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[rairo15.git] / preliminaries.tex

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-
-
-

 In what follows, we consider the Boolean algebra on the set 
 $\Bool=\{0,1\}$ with the classical operators of conjunction '.', 
 of disjunction '+', of negation '$\overline{~}$', and of 
 disjunctive union $\oplus$. 
 
 In what follows, we consider the Boolean algebra on the set 
 $\Bool=\{0,1\}$ with the classical operators of conjunction '.', 
 of disjunction '+', of negation '$\overline{~}$', and of 
 disjunctive union $\oplus$. 
 

-Let $n$ be a positive integer. A  {\emph{Boolean map} $f$ is 
-a function from the Boolean domain 
- to itself 
+Let us first introduce basic notations.
+Let $\mathsf{N}$ be a positive integer. The set $\{1, 2, \hdots , \mathsf{N}\}$
+of integers belonging between $1$ and $\mathsf{N}$
+is further denoted as $\llbracket 1, \mathsf{N} \rrbracket$.
+A  {\emph{Boolean map} $f$ is 
+a function from $\Bool^{\mathsf{N}}$  
+to itself 

 such that 
 such that 

-$x=(x_1,\dots,x_n)$ maps to $f(x)=(f_1(x),\dots,f_n(x))$.
+$x=(x_1,\dots,x_{\mathsf{N}})$ maps to $f(x)=(f_1(x),\dots,f_{\mathsf{N}}(x))$.
+In what follows, for any finite set $X$, $|X|$ denotes its cardinality and 
+$\lfloor y \rfloor$ is
+the largest integer lower than $y$.
+

 Functions are iterated as follows. 
 Functions are iterated as follows. 

-At the $t^{th}$ iteration, only the $s_{t}-$th component is
-``iterated'', where $s = \left(s_t\right)_{t \in \mathds{N}}$ is a sequence of indices taken in $\llbracket 1;n \rrbracket$ called ``strategy''. Formally,
-let $F_f: \llbracket1;n\rrbracket\times \Bool^{n}$ to $\Bool^n$ be defined by
+At the $t^{th}$ iteration, only the $s_{t}-$th component is said to be
+``iterated'', where $s = \left(s_t\right)_{t \in \mathds{N}}$ is a sequence of indices taken in $\llbracket 1;{\mathsf{N}} \rrbracket$ called ``strategy''. 
+Formally,
+let $F_f:  \Bool^{{\mathsf{N}}} \times \llbracket1;{\mathsf{N}}\rrbracket$ to $\Bool^{\mathsf{N}}$ be defined by

 \[
 \[

-F_f(i,x)=(x_1,\dots,x_{i-1},f_i(x),x_{i+1},\dots,x_n).
+F_f(x,i)=(x_1,\dots,x_{i-1},f_i(x),x_{i+1},\dots,x_{\mathsf{N}}).

 \]
 \]

-Then, let $x^0\in\Bool^n$ be an initial configuration
+Then, let $x^0\in\Bool^{\mathsf{N}}$ be an initial configuration

 and $s\in
 and $s\in

-\llbracket1;n\rrbracket^\Nats$ be a strategy, 
+\llbracket1;{\mathsf{N}}\rrbracket^\Nats$ be a strategy, 

 the dynamics are described by the recurrence
 \begin{equation}\label{eq:asyn}
 the dynamics are described by the recurrence
 \begin{equation}\label{eq:asyn}

-x^{t+1}=F_f(s_t,x^t).
+x^{t+1}=F_f(x^t,s_t).

 \end{equation}
 
 
 \end{equation}
 
 

+
+
+

 Let be given a Boolean map $f$. Its associated   
 {\emph{iteration graph}}  $\Gamma(f)$ is the
 directed graph such that  the set of vertices is
 Let be given a Boolean map $f$. Its associated   
 {\emph{iteration graph}}  $\Gamma(f)$ is the
 directed graph such that  the set of vertices is

-$\Bool^n$, and for all $x\in\Bool^n$ and $i\in \llbracket1;n\rrbracket$,
-the graph $\Gamma(f)$ contains an arc from $x$ to $F_f(i,x)$. 
+$\Bool^{\mathsf{N}}$, and for all $x\in\Bool^{\mathsf{N}}$ and $i\in \llbracket1;{\mathsf{N}}\rrbracket$,
+the graph $\Gamma(f)$ contains an arc from $x$ to $F_f(x,i)$.
+Each arc $(x,F_f(x,i))$ is labelled with $i$. 
+

 
 \begin{xpl}
 
 \begin{xpl}

-Let us consider for instance $n=3$.
+Let us consider for instance ${\mathsf{N}}=3$.

