Random properties of geometric under-sampled ring coupled map
Ina Taralova*, René Lozi†
Nowadays there exists an increasing demand for new and more efficient pseudo random
number generators (PRNG). These demands arise from different applications, such as multiagents competition, or secure information transmission [1]. In a previous work [2] we have
studied a ring-coupled symmetric tent maps by using the critical lines tool (the critical lines
being the first forward iterates of the lines long which the Jacobean is not defined, or
vanishes). It has been demonstrated that high order ( p  10 ) ring-coupled symmetric tent
maps could exhibit satisfactory random properties which have been validated by the NIST
tests for randomness. However, for low dimensions the chaotic attractor was not
equidistributed and contained holes. These holes have been shown to be delimited by the
critical lines and their forward iterates. Therefore, for a low system dimension, the
randomness properties were not satisfactory, until an additional parameterized undersampling had been applied to the output sequence. In order to improve the latter results, in
this paper an original geometrical under-sampling has been applied to the two-dimensional
system of ring-coupled symmetric tent maps. The critical lines tool has been used to define
analytically the frontiers of regions inside the chaotic attractor with different density. Using
the Markov matrix of transitions, a region with a constant density has been chosen. Subsequences belonging to this region have then been extracted. Is has been shown that the
expected density of the selected region can be experimentally validated. The applied NIST
tests have been successful, which validates the random properties of the sub-sequence. In
conclusion, the geometric under-sampling appears to be an efficient tool to improve the
randomness of the system even for low dimensional ring-coupled symmetric tent maps.
[1] Hénaff, S., Taralova, I., & Lozi, R., “Exact and Asymptotic Synchronization of a new
weakly coupled maps system”, Journal of Nonlinear Systems and Applications, 1, 87-95,
2010.
[2] Taralova, I., Espinel, A., & Lozi, R., “Dynamical and statistical analysis of a new Lozi
function for random numbers generation”, in Proceeding of Physcon 2011, León, Spain, 5-8
september, IPACS open Access Electronic Library.
†
Laboratoire J. A. Dieudonné, UMR CNRS 7351
Université de Nice Sophia-Antipolis, Parc Valrose,
06108 NICE Cedex 02, France
* L’UNAM, IRCCyN, UMR CNRS 6597,
Ecole Centrale de Nantes, 1, rue de la Noë, BP 92101,
44321 NANTES Cedex 3, France