-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).
-
-\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}.