 Let 
 $f^*: \Bool^3 \rightarrow \Bool^3$ be defined by
 Let 
 $f^*: \Bool^3 \rightarrow \Bool^3$ be defined by

-

 $f^*(x_1,x_2,x_3) = 
 (x_2 \oplus x_3, \overline{x_1}\overline{x_3} + x_1\overline{x_2},
 $f^*(x_1,x_2,x_3) = 
 (x_2 \oplus x_3, \overline{x_1}\overline{x_3} + x_1\overline{x_2},

-\overline{x_1}\overline{x_3} + x_1x_2)$
+\overline{x_1}\overline{x_3} + x_1x_2)$.

 The iteration graph $\Gamma(f^*)$ of this function is given in 
 Figure~\ref{fig:iteration:f*}.
 The iteration graph $\Gamma(f^*)$ of this function is given in 
 Figure~\ref{fig:iteration:f*}.

+\end{xpl}

 
 

-\vspace{-1em}

 \begin{figure}[ht]
 \begin{center}
 \begin{figure}[ht]
 \begin{center}

-\includegraphics[scale=0.5]{images/iter_f0b}
+\includegraphics[scale=0.5]{images/iter_f0c}

 \end{center}
 \end{center}

-\vspace{-0.5em}

 \caption{Iteration Graph $\Gamma(f^*)$ of the function $f^*$}\label{fig:iteration:f*}
 \end{figure}
 \caption{Iteration Graph $\Gamma(f^*)$ of the function $f^*$}\label{fig:iteration:f*}
 \end{figure}

-\end{xpl}
-
-\vspace{-0.5em}
-It is easy to associate a Markov Matrix $M$ to such a graph $G(f)$
-as follows:
-
-$M_{ij} = \frac{1}{n}$ if there is an edge from $i$ to $j$ in $\Gamma(f)$ and $i \neq j$;  $M_{ii} = 1 - \sum\limits_{j=1, j\neq i}^n M_{ij}$; and $M_{ij} = 0$ otherwise.
-
-\begin{xpl}
-The Markov matrix associated to the function $f^*$ is 

 
 

-\[
-M=\dfrac{1}{3} \left(
-\begin{array}{llllllll}
-1&1&1&0&0&0&0&0 \\
-1&1&0&0&0&1&0&0 \\
-0&0&1&1&0&0&1&0 \\
-0&1&1&1&0&0&0&0 \\
-1&0&0&0&1&0&1&0 \\
-0&0&0&0&1&1&0&1 \\
-0&0&0&0&1&0&1&1 \\
-0&0&0&1&0&1&0&1 
-\end{array}
-\right)
-\]
-
-
-
- 
-
-\end{xpl}
-
-
-It is usual to check whether rows of such kind of matrices
-converge to a specific 
-distribution. 
-Let us first recall the  \emph{Total Variation} distance $\tv{\pi-\mu}$,
-which is defined for two distributions $\pi$ and $\mu$ on the same set 
-$\Omega$  by:
-$$\tv{\pi-\mu}=\max_{A\subset \Omega} |\pi(A)-\mu(A)|.$$ 
-% It is known that
-% $$\tv{\pi-\mu}=\frac{1}{2}\sum_{x\in\Omega}|\pi(x)-\mu(x)|.$$
-
-Let then $M(x,\cdot)$ be the
-distribution induced by the $x$-th row of $M$. If the Markov chain
-induced by
-$M$ has a stationary distribution $\pi$, then we define
-$$d(t)=\max_{x\in\Omega}\tv{M^t(x,\cdot)-\pi}.$$
-Intuitively $d(t)$ is the largest deviation between
-the distribution $\pi$ and $M^t(x,\cdot)$, which 
-is the result of iterating $t$ times the function.
-Finally, let $\varepsilon$ be a positive number, the \emph{mixing time} 
-with respect to $\varepsilon$ is given by
-$$t_{\rm mix}(\varepsilon)=\min\{t \mid d(t)\leq \varepsilon\}.$$
-It defines the smallest iteration number 
-that is sufficient to obtain a deviation lesser than $\varepsilon$.
-% Notice that the upper and lower bounds of mixing times cannot    
-% directly be computed with eigenvalues formulae as expressed 
-% in~\cite[Chap. 12]{LevinPeresWilmer2006}. The authors of this latter work  
-% only consider reversible Markov matrices whereas we do no restrict our 
-% matrices to such a form.
-
-
-
-Let us finally present the pseudorandom number generator $\chi_{\textit{14Secrypt}}$
-which is based on random walks in $\Gamma(f)$. 
-More precisely, let be given a Boolean map $f:\Bool^n \rightarrow \Bool^n$,
-a PRNG \textit{Random},
-an integer $b$ that corresponds to an awaited mixing time, and 
+Let us finally recall the pseudorandom number generator $\chi_{\textit{14Secrypt}}$
+\cite{DBLP:conf/secrypt/CouchotHGWB14}
+formalized in Algorithm~\ref{CI Algorithm}.
+It is based on random walks in $\Gamma(f)$. 
+More precisely, let be given a Boolean map $f:\Bool^{\mathsf{N}} \rightarrow \Bool^{\mathsf{N}}$,
+an input PRNG \textit{Random},
+an integer $b$ that corresponds to a number of iterations, and 

