X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/prng_gpu.git/blobdiff_plain/8f1af7e18d4d59611a7b16178ac5f32cfe541056..9879779d913285ee14baad568f69be401dfd0fb3:/prng_gpu.tex?ds=inline

diff --git a/prng_gpu.tex b/prng_gpu.tex
index c5fbd5d..0c9f9c7 100644
--- a/prng_gpu.tex
+++ b/prng_gpu.tex
@@ -90,7 +90,13 @@ On the other side, speed is not the main requirement in cryptography: the great
 need is to define \emph{secure} generators able to withstand malicious
 attacks. Roughly speaking, an attacker should not be able in practice to make 
 the distinction between numbers obtained with the secure generator and a true random
-sequence. 
+sequence. \begin{color}{red} Or, in an equivalent formulation, he or she should not be
+able (in practice) to predict the next bit of the generator, having the knowledge of all the 
+binary digits that have been already released. ``Being able in practice'' refers here
+to the possibility to achieve this attack in polynomial time, and to the exponential growth
+of the difficulty of this challenge when the size of the parameters of the PRNG increases.
+\end{color}
+
 Finally, a small part of the community working in this domain focuses on a
 third requirement, that is to define chaotic generators.
 The main idea is to take benefits from a chaotic dynamical system to obtain a
@@ -124,10 +130,19 @@ statistical perfection refers to the ability to pass the whole
 {\it BigCrush} battery of tests, which is widely considered as the most
 stringent statistical evaluation of a sequence claimed as random.
 This battery can be found in the well-known TestU01 package~\cite{LEcuyerS07}.
+\begin{color}{red}
+More precisely, each time we performed a test on a PRNG, we ran it
+twice in order to observe if all $p-$values are inside [0.01, 0.99]. In
+fact, we observed that few $p-$values (less than ten) are sometimes
+outside this interval but inside [0.001, 0.999], so that is why a
+second run allows us to confirm that the values outside are not for
+the same test. With this approach all our PRNGs pass the {\it
+  BigCrush} successfully and all $p-$values are at least once inside
+[0.01, 0.99].
+\end{color}
 Chaos, for its part, refers to the well-established definition of a
 chaotic dynamical system proposed by Devaney~\cite{Devaney}.
 
-
 In a previous work~\cite{bgw09:ip,guyeux10} we have proposed a post-treatment on PRNGs making them behave
 as a chaotic dynamical system. Such a post-treatment leads to a new category of
 PRNGs. We have shown that proofs of Devaney's chaos can be established for this
@@ -157,8 +172,13 @@ The remainder of this paper  is organized as follows. In Section~\ref{section:re
   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{sec:efficient    PRNG}   presents   an   efficient
-implementation of  this chaotic PRNG  on a CPU, whereas   Section~\ref{sec:efficient PRNG
+\begin{color}{red}
+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 
+implementation on CPU.
+\end{color}
+ Section~\ref{sec:efficient PRNG
   gpu}   describes and evaluates theoretically  the  GPU   implementation. 
 Such generators are experimented in 
 Section~\ref{sec:experiments}.
@@ -167,7 +187,8 @@ generator is cryptographically secure, then it is the case too for the
 generator provided by the post-treatment.
 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
+in Section~\ref{sec:CSGPU}, \begin{color}{red} to a practical
+security evaluation in Section~\ref{sec:Practicak evaluation}, \end{color} and to an improvement of the
 Blum-Goldwasser protocol in Sect.~\ref{Blum-Goldwasser}.
 This research work ends by a conclusion section, in which the contribution is
 summarized and intended future work is presented.
@@ -175,7 +196,7 @@ summarized and intended future work is presented.
 
 
 
-\section{Related works on GPU based PRNGs}
+\section{Related work on GPU based PRNGs}
 \label{section:related works}
 
 Numerous research works on defining GPU based PRNGs have already been proposed  in the
@@ -480,7 +501,7 @@ We have proposed in~\cite{bgw09:ip} a new family of generators that receives
 two PRNGs as inputs. These two generators are mixed with chaotic iterations, 
 leading thus to a new PRNG that 
 \begin{color}{red}
-should improves the statistical properties of each
+should improve the statistical properties of each
 generator taken alone. 
 Furthermore, the generator obtained by this way possesses various chaos properties that none of the generators used as input
 present.
@@ -556,7 +577,7 @@ This new generator is designed by the following process.
 First of all, some chaotic iterations have to be done to generate a sequence 
 $\left(x^n\right)_{n\in\mathds{N}} \in \left(\mathds{B}^{32}\right)^\mathds{N}$ 
 of Boolean vectors, which are the successive states of the iterated system. 
-Some of these vectors will be randomly extracted and our pseudo-random bit 
+Some of these vectors will be randomly extracted and our pseudorandom bit 
 flow will be constituted by their components. Such chaotic iterations are 
 realized as follows. Initial state $x^0 \in \mathds{B}^{32}$ is a Boolean 
 vector taken as a seed and chaotic strategy $\left(S^n\right)_{n\in\mathds{N}}\in 
@@ -569,14 +590,14 @@ updated, as follows: $x_i^n = x_i^{n-1}$ if $i \neq S^n$, else $x_i^n = \overlin
 Such a procedure is equivalent to achieve chaotic iterations with
 the Boolean vectorial negation $f_0$ and some well-chosen strategies.
 Finally, some $x^n$ are selected
-by a sequence $m^n$ as the pseudo-random bit sequence of our generator.
+by a sequence $m^n$ as the pseudorandom bit sequence of our generator.
 $(m^n)_{n \in \mathds{N}} \in \mathcal{M}^\mathds{N}$ is computed from $PRNG_1$, where $\mathcal{M}\subset \mathds{N}^*$ is a finite nonempty set of integers.
 
