Gillat Kol

FOCS Conference 2025 Conference Paper

Constant Rate Codes for Adaptive Broadcasts Do Not Exist

• Klim Efremenko
• Gillat Kol
• Dmitry Paramonov
• Raghuvansh R. Saxena

Can the n-party broadcast channel, where any symbol sent by one party is received by all, be made resilient to noise with low overhead? Namely, is it possible to construct interactive error-correcting codes that convert any protocol designed for the noiseless broadcast channel into one that works over the noisy broadcast channel and is not much longer than the original protocol? [12, STOC 2018] showed that such interactive codes with constant multiplicative overhead are possible under the assumption that the noiseless protocol being simulated is non-adaptive, meaning that it is restricted to have a pre-determined order of turns. Their noise resilient simulating protocols, however, require adaptivity, where each party can decide whether or not to broadcast given all the information available to them, including their input and received transcript. The question of whether such a simulation is possible for general, potentially adaptive, noiseless protocols was left open. We resolve this question negatively, proving that any interactive code that converts adaptive noiseless broadcast protocols into adaptive broadcast protocols resilient to stochastic errors must incur a multiplicative overhead of Ω(log n/ log log n), which is nearly tight.

STOC Conference 2024 Conference Paper

Optimal Multi-pass Lower Bounds for MST in Dynamic Streams

• Sepehr Assadi
• Gillat Kol
• Zhijun Zhang 0007

SODA Conference 2023 Conference Paper

Interactive Coding with Small Memory

• Klim Efremenko
• Bernhard Haeupler
• Yael Tauman Kalai
• Gillat Kol
• Nicolas Resch
• Raghuvansh R. Saxena

In this work, we design an interactive coding scheme that converts any two party interactive protocol Π into another interactive protocol Π', such that even if errors are introduced during the execution of Π', the parties are able to determine what the outcome of running Π would be in an error-free setting. Importantly, our scheme preserves the space complexity of the protocol, in addition to the communication and computational complexities. Specifically, if the protocol Π has communication complexity T, computational complexity t, and space complexity s, the resulting protocol Π' is resilient to a constant ε > 0 fraction of adversarial errors, and has communication complexity approaching T as ε approaches 0, computational complexity poly( t ), and space complexity 𝒪( s log T ). Prior to this work, all known interactive coding schemes required the parties to use at least Ω( T ) space, as the parties were required to remember the transcript of the conversation thus far, or considered weaker error models.

STOC Conference 2023 Conference Paper

The Rate of Interactive Codes Is Bounded Away from 1

• Klim Efremenko
• Gillat Kol
• Dmitry Paramonov
• Raghuvansh R. Saxena

Kol and Raz [STOC 2013] showed how to simulate any alternating two-party communication protocol designed to work over the noiseless channel, by a protocol that works over a stochastic channel that corrupts each sent symbol with probability ‍є>0 independently, with only a 1+ O (√(є)) blowup to the communication. In particular, this implies that the maximum rate of such interactive codes approaches 1 as є goes to ‍0, as is also the case for the maximum rate of classical error correcting codes. Over the past decade, followup works have strengthened and generalized this result to other noisy channels, stressing on how fast the rate approaches 1 as є goes to 0, but retaining the assumption that the noiseless protocol is alternating. In this paper we consider the general case, where the noiseless protocols can have arbitrary orders of speaking . In contrast to Kol-Raz and to the followup results in this model, we show that the maximum rate of interactive codes that encode general protocols is upper bounded by a universal constant strictly smaller than 1. To put it differently, we show that there is an inherent blowup in communication when protocols with arbitrary orders of speaking are faced with any constant fraction of errors ‍є > 0. We mention that our result assumes a large alphabet set and resolves the (non-binary variant) of a conjecture by Haeupler [FOCS ‍2014].

