Communication Complexity of Permutation-Invariant Functions

Motivated by the quest for a broader understanding of communication complexity of simple functions, we introduce the class of "permutation-invariant" functions. A partial function $f:\{0,1\}^n \times \{0,1\}^n\to \{0,1,?\}$ is permutation-invariant if for every bijection $\pi:\{1,\ldots,n\} \to \{1,\ldots,n\}$ and every $\mathbf{x}, \mathbf{y} \in \{0,1\}^n$, it is the case that $f(\mathbf{x}, \mathbf{y}) = f(\mathbf{x}^{\pi}, \mathbf{y}^{\pi})$. Most of the commonly studied functions in communication complexity are permutation-invariant. For such functions, we present a simple complexity measure (computable in time polynomial in $n$ given an implicit description of $f$) that describes their communication complexity up to polynomial factors and up to an additive error that is logarithmic in the input size. This gives a coarse taxonomy of the communication complexity of simple functions. Our work highlights the role of the well-known lower bounds of functions such as 'Set-Disjointness' and 'Indexing', while complementing them with the relatively lesser-known upper bounds for 'Gap-Inner-Product' (from the sketching literature) and 'Sparse-Gap-Inner-Product' (from the recent work of Canonne et al. [ITCS 2015]). We also present consequences to the study of communication complexity with imperfectly shared randomness where we show that for total permutation-invariant functions, imperfectly shared randomness results in only a polynomial blow-up in communication complexity after an additive $O(\log \log n)$ overhead

New Bounds for the Garden-Hose Model

We show new results about the garden-hose model. Our main results include improved lower bounds based on non-deterministic communication complexity (leading to the previously unknown $\Theta(n)$ bounds for Inner Product mod 2 and Disjointness), as well as an $O(n\cdot \log^3 n)$ upper bound for the Distributed Majority function (previously conjectured to have quadratic complexity). We show an efficient simulation of formulae made of AND, OR, XOR gates in the garden-hose model, which implies that lower bounds on the garden-hose complexity $GH(f)$ of the order $\Omega(n^{2+\epsilon})$ will be hard to obtain for explicit functions. Furthermore we study a time-bounded variant of the model, in which even modest savings in time can lead to exponential lower bounds on the size of garden-hose protocols.Comment: In FSTTCS 201

Adversarial Wiretap Channel with Public Discussion

Wyner's elegant model of wiretap channel exploits noise in the communication channel to provide perfect secrecy against a computationally unlimited eavesdropper without requiring a shared key. We consider an adversarial model of wiretap channel proposed in [18,19] where the adversary is active: it selects a fraction $\rho_r$ of the transmitted codeword to eavesdrop and a fraction $\rho_w$ of the codeword to corrupt by "adding" adversarial error. It was shown that this model also captures network adversaries in the setting of 1-round Secure Message Transmission [8]. It was proved that secure communication (1-round) is possible if and only if $\rho_r + \rho_w <1$. In this paper we show that by allowing communicants to have access to a public discussion channel (authentic communication without secrecy) secure communication becomes possible even if $\rho_r + \rho_w >1$. We formalize the model of \awtppd protocol and for two efficiency measures, {\em information rate } and {\em message round complexity} derive tight bounds. We also construct a rate optimal protocol family with minimum number of message rounds. We show application of these results to Secure Message Transmission with Public Discussion (SMT-PD), and in particular show a new lower bound on transmission rate of these protocols together with a new construction of an optimal SMT-PD protocol