and on an iteration process called ``chaotic
iterations'' on which the post-treatment is based.
The proposed PRNG and its proof of chaos are given in Section~\ref{sec:pseudorandom}.
-
-Section~\ref{The generation of pseudorandom sequence} illustrates the statistical
-improvement related to the chaotic iteration based post-treatment, for
-our previously released PRNGs and a new efficient
+Section~\ref{sec:efficient PRNG} %{The generation of pseudorandom sequence} %illustrates the statistical
+%improvement related to the chaotic iteration based post-treatment, for
+%our previously released PRNGs and
+ contains a new efficient
implementation on CPU.
-
Section~\ref{sec:efficient PRNG
gpu} describes and evaluates theoretically the GPU implementation.
Such generators are experimented in
We show in Section~\ref{sec:security analysis} that, if the inputted
generator is cryptographically secure, then it is the case too for the
generator provided by the post-treatment.
-A practical
-security evaluation is also outlined in Section~\ref{sec:Practicak evaluation}.
+%A practical
+%security evaluation is also outlined in Section~\ref{sec:Practicak evaluation}.
Such a proof leads to the proposition of a cryptographically secure and
chaotic generator on GPU based on the famous Blum Blum Shub
in Section~\ref{sec:CSGPU} and to an improvement of the
\begin{theorem}
\label{t:chaos des general}
- The general chaotic iterations defined by
- \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}
+ The general chaotic iterations defined in Equation~\ref{eq:generalIC}
satisfy
the Devaney's property of chaos.
\end{theorem}
%%RAF : mis en supplementary
-\section{Statistical Improvements Using Chaotic Iterations}
-\label{The generation of pseudorandom sequence}
-The content is this section is given in Section~\ref{A-The generation of pseudorandom sequence} of the annex document.
-
+%\section{Statistical Improvements Using Chaotic Iterations}
+%\label{The generation of pseudorandom sequence}
+%The content is this section is given in Section~\ref{A-The generation of pseudorandom sequence} of the annex document.
+The reasons to desire chaos to achieve randomness are given in Annex~\ref{A-The generation of pseudorandom sequence}.
%% \label{The generation of pseudorandom sequence}
%% raise ambiguity.
-\subsection{First Efficient Implementation of a PRNG based on Chaotic Iterations}
+\section{First Efficient Implementation of a PRNG based on Chaotic Iterations}
\label{sec:efficient PRNG}
%
%Based on the proof presented in the previous section, it is now possible to
This section is dedicated to the security analysis of the
- proposed PRNGs, both from a theoretical and from a practical point of view.
+ proposed PRNGs.%, both from a theoretical and from a practical point of view.
-\subsection{Theoretical Proof of Security}
+%\subsection{Theoretical Proof of Security}
\label{sec:security analysis}
The standard definition
- of {\it indistinguishability} used is the classical one as defined for
+ of {\it indistinguishability} used here is the classical one as defined for
instance in~\cite[chapter~3]{Goldreich}.
This property shows that predicting the future results of the PRNG
cannot be done in a reasonable time compared to the generation time. It is important to emphasize that this
be broken in practice. But it also means that if the keys/seeds are large
enough, the system is secured.
As a complement, an example of a concrete practical evaluation of security
-is outlined in the next subsection.
+is outlined in Annex~\ref{A-sec:Practicak evaluation}.
In this section the concatenation of two strings $u$ and $v$ is classically
denoted by $uv$.
-\subsection{Practical Security Evaluation}
-\label{sec:Practicak evaluation}
-This subsection is given in Section~\ref{A-sec:Practicak evaluation} of the annex document.
+%\subsection{Practical Security Evaluation}
+%\label{sec:Practicak evaluation}
+%This subsection is given in Section
+A example of a practical security evaluation is outlined in
+Annex~\ref{A-sec:Practicak evaluation}.
%%RAF mis en annexe
proposed parameters, or if it is only a very fast
and statistically perfect generator on GPU, its
$(T,\varepsilon)-$security must be determined, and
-a formulation similar to Eq.\eqref{mesureConcrete}
+a formulation similar to Annex~\ref{A-sec:Practicak evaluation} %.Eq.\eqref{mesureConcrete}
must be established. Authors
hope to achieve this difficult task in a future
work.
