\newcommand{\alert}[1]{\begin{color}{blue}\textit{#1}\end{color}}
+
+\newcommand{\PCH}[1]{\begin{color}{blue}#1\end{color}}
+
\title{Efficient and Cryptographically Secure Generation of Chaotic Pseudorandom Numbers on GPU}
\begin{document}
\section{Security Analysis}
\label{sec:security analysis}
-
+\PCH{This section is dedicated to the analysis of the security of the
+ proposed PRNGs from a theoretical point of view. The standard definition
+ of {\it indistinguishability} used is the classical one as defined for
+ instance in~\cite[chapter~3]{Goldreich}. It is important to emphasize that
+ this property shows that predicting the future results of the PRNG's
+ cannot be done in a reasonable time compared to the generation time. This
+ is a relative notion between breaking time and the sizes of the
+ keys/seeds. Of course, if small keys or seeds are chosen, the system can
+ be broken in practice. But it also means that if the keys/seeds are large
+ enough, the system is secured.}
In this section the concatenation of two strings $u$ and $v$ is classically
denoted by $uv$.
question of key size.
\begin{color}{green}
-PCH, tu peux broder là-dessus?
+Most of theoretical cryptographic definitions are somehow an extension of the
+notion of one-way function. Intuitively a one way function is a function
+ easy to compute but which is practically impossible to
+inverse (i.e. from $f(x)$ it is not possible to compute $x$).
+Since the size of $x$ is known, it is always possible to use a brute force
+attack, that is computing $f(y)$ for all $y$'s of the good size until
+$f(y)\neq f(x)$. Informally, if a function is one-way, it means that every
+algorithm that can compute $x$ from $f(x)$ with a good probability requires
+a similar amount of time than the brute force attack. It is important to
+note that if the size of $x$ is small, then the brute force attack works in
+practice. The theoretical security properties don't guaranty that the system
+cannot be broken, it guaranty that if the keys are large enough, then the
+system still works (computing $f(x)$ can be done, even if $x$ is large), and
+cannot be broken in a reasonable time. The theoretical definition of a
+secure PRNG is more technical than the one on one-way function but the
+ideas are the same: a cryptographically secured PRNG can be broken
+ by a brute force prediction, but not in a reasonable time if the
+ keys/seeds are large enough.
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\bigskip