SODA Conference 2023 Conference Paper

Towards Multi-Pass Streaming Lower Bounds for Optimal Approximation of Max-Cut

• Lijie Chen 0001
• Gillat Kol
• Dmitry Paramonov
• Raghuvansh R. Saxena
• Zhao Song 0002
• Huacheng Yu

We consider the Max-Cut problem, asking how much space is needed by a streaming algorithm in order to estimate the value of the maximum cut in a graph. This problem has been extensively studied over the last decade, and we now have a near-optimal lower bound for one-pass streaming algorithms, showing that they require linear space to guarantee a better-than-2 approximation [50, 52]. This result relies on a lower bound for the cycle-finding problem, showing that it is hard for a one-pass streaming algorithm to find a cycle in a union of matchings. The end-goal of our research is to prove a similar lower bound for multi-pass streaming algorithms that guarantee a better-than-2 approximation for Max-Cut, a highly challenging open problem. In this paper, we take a significant step in this direction, showing that even o (log n )-pass streaming algorithms need n Ω(1) space to solve the cycle-finding problem. Our proof is quite involved, dividing the cycles in the graph into “short” and “long” cycles, and using tailor-made lower bound techniques to handle each case.

FOCS Conference 2022 Conference Paper

Binary Codes with Resilience Beyond 1/4 via Interaction

• Klim Efremenko
• Gillat Kol
• Raghuvansh R. Saxena
• Zhijun Zhang 0007

In the reliable transmission problem, a sender, Alice, wishes to transmit a bit-string x to a remote receiver, Bob, over a binary channel with adversarial noise. The solution to this problem is to encode x using an error correcting code. As it is long known that the distance of binary codes is at most 1/2, reliable transmission is possible only if the channel corrupts (flips) at most a 1/4-fraction of the communicated bits. We revisit the reliable transmission problem in the two-way setting, where both Alice and Bob can send bits to each other. Our main result is the construction of two-way error correcting codes that are resilient to a constant fraction of corruptions strictly larger than 1/4. Moreover, our code has constant rate and requires Bob to only send one short message. We mention that our result resolves an open problem by Haeupler, Kamath, and Velingker [APPROX-RANDOM, 2015] and by Gupta, Kalai, and Zhang [STOC, 2022]. Curiously, our new two-way code requires a fresh perspective on classical error correcting codes: While classical codes have only one distance guarantee for all pairs of codewords (i. e. , the minimum distance), we construct codes where the distance between a pair of codewords depends on the “compatibility” of the messages they encode. We also prove that such codes are necessary for our result.

STOC Conference 2022 Conference Paper

Circuits resilient to short-circuit errors

• Klim Efremenko
• Bernhard Haeupler
• Yael Tauman Kalai
• Pritish Kamath
• Gillat Kol
• Nicolas Resch
• Raghuvansh R. Saxena

Given a Boolean circuit C , we wish to convert it to a circuit C ′ that computes the same function as C even if some of its gates suffer from adversarial short circuit errors, i.e., their output is replaced by the value of one of their inputs. Can we design such a resilient circuit C ′ whose size is roughly comparable to that of C ? Prior work gave a positive answer for the special case where C is a formula.

SODA Conference 2022 Conference Paper

Distributed Zero-Knowledge Proofs Over Networks

• Aviv Bick
• Gillat Kol
• Rotem Oshman

Zero knowledge proofs are one of the most influential concepts in theoretical computer science. In the seminal definition due to Goldwasser, Micali and Rackoff dating back to the 1980s, a computationally-bounded verifier interacts with a powerful but untrusted prover, with the goal of becoming convinced that the input is in some language. In addition to the usual requirements of completeness and soundness, in a zero knowledge proof, we protect the prover's knowledge: assuming the prover is honest, anything that the verifier can deduce after interacting with the prover, it could have deduced by itself. Zero knowledge proofs have found many applications within theoretical computer science and beyond, e. g. , in cryptography, client-cloud computing, blockchains and cryptocurrencies, electronic voting and auctions, and in the financial industry. We define and study the notion of distributed zero knowledge proofs, reconciling the computational notion of zero-knowledge with the communication-based paradigm of distributed graph algorithms. In our setting, a network of verifiers interacts with an untrusted prover to decide some distributed language. As is usually the case in distributed graph algorithms, we assume that the verifiers have local views of the network and each only knows its neighbors. The prover, on the other hand, is assumed to know the entire network graph, as well as any input that the verifier may possess. As in the computational centralized setting, the protocol we design should protect this knowledge. In particular, due to the dual role of the underlying graph in distributed graph algorithms, serving as both the communication topology and the input to the problem, our protocol must protect the graph itself. We construct communication-efficient distributed zero knowledge proofs for two central problems: the 3-coloring problem, one of the poster children of computational zero-knowledge, and for the spanning-tree verification problem, a fundamental building block for designing graph algorithms. We also give a general scheme for converting proof labeling-schemes to distributed zero-knowledge protocols with related parameters. Our protocols combine ideas from computational complexity, distributed computing, and cryptography.

