Obviously, replacing the previous CI PRNG Algorithms by
Equation~\ref{equation Oplus0}, which is possible when the iteration function is
-the vectorial negation, leads to a speed improvement. However, proofs
+the vectorial negation, leads to a speed improvement
+(the resulting generator will be referred as ``Xor CI PRNG''
+in what follows).
+However, proofs
of chaos obtained in~\cite{bg10:ij} have been established
only for chaotic iterations of the form presented in Definition
\ref{Def:chaotic iterations}. The question is now to determine whether the
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}
-
-
-We have performed statistical analysis of each of the aforementioned CIPRNGs.
-The results are reproduced in Tables~\ref{NIST and DieHARD tests suite passing rate the for PRNGs without CI} and \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.
+%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 ``New CI'' generators.
+Concerning the ``Xor CI PRNG'', the speed improvement makes that statistical
+results are not as good as for the two other versions of these CIPRNGs.
-\subsubsection{Tests based on the Single CIPRNG}
\begin{table*}
\renewcommand{\arraystretch}{1.3}
\end{tabular}
\end{table*}
-The statistical tests results of the PRNGs using the single CIPRNG method are given in Table~\ref{NIST and DieHARD tests suite passing rate the for single CIPRNGs}.
-We can observe that, except for the Xor CIPRNG, all of the CIPRNGs have passed the 15 tests of the NIST battery and the 18 tests of the DieHARD one.
-Moreover, considering these scores, we can deduce that both the single Old CIPRNG and the single New CIPRNG are relatively steadier than the single Xor CIPRNG approach, when applying them to different PRNGs.
-However, the Xor CIPRNG is obviously the fastest approach to generate a CI random sequence, and it still improves the statistical properties relative to each generator taken alone, although the test values are not as good as desired.
-
-Therefore, all of these three ways are interesting, for different reasons, in the production of pseudorandom numbers and,
-on the whole, the single CIPRNG method can be considered to adapt to or improve all kinds of PRNGs.
-
-To have a realization of the Xor CIPRNG that can pass all the tests embedded into the NIST battery, the Xor CIPRNG with multiple functional powers are investigated in Section~\ref{Tests based on Multiple CIPRNG}.
-
-
-\subsubsection{Tests based on the Mixed CIPRNG}
-
-To compare the previous approach with the CIPRNG design that uses a Mixed CIPRNG, we have taken into account the same inputted generators than in the previous section.
-These inputted couples $(PRNG_1,PRNG_2)$ of PRNGs are used in the Mixed approach as follows:
-\begin{equation}
-\left\{
-\begin{array}{l}
-x^0 \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket, S \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket^\mathds{N} \\
-\forall n \in \mathds{N}^*, x^n = x^{n-1} \oplus PRNG_1\oplus PRNG_2,
-\end{array}
-\right.
-\label{equation Oplus}
-\end{equation}
-With this Mixed CIPRNG approach, both the Old CIPRNG and New CIPRNG continue to pass all the NIST and DieHARD suites.
-In addition, we can see that the PRNGs using a Xor CIPRNG approach can pass more tests than previously.
-The main reason of this success is that the Mixed Xor CIPRNG has a longer period.
-Indeed, let $n_{P}$ be the period of a PRNG $P$, then the period deduced from the single Xor CIPRNG approach is obviously equal to:
-\begin{equation}
-n_{SXORCI}=
-\left\{
-\begin{array}{ll}
-n_{P}&\text{if~}x^0=x^{n_{P}}\\
-2n_{P}&\text{if~}x^0\neq x^{n_{P}}.\\
-\end{array}
-\right.
-\label{equation Oplus}
-\end{equation}
-
-Let us now denote by $n_{P1}$ and $n_{P2}$ the periods of respectively the $PRNG_1$ and $PRNG_2$ generators, then the period of the Mixed Xor CIPRNG will be:
-\begin{equation}
-n_{XXORCI}=
-\left\{
-\begin{array}{ll}
-LCM(n_{P1},n_{P2})&\text{if~}x^0=x^{LCM(n_{P1},n_{P2})}\\
-2LCM(n_{P1},n_{P2})&\text{if~}x^0\neq x^{LCM(n_{P1},n_{P2})}.\\
-\end{array}
-\right.
-\label{equation Oplus}
-\end{equation}
-
-In Table~\ref{DieHARD fail mixex CIPRNG}, we only show the results for the Mixed CIPRNGs that cannot pass all DieHARD suites (the NIST tests are all passed). It demonstrates that Mixed Xor CIPRNG involving LCG, MRG, LCG2, LCG3, MRG2, or INV cannot pass the two following tests, namely the ``Matrix Rank 32x32'' and the ``COUNT-THE-1's'' tests contained into the DieHARD battery. Let us recall their definitions:
-
-\begin{itemize}
- \item \textbf{Matrix Rank 32x32.} A random 32x32 binary matrix is formed, each row having a 32-bit random vector. Its rank is an integer that ranges from 0 to 32. Ranks less than 29 must be rare, and their occurences must be pooled with those of rank 29. To achieve the test, ranks of 40,000 such random matrices are obtained, and a chisquare test is performed on counts for ranks 32,31,30 and for ranks $\leq29$.
