pch : mise a jour de reponse pour les refs (une derniere)

[prng_gpu.git] / supplementary.tex

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\newcommand{\X}{\mathcal{X}}
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\title{Annex document of ``Efficient and Cryptographically Secure Generation of Chaotic Pseudorandom Numbers on GPU''}
\begin{document}

\author{Jacques M. Bahi, Rapha\"{e}l Couturier,  Christophe
Guyeux, and Pierre-Cyrille Héam\thanks{Authors in alphabetic order}}
   

\maketitle

\IEEEdisplaynotcompsoctitleabstractindextext
\IEEEpeerreviewmaketitle


\section{Proofs of Chaos of the General Formulation of the Chaotic Iterations}
\label{deuxième def}
Let us consider the discrete dynamical systems in chaotic iterations having 
the general form: $\forall    n\in     \mathds{N}^{\ast     }$, $  \forall     i\in
\llbracket1;\mathsf{N}\rrbracket $,

\begin{equation}
  x_i^n=\left\{
\begin{array}{ll}
  x_i^{n-1} &  \text{ if  } i \notin \mathcal{S}^n \\
  \left(f(x^{n-1})\right)_{S^n} & \text{ if }i \in \mathcal{S}^n.
\end{array}\right.
\label{general CIs}
\end{equation}

In other words, at the $n^{th}$ iteration, only the cells whose id is
contained into the set $S^{n}$ are iterated.

Let us now rewrite these general chaotic iterations as usual discrete dynamical
system of the form $X^{n+1}=f(X^n)$ on an ad hoc metric space. Such a formulation
is required in order to study the topological behavior of the system.

Let us introduce the following function:
\begin{equation}
\begin{array}{cccc}
 \chi: & \llbracket 1; \mathsf{N} \rrbracket \times \mathcal{P}\left(\llbracket 1; \mathsf{N} \rrbracket\right) & \longrightarrow & \mathds{B}\\
         & (i,X) & \longmapsto  & \left\{ \begin{array}{ll} 0 & \textrm{if }i \notin X, \\ 1 & \textrm{if }i \in X,  \end{array}\right.
\end{array} 
\end{equation}
where $\mathcal{P}\left(X\right)$ is for the powerset of the set $X$, that is, $Y \in \mathcal{P}\left(X\right) \Longleftrightarrow Y \subset X$.

Given a function $f:\mathds{B}^\mathsf{N} \longrightarrow \mathds{B}^\mathsf{N} $, define the function:
$F_{f}:  \mathcal{P}\left(\llbracket1;\mathsf{N}\rrbracket \right) \times \mathds{B}^{\mathsf{N}} 
\longrightarrow \mathds{B}^{\mathsf{N}}$
\begin{equation*}
\begin{array}{rll}
 (P,E) & \longmapsto & \left( E_{j}.\chi (j,P)+f(E)_{j}.\overline{\chi(j,P)}\right) _{j\in \llbracket1;\mathsf{N}\rrbracket}%
\end{array}%
\end{equation*}%
where + and . are the Boolean addition and product operations, and $\overline{x}$ 
is the negation of the Boolean $x$.
Consider the phase space:
\begin{equation}
\mathcal{X} = \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)^\mathds{N} \times
\mathds{B}^\mathsf{N},
\end{equation}
\noindent and the map defined on $\mathcal{X}$:
\begin{equation}
G_f\left(S,E\right) = \left(\sigma(S), F_f(i(S),E)\right), %\label{Gf} %%RAPH, j'ai viré ce label qui existe déjà avant...
\end{equation}
\noindent where $\sigma$ is the \emph{shift} function defined by $\sigma
(S^{n})_{n\in \mathds{N}}\in \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)^\mathds{N}\longrightarrow (S^{n+1})_{n\in
\mathds{N}}\in \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)^\mathds{N}$ and $i$ is the \emph{initial function} 
$i:(S^{n})_{n\in \mathds{N}} \in \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)^\mathds{N}\longrightarrow S^{0}\in \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)$. 
Then the general chaotic iterations defined in Equation \ref{general CIs} can 
be described by the following discrete dynamical system:
\begin{equation}
\left\{
\begin{array}{l}
X^0 \in \mathcal{X} \\
X^{k+1}=G_{f}(X^k).%
\end{array}%
\right.
\end{equation}%

Once more, a shift function appears as a component of these general chaotic 
iterations. 

