Randomness recycling in constant-round private computations

Abstract

In this paper we study the randomness complexity needed to distributively perform k XOR computation in a t-private way using constant-round protocols. We show that cover-free families allow the recycling of random bits for constant-round private protocols. More precisely, we show that after an 1-round initialization phase during which random bits are distributed among the players, it is possible to perform each of k XOR computations using 2-rounds of communication. In each phase the random bits are used according to a cover-free family and this allows to use each random bit for more than one computation. For t = 2, we design a protocol that uses O(n log k) random bits instead of O(nk) bits if no recycling is performed. More generally, if t > 1 then O(kt2 log n) random bits are sufficient to accomplisht his task, for t = O(n1/2−∊_) for constant ∊ > 0.