Converses for Secret Key Agreement and Secure Computing

We consider information theoretic secret key agreement and secure function computation by multiple parties observing correlated data, with access to an interactive public communication channel. Our main result is an upper bound on the secret key length, which is derived using a reduction of binary hypothesis testing to multiparty secret key agreement. Building on this basic result, we derive new converses for multiparty secret key agreement. Furthermore, we derive converse results for the oblivious transfer problem and the bit commitment problem by relating them to secret key agreement. Finally, we derive a necessary condition for the feasibility of secure computation by trusted parties that seek to compute a function of their collective data, using an interactive public communication that by itself does not give away the value of the function. In many cases, we strengthen and improve upon previously known converse bounds. Our results are single-shot and use only the given joint distribution of the correlated observations. For the case when the correlated observations consist of independent and identically distributed (in time) sequences, we derive strong versions of previously known converses

Making Asynchronous Distributed Computations Robust to Channel Noise

We consider the problem of making distributed computations robust to noise, in particular to worst-case (adversarial) corruptions of messages. We give a general distributed interactive coding scheme which simulates any asynchronous distributed protocol while tolerating a maximal corruption level of Theta(1/n)-fraction of all messages. Our noise tolerance is optimal and is obtained with only a moderate overhead in the number of messages. Our result is the first fully distributed interactive coding scheme in which the topology of the communication network is not known in advance. Prior work required either a coordinating node to be connected to all other nodes in the network or assumed a synchronous network in which all nodes already know the complete topology of the network. Overcoming this more realistic setting of an unknown topology leads to intriguing distributed problems, in which nodes try to learn sufficient information about the network topology in order to perform efficient coding and routing operations for coping with the noise. What makes these problems hard is that these topology exploration computations themselves must already be robust to noise

Noisy Beeping Networks

We introduce noisy beeping networks, where nodes have limited communication capabilities, namely, they can only emit energy or sense the channel for energy. Furthermore, imperfections may cause devices to malfunction with some fixed probability when sensing the channel, which amounts to deducing a noisy received transmission. Such noisy networks have implications for ultra-lightweight sensor networks and biological systems. We show how to compute tasks in a noise-resilient manner over noisy beeping networks of arbitrary structure. In particular, we transform any algorithm that assumes a noiseless beeping network (of size $n$) into a noise-resilient version while incurring a multiplicative overhead of only $O(\log n)$ in its round complexity, with high probability. We show that our coding is optimal for some tasks, such as node-coloring of a clique. We further show how to simulate a large family of algorithms designed for distributed networks in the CONGEST($B$) model over a noisy beeping network. The simulation succeeds with high probability and incurs an asymptotic multiplicative overhead of $O(B\cdot \Delta \cdot \min(n,\Delta^2))$ in the round complexity, where $\Delta$ is the maximal degree of the network. The overhead is tight for certain graphs, e.g., a clique. Further, this simulation implies a constant overhead coding for constant-degree networks

Interactive Coding Resilient to an Unknown Number of Erasures

We consider distributed computations between two parties carried out over a noisy channel that may erase messages. Following a noise model proposed by Dani et al. (2018), the noise level observed by the parties during the computation in our setting is arbitrary and a priori unknown to the parties. We develop interactive coding schemes that adapt to the actual level of noise and correctly execute any two-party computation. Namely, in case the channel erases T transmissions, the coding scheme will take N+2T transmissions using an alphabet of size 4 (alternatively, using 2N+4T transmissions over a binary channel) to correctly simulate any binary protocol that takes N transmissions assuming a noiseless channel. We can further reduce the communication to N+T by relaxing the communication model and allowing parties to remain silent rather than forcing them to communicate in every round of the coding scheme. Our coding schemes are efficient, deterministic, have linear overhead both in their communication and round complexity, and succeed (with probability 1) regardless of the number of erasures T

Protecting Single-Hop Radio Networks from Message Drops

Single-hop radio networks (SHRN) are a well studied abstraction of communication over a wireless channel. In this model, in every round, each of the n participating parties may decide to broadcast a message to all the others, potentially causing collisions. We consider the SHRN model in the presence of stochastic message drops (i.e., erasures), where in every round, the message received by each party is erased (replaced by ?) with some small constant probability, independently. Our main result is a constant rate coding scheme, allowing one to run protocols designed to work over the (noiseless) SHRN model over the SHRN model with erasures. Our scheme converts any protocol ? of length at most exponential in n over the SHRN model to a protocol ?\u27 that is resilient to constant fraction of erasures and has length linear in the length of ?. We mention that for the special case where the protocol ? is non-adaptive, i.e., the order of communication is fixed in advance, such a scheme was known. Nevertheless, adaptivity is widely used and is known to hugely boost the power of wireless channels, which makes handling the general case of adaptive protocols ? both important and more challenging. Indeed, to the best of our knowledge, our result is the first constant rate scheme that converts adaptive protocols to noise resilient ones in any multi-party model

The Adversarial Noise Threshold for Distributed Protocols

We consider the problem of implementing distributed protocols, despite adversarial channel errors, on synchronous-messaging networks with arbitrary topology. In our first result we show that any $n$-party $T$-round protocol on an undirected communication network $G$ can be compiled into a robust simulation protocol on a sparse ($\mathcal{O}(n)$ edges) subnetwork so that the simulation tolerates an adversarial error rate of $\Omega\left(\frac{1}{n}\right)$; the simulation has a round complexity of $\mathcal{O}\left(\frac{m \log n}{n} T\right)$, where $m$ is the number of edges in $G$. (So the simulation is work-preserving up to a $\log$ factor.) The adversary's error rate is within a constant factor of optimal. Given the error rate, the round complexity blowup is within a factor of $\mathcal{O}(k \log n)$ of optimal, where $k$ is the edge connectivity of $G$. We also determine that the maximum tolerable error rate on directed communication networks is $\Theta(1/s)$ where $s$ is the number of edges in a minimum equivalent digraph. Next we investigate adversarial per-edge error rates, where the adversary is given an error budget on each edge of the network. We determine the exact limit for tolerable per-edge error rates on an arbitrary directed graph. However, the construction that approaches this limit has exponential round complexity, so we give another compiler, which transforms $T$-round protocols into $\mathcal{O}(mT)$-round simulations, and prove that for polynomial-query black box compilers, the per-edge error rate tolerated by this last compiler is within a constant factor of optimal.Comment: 23 pages, 2 figures. Fixes mistake in theorem 6 and various typo