To study the Devaney's chaos property, a distance between two points 
$X = (S,E), Y = (\check{S},\check{E})$ of $\mathcal{X}$ must be defined.
Let us introduce:
\begin{equation}
d(X,Y)=d_{e}(E,\check{E})+d_{s}(S,\check{S}),
\label{nouveau d}
\end{equation}
\noindent where $ \displaystyle{d_{e}(E,\check{E})} = \displaystyle{\sum_{k=1}^{\mathsf{N}%
 }\delta (E_{k},\check{E}_{k})}$  is once more the Hamming distance, and
$  \displaystyle{d_{s}(S,\check{S})}  =  \displaystyle{\dfrac{9}{\mathsf{N}}%
 \sum_{k=1}^{\infty }\dfrac{|S^k\Delta {S}^k|}{10^{k}}}$,
%%RAPH : ici, j'ai supprimé tous les sauts à la ligne
%% \begin{equation}
%% \left\{
%% \begin{array}{lll}
%% \displaystyle{d_{e}(E,\check{E})} & = & \displaystyle{\sum_{k=1}^{\mathsf{N}%
%% }\delta (E_{k},\check{E}_{k})} \textrm{ is once more the Hamming distance}, \\
%% \displaystyle{d_{s}(S,\check{S})} & = & \displaystyle{\dfrac{9}{\mathsf{N}}%
%% \sum_{k=1}^{\infty }\dfrac{|S^k\Delta {S}^k|}{10^{k}}}.%
%% \end{array}%
%% \right.
%% \end{equation}
where $|X|$ is the cardinality of a set $X$ and $A\Delta B$ is for the symmetric difference, defined for sets A, B as
$A\,\Delta\,B = (A \setminus B) \cup (B \setminus A)$.


\begin{proposition}
The function $d$ defined in Eq.~\ref{nouveau d} is a metric on $\mathcal{X}$.
\end{proposition}

\begin{proof}
 $d_e$ is the Hamming distance. We will prove that $d_s$ is a distance
too, thus $d$, as being the sum of two distances, will also be a distance.
 \begin{itemize}
\item Obviously, $d_s(S,\check{S})\geqslant 0$, and if $S=\check{S}$, then 
$d_s(S,\check{S})=0$. Conversely, if $d_s(S,\check{S})=0$, then 
$\forall k \in \mathds{N}, |S^k\Delta {S}^k|=0$, and so $\forall k, S^k=\check{S}^k$.
 \item $d_s$ is symmetric 
($d_s(S,\check{S})=d_s(\check{S},S)$) due to the commutative property
of the symmetric difference. 
\item Finally, $|S \Delta S''| = |(S \Delta \varnothing) \Delta S''|= |S \Delta (S'\Delta S') \Delta S''|= |(S \Delta S') \Delta (S' \Delta S'')|\leqslant |S \Delta S'| + |S' \Delta S''|$, 
and so for all subsets $S,S',$ and $S''$ of $\llbracket 1, \mathsf{N} \rrbracket$, 
we have $d_s(S,S'') \leqslant d_e(S,S')+d_s(S',S'')$, and the triangle
inequality is obtained.
 \end{itemize}
\end{proof}


Before being able to study the topological behavior of the general 
chaotic iterations, we must first establish that:

\begin{proposition}
 For all $f:\mathds{B}^\mathsf{N} \longrightarrow \mathds{B}^\mathsf{N} $, the function $G_f$ is continuous on 
$\left( \mathcal{X},d\right)$.
\end{proposition}


\begin{proof}
We use the sequential continuity.
Let $(S^n,E^n)_{n\in \mathds{N}}$ be a sequence of the phase space $%
\mathcal{X}$, which converges to $(S,E)$. We will prove that $\left(
G_{f}(S^n,E^n)\right) _{n\in \mathds{N}}$ converges to $\left(
G_{f}(S,E)\right) $. Let us remark that for all $n$, $S^n$ is a strategy,
thus, we consider a sequence of strategies (\emph{i.e.}, a sequence of
sequences).\newline
As $d((S^n,E^n);(S,E))$ converges to 0, each distance $d_{e}(E^n,E)$ and $d_{s}(S^n,S)$ converges
to 0. But $d_{e}(E^n,E)$ is an integer, so $\exists n_{0}\in \mathds{N},$ $%
d_{e}(E^n,E)=0$ for any $n\geqslant n_{0}$.\newline
In other words, there exists a threshold $n_{0}\in \mathds{N}$ after which no
cell will change its state:
$\exists n_{0}\in \mathds{N},n\geqslant n_{0}\Rightarrow E^n = E.$

In addition, $d_{s}(S^n,S)\longrightarrow 0,$ so $\exists n_{1}\in %
\mathds{N},d_{s}(S^n,S)<10^{-1}$ for all indexes greater than or equal to $%
n_{1}$. This means that for $n\geqslant n_{1}$, all the $S^n$ have the same
first term, which is $S^0$: $\forall n\geqslant n_{1},S_0^n=S_0.$

