A Santha--Vazirani (SV) source is a sequence of random bits where the conditional distribution of each bit, given the previous bits, can be partially controlled by an adversary. Santha and Vazirani show that deterministic randomness extraction from these sources is impossible. In this paper, we study the generalization of SV sources for nonbinary sequences. We show that unlike the binary setup of Santha and Vazirani, deterministic randomness extraction in the generalized case is sometimes possible. In particular, if the adversary has access to $s$ “nondegenerate” dice that are $c$ -sided and can choose one die to throw based on the previous realizations of the dice, then deterministic randomness extraction is possible if $s<c$ . We present a necessary condition and a sufficient condition for the possibility of deterministic randomness extraction. These two conditions complement each other in the nondegenerate cases. Next, we turn to a distributed setting. In this setting the SV source consists of a random sequence of pairs $(a_1, b_1), (a_2, b_2), \ldots$ distributed between two parties, where the first party receives $a_i$ 's and the second one receives $b_i$ 's. The goal of the two parties is to extract common randomness without communication. Using the notion of maximal correlation, we prove a necessary condition and a sufficient condition for the possibility of common randomness extraction from these sources. Based on these two conditions, the problem of common randomness extraction essentially reduces to the problem of randomness extraction from (nondistributed) SV sources. This result generalizes results of Gács and Körner, and Witsenhausen about common randomness extraction from independently and identically distributed sources to adversarial sources.A Santha--Vazirani (SV) source is a sequence of random bits where the conditional distribution of each bit, given the previous bits, can be partially controlled by an adversary. Santha and Vazirani show that deterministic randomness extraction from these sources is impossible. In this paper, we study the generalization of SV sources for nonbinary sequences. We show that unlike the binary setup of Santha and Vazirani, deterministic randomness extraction in the generalized case is sometimes possible. In particular, if the adversary has access to $s$ “nondegenerate” dice that are $c$-sided and can choose one die to throw based on the previous realizations of the dice, then deterministic randomness extraction is possible if $s<c$. We present a necessary condition and a sufficient condition for the possibility of deterministic randomness extraction. These two conditions complement each other in the nondegenerate cases. Next, we turn to a distributed setting. In this setting the SV source consists of a random sequence of pairs $(a_1, b_1), (a_2, b_2), \ldots$ distributed between two parties, where the first party receives $a_i$'s and the second one receives $b_i$'s. The goal of the two parties is to extract common randomness without communication. Using the notion of maximal correlation, we prove a necessary condition and a sufficient condition for the possibility of common randomness extraction from these sources. Based on these two conditions, the problem of common randomness extraction essentially reduces to the problem of randomness extraction from (nondistributed) SV sources. This result generalizes results of Gács and Körner, and Witsenhausen about common randomness extraction from independently and identically distributed sources to adversarial sources.

SIAM Journal on ComputingJan 1, 20178 citations

The Novel Multi Source Method for the Randomness Extraction

Maksim Iavich, Tamari Kuchukhidze

Lecture notes on data engineering and communications technologiesJan 1, 20230 citations

Quantum-Resilient Randomness Extraction

Renato Renner

Lecture notes in computer scienceJan 1, 20110 citations

Deterministic Randomness Extraction from Generalized and Distributed Santha-Vazirani Sources