 The basic design procedure of the New CI generator is summarized in Algorithm~\ref{Chaotic iteration1}.
 The internal state is $x$, the output state is $r$. $a$ and $b$ are those computed by the two input
 PRNGs. Lastly, the value $g(a)$ is an integer defined as in Eq.~\ref{Formula}.
-This function is required to make the outputs uniform in $\llbracket 0, 2^\mathsf{N}-1 \rrbracket$
-(the reader is referred to~\cite{bg10:ip} for more information).
+This function must be chosen such that the outputs of the resulted PRNG is uniform in $\llbracket 0, 2^\mathsf{N}-1 \rrbracket$. Function of \eqref{Formula} achieves this
+goal (other candidates and more information can be found in ~\cite{bg10:ip}).
 
 \begin{equation}
 \label{Formula}
@@ -602,8 +623,7 @@ N \text{ if }\sum_{i=0}^{N-1}{C^i_{32}}\leqslant{y^n}<1.\\
 }
 \ENDFOR
 \STATE$a\leftarrow{PRNG_1()}$\;
-\STATE$m\leftarrow{g(a)}$\;
-\STATE$k\leftarrow{m}$\;
+\STATE$k\leftarrow{g(a)}$\;
 \WHILE{$i=0,\dots,k$}
 
 \STATE$b\leftarrow{PRNG_2()~mod~\mathsf{N}}$\;
@@ -643,7 +663,7 @@ x^0 \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket, S \in \llbracket 0, 2^\mathsf{N
 \forall n \in \mathds{N}^*, x^n = x^{n-1} \oplus S^n,
 \end{array}
 \right.
-\label{equation Oplus0}
+\label{equation Oplus}
 \end{equation}
 where $\oplus$ is for the bitwise exclusive or between two integers. 
 This rewriting can be understood as follows. The $n-$th term $S^n$ of the
@@ -653,7 +673,7 @@ as an integer having $\mathsf{N}$ bits too). More precisely, the $k-$th
 component of this state (a binary digit) changes if and only if the $k-$th 
 digit in the binary decomposition of $S^n$ is 1.
 
-The single basic component presented in Eq.~\ref{equation Oplus0} is of 
+The single basic component presented in Eq.~\ref{equation Oplus} is of 
 ordinary use as a good elementary brick in various PRNGs. It corresponds
 to the following discrete dynamical system in chaotic iterations:
 
@@ -675,7 +695,7 @@ we select a subset of components to change.
 
 
 Obviously, replacing the previous CI PRNG Algorithms by 
-Equation~\ref{equation Oplus0}, which is possible when the iteration function is
+Equation~\ref{equation Oplus}, which is possible when the iteration function is
 the vectorial negation, leads to a speed improvement 
 (the resulting generator will be referred as ``Xor CI PRNG''
 in what follows).
@@ -935,7 +955,7 @@ have $d((S,E),(\tilde S,E))<\epsilon$.
 \begin{color}{red}
 \section{Statistical Improvements Using Chaotic Iterations}
 
-\label{The generation of pseudo-random sequence}
+\label{The generation of pseudorandom sequence}
 
 
 Let us now explain why we are reasonable grounds to believe that chaos 
@@ -1213,7 +1233,7 @@ raise ambiguity.
 
 
 
-\lstset{language=C,caption={C code of the sequential PRNG based on chaotic iterations},label=algo:seqCIPRNG}
+\lstset{language=C,caption={C code of the sequential PRNG based on chaotic iterations},label={algo:seqCIPRNG}}
 \begin{small}
 \begin{lstlisting}
 
@@ -1723,6 +1743,7 @@ secure.
 
 \begin{color}{red}
 \subsection{Practical Security Evaluation}
+\label{sec:Practicak evaluation}
 
 Suppose now that the PRNG will work during 
 $M=100$ time units, and that during this period,
