The Multiparty Communication Complexity of Interleaved Group Products

Abstract

Party A_i of k parties A₁,...,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 a₁₁...a_k1a₁₂...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 G², then a pointwise product of s independent copies of X is nearly uniform in G^m, where s depends on m only.