Salman Beigi, Omid Etesami, Amin Gohari

A Santha-Vazirani (SV) source is a sequence of random bits where the conditional distribution of each bit, given the previous bits, can be partially controlled by an adversary. Santha and Vazirani show that deterministic randomness extraction from these sources is impossible. In this paper, we study the generalization of SV sources for non-binary sequences. We show that unlike the binary case, deterministic randomness extraction in the generalized case is sometimes possible. We present a necessary condition and a sufficient condition for the possibility of deterministic randomness extraction. These two conditions coincide in "non-degenerate" cases. Next, we turn to a distributed setting. In this setting the SV source consists of a random sequence of pairs $(a_1, b_1), (a_2, b_2), \ldots$ distributed between two parties, where the first party receives $a_i$ 's and the second one receives $b_i$ 's. The goal of the two parties is to extract common randomness without communication. Using the notion of maximal correlation, we prove a necessary condition and a sufficient condition for the possibility of common randomness extraction from these sources. Based on these two conditions, the problem of common randomness extraction essentially reduces to the problem of randomness extraction from (non-distributed) SV sources. This result generalizes results of Gács and Körner, and Witsenhausen about common randomness extraction from i.i.d. sources to adversarial sources.A Santha-Vazirani (SV) source is a sequence of random bits where the conditional distribution of each bit, given the previous bits, can be partially controlled by an adversary. Santha and Vazirani show that deterministic randomness extraction from these sources is impossible. In this paper, we study the generalization of SV sources for non-binary sequences. We show that unlike the binary case, deterministic randomness extraction in the generalized case is sometimes possible. We present a necessary condition and a sufficient condition for the possibility of deterministic randomness extraction. These two conditions coincide in "non-degenerate" cases. Next, we turn to a distributed setting. In this setting the SV source consists of a random sequence of pairs $(a_1, b_1), (a_2, b_2), \ldots$ distributed between two parties, where the first party receives $a_i$'s and the second one receives $b_i$'s. The goal of the two parties is to extract common randomness without communication. Using the notion of maximal correlation, we prove a necessary condition and a sufficient condition for the possibility of common randomness extraction from these sources. Based on these two conditions, the problem of common randomness extraction essentially reduces to the problem of randomness extraction from (non-distributed) SV sources. This result generalizes results of Gács and Körner, and Witsenhausen about common randomness extraction from i.i.d. sources to adversarial sources.

arXiv:1412.6641Electronic colloquium on computational complexityDec 20, 20141 citation

Extraction of quantum randomness

Igor M. Arbekov, S. N. Molotkov

Uspekhi Fizicheskih NaukNov 1, 202016 citations

Free randomness amplification using bipartite chain correlations

Andrzej Grudka, Karol Horodecki, Michał Horodecki +3 more

A direct analysis of the task of randomness amplification from Santha-Vazirani sources using the violation of the chained Bell inequality is performed in terms of the convex combination of no-signaling boxes required to simulate quantum violation of the inequality. This analysis is used to find the exact threshold value of the initial randomness parameter from which perfect randomness can be extracted in the asymptotic limit of a large number of measurement settings. As a byproduct, we provide a tool for the analysis of randomness amplification protocols, namely a general characterization of the probability distributions of bits generated by Santha-Vazirani sources, which are shown to be mixtures of specific permutations of Bernoulli distributions with a parameter defined by the source.A direct analysis of the task of randomness amplification from Santha-Vazirani sources using the violation of the chained Bell inequality is performed in terms of the convex combination of no-signaling boxes required to simulate quantum violation of the inequality. This analysis is used to find the exact threshold value of the initial randomness parameter from which perfect randomness can be extracted in the asymptotic limit of a large number of measurement settings. As a byproduct, we provide a tool for the analysis of randomness amplification protocols, namely a general characterization of the probability distributions of bits generated by Santha-Vazirani sources, which are shown to be mixtures of specific permutations of Bernoulli distributions with a parameter defined by the source.

arXiv:1303.5591Physical Review ASep 22, 201431 citations

s

“nondegenerate” dice that are

c

-sided and can choose one die to throw based on the previous realizations of the dice, then deterministic randomness extraction is possible if

s<c

. We present a necessary condition and a sufficient condition for the possibility of deterministic randomness extraction. These two conditions complement each other in the nondegenerate cases. Next, we turn to a distributed setting. In this setting the SV source consists of a random sequence of pairs

(a_1, b_1), (a_2, b_2), \ldots

distributed between two parties, where the first party receives

a_i

's and the second one receives

b_i

's. The goal of the two parties is to extract common randomness without communication. Using the notion of maximal correlation, we prove a necessary condition and a sufficient condition for the possibility of common randomness extraction from these sources. Based on these two conditions, the problem of common randomness extraction essentially reduces to the problem of randomness extraction from (nondistributed) SV sources. This result generalizes results of Gács and Körner, and Witsenhausen about common randomness extraction from independently and identically distributed sources to adversarial sources.

(a_1, b_1), (a_2, b_2), \ldots

distributed between two parties, where the first party receives

a_i

's and the second one receives

b_i

's. The goal of the two parties is to extract common randomness without communication. Using the notion of maximal correlation, we prove a necessary condition and a sufficient condition for the possibility of common randomness extraction from these sources. Based on these two conditions, the problem of common randomness extraction essentially reduces to the problem of randomness extraction from (non-distributed) SV sources. This result generalizes results of Gács and Körner, and Witsenhausen about common randomness extraction from i.i.d. sources to adversarial sources.