Thus, after the $max(n_{0},n_{1})^{th}$ term, states of $E^n$ and $E$ are
identical and strategies $S^n$ and $S$ start with the same first term.\newline
Consequently, states of $G_{f}(S^n,E^n)$ and $G_{f}(S,E)$ are equal,
so, after the $max(n_0, n_1)^{th}$ term, the distance $d$ between these two points is strictly less than 1.\newline
\noindent We now prove that the distance between $\left(
G_{f}(S^n,E^n)\right) $ and $\left( G_{f}(S,E)\right) $ is convergent to
0. Let $\varepsilon >0$. \medskip
\begin{itemize}
\item If $\varepsilon \geqslant 1$, we see that the distance
between $\left( G_{f}(S^n,E^n)\right) $ and $\left( G_{f}(S,E)\right) $ is
strictly less than 1 after the $max(n_{0},n_{1})^{th}$ term (same state).
\medskip
\item If $\varepsilon <1$, then $\exists k\in \mathds{N},10^{-k}\geqslant
\varepsilon > 10^{-(k+1)}$. But $d_{s}(S^n,S)$ converges to 0, so
\begin{equation*}
\exists n_{2}\in \mathds{N},\forall n\geqslant
n_{2},d_{s}(S^n,S)<10^{-(k+2)},
\end{equation*}%
thus after $n_{2}$, the $k+2$ first terms of $S^n$ and $S$ are equal.
\end{itemize}
\noindent As a consequence, the $k+1$ first entries of the strategies of $%
G_{f}(S^n,E^n)$ and $G_{f}(S,E)$ are the same ($G_{f}$ is a shift of strategies) and due to the definition of $d_{s}$, the floating part of
the distance between $(S^n,E^n)$ and $(S,E)$ is strictly less than $%
10^{-(k+1)}\leqslant \varepsilon $.

In conclusion,
%%RAPH : ici j'ai rajouté une ligne
%%TOF : ici j'ai rajouté un commentaire
%%TOF : ici aussi
$
\forall \varepsilon >0,$ $\exists N_{0}=max(n_{0},n_{1},n_{2})\in \mathds{N}
,$ $\forall n\geqslant N_{0},$
$ d\left( G_{f}(S^n,E^n);G_{f}(S,E)\right)
\leqslant \varepsilon .
$
$G_{f}$ is consequently continuous.
\end{proof}


It is now possible to study the topological behavior of the general chaotic
iterations. We will prove that,

\begin{theorem}
\label{t:chaos des general}
 The general chaotic iterations defined on Equation~\ref{general CIs} satisfy
the Devaney's property of chaos.
\end{theorem}

Let us firstly prove the following lemma.

\begin{lemma}[Strong transitivity]
\label{strongTrans}
 For all couples $X,Y \in \mathcal{X}$ and any neighborhood $V$ of $X$, we can 
find $n \in \mathds{N}^*$ and $X' \in V$ such that $G^n(X')=Y$.
\end{lemma}

\begin{proof}
 Let $X=(S,E)$, $\varepsilon>0$, and $k_0 = \lfloor log_{10}(\varepsilon)+1 \rfloor$. 
Any point $X'=(S',E')$ such that $E'=E$ and $\forall k \leqslant k_0, S'^k=S^k$, 
are in the open ball $\mathcal{B}\left(X,\varepsilon\right)$. Let us define 
$\check{X} = \left(\check{S},\check{E}\right)$, where $\check{X}= G^{k_0}(X)$.
We denote by $s\subset \llbracket 1; \mathsf{N} \rrbracket$ the set of coordinates
that are different between $\check{E}$ and the state of $Y$. Thus each point $X'$ of
the form $(S',E')$ where $E'=E$ and $S'$ starts with 
$(S^0, S^1, \hdots, S^{k_0},s,\hdots)$, verifies the following properties:
\begin{itemize}
 \item $X'$ is in $\mathcal{B}\left(X,\varepsilon\right)$,
 \item the state of $G_f^{k_0+1}(X')$ is the state of $Y$.
\end{itemize}
Finally the point $\left(\left(S^0, S^1, \hdots, S^{k_0},s,s^0, s^1, \hdots\right); E\right)$, 
where $(s^0,s^1, \hdots)$ is the strategy of $Y$, satisfies the properties
claimed in the lemma.
\end{proof}

We can now prove the Theorem~\ref{t:chaos des general}.

\begin{proof}[Theorem~\ref{t:chaos des general}]
Firstly, strong transitivity implies transitivity.

Let $(S,E) \in\mathcal{X}$ and $\varepsilon >0$. To
prove that $G_f$ is regular, it is sufficient to prove that
there exists a strategy $\tilde S$ such that the distance between
$(\tilde S,E)$ and $(S,E)$ is less than $\varepsilon$, and such that
$(\tilde S,E)$ is a periodic point.

Let $t_1=\lfloor-\log_{10}(\varepsilon)\rfloor$, and let $E'$ be the
configuration that we obtain from $(S,E)$ after $t_1$ iterations of
$G_f$. As $G_f$ is strongly transitive, there exists a strategy $S'$ 
and $t_2\in\mathds{N}$ such
that $E$ is reached from $(S',E')$ after $t_2$ iterations of $G_f$.

