In this work we propose to take certain chaotic systems as a reference for the design of PRNGs based on nonlinear congruences. In detail, in Section 2 we report a brief comparison between linear and nonlinear PRNGs. Since our aim is to derive nonlinear congruential generators from certain chaotic maps, in Section 3 we overview some theoretical fundamentals about TRNGs based on statistically stable mixing dynamical systems, focusing on the family of the Rényi maps. In Section 4 we discuss the link that exists between the dynamics of chaotic and pseudo-chaotic systems: to explain how the two dynamics are related it is necessary to project some results achieved within the Ergodic Theory (valid for chaotic systems) on the world of digital pseudo-chaos. To this aim, we have proposed a weaker and more general interpretation of the Shadowing Theory proposed by Coomes et al., focusing on probability measures, rather than on single chaotic trajectories. In Section 5 we study how to digitize the R´enyi maps, discussing how to set a minimum period length of the digitized trajectories. In Section 6 we present two alternative methods for the design of PRNGs based on nonlinear recurrences derived from the Renyi map, reporting the results of the NIST SP800.22 standard statistical test suite.

Addabbo, T., Fort, A., Rocchi, S., & Vignoli, V. (2011). Digitized Chaos for Pseudo-random Number Generation in Cryptography. In Chaos-Based Cryptography (pp. 67-97). Berlin / Heidelberg : Springer [10.1007/978-3-642-20542-2_3].

Digitized Chaos for Pseudo-random Number Generation in Cryptography

ADDABBO, TOMMASO;FORT, ADA;ROCCHI, SANTINA;VIGNOLI, VALERIO
2011

Abstract

In this work we propose to take certain chaotic systems as a reference for the design of PRNGs based on nonlinear congruences. In detail, in Section 2 we report a brief comparison between linear and nonlinear PRNGs. Since our aim is to derive nonlinear congruential generators from certain chaotic maps, in Section 3 we overview some theoretical fundamentals about TRNGs based on statistically stable mixing dynamical systems, focusing on the family of the Rényi maps. In Section 4 we discuss the link that exists between the dynamics of chaotic and pseudo-chaotic systems: to explain how the two dynamics are related it is necessary to project some results achieved within the Ergodic Theory (valid for chaotic systems) on the world of digital pseudo-chaos. To this aim, we have proposed a weaker and more general interpretation of the Shadowing Theory proposed by Coomes et al., focusing on probability measures, rather than on single chaotic trajectories. In Section 5 we study how to digitize the R´enyi maps, discussing how to set a minimum period length of the digitized trajectories. In Section 6 we present two alternative methods for the design of PRNGs based on nonlinear recurrences derived from the Renyi map, reporting the results of the NIST SP800.22 standard statistical test suite.
9783642205415
Addabbo, T., Fort, A., Rocchi, S., & Vignoli, V. (2011). Digitized Chaos for Pseudo-random Number Generation in Cryptography. In Chaos-Based Cryptography (pp. 67-97). Berlin / Heidelberg : Springer [10.1007/978-3-642-20542-2_3].
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11365/21707
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