We study the round complexity of two-party protocols for generating a random n-bit string such that the output is guaranteed to have bounded “bias ” even if one of the two parties deviates from the protocol (possibly using unlimited computational resources). Specifically, we require that the output’s statistical difference from the uniform distribution on {0, 1} n is bounded by a constant less than 1. We present a protocol for the above problem that has 2 log ∗ n + O(1) rounds, improving a previous 2n-round protocol of Goldreich, Goldwasser, and Linial (FOCS ’91). Like the GGL Protocol, our protocol actually provides a stronger guarantee, ensuring that the output lands in any set T ⊆ {0, 1} n of density µ with probability at most O ( √ µ + δ), where δ may be an arbitrarily small constant. We then prove a nearly matching lower bound, showing that any protocol guaranteeing bounded statistical difference requires at least log ∗ n − log ∗ log ∗ n − O(1) rounds. We also prove several results for the case when the output’s bias is measured by the maximum multiplicative factor by which a party can increase the probability of a set T ⊆ {0, 1} n.
We study the round complexity of two-party protocols for generating a random n-bit string such that the output is guaranteed to have bounded “bias ” even if one of the two parties deviates from the protocol (possibly using unlimited computational resources). Specifically, we require that the output’s statistical difference from the uniform distribution on {0, 1} n is bounded by a constant less than 1. We present a protocol for the above problem that has 2 log ∗ n + O(1) rounds, improving a previous 2n-round protocol of Goldreich, Goldwasser, and Linial (FOCS ’91). Like the GGL Protocol, our protocol actually provides a stronger guarantee, ensuring that the output lands in any set T ⊆ {0, 1} n of density µ with probability at most O ( √ µ + δ), where δ may be an arbitrarily small constant. We then prove a nearly matching lower bound, showing that any protocol guaranteeing bounded statistical difference requires at least log ∗ n − log ∗ log ∗ n − O(1) rounds. We also prove several results for the case when the output’s bias is measured by the maximum multiplicative factor by which a party can increase the probability of a set T ⊆ {0, 1} n.
We study the round complexity of two-party protocols for generating a random n-bit string such that the output is guaranteed to have bounded “bias,” even if one of the two parties deviates from the protocol (possibly using unlimited computational resources). Specifically, we require that the output's statistical difference from the uniform distribution on {0,1}n is bounded by a constant less than 1. We present a protocol for the above problem that has 2log∗n+O(1) rounds, improving a previous 2n-round protocol of Goldreich, Goldwasser, and Linial (FOCS '91). Like the GGL Protocol, our protocol actually provides a stronger guarantee, ensuring that the output lands in any set T⊆{0,1}n of density μ with probability at most O(μ+δ), where δ may be an arbitrarily small constant. We then prove a nearly matching lower bound, showing that any protocol guaranteeing bounded statistical difference requires at least log∗n−log∗log∗n−O(1) rounds. We also prove several results for the case when the output's bias is measured by the maximum multiplicative factor by which a party can increase the probability of a set T⊆{0,1}n.
We study the round complexity of two-party protocols for generating a random n-bit string such that the output is guaranteed to have bounded bias (according to some measure) even if one of the two parties deviates from the protocol (even using unlimited computational resources). Specifically, we require that the output's statistical difference from the uniform distribution on zon is bounded by a constant less than 1.We present a protocol for the above problem that has 2 log*n+O(1) rounds, improving a 2n-round protocol that follows from the work of Goldreich, Goldwasser, and Linial (FOCS'91). Like the GGL protocol, our protocol actually provides a stronger guarantee, ensuring that the output lands in any set T⊆zon of density μ with probability at most O(√μ+δ), where δ is an arbitarily small constant.We then prove a matching lower bound, showing that any protocol guaranteeing bounded statistical difference requires at least log*n - log* log*n-O(1) rounds. As far as we know, this is the first nontrivial lower bound on the round complexity of random selection protocols (of any type) that does not impose additional constraints (e.g. on communication or "simulatability").We also state several results for the case when the output's bias is measured by the maximum multiplicative factor by which a party can increase the probability of a set T ⊆ zon.