-
- \item \textbf{COUNT-THE-1's TEST} Consider the file under test as a stream of bytes (four per 2 bit integer). Each byte can contain from 0 to 8 1's, with probabilities 1,8,28,56,70,56,28,8,1 over 256. Now let the stream of bytes provide a string of overlapping 5-letter words, each ``letter'' taking values A,B,C,D,E. The letters are determined by the number of 1's in a byte: 0,1, or 2 yield A, 3 yields B, 4 yields C, 5 yields D and 6,7, or 8 yield E. Thus we have a monkey at a typewriter hitting five keys with various probabilities (37,56,70,56,37 over 256). There are $5^5$ possible 5-letter words, and from a string of 256,000 (over-lapping) 5-letter words, counts are made on the frequencies for each word. The quadratic form in the weak inverse of the covariance matrix of the cell counts provides a chisquare test: Q5-Q4, the difference of the naive Pearson sums of $(OBS-EXP)^2/EXP$ on counts for 5- and 4-letter cell counts.
-\end{itemize}
-
-The reason of these fails is that the output of LCG, LCG2, LCG3, MRG, and MRG2 under the experiments are in 31-bit. Compare with the Single CIPRNG, using different PRNGs to build CIPRNG seems more efficient in improving random number quality (mixed Xor CI can 100\% pass NIST, but single cannot).
+We have then investigate in~\cite{bfg12a:ip} if it is possible to improve
+the statistical behavior of the Xor CI version by combining more than one
+$\oplus$ operation. Results are summarized in~\ref{threshold}, showing
+that 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{Scores of mixed Xor CIPRNGs when considering the DieHARD battery}
-\label{DieHARD fail mixex CIPRNG}
-\centering
- \begin{tabular}{|l||c|c|c|c|c|c|}
- \hline
-\backslashbox{\textbf{$PRNG_1$}} {\textbf{$PRNG_0$}} & LCG & MRG & INV & LCG2 & LCG3 & MRG2 \\ \hline\hline
-LCG &\backslashbox{} {} &16/18&16/18 &16/18 &16/18 &16/18\\ \hline
-MRG &16/18 &\backslashbox{} {} &16/18&16/18 &16/18 &16/18\\ \hline
-INV &16/18 &16/18&\backslashbox{} {} &16/18 &16/18&16/18 \\ \hline
-LCG2 &16/18 &16/18 &16/18 &\backslashbox{} {} &16/18&16/18\\ \hline
-LCG3 &16/18 &16/18 &16/18&16/18&\backslashbox{} {} &16/18\\ \hline
-MRG2 &16/18 &16/18 &16/18&16/18 &16/18 &\backslashbox{} {} \\ \hline
-\end{tabular}
-\end{table*}
-
-\subsubsection{Tests based on the Multiple CIPRNG}
-\label{Tests based on Multiple CIPRNG}
-
-Until now, the combination of at most two input PRNGs has been investigated.
-We now regard the possibility to use a larger number of generators to improve the statistics of the generated pseudorandom numbers, leading to the multiple functional power approach.
-For the CIPRNGs which have already pass both the NIST and DieHARD suites with 2 inputted PRNGs (all the Old and New CIPRNGs, and some of the Xor CIPRNGs), it is not meaningful to consider their adaption of this multiple CIPRNG method, hence only the Multiple Xor CIPRNGs, having the following form, will be investigated.
-\begin{equation}
-\left\{
-\begin{array}{l}
-x^0 \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket, S \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket^\mathds{N} \\
-\forall n \in \mathds{N}^*, x^n = x^{n-1} \oplus S^{nm}\oplus S^{nm+1}\ldots \oplus S^{nm+m-1} ,
-\end{array}
-\right.
-\label{equation Oplus}
-\end{equation}
-
-The question is now to determine the value of the threshold $m$ (the functional power) making the multiple CIPRNG being able to pass the whole NIST battery.
-Such a question is answered in Table~\ref{threshold}.
-
-
-\begin{table*}
-\renewcommand{\arraystretch}{1.3}
-\caption{Functional power $m$ making it possible to pass the whole NIST battery}
+\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|}
\end{tabular}
\end{table*}
-\subsubsection{Results Summary}
-
-We can summarize the obtained results as follows.
-\begin{enumerate}
-\item The CIPRNG method is able to improve the statistical properties of a large variety of PRNGs.
-\item Using different PRNGs in the CIPRNG approach is better than considering several instances of one unique PRNG.
-\item The statistical quality of the outputs increases with the functional power $m$.
-\end{enumerate}
-
+Next subsection gives a concrete implementation of this Xor CI PRNG, which will
+new be simply called CIPRNG, or ``the proposed PRNG'', if this statement does not
+raise ambiguity.
\end{color}
-\section{Efficient PRNG based on Chaotic Iterations}
+\subsection{Efficient PRNG based on Chaotic Iterations}
\label{sec:efficient PRNG}
Based on the proof presented in the previous section, it is now possible to
