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CivilComp Proceedings
ISSN 17593433 CCP: 112
PROCEEDINGS OF THE SIXTH INTERNATIONAL CONFERENCE ON PARALLEL, DISTRIBUTED, GPU AND CLOUD COMPUTING FOR ENGINEERING Edited by: P. Iványi and B.H.V. Topping
Paper 16
An update on the topological and ergodic properties of asynchronous iterations C. Guyeux
FemtoST institute, Universite de Bourgogne FrancheComte, France C. Guyeux, "An update on the topological and ergodic properties of
asynchronous iterations", in P. Iványi, B.H.V. Topping, (Editors), "Proceedings of the Sixth International Conference on Parallel, Distributed, GPU and Cloud Computing for Engineering", CivilComp Press, Stirlingshire, UK, Paper 16, 2019. doi:10.4203/ccp.112.16
Keywords: discrete dynamical system, chaos theory, ergodicity, information security..
Summary
Asynchronous delay iterations from discrete mathematics have long been used to accelerate
convergence in highperformance computing. They have the particularity of being defined on
an infinite (discrete) set that can be seen as a Cartesian product: at each iteration, a vector
of fixedsize bits is updated from a function whose return also depends on a parameter, itself
obtained from the first term of a sequence that is iterated at each iteration. The resulting
product space is infinite but only handles bounded integers. In addition, only the bit vector
must be stored in the finite state machine, a new term of the infinite sequence can be read at
each iteration. In doing so, we obtain a discrete dynamic system that can be implemented as
is on our computers, but which, from the moment they receive new data at each clock stroke,
iterates over an infinite set. Any algorithm performing the function mentioned above, and
iterating in order to update the memory, can thus be studied for its chaotic properties, and
when such properties are proven for the dynamical system, the associated computer program
is still chaotic in the most rigorous acceptation of this term, while implemented in finite state
machines.
Many advances in the theoretical study of the randomness of such asynchronous iterations have been achieved in the past decade, and these results have been successfully applied in various areas of IT security. The objective of this article is to review these various advances in the study of the disorder of asynchronous iterations, both theoretically and practically, and to present new avenues of research and the latest results on ergodicity cases. In detail, we will present the link between asynchronous iterations and a certain category of connected graphs, and we will deduce a characterization of the chaos of such iterations, asmathematically defined by Devaney, Knudsen, in the sense of topological and metric entropies, and for Lyapounov’s exponent. New results from measurement theory, including the ergodicity of these iterations, will then be introduced, and we will then provide an overview of the applications of these results in computer security, focusing in particular on cryptographically secure pseudorandom generation and symmetric encryption. purchase the fulltext of this paper (price £22)
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