Before this work,  the second class of
chaos-based PRNG $\textit{CIPRNG}_f^2$ was robust against 
batteries of statistical tests and was abusively said to be chaotic.
This work has formally established the proof that
the $\textit{CIPRNG}_f^2$  is chaotic according to the Devaney 
definition for some well-chosen functions $f$.
The chaos condition is expressed as a necessary and sufficient condition 
on a graph of iterations: this one has to be strongly connected.  

It has thus bridged the gap between the need of true chaos for some
applications and the practical efficiency. 
In a future work, we plan to study sufficient conditions on $f$ to
observe this 
connectivity
and to study computed functions in the perspective
of mixing time to improve their practical efficiency. 