Consider the strategy $\tilde S$ that alternates the first $t_1$ terms
of $S$ and the first $t_2$ terms of $S'$: 
%%RAPH : j'ai coupé la ligne en 2
$$\tilde
S=(S_0,\dots,S_{t_1-1},S'_0,\dots,S'_{t_2-1},S_0,$$$$\dots,S_{t_1-1},S'_0,\dots,S'_{t_2-1},S_0,\dots).$$ It
is clear that $(\tilde S,E)$ is obtained from $(\tilde S,E)$ after
$t_1+t_2$ iterations of $G_f$. So $(\tilde S,E)$ is a periodic
point. Since $\tilde S_t=S_t$ for $t<t_1$, by the choice of $t_1$, we
have $d((S,E),(\tilde S,E))<\epsilon$.
\end{proof}





\section{Statistical Improvements Using Chaotic Iterations}

\label{The generation of pseudorandom sequence}


Let us explain in this annex why we have reasonable ground to believe that chaos 
can improve statistical properties.
We will show in this section that chaotic properties as defined in the
mathematical theory of chaos are related to some statistical tests that can be found
in the NIST battery. Furthermore, we will check that, when mixing defective PRNGs with
chaotic iterations, the new generator presents better statistical properties
(this section summarizes and extends the work of~\cite{bfg12a:ip}).



\subsection{Qualitative relations between topological properties and statistical tests}


There are various relations between topological properties that describe an unpredictable behavior for a discrete 
dynamical system on the one
hand, and statistical tests to check the randomness of a numerical sequence
on the other hand. These two mathematical disciplines follow a similar 
objective in case of a recurrent sequence (to characterize an intrinsically complicated behavior for a
recurrent sequence), with two different but complementary approaches.
It is true that the following illustrative links give only qualitative arguments, 
and proofs should be provided later to make such arguments irrefutable. However 
they give a first understanding of the reason why we think that chaotic properties should tend
to improve the statistical quality of PRNGs.
%
Let us now list some of these relations between topological properties defined in the mathematical
theory of chaos and tests embedded into the NIST battery. %Such relations need to be further 
%investigated, but they presently give a first illustration of a trend to search similar properties in the 
%two following fields: mathematical chaos and statistics.


\begin{itemize}
    \item \textbf{Regularity}. As stated in Section~\ref{M-subsec:Devaney} of the main document, a chaotic dynamical system must 
have an element of regularity. Depending on the chosen definition of chaos, this element can be the existence of
a dense orbit, the density of periodic points, etc. The key idea is that a dynamical system with no periodicity
is not as chaotic as a system having periodic orbits: in the first situation, we can predict something and gain a
knowledge about the behavior of the system, that is, it never enters into a loop. A similar importance for periodicity is emphasized in
the two following NIST tests~\cite{Nist10}:
    \begin{itemize}
        \item \textbf{Non-overlapping Template Matching Test}. Detect generators that produce too many occurrences of a given non-periodic (aperiodic) pattern.
        \item \textbf{Discrete Fourier Transform (Spectral) Test}. Detect periodic features (i.e., repetitive patterns that are close one to another) in the tested sequence that would indicate a deviation from the assumption of randomness.
    \end{itemize}

\item \textbf{Transitivity}. This topological property previously introduced  states that the dynamical system is intrinsically complicated: it cannot be simplified into 
two subsystems that do not interact, as we can find in any neighborhood of any point another point whose orbit visits the whole phase space. 
This focus on the places visited by the orbits of the dynamical system takes various nonequivalent formulations in the mathematical theory
of chaos, namely: transitivity, strong transitivity, total transitivity, topological mixing, and so on~\cite{bg10:ij}. A similar attention 
is brought on the states visited during a random walk in the two tests below~\cite{Nist10}:
    \begin{itemize}
        \item \textbf{Random Excursions Variant Test}. Detect deviations from the expected number of visits to various states in the random walk.
        \item \textbf{Random Excursions Test}. Determine if the number of visits to a particular state within a cycle deviates from what one would expect for a random sequence.
    \end{itemize}

