The Multiparty Communication Complexity of Interleaved Group Products

Party A
_i
of k parties A
₁
,...,A
_k
receives on its forehead a t-tuple (a
_i1
,...,a
_it
) of elements from the group G = SL(2, q). The parties are promised that the interleaved product a
₁₁
...a
_k1
a
₁₂
...a
_k2
...a
_1t
...a
_kt
is equal either to the identity e or to some other fixed element g ∈ G. Their goal is to determine which of e and g the interleaved product is equal to, using the least amount of communication. We show that for all fixed k and all sufficiently large t the communication is Ω(t log |G|), which is tight. As an application, we establish the security of the leakage-resilient circuits studied by Miles and Viola (STOC 2013) in the "only computation leaks" model. Our main technical contribution is of independent interest. We show that if X is a probability distribution on G
^m
such that any two coordinates are uniform in G
²
, then a pointwise product of s independent copies of X is nearly uniform in G
^m
, where s depends on m only.