 an initial configuration $x^0$. 
 Starting from $x^0$, the algorithm repeats $b$ times 
 a random choice of which edge to follow and traverses this edge.
 The final configuration is thus outputted.
 an initial configuration $x^0$. 
 Starting from $x^0$, the algorithm repeats $b$ times 
 a random choice of which edge to follow and traverses this edge.
 The final configuration is thus outputted.

-This PRNG is formalized in Algorithm~\ref{CI Algorithm}.

 
 
 
 

-
-\vspace{-1em}

 \begin{algorithm}[ht]
 \begin{algorithm}[ht]

-%\begin{scriptsize}
-\KwIn{a function $f$, an iteration number $b$, an initial configuration $x^0$ ($n$ bits)}
-\KwOut{a configuration $x$ ($n$ bits)}
+\begin{scriptsize}
+%\JFC{Mettre ceci dans une boite flottante}
+\KwIn{a function $f$, an iteration number $b$, an initial configuration $x^0$ (${\mathsf{N}}$ bits)}
+\KwOut{a configuration $x$ (${\mathsf{N}}$ bits)}

 $x\leftarrow x^0$\;
 \For{$i=0,\dots,b-1$}
 {
 $x\leftarrow x^0$\;
 \For{$i=0,\dots,b-1$}
 {

-$s\leftarrow{\textit{Random}(n)}$\;
-$x\leftarrow{F_f(s,x)}$\;
+$s\leftarrow{\textit{Random}({\mathsf{N}})}$\;
+$x\leftarrow{F_f(x,s)}$\;

 }
 return $x$\;
 }
 return $x$\;

-%\end{scriptsize}
+\end{scriptsize}

 \caption{Pseudo Code of the $\chi_{\textit{14Secrypt}}$ PRNG}
 \label{CI Algorithm}
 \end{algorithm}
 \caption{Pseudo Code of the $\chi_{\textit{14Secrypt}}$ PRNG}
 \label{CI Algorithm}
 \end{algorithm}

-\vspace{-0.5em}
-This PRNG is a particularized version of Algorithm given in~\cite{BCGR11}.
-Compared to this latter, the length of the random 
-walk of our algorithm is always constant (and is equal to $b$) whereas it 
-was given by a second PRNG in this latter.
-However, all the theoretical results that are given in~\cite{BCGR11} remain
-true since the proofs do not rely on this fact. 
-
-Let $f: \Bool^{n} \rightarrow \Bool^{n}$.
-It has been shown~\cite[Th. 4, p. 135]{BCGR11}} that 
-if its iteration graph is strongly connected, then 
-the output of $\chi_{\textit{14Secrypt}}$ follows 
-a law that tends to the uniform distribution 
-if and only if its Markov matrix is a doubly stochastic matrix.
-  
-Let us now present  a method to
-generate  functions
-with Doubly Stochastic matrix and Strongly Connected iteration graph,
- denoted as DSSC matrix.   
+
+
+
+With all this material, we can study the chaos properties of these 
+function.
+This is the aims of the next section. 
+