\item \textbf{Chaos according to Li and Yorke}. Two points of the phase space $(x,y)$ define a couple of Li-Yorke when $\limsup_{n \rightarrow +\infty} d(f^{(n)}(x), f^{(n)}(y))>0$ et $\liminf_{n \rightarrow +\infty} d(f^{(n)}(x), f^{(n)}(y))=0$, meaning that their orbits always oscillate as the iterations pass. When a system is compact and contains an uncountable set of such points, it is claimed as chaotic according
to Li-Yorke~\cite{Li75,Ruette2001}. A similar property is regarded in the following NIST test~\cite{Nist10}.
    \begin{itemize}
        \item \textbf{Runs Test}. To determine whether the number of runs of ones and zeros of various lengths is as expected for a random sequence. In particular, this test determines whether the oscillation between such zeros and ones is too fast or too slow.
    \end{itemize}
    \item \textbf{Topological entropy}. The desire to formulate an equivalency of the thermodynamics entropy
has emerged both in the topological and statistical fields. Once again, a similar objective has led to two different
rewritting of an entropy based disorder: the famous Shannon definition of entropy is approximated in the statistical approach, 
whereas topological entropy is defined as follows:
$x,y \in \mathcal{X}$ are $\varepsilon-$\emph{separated in time $n$} if there exists $k \leqslant n$ such that $d\left(f^{(k)}(x),f^{(k)}(y)\right)>\varepsilon$. Then $(n,\varepsilon)-$separated sets are sets of points that are all $\varepsilon-$separated in time $n$, which
leads to the definition of $s_n(\varepsilon,Y)$, being the maximal cardinality of all $(n,\varepsilon)-$separated sets. Using these notations, 
the topological entropy is defined as follows: $$h_{top}(\mathcal{X},f)  = \displaystyle{\lim_{\varepsilon \rightarrow 0} \Big[ \limsup_{n \rightarrow +\infty} \dfrac{1}{n} \log s_n(\varepsilon,\mathcal{X})\Big]}.$$
This value measures the average exponential growth of the number of distinguishable orbit segments. 
In this sense, it measures the complexity of the topological dynamical system, whereas 
the Shannon approach comes to mind when defining the following test~\cite{Nist10}:
    \begin{itemize}
\item \textbf{Approximate Entropy Test}. Compare the frequency of the overlapping blocks of two consecutive/adjacent lengths ($m$ and $m+1$) against the expected result for a random sequence.
    \end{itemize}

    \item \textbf{Non-linearity, complexity}. Finally, let us remark that non-linearity and complexity are 
not only sought in general to obtain chaos, but they are also required for randomness, as illustrated by the two tests below~\cite{Nist10}.
    \begin{itemize}
\item \textbf{Binary Matrix Rank Test}. Check for linear dependence among fixed length substrings of the original sequence.
\item \textbf{Linear Complexity Test}. Determine whether or not the sequence is complex enough to be considered random.
      \end{itemize}
\end{itemize}


We have proven in our previous works~\cite{guyeux12:bc} that chaotic iterations satisfying Theorem~\ref{M-Th:Caractérisation   des   IC   chaotiques} of the main document are, among other
things, strongly transitive, topologically mixing, chaotic as defined by Li and Yorke,
and that they have a topological entropy and an exponent of Lyapunov both equal to $ln(\mathsf{N})$,
where $\mathsf{N}$ is the size of the iterated vector.
These topological properties make that we are ground to believe that a generator based on chaotic
iterations will probably be able to pass all the existing statistical batteries for pseudorandomness like
the NIST one. The following subsections, in which we prove that defective generators have their
statistical properties improved by chaotic iterations, show that such an assumption is true.

\subsection{Details of some Existing Generators}

The list of defective PRNGs we will use 
as inputs for the statistical tests to come is introduced here.

Firstly, the simple linear congruency generators (LCGs) will be used. 
They are defined by the following recurrence:
\begin{equation}
x^n = (ax^{n-1} + c)~mod~m,
\label{LCG}
\end{equation}
where $a$, $c$, and $x^0$ must be, among other things, non-negative and inferior to 
$m$~\cite{LEcuyerS07}. In what follows, 2LCGs and 3LCGs refer to two (resp. three) 
combinations of such LCGs. For further details, see~\cite{bfg12a:ip,combined_lcg}.

Secondly, the multiple recursive generators (MRGs) which will be used,
are based on a linear recurrence of order 
$k$, modulo $m$~\cite{LEcuyerS07}:
\begin{equation}
x^n = (a^1x^{n-1}+~...~+a^kx^{n-k})~mod~m .
\label{MRG}
\end{equation}
The combination of two MRGs (referred as 2MRGs) is also used in these experiments.

Generators based on linear recurrences with carry will be regarded too.
This family of generators includes the add-with-carry (AWC) generator, based on the recurrence:
\begin{equation}
\label{AWC}
\begin{array}{l}
x^n = (x^{n-r} + x^{n-s} + c^{n-1})~mod~m, \\
c^n= (x^{n-r} + x^{n-s} + c^{n-1}) / m, \end{array}\end{equation}
the SWB generator, having the recurrence:
\begin{equation}
\label{SWB}
\begin{array}{l}
x^n = (x^{n-r} - x^{n-s} - c^{n-1})~mod~m, \\
c^n=\left\{
\begin{array}{l}
1 ~~~~~\text{if}~ (x^{i-r} - x^{i-s} - c^{i-1})<0\\
0 ~~~~~\text{else},\end{array} \right. \end{array}\end{equation}
and the SWC generator, which is based on the following recurrence:
\begin{equation}
\label{SWC}
\begin{array}{l}
x^n = (a^1x^{n-1} \oplus ~...~ \oplus a^rx^{n-r} \oplus c^{n-1}) ~ mod ~ 2^w, \\
c^n = (a^1x^{n-1} \oplus ~...~ \oplus a^rx^{n-r} \oplus c^{n-1}) ~ / ~ 2^w. \end{array}\end{equation}

