Variations on Muchnik’s Conditional Complexity Theorem

Musatov, Daniil; Romashchenko, Andrei; Shen, Alexander

doi:10.1007/s00224-011-9321-z

Cited by 14 publications

(24 citation statements)

References 14 publications

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“…Эта идея была предложена (в другой ситуации и ещё до Мучника) в статье [14], и применена к доказательству теоремы Мучника в [125]. Преимущество такого подхода в том, что можно использовать известные явные конструкции экстракторов и другую тех-нику теории сложности (например, псевдослучайные генераторы, как предложил А. Ромащенко) для доказательства варианта теоремы Мучника, где используется сложность с ограничением на используемую память [123,124].…”

Section: условное кодированиеunclassified

Kolmogorov Complexity and Algorithmic Randomness

Shen

Uspensky

Vereshchagin

2017

Mathematical Surveys And Monographs

101

122

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Section: условное кодированиеunclassified

Kolmogorov Complexity and Algorithmic Randomness

Shen

Uspensky

Vereshchagin

2017

Mathematical Surveys And Monographs

101

122

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“…Both ideas have been used in other secret key agreement protocols. However, their technical implementation in the AIT framework requires specific constructions needed for this setting, such as, to give just one example, prefix extractors, which were first introduced explicitly in [RRV02], and in AIT in [MRS11].…”

Section: And (3) Then With Probability At Leastmentioning

confidence: 99%

An Operational Characterization of Mutual Information in Algorithmic Information Theory

Romashchenko

Zimand

2019

J. ACM

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We show that the mutual information, in the sense of Kolmogorov complexity, of any pair of strings x and y is equal, up to logarithmic precision, to the length of the longest shared secret key that two parties, one having x and the complexity profile of the pair and the other one having y and the complexity profile of the pair, can establish via a probabilistic protocol with interaction on a public channel. For ℓ > 2, the longest shared secret that can be established from a tuple of strings (x1, ..., x ℓ ) by ℓ parties, each one having one component of the tuple and the complexity profile of the tuple, is equal, up to logarithmic precision, to the complexity of the tuple minus the minimum communication necessary for distributing the tuple to all parties. We establish the communication complexity of secret key agreement protocols that produce a secret key of maximal length, for protocols with public randomness. We also show that if the communication complexity drops below the established threshold, then only very short secret keys can be obtained. supported in part by the National Science Foundation through grant 1811729. No protocol can produce a longer shared secret key (up to an O(log n) additive term).Secret key agreement for three or more parties. Mutual information is only defined for two strings, but secret key agreement can be explored for the case of more strings. Let us consider again an example. Suppose that each of Alice, Bob, and Charlie have a point in the affine plane over the finite field with 2 n elements, and that the three points, which we call A, B, C, are collinear. Thus, each party has 2n bits of information, but together they have 5n bits of information, because given two points, the third one can be described with n bits. The parties want to establish a common secret key, but they can only communicate by broadcasting messages over a public channel. They can proceed as follows. Alice will broadcast a string p A , Bob a string p B , and Charlie a string p C , such that each party using his/her point and the received information will reconstruct the three collinear points A, B, C. A protocol that achieves this is called an omniscience protocol because it spreads to everyone the information possessed at the beginning individually by each party. In the next step, each party will compress the 5n bits, comprising the three points, to a string that is random given p A , p B , p C . The compressed string is the common secret key. We will see that up to logarithmic precision it has length 5n − (|p A | + |p B | + |p C |). Assuming we know how to do the omniscience protocol and the compression step, this protocol produces a common secret key of length 5n − CO(A, B, C), where CO (A, B, C) is the minimum communication for the omniscience task for the points A, B, C. In our example, it is clear that each one of p A , p B , p C must be at least n bits long, and that any two of these strings must contain together at least 3n bits. Using some recent results from the reference [Zim17], it can be shown that an...

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“…For every K, n and constant δ , there exists explicit (K, δ )-dispersers G = (L = {0, 1} n , R = {0, 1} m , E ⊆ L × R) in which every node in L has degree D = n2 O((log log n) 2 ) and |R| = αKD n 3 , for some constant α. 3 The key combinatorial object that we use is provided in the following lemma. Lemma 4.…”

Section: Definitionmentioning

confidence: 99%

“…Now, why are dispersers sufficient? The answer, inspired by [3], stems from the idea from [1] to use for this kind of compression graphs that allow on-line matching. These are unbalanced bipartite graphs, which, in their simplest form, have LEFT = {0, 1} n , RIGHT = {0, 1} k+small overhead , and left degree = poly(n), and which permit on-line matching up to size K = 2 k .…”

Section: Introductionmentioning

confidence: 99%