FOCS Conference 2022 Conference Paper

Rounds vs Communication Tradeoffs for Maximal Independent Sets

• Sepehr Assadi
• Gillat Kol
• Zhijun Zhang 0007

We consider the problem of finding a maximal independent set (MIS) in the shared blackboard communication model with vertex-partitioned inputs. There are n players corresponding to vertices of an undirected graph, and each player sees the edges incident on its vertex – this way, each edge is known by both its endpoints and is thus shared by two players. The players communicate in simultaneous rounds by posting their messages on a shared blackboard visible to all players, with the goal of computing an MIS of the graph. While the MIS problem is well studied in other distributed models, and while shared blackboard is, perhaps, the simplest broadcast model, lower bounds for our problem were only known against one-round protocols. We present a lower bound on the round-communication tradeoff for computing an MIS in this model. Specifically, we show that when r rounds of interaction are allowed, at least one player needs to communicate $\Omega(n^{1/20^{r+1}})$ bits. In particular, with logarithmic bandwidth, finding an MIS requires $\Omega(\log\log n)$ rounds. This lower bound can be compared with the algorithm of Ghaffari, Gouleakis, Konrad, Mitrović, and Rubinfeld [PODC 2018] that solves MIS in $O(\log\log n)$ rounds but with a logarithmic bandwidth for an average player. Additionally, our lower bound further extends to the closely related problem of maximal bipartite matching. The presence of edge-sharing gives the algorithms in our model a surprising power and numerous algorithmic results exploiting this power are known. For a similar reason, proving lower bounds in this model is much more challenging, as this sharing in the players’ inputs prohibits the use of standard number-in-hand communication complexity arguments. Thus, to prove our results, we devise a new round elimination framework, which we call partial-input embedding, that may also be useful in future work for proving round-sensitive lower bounds in the presence of shared inputs. Finally, we discuss several implications of our results to multi-round (adaptive) distributed sketching algorithms, broadcast congested clique, and to the welfare maximization problem in two-sided matching markets.

STOC Conference 2021 Conference Paper

Almost optimal super-constant-pass streaming lower bounds for reachability

• Lijie Chen 0001
• Gillat Kol
• Dmitry Paramonov
• Raghuvansh R. Saxena
• Zhao Song 0002
• Huacheng Yu

We give an almost quadratic n 2− o (1) lower bound on the space consumption of any o (√log n )-pass streaming algorithm solving the (directed) s - t reachability problem. This means that any such algorithm must essentially store the entire graph. As corollaries, we obtain almost quadratic space lower bounds for additional fundamental problems, including maximum matching, shortest path, matrix rank, and linear programming. Our main technical contribution is the definition and construction of set hiding graphs, that may be of independent interest: we give a general way of encoding a set S ⊆ [ k ] as a directed graph with n = k 1 + o ( 1 ) vertices, such that deciding whether i ∈ S boils down to deciding if t i is reachable from s i , for a specific pair of vertices ( s i , t i ) in the graph. Furthermore, we prove that our graph “hides” S , in the sense that no low-space streaming algorithm with a small number of passes can learn (almost) anything about S .