Then the generalized feedback shift register (GFSR) generator has been implemented, that is:
\begin{equation}
x^n = x^{n-r} \oplus x^{n-k} .
\label{GFSR}
\end{equation}


Finally, the nonlinear inversive (INV) generator~\cite{LEcuyerS07} has been studied, which is:

\begin{equation}
\label{INV}
\begin{array}{l}
x^n=\left\{
\begin{array}{ll}
(a^1 + a^2 / z^{n-1})~mod~m & \text{if}~ z^{n-1} \neq 0 \\
a^1 & \text{if}~  z^{n-1} = 0 .\end{array} \right. \end{array}\end{equation}



\begin{table}
\renewcommand{\arraystretch}{1.3}
\caption{TestU01 Statistical Test Failures}
\label{TestU011}
\centering
  \begin{tabular}{lccccc}
    \toprule
Test name &Tests& Logistic              & XORshift      & ISAAC\\
Rabbit                          &       38      &21             &14     &0       \\
Alphabit                        &       17      &16             &9      &0       \\
Pseudo DieHARD                  &126    &0              &2      &0      \\
FIPS\_140\_2                    &16     &0              &0      &0      \\
SmallCrush                      &15     &4              &5      &0       \\
Crush                           &144    &95             &57     &0       \\
Big Crush                       &160    &125            &55     &0       \\ \hline
Failures                &       &261            &146    &0       \\
\bottomrule
  \end{tabular}
\end{table}



\begin{table}
\renewcommand{\arraystretch}{1.3}
\caption{TestU01 Statistical Test Failures for Old CI algorithms ($\mathsf{N}=4$)}
\label{TestU01 for Old CI}
\centering
  \begin{tabular}{lcccc}
    \toprule
\multirow{3}*{Test name} & \multicolumn{4}{c}{Old CI}\\
&Logistic& XORshift& ISAAC&ISAAC  \\ 
&+& +& + & + \\ 
&Logistic& XORshift& XORshift&ISAAC  \\ \cmidrule(r){2-5}
Rabbit                                  &7      &2      &0      &0       \\
Alphabit                                & 3     &0      &0      &0       \\
DieHARD                         &0      &0      &0      &0      \\
FIPS\_140\_2                    &0      &0      &0      &0      \\
SmallCrush                              &2      &0      &0      &0       \\
Crush                                   &47     &4      &0      &0       \\
Big Crush                               &79     &3      &0      &0       \\ \hline
Failures                                &138    &9      &0      &0       \\
\bottomrule
  \end{tabular}
\end{table}





\subsection{Statistical tests}
\label{Security analysis}

Three batteries of tests are reputed and regularly used
to evaluate the statistical properties of newly designed pseudorandom
number generators. These batteries are named DieHard~\cite{Marsaglia1996},
the NIST suite~\cite{ANDREW2008}, and the most stringent one called
TestU01~\cite{LEcuyerS07}, which encompasses the two other batteries.



\label{Results and discussion}
\begin{table*}
\renewcommand{\arraystretch}{1.3}
\caption{NIST and DieHARD tests suite passing rates for PRNGs without CI}
\label{NIST and DieHARD tests suite passing rate the for PRNGs without CI}
\centering
  \begin{tabular}{|l||c|c|c|c|c|c|c|c|c|c|}
    \hline\hline
Types of PRNGs & \multicolumn{2}{c|}{Linear PRNGs} & \multicolumn{4}{c|}{Lagged PRNGs} & \multicolumn{1}{c|}{ICG PRNGs} & \multicolumn{3}{c|}{Mixed PRNGs}\\ \hline
\backslashbox{\textbf{$Tests$}} {\textbf{$PRNG$}} & LCG& MRG& AWC & SWB  & SWC & GFSR & INV & LCG2& LCG3& MRG2 \\ \hline
NIST & 11/15 & 14/15 &\textbf{15/15} & \textbf{15/15}   & 14/15 & 14/15  & 14/15 & 14/15& 14/15& 14/15 \\ \hline
DieHARD & 16/18 & 16/18 & 15/18 & 16/18 & \textbf{18/18} & 16/18 & 16/18 & 16/18& 16/18& 16/18\\ \hline
\end{tabular}
\end{table*}

