ENOC 2008, Saint Petersburg, Russia, June, 30-July, 4 2008 CHAOTIC SAMPLING, VERY WEAKLY COUPLING, AND CHAOTING MIXING: THREE SIMPLE SYNERGISTIC MECHANISMS TO MAKE NEW FAMILIES OF CHAOTIC PSEUDO RANDOM NUMBER GENERATORS René Lozi Laboratoire J.A. Dieudonné – UMR du CNRS N° 6621, University of Nice-Sophia-Antipolis, Parc Valrose, 06108 Nice Cedex 02, France and Institut Universitaire de Formation des Maîtres Célestin Freinet-académie de Nice, University of Nice-Sophia-Antipolis, 89 avenue George V, 06046 Nice Cedex 1, France rlozi@unice.fr Abstract We introduce and combine synergistically three simple new mechanisms: very weakly coupling of chaotic maps, chaotic sampling and chaotic mixing of iterated points in order to make new families of enhanced Chaotic Pseudo Random Number Generators (CPRNG). The key feature of these CPRNG is that they use chaotic numbers themselves in order to sample and to mix chaotically several subsequences of chaotic numbers. We analyze numerically the properties of these new families and underline their very high qualities and usefulness as CPRNG when series are computed up to 10 13 iterations. Key words Chaos, pseudo random numbers, coupled maps. 1. Introduction When a dynamical system is realized on a computer using floating point or double precision numbers, the computation is of a discretization, where finite machine arithmetic replaces continuum state space. For chaotic dynamical systems, the discretization often has collapsing effects to a fixed point or to short cycles [Lanford III, 1998; Gora, Boyarsky, Islam and Bahsoun, 2006]. In order to preserve the chaotic properties of the continuous models in numerical experiments we have introduced as a first one mechanism the very weak multidimensional coupling of p one- dimensional dynamical systems which is noteworthy [Lozi, 2006]. Moreover each component of these numbers belonging to p ℝ are equally distributed over a given finite interval J ⊂ ℝ . Numerical computations show that this distribution is obtained with a very good approximation. They have also the property that the length of the periods of the numerically observed orbits is very large. However chaotic numbers are not pseudo- random numbers because the plot of the couples of iterated points (x n , x n+1 ) in the phase plane shows up the map f used as one-dimensional dynamical systems to generate them. A second simple mechanism is then used to hide the graph of this genuine function f in the phase space ( l n l n x x 1 , + . The pivotal idea of this mechanism is to sample chaotically the sequence ( … … , , , , , , 1 2 1 0 l n l n l l l x x x x x + selecting l n x every time the value of m n x is strictly greater than a threshold T ∈ J, with l ≠ m, for 1 ≤ l, m ≤ p . A third mechanism can improve the unpredictability of the chaotic sequence generated as above, using synergistically all the components of the vector X, instead of two. This simple third mechanism is based on the chaotic mixing of the p-1 sequences ( 29 … … , , , , , , 1 1 1 1 2 1 1 1 0 + n n x x x x x , ( 29 … … , , , , , , 2 1 2 2 2 2 1 2 0 + n n x x x x x ,…, ( 29 … … , , , , , , 1 1 1 1 2 1 1 1 0 - + - - - - p n p n p p p x x x x x using the last one ( … … , , , , , , 1 2 1 0 p n p n p p p x x x x x + with respect to a given partition r 1 , r 2 , …, r p-1 of J , to distribute the iterated points. In this paper we explore numerically the properties of these new families and underline their very high qualities and usefulness as CPRNG when series are computed up to 10 13 iterations. Generation of random or pseudorandom numbers, nowadays, is a key feature of industrial mathematics. Pseudorandom or chaotic numbers are used in many areas of contemporary technology such as modern communication systems and engineering applications. Everything we do to achieve privacy and security in the computer age depends on random numbers. More and more European or US patents using discrete mappings for this purpose are obtained by researchers of discrete dynamical systems [Petersen and Sorensen, 2007; Ruggiero, Mascolo, Pedaci and Amato, 2006]. The idea of construction of chaotic pseudorandom number generators (CPRNG) applying discrete chaotic dynamical systems,           = p x x X ⋮ 1