Pseudo-Random Number Generators and Chaos
Summary :
Chaotic response of nonlinear deterministic systems has recently attracted considerable interest of researchers of various branches of science. This paper proposes an application of chaotic phenomena in the field of numerical analysis. Namely, a new method is proposed for designing pseudo-random number generators by making use of chaotic first order nonlinear difference equations. In this paper, in the first place, this design problem is formulated mathematically as an inverse problem of the Perron-Frobenius equation. Here, the Perron-Frobenius equation is the one expressing relationship between a nonlinear transformation generating a certain type of nonlinear difference equations and a density of a distribution function for solutions of such a difference equation, and the inverse problem of the Perron-Frobenius equation is a problem of solving the Perron-Frobenius equation for a nonlinear transformation provided that a density of a distribution function is given. This inverse problem of the Perron-Frobenius equation is then solved under the conditions that the density function is a step function and the nonlinear transformation is a piecewise linear transformation satisfying certain conditions. As an application of this result, it is shown that a uniform random number generator with an arbitrary Kolmogorov's entropy can be constructed. Moreover, a new efficient method is proposed for generating pseudo-random numbers which distribute according to an arbitrary step function type density of a distribution function.
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- IEICE TRANSACTIONS on transactions Vol.E65-E No.9 pp.534-541
- Publication Date
- 1982/09/25
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- Type of Manuscript
- PAPER
- Category
- Mathematics
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Shin'ichi OISHI, Hajime INOUE, "Pseudo-Random Number Generators and Chaos" in IEICE TRANSACTIONS on transactions,
vol. E65-E, no. 9, pp. 534-541, September 1982, doi: .
Abstract: Chaotic response of nonlinear deterministic systems has recently attracted considerable interest of researchers of various branches of science. This paper proposes an application of chaotic phenomena in the field of numerical analysis. Namely, a new method is proposed for designing pseudo-random number generators by making use of chaotic first order nonlinear difference equations. In this paper, in the first place, this design problem is formulated mathematically as an inverse problem of the Perron-Frobenius equation. Here, the Perron-Frobenius equation is the one expressing relationship between a nonlinear transformation generating a certain type of nonlinear difference equations and a density of a distribution function for solutions of such a difference equation, and the inverse problem of the Perron-Frobenius equation is a problem of solving the Perron-Frobenius equation for a nonlinear transformation provided that a density of a distribution function is given. This inverse problem of the Perron-Frobenius equation is then solved under the conditions that the density function is a step function and the nonlinear transformation is a piecewise linear transformation satisfying certain conditions. As an application of this result, it is shown that a uniform random number generator with an arbitrary Kolmogorov's entropy can be constructed. Moreover, a new efficient method is proposed for generating pseudo-random numbers which distribute according to an arbitrary step function type density of a distribution function.
URL: https://global.ieice.org/en_transactions/transactions/10.1587/e65-e_9_534/_p
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@ARTICLE{e65-e_9_534,
author={Shin'ichi OISHI, Hajime INOUE, },
journal={IEICE TRANSACTIONS on transactions},
title={Pseudo-Random Number Generators and Chaos},
year={1982},
volume={E65-E},
number={9},
pages={534-541},
abstract={Chaotic response of nonlinear deterministic systems has recently attracted considerable interest of researchers of various branches of science. This paper proposes an application of chaotic phenomena in the field of numerical analysis. Namely, a new method is proposed for designing pseudo-random number generators by making use of chaotic first order nonlinear difference equations. In this paper, in the first place, this design problem is formulated mathematically as an inverse problem of the Perron-Frobenius equation. Here, the Perron-Frobenius equation is the one expressing relationship between a nonlinear transformation generating a certain type of nonlinear difference equations and a density of a distribution function for solutions of such a difference equation, and the inverse problem of the Perron-Frobenius equation is a problem of solving the Perron-Frobenius equation for a nonlinear transformation provided that a density of a distribution function is given. This inverse problem of the Perron-Frobenius equation is then solved under the conditions that the density function is a step function and the nonlinear transformation is a piecewise linear transformation satisfying certain conditions. As an application of this result, it is shown that a uniform random number generator with an arbitrary Kolmogorov's entropy can be constructed. Moreover, a new efficient method is proposed for generating pseudo-random numbers which distribute according to an arbitrary step function type density of a distribution function.},
keywords={},
doi={},
ISSN={},
month={September},}
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TY - JOUR
TI - Pseudo-Random Number Generators and Chaos
T2 - IEICE TRANSACTIONS on transactions
SP - 534
EP - 541
AU - Shin'ichi OISHI
AU - Hajime INOUE
PY - 1982
DO -
JO - IEICE TRANSACTIONS on transactions
SN -
VL - E65-E
IS - 9
JA - IEICE TRANSACTIONS on transactions
Y1 - September 1982
AB - Chaotic response of nonlinear deterministic systems has recently attracted considerable interest of researchers of various branches of science. This paper proposes an application of chaotic phenomena in the field of numerical analysis. Namely, a new method is proposed for designing pseudo-random number generators by making use of chaotic first order nonlinear difference equations. In this paper, in the first place, this design problem is formulated mathematically as an inverse problem of the Perron-Frobenius equation. Here, the Perron-Frobenius equation is the one expressing relationship between a nonlinear transformation generating a certain type of nonlinear difference equations and a density of a distribution function for solutions of such a difference equation, and the inverse problem of the Perron-Frobenius equation is a problem of solving the Perron-Frobenius equation for a nonlinear transformation provided that a density of a distribution function is given. This inverse problem of the Perron-Frobenius equation is then solved under the conditions that the density function is a step function and the nonlinear transformation is a piecewise linear transformation satisfying certain conditions. As an application of this result, it is shown that a uniform random number generator with an arbitrary Kolmogorov's entropy can be constructed. Moreover, a new efficient method is proposed for generating pseudo-random numbers which distribute according to an arbitrary step function type density of a distribution function.
ER -
English
