We study the role of interaction in the Common Randomness Generation (CRG) and Secret Key Generation (SKG) problems. In the CRG problem, two players, Alice and Bob, respectively get samples

X_1,X_2,\dots

and

Y_1,Y_2,\dots

with the pairs

(X_1,Y_1)

(X_2, Y_2)

\dots

being drawn independently from some known probability distribution

μ

. They wish to communicate so as to agree on

L

bits of randomness. The SKG problem is the restriction of the CRG problem to the case where the key is required to be close to random even to an eavesdropper who can listen to their communication (but does not have access to the inputs of Alice and Bob). In this work, we study the relationship between the amount of communication and the number of rounds of interaction in both the CRG and the SKG problems. Specifically, we construct a family of distributions

μ= μ_{r, n,L}

, parametrized by integers

r

n

and

L

, such that for every

r

there exists a constant

b = b(r)

for which CRG (respectively SKG) is feasible when

(X_i,Y_i) \sim μ_{r,n,L}

with

r+1

rounds of communication, each consisting of

O(\log n)

bits, but when restricted to

r/2 - 3

rounds of interaction, the total communication must exceed

Ω(n/\log^{b}(n))

bits. Prior to our work no separations were known for

r \geq 2

DOI: 10.48550/arxiv.1808.08907

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