STOC Conference 2021 Conference Paper

Optimal error resilience of adaptive message exchange

• Klim Efremenko
• Gillat Kol
• Raghuvansh R. Saxena

We study the error resilience of the message exchange task: Two parties, each holding a private input, want to exchange their inputs. However, the channel connecting them is governed by an adversary that may corrupt a constant fraction of the transmissions. What is the maximum fraction of corruptions that still allows the parties to exchange their inputs?

FOCS Conference 2021 Conference Paper

Statistically Near-Optimal Hypothesis Selection

• Olivier Bousquet
• Mark Braverman
• Gillat Kol
• Klim Efremenko
• Shay Moran

Hypothesis Selection is a fundamental distribution learning problem where given a comparator-class $\mathcal{Q}=\{q_{1}, \ldots, q_{n}\}$ of distributions, and a sampling access to an unknown target distribution $p$, the goal is to output a distribution $q$ such that $\mathsf{TV}(p, q)$ is close to opt, where $\mathsf{opt}=\min\nolimits_{i}\{\mathsf{TV}(p, q_{i})\}$ and TV (. ,.) denotes the total-variation distance. Despite the fact that this problem has been studied since the 19th century, its complexity in terms of basic resources, such as number of samples and approximation guarantees, remains unsettled (this is discussed, e. g. , in the charming book by Devroye and Lugosi '00). This is in stark contrast with other (younger) learning settings, such as PAC learning, for which these complexities are well understood. We derive an optimal 2-approximation learning strategy for the Hypothesis Selection problem, outputting $q$ such that, $\mathsf{TV}(p, q)\leq 2\cdot\text{opt}+\varepsilon$, with a (nearly) optimal sample complexity of $\tilde{O}(\log n/\varepsilon^{2})$. This is the first algorithm that simultaneously achieves the best approximation factor and sample complexity: previously, Bousquet, Kane, and Moran (COLT ‘19) gave a learner achieving the optimal 2-approximation, but with an exponentially worse sample complexity of $\tilde{O}(\sqrt{n}/\varepsilon^{2. 5})$, and Yatracos (Annals of Statistics '85) gave a learner with optimal sample complexity of $O(\log n/\varepsilon^{2})$ but with a sub-optimal approximation factor of 3. We mention that many works in the Density Estimation (a. k. a. , Distribution Learning) literature use Hypothesis Selection as a black box subroutine. Our result therefore implies an improvement on the approximation factors obtained by these works, while keeping their sample complexity intact. For example, our result improves the approximation factor of the algorithm of Ashtiani, Ben-David, Harvey, Liaw, and Mehrabian (JACM '20) for agnostic learning of mixtures of gaussians from 9 to 6, while maintaining its nearly-tight sample complexity.

FOCS Conference 2021 Conference Paper

Tight Bounds for General Computation in Noisy Broadcast Networks

• Klim Efremenko
• Gillat Kol
• Dmitry Paramonov
• Raghuvansh R. Saxena

Let II be a protocol over the n-party broadcast channel, where in each round, a pre-specified party broadcasts a symbol to all other parties. We wish to design a scheme that takes such a protocol II as input and outputs a noise resilient protocol II’ that simulates II over the noisy broadcast channel, where each received symbol is flipped with a fixed constant probability, independently. What is the minimum overhead in the number of rounds that is incurred by any such simulation scheme? A classical result by Gallager from the 80's shows that non-interactive T-round protocols, where the bit communicated in every round is independent of the communication history, can be converted to noise resilient ones with only an $\mathrm{O}(\log\log T$ ) multiplicative overhead in the number of rounds. Can the same be proved for any protocol? Or, are there protocols whose simulation requires an $\Omega(\log T)$ overhead (which always suffices)? We answer both the above questions in the negative: We give a simulation scheme with an $\tilde{O}(\sqrt{\log T})$ overhead for every protocol and channel alphabet. We also prove an (almost) matching lower bound of $\Omega(\sqrt{\log T})$ on the overhead required to simulate the pointer chasing protocol with T = n and polynomial alphabet.

