
From the last two decades,  many researchers have investigated the use
of       chaotic       systems       to       build       pseudorandom
sequences~\cite{915396,915385,5376454}.   Two    main   features   can
intuitively explain this interest:  first of all, chaotic systems have
unpredictable or disorder-like behavior  which is needed for producing
complex sequences.  Furthermore, such systems are extremely sensitive
to  the initial states  too: a  tiny difference in the input 
may lead  to dramatic changes in output.

Moreover,  the  property  of  chaos  is often  reduced  to  iterate  a
\emph{chaotic} function, namely the logistic map\footnote{The logistic
map is  defined by $X^0 \in  [0,1], X^{n+1} = \mu  X^n(1-X^n), \mu \in
[0,4]$.}~\cite{915396,915385},  the Arnolds  cat map\footnote{Discrete
Arnolds cat map is defined by  $(X^0, Y^0)\in [0,N]^2, X^{n+1} = 2 Y^n
+ X^n$ mod $N$, $Y^{n+1} = X^n+Y^n$ mod $N$.}~\cite{5376454}\ldots
Au\-thors
are thus focused on finding parameters of such kind of functions which
are  the most  suitable  for generating  random-like sequences.   More
precisely,  they have  thus  to find  function  parameters that  avoid
parasitic attractors  and that lead  to a uniform distribution  of the
output.     To    check    how    accurate   are    their    generated
sequences~\cite{bfgw13:ij},  authors  are then  left  to submit  their
chaos-based  PRNG with  good  parameters on  statistical batteries  of
tests~\cite{DBLP:journals/corr/abs-1112-5239},                  namely:
DieHARD~\cite{Marsaglia1996},          NIST~\cite{Nist10},         and
TestU01~\cite{LEcuyerS07}.



Devaney  has formalized~\cite{Devaney}  the fundamental  properties of
the chaotic maps, namely:  sensitive dependence on initial conditions,
transitivity, and density of periodic  points.  For short, a system is
sensitive  to  initial  conditions  if  any  point  contains,  in  any
neighborhood,  another  point   with  a  completely  different  future
trajectory.   Topological transitivity  is established  when,  for any
element, any neighborhood of  its future evolution eventually overlaps
with any  other open set.   On the contrary,  a dense set  of periodic
points is an element of regularity that a chaotic dynamical system has
to exhibit.   However, because  of the finiteness  of the memory  of a
computer, only  a kind  of discrete chaos  is generated.  This induces
that chaotic  properties, which  could have been  proven on  $\Reels$, can
however be lost on floating point numbers, which is the interpretation
domain of $\Reels$.

To  avoid this  loss of  chaos, we  had constructed  chaos-based PRNGs
that  iterate a continuous  functions $G_f$  defined on  the discrete
domain $\llbracket  1 ;  n \rrbracket^{\Nats} \times  \{0,1\}^n$ where
$f$ is  a Boolean function (\textit{i.e.}, $f  : \{0,1\}^n \rightarrow
\{0,1\}^n$).           These          PRNGs         are          named
$\textit{CIPRNG}_f^1(u)$~\cite{bgw09:ip,guyeuxTaiwan10},
$\textit{CIPRNG}_f^2(u,v)$~\cite{wbg10:ip},                         and
$\textit{XOR~CIPRNG}(S)$~\cite{DBLP:journals/corr/abs-1112-5239},
where \textit{CI} stands for  \emph{Chaotic Iterations}, which have
been particularized according to the function they iterate.

Chaotic   properties  have   been  well   established  for   both  the
$\textit{XOR~CIPRNG}$   and  $\textit{CIPRNG}_f^1(u)$   under  certain
conditions  for  the iteration  function  $f$,  it  has been  formerly
deduced that it  was the case too for  the whole $\textit{CIPRNG}_f^2$
category of generators.  However, contrarily to what has been too much
rapidly deduced, this claim is not obvious (a subsequence of a chaotic
sequence  is   not  necessarily  a   chaotic  sequence  too)   and  it
necessitates a rigorous proof, which is the aim of this article.

The    remainder    of    the    paper    is    organized    as    follows.
Section~\ref{sec:notations}  recalls definitions  of  chaos properties
and  of the  family  of \textit{CIPRNG}.  Section~\ref{sec:wellchosen}
gives  the  complete  proof  of chaos  for  the  $\textit{CIPRNG}_f^2$
category.  This is the main  contribution of this work. The paper ends
with a conclusion section where intended future work is presented.
