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Statistically Close
The variation distance of two distributions X and Y over a finite domain D, (often referred to as ''statistical difference'' or ''statistical distance'' Reyzin, Leo. (Lecture NotesExtractors and the Leftover Hash Lemma in cryptography) is defined as \Delta(X,Y)=\frac \sum _ , \Pr =\alpha- \Pr =\alpha, . We say that two probability ensembles \_ and \_ are statistically close if \Delta(X_k,Y_k) is a negligible function in k. References See also * Zero-knowledge proof * Randomness extractor A randomness extractor, often simply called an "extractor", is a function, which being applied to output from a weakly random information entropy, entropy source, together with a short, uniformly random seed, generates a highly random output that ap ... Cryptography {{crypto-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Total Variation Distance Of Probability Measures
In probability theory, the total variation distance is a distance measure for probability distributions. It is an example of a statistical distance metric, and is sometimes called the statistical distance, statistical difference or variational distance. Definition Consider a measurable space (\Omega, \mathcal) and probability measures P and Q defined on (\Omega, \mathcal). The total variation distance between P and Q is defined as: :\delta(P,Q)=\sup_\left, P(A)-Q(A)\. This is the largest absolute difference between the probabilities that the two probability distributions assign to the same event. Properties The total variation distance is an F-divergence, ''f''-divergence and an integral probability metric. Relation to other distances The total variation distance is related to the Kullback–Leibler divergence by Pinsker's inequality, Pinsker’s inequality: :\delta(P,Q) \le \sqrt. One also has the following inequality, due to Bretagnolle–Huber inequality, Bretagnolle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Ensembles
In cryptography, a distribution ensemble or probability ensemble is a family of distributions or random variables X = \_ where I is a (countable) index set, and each X_i is a random variable, or probability distribution. Often I=\N and it is required that each X_n have a certain property for ''n'' sufficiently large. For example, a uniform ensemble U = \_ is a distribution ensemble where each U_n is Uniform distribution (discrete), uniformly distributed over strings of length ''n''. In fact, many applications of probability ensembles implicitly assume that the probability spaces for the random variables all coincide in this way, so every probability ensemble is also a Stochastic_process#Definitions, stochastic process. See also * Provable security * Statistically close * Pseudorandom ensemble * Computational indistinguishability References * Goldreich, Oded (2001). ''Foundations of Cryptography: Volume 1, Basic Tools''. Cambridge University Press. . Fragments available at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Negligible Function
In mathematics, a negligible function is a function \mu:\mathbb\to\mathbb such that for every positive integer ''c'' there exists an integer ''N''''c'' such that for all ''x'' > ''N''''c'', :, \mu(x), 0 such that for all ''x'' > ''N''poly : , \mu(x), 0, there exists a positive number \delta>0 such that , x-x_0, N_\varepsilon ::, \mu(x), 0 by the functions 1/x^c where c>0 or by 1/\operatorname(x) where \operatorname(x) is a positive polynomial. This leads to the definitions of negligible functions given at the top of this article. Since the constants \varepsilon>0 can be expressed as 1/\operatorname(x) with a constant polynomial this shows that negligible functions are a subset of the infinitesimal functions. Use in cryptography In complexity-based modern cryptography, a security scheme is ''provably secure'' if the probability of security failure (e.g., inverting a one-way function, distinguishing cryptographically strong pseudorandom bits from truly ran ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zero-knowledge Proof
In cryptography, a zero-knowledge proof or zero-knowledge protocol is a method by which one party (the prover) can prove to another party (the verifier) that a given statement is true while the prover avoids conveying any additional information apart from the fact that the statement is indeed true. The essence of zero-knowledge proofs is that it is trivial to prove that one possesses knowledge of certain information by simply revealing it; the challenge is to prove such possession without revealing the information itself or any additional information. If proving a statement requires that the prover possess some secret information, then the verifier will not be able to prove the statement to anyone else without possessing the secret information. The statement being proved must include the assertion that the prover has such knowledge, but without including or transmitting the knowledge itself in the assertion. Otherwise, the statement would not be proved in zero-knowledge because it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Randomness Extractor
A randomness extractor, often simply called an "extractor", is a function, which being applied to output from a weakly random information entropy, entropy source, together with a short, uniformly random seed, generates a highly random output that appears Independent and identically distributed random variables, independent from the source and Uniform distribution (discrete), uniformly distributed. Examples of weakly random sources include radioactive decay or thermal noise; the only restriction on possible sources is that there is no way they can be fully controlled, calculated or predicted, and that a lower bound on their entropy rate can be established. For a given source, a randomness extractor can even be considered to be a true random number generator (Hardware_random_number_generator, TRNG); but there is no single extractor that has been proven to produce truly random output from any type of weakly random source. Sometimes the term "bias" is used to denote a weakly random sou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