Table~\ref{NIST and DieHARD tests suite passing rate the for PRNGs without CI} shows the 
results on the two first batteries recalled above, indicating that all the PRNGs presented
in the previous section
cannot pass all these tests. In other words, the statistical quality of these PRNGs cannot 
fulfill the up-to-date standards presented previously. We have shown in~\cite{bfg12a:ip} that the use of chaotic
iterations can solve this issue.
%More precisely, to
%illustrate the effects of chaotic iterations on these defective PRNGs, experiments have been divided in three parts~\cite{bfg12a:ip}:
%\begin{enumerate}
%  \item \textbf{Single CIPRNG}: The PRNGs involved in CI computing are of the same category.
%  \item \textbf{Mixed CIPRNG}: Two different types of PRNGs are mixed during the chaotic iterations process.
%  \item \textbf{Multiple CIPRNG}: The generator is obtained by repeating the composition of the iteration function as follows: $x^0\in \mathds{B}^{\mathsf{N}}$, and $\forall n\in \mathds{N}^{\ast },\forall i\in \llbracket1;\mathsf{N}\rrbracket, x_i^n=$
%\begin{equation}
%\begin{array}{l}
%\left\{
%\begin{array}{l}
%x_i^{n-1}~~~~~\text{if}~S^n\neq i \\
%\forall j\in \llbracket1;\mathsf{m}\rrbracket,f^m(x^{n-1})_{S^{nm+j}}~\text{if}~S^{nm+j}=i.\end{array} \right. \end{array}
%\end{equation}
%$m$ is called the \emph{functional power}.
%\end{enumerate}
%
The obtained results are reproduced in Table
\ref{NIST and DieHARD tests suite passing rate the for single CIPRNGs}.
The scores written in boldface indicate that all the tests have been passed successfully, whereas an 
asterisk ``*'' means that the considered passing rate has been improved.
The improvements are obvious for both the ``Old CI'' and the ``New CI'' generators.
Concerning the ``Xor CI PRNG'', the score is less spectacular. Because of a large speed improvement, the statistics
 are not as good as for the two other versions of these CIPRNGs.
However 8 tests have been improved (with no deflation for the other results).


\begin{table*}
\renewcommand{\arraystretch}{1.3}
\caption{NIST and DieHARD tests suite passing rates for PRNGs with CI}
\label{NIST and DieHARD tests suite passing rate the for single CIPRNGs}
\centering
  \begin{tabular}{|l||c|c|c|c|c|c|c|c|c|c|c|c|}
    \hline
Types of PRNGs & \multicolumn{2}{c|}{Linear PRNGs} & \multicolumn{4}{c|}{Lagged PRNGs} & \multicolumn{1}{c|}{ICG PRNGs} & \multicolumn{3}{c|}{Mixed PRNGs}\\ \hline
\backslashbox{\textbf{$Tests$}} {\textbf{$Single~CIPRNG$}} & LCG  & MRG & AWC & SWB & SWC & GFSR & INV& LCG2 & LCG3& MRG2 \\ \hline\hline
Old CIPRNG\\ \hline \hline
NIST & \textbf{15/15} *  & \textbf{15/15} * & \textbf{15/15}   & \textbf{15/15}   & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} *& \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} \\ \hline
DieHARD & \textbf{18/18} *  & \textbf{18/18} * & \textbf{18/18} *  & \textbf{18/18} *  & \textbf{18/18}  & \textbf{18/18} * & \textbf{18/18} *& \textbf{18/18} * & \textbf{18/18} *& \textbf{18/18} * \\ \hline
New CIPRNG\\ \hline \hline
NIST & \textbf{15/15} *  & \textbf{15/15} * & \textbf{15/15}   & \textbf{15/15}  & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} *& \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} \\ \hline
DieHARD & \textbf{18/18} *  & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18}  & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} *& \textbf{18/18} *\\ \hline
Xor CIPRNG\\ \hline\hline
NIST & 14/15*& \textbf{15/15} *   & \textbf{15/15}   & \textbf{15/15}   & 14/15 & \textbf{15/15} * & 14/15& \textbf{15/15} * & \textbf{15/15} *& \textbf{15/15}  \\ \hline
DieHARD & 16/18 & 16/18 & 17/18* & \textbf{18/18} * & \textbf{18/18}  & \textbf{18/18} * & 16/18 & 16/18 & 16/18& 16/18\\ \hline
\end{tabular}
\end{table*}


We have then investigated in~\cite{bfg12a:ip} if it were possible to improve
the statistical behavior of the Xor CI version by combining more than one 
$\oplus$ operation. Results are summarized in Table~\ref{threshold}, illustrating
the progressive increasing effects of chaotic iterations, when giving time to chaos to get settled in.
Thus rapid and perfect PRNGs, regarding the NIST and DieHARD batteries, can be obtained 
using chaotic iterations on defective generators.