FOCS Conference 2020 Conference Paper

Binary Interactive Error Resilience Beyond ${{}^{1}}\! /\! _{8}$ (or why $({{}^{1}}\! /\! _{2})^{3} > {{}^{1}}\! /\! _{8})$

• Klim Efremenko
• Gillat Kol
• Raghuvansh R. Saxena

Interactive error correcting codesInteractive error correcting codes are codes that encode a two party communication protocol to an error-resilient protocol that succeeds even if a constant fraction of the communicated symbols are adversarially corrupted, at the cost of increasing the communication by a constant factor. What is the largest fraction of corruptions that such codes can protect against? If the error-resilient protocol is allowed to communicate large (constant sized) symbols, Braverman and Rao (STOC, 2011) show that the maximum rate of corruptions that can be tolerated is 1 /4. They also give a binary interactive error correcting protocol that only communicates bits and is resilient to 1 /2 fraction of errors, but leave the optimality of this scheme as an open problem. We answer this question in the negative, breaking the 1 /8 barrier. Specifically, we give a binary interactive error correcting scheme that is resilient to 5 /39 > 1 /8 fraction of adversarial errors. Our scheme builds upon a novel construction of binary list-decodable interactive codes with small list size.

STOC Conference 2020 Conference Paper

Interactive error resilience beyond 2/7

• Klim Efremenko
• Gillat Kol
• Raghuvansh R. Saxena

FOCS Conference 2020 Conference Paper

Multi-Pass Graph Streaming Lower Bounds for Cycle Counting, MAX-CUT, Matching Size, and Other Problems

• Sepehr Assadi
• Gillat Kol
• Raghuvansh R. Saxena
• Huacheng Yu

Consider the following gap cycle counting problem in the streaming model: The edges of a 2-regular n-vertex graph G are arriving one-by-one in a stream and we are promised that G is a disjoint union of either k-cycles or 2k-cycles for some small k; the goal is to distinguish between these two cases using a limited memory. Verbin and Yu [SODA 2011] introduced this problem and showed that any single-pass streaming algorithm solving it requires n 1-Ω(1/k) space. This result and the proof technique behind it-the Boolean Hidden Hypermatching communication problem-has since been used extensively for proving streaming lower bounds for various problems, including approximating MAX-CUT, matching size, property testing, matrix rank and Schatten norms, streaming unique games and CSPs, and many others. Despite its significance and broad range of applications, the lower bound technique of Verbin and Yu comes with a key weakness that is also inherited by all subsequent results: the Boolean Hidden Hypermatching problem is hard only if there is exactly one round of communication and, in fact, can be solved with logarithmic communication in two rounds. Therefore, all streaming lower bounds derived from this problem only hold for single-pass algorithms. Our goal in this paper is to remedy this state-of-affairs. We prove the first multi-pass lower bound for the gap cycle counting problem: Any p-pass streaming algorithm that can distinguish between disjoint union of k-cycles vs 2k-cycles-or even k-cycles vs one Hamiltonian cycle-requires n 1-1/kΩ(1/p) space. This makes progress on multiple open questions in this line of research dating back to the work of Verbin and Yu. As a corollary of this result and by simple (or even no) modification of prior reductions, we can extend many of previous lower bounds to multi-pass algorithms. For instance, we can now prove that any streaming algorithm that ( 1+ε) -approximates the value of MAX-CUT, maximum matching size, or rank of an n-by- n matrix, requires either n Ω(1) space or Ω(log( 1 /ε)) passes. For all these problems, prior work left open the possibility of even an O(logn) space algorithm in only two passes.

FOCS Conference 2019 Conference Paper

Radio Network Coding Requires Logarithmic Overhead

• Klim Efremenko
• Gillat Kol
• Raghuvansh R. Saxena

We consider the celebrated radio network model for abstracting communication in wireless networks. In this model, in any round, each node in the network may broadcast a message to all its neighbors. However, a node is able to hear a message broadcast by a neighbor only if no collision occurred, meaning that it was the only neighbor broadcasting. While the (noiseless) radio network model received a lot of attention over the last few decades, the effect of noise on radio networks is still not well understood. In this paper, we take a step forward and show that making radio network protocols resilient to noise may require a substantial performance overhead. Specifically, we construct a multi-hop network and a communication protocol over this network that works in T rounds when there is no noise. We prove that any scheme that simulates our protocol and is resilient to stochastic noise, requires at least cT log(n) rounds, for some constant c. This stands in contrast to our previous result (STOC, 2018), showing that protocols over the single-hop (clique) network can be made noise resilient with only a constant overhead. Our result also settles a recent conjecture by Censor-Hillel, Haeupler, Hershkowitz, Zuzic (2018). We complement the above result by giving a scheme to simulate any protocol with a fixed order of transmissions with only an O(log (n)) overhead.

