The Role of Interaction in Common Randomness and Secret Key Generation

Abstract

In this work we study the problems of common randomness generation (CRG) and secret key generation (SKG). In the CRG problem, two parties, Alice and Bob, receive samples X and Y, respectively, from some joint source distribution μ. The two parties wish to agree on a key consisting of many bits of randomness, by exchanging messages that depend on each party’s respective input and the previous messages. The SKG problem is the same as CRG, except that an eavesdropper who observes the messages must not be able to determine much information about the key. We study the tradeoff between the minimum total length of all messages for a protocol generating a given number of bits of randomness and the minimum possible number of rounds in such a protocol. We construct a source distribution parametrized by positive integers r, n, and l, achieving such a tradeoff in a strong sense: when Alice and Bob can use r + 2 rounds of communication and l ≥ n, they can agree on l bits of entropy by communicating only O(log n) bits, but when they are restricted to r rounds of communication, they require communication of Ω(n^(1/2) / poly log(n)) bits to agree on l bits of entropy. We also prove an analogous result for the setting in which Alice and Bob can amortize, meaning that they receive N i.i.d. samples of (X, Y) from μ, and the communication and key length, respectively, are measured by the ratio of the actual number of bits communicated and the actual key length, respectively, to N.