We study the two party problem of randomly selecting a string among all the strings of length n. We want the protocol to have the property that the output distribution has high entropy, even when one of the two parties is dishonest and deviates from the protocol. We develop protocols that achieve high, close to n, entropy. In the literature the randomness guarantee is usually expressed as being close to the uniform distribution or in terms of resiliency. The notion of entropy is not directly comparable to that of resiliency, but we establish a connection between the two that allows us to compare our protocols with the existing ones. We construct an explicit protocol that yields entropy n - O(1) and has 4log^* n rounds, improving over the protocol of Goldwasser et al. that also achieves this entropy but needs O(n) rounds. Both these protocols need O(n^2) bits of communication. Next we reduce the communication in our protocols. We show the existence, non-explicitly, of a protocol that has 6-rounds, 2n + 8log n bits of communication and yields entropy n- O(log n) and min-entropy n/2 - O(log n). Our protocol achieves the same entropy bound as the recent, also non-explicit, protocol of Gradwohl et al., however achieves much higher min-entropy: n/2 - O(log n) versus O(log n). Finally we exhibit very simple explicit protocols. We connect the security parameter of these geometric protocols with the well studied Kakeya problem motivated by harmonic analysis and analytical number theory. We are only able to prove that these protocols have entropy 3n/4 but still n/2 - O(log n) min-entropy. Therefore they do not perform as well with respect to the explicit constructions of Gradwohl et al. entropy-wise, but still have much better min-entropy. We conjecture that these simple protocols achieve n -o(n) entropy. Our geometric construction and its relation to the Kakeya problem follows a new and different approach to the random selection problem than any of the previously known protocols.
Abstract. We study the two party problem of randomly selecting a string among all the strings of length n. We want the protocol to have the property that the output distribution has high entropy, even when one of the two parties is dishonest and deviates from the protocol. We develop protocols that achieve high, close to n, entropy. In the literature the randomness guarantee is usually expressed as being close to the uniform distribution or in terms of resiliency. The notion of entropy is not directly comparable to that of resiliency, but we establish a connection between the two that allows us to compare our protocols with the existing ones. We construct an explicit protocol that yields entropy n − O(1) and has 4log ∗ n rounds, improving over the protocol of Goldreich et al. [3] that also achieves this entropy but needs O(n) rounds. Both these protocols need O(n 2) bits of communication. Next we reduce the communication in our protocols. We show the existence, nonexplicitly, of a protocol that has 6 rounds, 2n + 8log n bits of communication
In this work we study the effect of rounds of interaction on the common randomness generation (CRG) problem. In the CRG problem, two parties, Alice and Bob, receive samples Xi and Yi, respectively, where (Xi, Yi) are drawn jointly from a source distribution μ. The two parties wish to agree on a common random key consisting of many bits of randomness, by exchanging messages that depend on each party's respective input and the previous messages. In this work we study the amortized version of the problem, i.e., the number of bits of communication needed per random bit output by Alice and Bob, in the limit as the number of bits generated tends to infinity. The amortized version of the CRG problem has been extensively studied in the information theory literature, though very little was known about the effect of interaction on this problem. Recently Bafna et al. (SODA 2019) considered the non-amortized version of the problem (so here the goal of the interaction is to generate a fixed number of random bits): they gave a family of sources μr,n parameterized by r,n ϵ ℕ, such that with r + 2 rounds of communication one can generate n bits of common randomness with this source with O(r log n) communication, whereas with roughly r/2 rounds the communication complexity is Ω(n/ poly log n). Note in particular that their source is designed with the target number of bits in mind and hence the result does not apply to the amortized setting. In this work we strengthen the work of Bafna et al. in two ways: First we show that the results extend to the classical amortized setting. We also reduce the gap between the round complexity in the upper and lower bounds to an additive constant. Specifically we show that for every pair r, n ϵ ℕ the (amortized) communication complexity to generate Ω(n) bits of common randomness from the source μr,n using r + 2 rounds of communication is O(r log n) whereas the amortized communication required to generate the same amount of randomness from r rounds is . Our techniques exploit known connections between information complexity and CRG, and the main novelty is our ability to analyze the information complexity of protocols getting inputs from the source μr,n.
In a distributed coin-flipping protocol, Blum [ACM Transactions on Computer Systems ’83], the parties try to output a common (close to) uniform bit, even when some adversarially chosen parties try to bias the common output. In an adaptively secure full-information coin flip, Ben-Or and Linial [FOCS ’85], the parties communicate over a broadcast channel, and a computationally unbounded adversary can choose which parties to corrupt along the protocol execution. Ben-Or and Linial proved that the n -party majority protocol is resilient to O(n) corruptions, and conjectured this is a tight upper bound for any n -party protocol (of any round complexity). Their conjecture was proved to be correct, up to polylogarithmic factors, for single-turn (each party sends a single message) single-bit (a message is one bit) protocols Lichtenstein et al. [Combinatorica ’89], symmetric protocols Goldwasser et al. [ICALP ’15], and recently for (arbitrary message length) single-turn protocols Tauman Kalai et al. [DISC ’18]. Yet, the question of many-turn protocols was left entirely open. In this work, we close the above gap, proving that no n -party protocol (of any round complexity) is resilient to Ω(n⋅log3n) adaptive corruptions. Namely, majority is the optimal coin-flipping protocol against adaptive adversaries (up to polylogarithmic factors).