STOC Conference 2018 Conference Paper

Interactive coding over the noisy broadcast channel

• Klim Efremenko
• Gillat Kol
• Raghuvansh R. Saxena

A set of n players, each holding a private input bit, communicate over a noisy broadcast channel. Their mutual goal is for all players to learn all inputs. At each round one of the players broadcasts a bit to all the other players, and the bit received by each player is flipped with a fixed constant probability (independently for each recipient). How many rounds are needed?

STOC Conference 2018 Conference Paper

Interactive compression to external information

• Mark Braverman
• Gillat Kol

We describe a new way of compressing two-party communication protocols to get protocols with potentially smaller communication. We show that every communication protocol that communicates C bits and reveals I bits of information about the participants’ private inputs to an observer that watches the communication, can be simulated by a new protocol that communicates at most poly ( I ) · loglog( C ) bits. Our result is tight up to polynomial factors, as it matches the recent work separating communication complexity from external information cost.

STOC Conference 2017 Conference Paper

Time-space hardness of learning sparse parities

• Gillat Kol
• Ran Raz
• Avishay Tal

We define a concept class ℱ to be time-space hard (or memory-samples hard) if any learning algorithm for ℱ requires either a memory of size super-linear in n or a number of samples super-polynomial in n , where n is the length of one sample. A recent work shows that the class of all parity functions is time-space hard [Raz, FOCS'16]. Building on [Raz, FOCS'16], we show that the class of all sparse parities of Hamming weight ℓ is time-space hard, as long as ℓ ≥ ω(log n / loglog n ). Consequently, linear-size DNF Formulas, linear-size Decision Trees and logarithmic-size Juntas are all time-space hard. Our result is more general and provides time-space lower bounds for learning any concept class of parity functions. We give applications of our results in the field of bounded-storage cryptography. For example, for every ωlog n ) ≤ k ≤ n , we obtain an encryption scheme that requires a private key of length k , and time complexity of n per encryption/decryption of each bit, and is provably and unconditionally secure as long as the attacker uses at most o ( nk ) memory bits and the scheme is used at most 2 o ( k ) times. Previously, this was known only for k = n [Raz, FOCS'16].

STOC Conference 2016 Conference Paper

Exponential separation of communication and external information

• Anat Ganor
• Gillat Kol
• Ran Raz

We show an exponential gap between communication complexity and external information complexity, by analyzing a communication task suggested as a candidate by Braverman. Previously, only a separation of communication complexity and internal information complexity was known.

STOC Conference 2016 Conference Paper

Interactive compression for product distributions

• Gillat Kol

We study the interactive compression problem: Given a two-party communication protocol with small information cost, can it be compressed so that the total number of bits communicated is also small? We consider the case where the parties have inputs that are independent of each other, and give a simulation protocol that communicates I^2 * polylog(I) bits, where I is the information cost of the original protocol. Our protocol is the first simulation protocol whose communication complexity is bounded by a polynomial in the information cost of the original protocol.

