1,330 research outputs found
Almost-perfect secret sharing
Splitting a secret s between several participants, we generate (for each
value of s) shares for all participants. The goal: authorized groups of
participants should be able to reconstruct the secret but forbidden ones get no
information about it. In this paper we introduce several notions of non-
perfect secret sharing, where some small information leak is permitted. We
study its relation to the Kolmogorov complexity version of secret sharing
(establishing some connection in both directions) and the effects of changing
the secret size (showing that we can decrease the size of the secret and the
information leak at the same time).Comment: Acknowledgments adde
Shannon Information and Kolmogorov Complexity
We compare the elementary theories of Shannon information and Kolmogorov
complexity, the extent to which they have a common purpose, and where they are
fundamentally different. We discuss and relate the basic notions of both
theories: Shannon entropy versus Kolmogorov complexity, the relation of both to
universal coding, Shannon mutual information versus Kolmogorov (`algorithmic')
mutual information, probabilistic sufficient statistic versus algorithmic
sufficient statistic (related to lossy compression in the Shannon theory versus
meaningful information in the Kolmogorov theory), and rate distortion theory
versus Kolmogorov's structure function. Part of the material has appeared in
print before, scattered through various publications, but this is the first
comprehensive systematic comparison. The last mentioned relations are new.Comment: Survey, LaTeX 54 pages, 3 figures, Submitted to IEEE Trans
Information Theor
Application of Kolmogorov complexity and universal codes to identity testing and nonparametric testing of serial independence for time series
We show that Kolmogorov complexity and such its estimators as universal codes
(or data compression methods) can be applied for hypotheses testing in a
framework of classical mathematical statistics. The methods for identity
testing and nonparametric testing of serial independence for time series are
suggested.Comment: submitte
Quantum Kolmogorov Complexity
In this paper we give a definition for quantum Kolmogorov complexity. In the
classical setting, the Kolmogorov complexity of a string is the length of the
shortest program that can produce this string as its output. It is a measure of
the amount of innate randomness (or information) contained in the string.
We define the quantum Kolmogorov complexity of a qubit string as the length
of the shortest quantum input to a universal quantum Turing machine that
produces the initial qubit string with high fidelity. The definition of Vitanyi
(Proceedings of the 15th IEEE Annual Conference on Computational Complexity,
2000) measures the amount of classical information, whereas we consider the
amount of quantum information in a qubit string. We argue that our definition
is natural and is an accurate representation of the amount of quantum
information contained in a quantum state.Comment: 14 pages, LaTeX2e, no figures, \usepackage{amssymb,a4wide}. To appear
in the Proceedings of the 15th IEEE Annual Conference on Computational
Complexit
Quantum Kolmogorov Complexity Based on Classical Descriptions
We develop a theory of the algorithmic information in bits contained in an
individual pure quantum state. This extends classical Kolmogorov complexity to
the quantum domain retaining classical descriptions. Quantum Kolmogorov
complexity coincides with the classical Kolmogorov complexity on the classical
domain. Quantum Kolmogorov complexity is upper bounded and can be effectively
approximated from above under certain conditions. With high probability a
quantum object is incompressible. Upper- and lower bounds of the quantum
complexity of multiple copies of individual pure quantum states are derived and
may shed some light on the no-cloning properties of quantum states. In the
quantum situation complexity is not sub-additive. We discuss some relations
with ``no-cloning'' and ``approximate cloning'' properties.Comment: 17 pages, LaTeX, final and extended version of quant-ph/9907035, with
corrections to the published journal version (the two displayed equations in
the right-hand column on page 2466 had the left-hand sides of the displayed
formulas erroneously interchanged
Communication Complexity of the Secret Key Agreement in Algorithmic Information Theory
It is known that the mutual information, in the sense of Kolmogorov
complexity, of any pair of strings x and y is equal to the length of the
longest shared secret key that two parties can establish via a probabilistic
protocol with interaction on a public channel, assuming that the parties hold
as their inputs x and y respectively. We determine the worst-case communication
complexity of this problem for the setting where the parties can use private
sources of random bits. We show that for some x, y the communication complexity
of the secret key agreement does not decrease even if the parties have to agree
on a secret key whose size is much smaller than the mutual information between
x and y. On the other hand, we discuss examples of x, y such that the
communication complexity of the protocol declines gradually with the size of
the derived secret key. The proof of the main result uses spectral properties
of appropriate graphs and the expander mixing lemma, as well as information
theoretic techniques.Comment: 33 pages, 6 figures. v3: the full version of the MFCS 2020 pape
- …