicit protocols. We connect the security parameter of these geometric protocols with the well studied Kakeya problem motivated by harmonic analysis and analytical number theory. We are only able to prove that these protocols have entropy \(3n/4\) but still \(n/2 - O(\log n)\) min-entropy. Therefore they do not perform as well with respect to the explicit constructions of Gradwohl et al. [loc. cit.] entropy-wise, but still have much better min-entropy. We conjecture that these simple protocols achieve \(n - o(n)\) entropy. Our geometric construction and its relation to the Kakeya problem follows a new and different approach to the random selection problem than any of the previously known protocols.
For the entire collection see [Zbl 1123.68005].

MSC:

68P30 Coding and information theory (compaction, compression, models of communication, encoding schemes, etc.) (aspects in computer science)
68Q10 Modes of computation (nondeterministic, parallel, interactive, probabilistic, etc.)
68Q11 Communication complexity, information complexity
68W20 Randomized algorithms
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