Communication Complexity with Small Advantage

We study problems in randomized communication complexity when the protocol is only required to attain some small advantage over purely random guessing, i.e., it produces the correct output with probability at least epsilon greater than one over the codomain size of the function. Previously, Braverman and Moitra (STOC 2013) showed that the set-intersection function requires Theta(epsilon n) communication to achieve advantage epsilon. Building on this, we prove the same bound for several variants of set-intersection: (1) the classic "tribes" function obtained by composing with And (provided 1/epsilon is at most the width of the And), and (2) the variant where the sets are uniquely intersecting and the goal is to determine partial information about (say, certain bits of the index of) the intersecting coordinate

A Lower Bound for Sampling Disjoint Sets

Suppose Alice and Bob each start with private randomness and no other input, and they wish to engage in a protocol in which Alice ends up with a set x subseteq[n] and Bob ends up with a set y subseteq[n], such that (x,y) is uniformly distributed over all pairs of disjoint sets. We prove that for some constant beta0 of the uniform distribution over all pairs of disjoint sets of size sqrt{n}

On the Streaming Indistinguishability of a Random Permutation and a Random Function

An adversary with $S$ bits of memory obtains a stream of $Q$ elements that are uniformly drawn from the set $\{1,2,\ldots,N\}$, either with or without replacement. This corresponds to sampling $Q$ elements using either a random function or a random permutation. The adversary\u27s goal is to distinguish between these two cases. This problem was first considered by Jaeger and Tessaro (EUROCRYPT 2019), which proved that the adversary\u27s advantage is upper bounded by $\sqrt{Q \cdot S/N}$. Jaeger and Tessaro used this bound as a streaming switching lemma which allowed proving that known time-memory tradeoff attacks on several modes of operation (such as counter-mode) are optimal up to a factor of $O(\log N)$ if $Q \cdot S \approx N$. However, the bound\u27s proof assumed an unproven combinatorial conjecture. Moreover, if $Q \cdot S \ll N$ there is a gap between the upper bound of $\sqrt{Q \cdot S/N}$ and the $Q \cdot S/N$ advantage obtained by known attacks. In this paper, we prove a tight upper bound (up to poly-logarithmic factors) of $O(\log Q \cdot Q \cdot S/N)$ on the adversary\u27s advantage in the streaming distinguishing problem. The proof does not require a conjecture and is based on a hybrid argument that gives rise to a reduction from the unique-disjointness communication complexity problem to streaming

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum

Communication Complexity with Small Advantage

We study problems in randomized communication complexity when theprotocol is only required to attain some small advantage over purelyrandom guessing, i.e., it produces the correct output withprobability at least ϵ greater than one over the codomainsize of the function. Previously, Braverman and Moitra (in:Proceedings of the 45th symposium on theory of computing (STOC),ACM, pp 161–170, 2013) showed that the set-intersection functionrequires Θ (ϵn) communication to achieve advantageϵ. Building on this, we prove the same bound for severalvariants of set-intersection: (1) the classic “tribes” functionobtained by composing with And (provided 1 / ϵ is at mostthe width of the And), and (2) the variant where the sets areuniquely intersecting and the goal is to determine partialinformation about (say, certain bits of the index of) theintersecting coordinate

Correction to: Communication complexity with small advantage (computational complexity, (2020), 29, 1, (2), 10.1007/s00037-020-00192-w)

Authors would like to correct the incorrect author references in theonline published article