\begin{table*}
\renewcommand{\arraystretch}{1.3}
\caption{Number of $\oplus$ operations to pass the whole NIST and DieHARD batteries}
\label{threshold}
\centering
  \begin{tabular}{|l||c|c|c|c|c|c|c|c|}
    \hline
Inputted $PRNG$ & LCG & MRG & SWC & GFSR & INV& LCG2 & LCG3  & MRG2 \\ \hline\hline
Threshold  value $m$& 19 & 7  & 2& 1 & 11& 9& 3& 4\\ \hline\hline
\end{tabular}
\end{table*}

Finally, the TestU01 battery has been launched on three well-known generators 
(a logistic map, a simple XORshift, and the cryptographically secure ISAAC, 
see Table~\ref{TestU011}). These results can be compared with 
Table~\ref{TestU01 for Old CI}, which gives the scores obtained by the
Old CI PRNG that has received these generators.
The obvious improvement speaks for itself, and together with the other
results recalled in this section, it reinforces the opinion that a strong
correlation between topological properties and statistical behavior exists.


The next subsection will now give a concrete original implementation of the Xor CI PRNG, the
fastest generator in the chaotic iteration based family. In the remainder,
this generator will be simply referred to as CIPRNG, or ``the proposed PRNG'', if this statement does not
raise ambiguity.



\section{Practical Security Evaluation}
\label{sec:Practicak evaluation}

Pseudorandom generators based on Eq.~\eqref{M-equation Oplus} of the main document are thus cryptographically secure when
they are XORed with an already cryptographically
secure PRNG. But, as stated previously,
such a property does not mean that, whatever the
key size, no attacker can predict the next bit
knowing all the previously released ones.
However, given a key size, it is possible to 
measure in practice the minimum duration needed
for an attacker to break a cryptographically
secure PRNG, if we know the power of his/her
machines. Such a concrete security evaluation 
is related to the $(T,\varepsilon)-$security
notion, which is recalled and evaluated in what 
follows, for the sake of completeness.

Let us firstly recall that,
\begin{definition}
Let $\mathcal{D} : \mathds{B}^M \longrightarrow \mathds{B}$ be a probabilistic algorithm that runs
in time $T$. 
Let $\varepsilon > 0$. 
$\mathcal{D}$ is called a $(T,\varepsilon)-$distinguishing attack on pseudorandom
generator $G$ if

\begin{flushleft}
$\left| Pr[\mathcal{D}(G(k)) = 1 \mid k \in_R \{0,1\}^\ell ]\right.$
\end{flushleft}

\begin{flushright}
$ - \left. Pr[\mathcal{D}(s) = 1 \mid s \in_R \mathds{B}^M ]\right| \geqslant \varepsilon,$
\end{flushright}

\noindent where the probability is taken over the internal coin flips of $\mathcal{D}$, and the notation
``$\in_R$'' indicates the process of selecting an element at random and uniformly over the
corresponding set.
\end{definition}

Let us recall that the running time of a probabilistic algorithm is defined to be the
maximum of the expected number of steps needed to produce an output, maximized
over all inputs; the expected number is averaged over all coin flips made by the algorithm~\cite{Knuth97}.
We are now able to define the notion of cryptographically secure PRNGs:

\begin{definition}
A pseudorandom generator is $(T,\varepsilon)-$secure if there exists no $(T,\varepsilon)-$distinguishing attack on this pseudorandom generator.
\end{definition}







Suppose now that the PRNG of Eq.~\eqref{M-equation Oplus} of the main document will work during 
$M=100$ time units, and that during this period,
an attacker can realize $10^{12}$ clock cycles.
We thus wonder whether, during the PRNG's 
lifetime, the attacker can distinguish this 
sequence from a truly random one, with a probability
greater than $\varepsilon = 0.2$.
We consider that $N$ has 900 bits.

Predicting the next generated bit knowing all the
previously released ones by Eq.~\eqref{M-equation Oplus} of the main document is obviously equivalent to predicting the
next bit in the BBS generator, which
is cryptographically secure. More precisely, it
is $(T,\varepsilon)-$secure: no 
$(T,\varepsilon)-$distinguishing attack can be
successfully realized on this PRNG, if~\cite{Fischlin}
\begin{equation}
T \leqslant \dfrac{L(N)}{6 N (log_2(N))\varepsilon^{-2}M^2}-2^7 N \varepsilon^{-2} M^2 log_2 (8 N \varepsilon^{-1}M)
\label{mesureConcrete}
\end{equation}
where $M$ is the length of the output ($M=100$ in
our example), and $L(N)$ is equal to
$$
2.8\times 10^{-3} exp \left(1.9229 \times (N ~ln~ 2)^\frac{1}{3} \times (ln(N~ln~  2))^\frac{2}{3}\right)
$$
is the number of clock cycles to factor a $N-$bit
integer.




A direct numerical application shows that this attacker 
cannot achieve his $(10^{12},0.2)$ distinguishing
attack in that context.






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