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High Entropy Random Selection Protocols (Extended Abstract)

Harry Buhrman, Matthias Christandl, Michal Kouck, Zvi Lotker, Boaz Patt-shamir B, Nikolay Vereshchagin

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

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

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High Entropy Random Selection Protocols

Nikolay Vereshchagin, Harry Buhrman, Cristandl, Matthias +3 more

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.

Dagstuhl Seminar Proceedings•Jan 1, 2008•0 citations

The round complexity of two-party random selection

Saurabh Sanghvi, Salil Vadhan

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.

ACM Symposium on Theory of Computing (STOC)•May 22, 2005•13 citations

The Round Complexity of Two-Party Random Selection

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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 $T \subseteq \{0,1\}^n$ of density $\mu$ with probability at most $O(\sqrt{\mu + \delta})$, where $\delta$ 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 $T \subseteq \{0,1\}^n$.

SIAM Journal on Computing•Jan 1, 2008•7 citations

The round complexity of two-party random selection

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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.

•Jan 1, 2005•0 citations

Secure Multi-Party Shuffling.

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Abstract. In secure multi-party shuffling, multiple parties, each holding an input, want to agree on a random permutation of their inputs while keeping the permutation secret. This problem is important as a primitive in many privacy-preserving applications such as anonymous communication, location-based services, and electronic voting. Known techniques for solving this problem suffer from poor scalability, load-balancing issues, trusted party assumptions, and/or weak security guarantees. In this paper, we propose an unconditionally-secure protocol for multi-party shuffling that scales well with the number of parties and is load-balanced. In particular, we require each party to send only a polylogarithmic number of bits and perform a polylogarithmic number of operations while incurring only a logarithmic round complexity. We show security under universal composability against up to about n/3 fully-malicious parties. We also provide simulation results showing that our protocol im-proves significantly over previous work. For example, for one million parties, when compared to the state of the art, our protocol reduces the communication and computation costs by at least three orders of magnitude and slightly decreases the number of communication rounds. 1

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Protocols for Multiparty Coin Toss With Dishonest Majority.

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Abstract. Coin-tossing protocols are protocols that generate a random bit with uniform distribution. These protocols are used as a building block in many cryptographic protocols. Cleve [STOC 1986] has shown that if at least half of the parties can be malicious, then, in any r-round coin-tossing protocol, the malicious parties can cause a bias of Ω(1/r) to the bit that the honest parties output. However, for more than two t, where t is the number

•Jan 1, 2012•2 citations
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
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.
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 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\{0,1\}^n{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)2 \log^* n + O(1)2log∗n+O(1) rounds, improving a previous 2n2n2n-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}nT \subseteq \{0,1\}^nT⊆{0,1}n of density μ\muμ with probability at most O(μ+δ)O(\sqrt{\mu + \delta})O(μ+δ​), where δ\deltaδ 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)\log^* n - \log^* \log^* n - O(1)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}nT \subseteq \{0,1\}^nT⊆{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.
Abstract. In secure multi-party shuffling, multiple parties, each holding an input, want to agree on a random permutation of their inputs while keeping the permutation secret. This problem is important as a primitive in many privacy-preserving applications such as anonymous communication, location-based services, and electronic voting. Known techniques for solving this problem suffer from poor scalability, load-balancing issues, trusted party assumptions, and/or weak security guarantees. In this paper, we propose an unconditionally-secure protocol for multi-party shuffling that scales well with the number of parties and is load-balanced. In particular, we require each party to send only a polylogarithmic number of bits and perform a polylogarithmic number of operations while incurring only a logarithmic round complexity. We show security under universal composability against up to about n/3 fully-malicious parties. We also provide simulation results showing that our protocol im-proves significantly over previous work. For example, for one million parties, when compared to the state of the art, our protocol reduces the communication and computation costs by at least three orders of magnitude and slightly decreases the number of communication rounds. 1