SODA Conference 2016 Conference Paper

Towards Optimal Deterministic Coding for Interactive Communication

• Ran Gelles
• Bernhard Haeupler
• Gillat Kol
• Noga Ron-Zewi
• Avi Wigderson

We study efficient, deterministic interactive coding schemes that simulate any interactive protocol both under random and adversarial errors, and can achieve a constant communication rate independent of the protocol length. For channels that flip bits independently with probability ∊ < 1/2, our coding scheme achieves a communication rate of and a failure probability of exp(− n /log n ) in length n protocols. Prior to our work, all nontrivial deterministic schemes (either efficient or not) had a rate bounded away from 1. Furthermore, the best failure probability achievable by an efficient deterministic coding scheme with constant rate was only quasi-polynomial, i. e. , of the form exp(− log O (1) n ) (Braverman, ITCS 2012). For channels in which an adversary controls the noise pattern our coding scheme can tolerate Ω(1/log n ) fraction of errors with rate approaching 1. Once more, all previously known nontrivial deterministic schemes (either efficient or not) in the adversarial setting had a rate bounded away from 1, and no nontrivial efficient deterministic coding schemes were known with any constant rate. Essential to both results is an explicit, efficiently encodable and decodable systematic tree code of length n that has relative distance Ω(1/log n ) and rate approaching 1, defined over an O (log n )-bit alphabet. No nontrivial tree code (either efficient or not) was known to approach rate 1, and no nontrivial distance bound was known for any efficient constant rate tree code. The fact that our tree code is systematic, turns out to play an important role in obtaining rate in the random error model, and approaching rate 1 in the adversarial error model.

STOC Conference 2015 Conference Paper

Exponential Separation of Information and Communication for Boolean Functions

• Anat Ganor
• Gillat Kol
• Ran Raz

We show an exponential gap between communication complexity and information complexity for boolean functions, by giving an explicit example of a partial function with information complexity ≤ O(k), and distributional communication complexity ≥ 2 k . This shows that a communication protocol for a partial boolean function cannot always be compressed to its internal information. By a result of Braverman [Bra12], our gap is the largest possible. By a result of Braverman and Rao [BR11], our example shows a gap between communication complexity and amortized communication complexity, implying that a tight direct sum result for distributional communication complexity of boolean functions cannot hold, answering a long standing open problem. Our techniques build on [GKR14], that proved a similar result for relations with very long outputs (double exponentially long in k). In addition to the stronger result, the current work gives a simpler proof, benefiting from the short output length of boolean functions.

FOCS Conference 2014 Conference Paper

Exponential Separation of Information and Communication

• Anat Ganor
• Gillat Kol
• Ran Raz

We show an exponential gap between communication complexity and information complexity, by giving an explicit example for a communication task (relation), with information complexity ≤ O(k), and distributional communication complexity ≥2k. This shows that a communication protocol cannot always be compressed to its internal information. By a result of Braverman [1], our gap is the largest possible. By a result of Braverman and Rao [2], our example shows a gap between communication complexity and amortized communication complexity, implying that a tight direct sum result for distributional communication complexity cannot hold.

STOC Conference 2013 Conference Paper

Interactive channel capacity

• Gillat Kol
• Ran Raz

We study the interactive channel capacity of an ε-noisy channel. The interactive channel capacity C(ε) is defined as the minimal ratio between the communication complexity of a problem (over a non-noisy channel), and the communication complexity of the same problem over the binary symmetric channel with noise rate ε, where the communication complexity tends to infinity.

STOC Conference 2008 Conference Paper

Games for exchanging information

• Gillat Kol
• Moni Naor

We consider the rational versions of two of the classical problems in foundations of cryptography: secret sharing and multiparty computation, suggested by Halpern and Teague (STOC 2004). Our goal is to design games and fair strategies that encourage rational participants to exchange information about their inputs for their mutual benefit, when the only mean of communication is a broadcast channel. We show that protocols for the above information exchanging tasks, where players' values come from a bounded domain, cannot satisfy some of the most desirable properties. In contrast, we provide a rational secret sharing scheme with simultaneous broadcast channel in which shares are taken from an unbounded domain, but have finite (and polynomial sized) expectation. Previous schemes (mostly cryptographic) have required computational assumptions, making them inexact and susceptible to backward induction, or used stronger communication channels. Our scheme is non-cryptographic, immune to backward induction, and satisfies a stronger rationality concept (strict Nash equilibrium). We show that our solution can also be used to construct an ε-Nash equilibrium secret sharing scheme for the case of a non-simultaneous broadcast channel.