avancées dans la réécriture

[prng_gpu.git] / prng_gpu.tex

diff --git a/prng_gpu.tex b/prng_gpu.tex
index bc40b2d2670c13cc3b9b1978f9a8baf155ee0ff0..0c9f9c742e91de13b4286de29106cf7a76abfa8f 100644 (file)
--- a/prng_gpu.tex
+++ b/prng_gpu.tex
@@ -13,6 +13,9 @@
 \usepackage[standard]{ntheorem}
 \usepackage{algorithmic}
 \usepackage{slashbox}
 \usepackage[standard]{ntheorem}
 \usepackage{algorithmic}
 \usepackage{slashbox}

+\usepackage{ctable}
+\usepackage{tabularx}
+\usepackage{multirow}

 
 % Pour mathds : les ensembles IR, IN, etc.
 \usepackage{dsfont}
 
 % Pour mathds : les ensembles IR, IN, etc.
 \usepackage{dsfont}


@@ -87,7 +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
 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
 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


@@ -121,10 +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}.
 {\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}.
 
 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
 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


@@ -154,8 +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}.
   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}.
   gpu}   describes and evaluates theoretically  the  GPU   implementation. 
 Such generators are experimented in 
 Section~\ref{sec:experiments}.


@@ -164,7 +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
 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.
 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.


@@ -172,7 +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
 \label{section:related works}
 
 Numerous research works on defining GPU based PRNGs have already been proposed  in the


@@ -477,7 +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}
 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.
 generator taken alone. 
 Furthermore, the generator obtained by this way possesses various chaos properties that none of the generators used as input
 present.


@@ -553,7 +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. 
 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 
 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 


@@ -566,14 +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
 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}.
 $(m^n)_{n \in \mathds{N}} \in \mathcal{M}^\mathds{N}$ 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}
 
 \begin{equation}
 \label{Formula}


@@ -599,8 +623,7 @@ N \text{ if }\sum_{i=0}^{N-1}{C^i_{32}}\leqslant{y^n}<1.\\
 }
 \ENDFOR
 \STATE$a\leftarrow{PRNG_1()}$\;
 }
 \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$b\leftarrow{PRNG_2()~mod~\mathsf{N}}$\;
 \WHILE{$i=0,\dots,k$}
 
 \STATE$b\leftarrow{PRNG_2()~mod~\mathsf{N}}$\;


@@ -640,7 +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.
 \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
 \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


@@ -650,7 +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.
 
 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:
 
 ordinary use as a good elementary brick in various PRNGs. It corresponds
 to the following discrete dynamical system in chaotic iterations:
 


@@ -672,7 +695,7 @@ we select a subset of components to change.
 
 
 Obviously, replacing the previous CI PRNG Algorithms by 
 
 
 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).
 the vectorial negation, leads to a speed improvement 
 (the resulting generator will be referred as ``Xor CI PRNG''
 in what follows).


@@ -932,7 +955,7 @@ have $d((S,E),(\tilde S,E))<\epsilon$.
 \begin{color}{red}
 \section{Statistical Improvements Using Chaotic Iterations}
 
 \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 
 
 
 Let us now explain why we are reasonable grounds to believe that chaos 


@@ -1007,6 +1030,53 @@ a^1 & \text{if}~  z^{n-1} = 0 .\end{array} \right. \end{array}\end{equation}
 
 
 
 
 
 

+\begin{table}
+\renewcommand{\arraystretch}{1.3}
+\caption{TestU01 Statistical Test}
+\label{TestU011}
+\centering
+  \begin{tabular}{lccccc}
+    \toprule
+Test name &Tests& Logistic             & XORshift      & ISAAC\\
+Rabbit                                 &       38      &21             &14     &0       \\
+Alphabit                       &       17      &16             &9      &0       \\
+Pseudo DieHARD                         &126    &0              &2      &0      \\
+FIPS\_140\_2                   &16     &0              &0      &0      \\
+SmallCrush                     &15     &4              &5      &0       \\
+Crush                          &144    &95             &57     &0       \\
+Big Crush                      &160    &125            &55     &0       \\ \hline
+Failures               &       &261            &146    &0       \\
+\bottomrule
+  \end{tabular}
+\end{table}
+
+
+
+\begin{table}
+\renewcommand{\arraystretch}{1.3}
+\caption{TestU01 Statistical Test for Old CI algorithms ($\mathsf{N}=4$)}
+\label{TestU01 for Old CI}
+\centering
+  \begin{tabular}{lcccc}
+    \toprule
+\multirow{3}*{Test name} & \multicolumn{4}{c}{Old CI}\\
+&Logistic& XORshift& ISAAC&ISAAC  \\ 
+&+& +& + & + \\ 
+&Logistic& XORshift& XORshift&ISAAC  \\ \cmidrule(r){2-5}
+Rabbit                                         &7      &2      &0      &0       \\
+Alphabit                               & 3     &0      &0      &0       \\
+DieHARD                        &0      &0      &0      &0      \\
+FIPS\_140\_2                   &0      &0      &0      &0      \\
+SmallCrush                             &2      &0      &0      &0       \\
+Crush                                  &47     &4      &0      &0       \\
+Big Crush                              &79     &3      &0      &0       \\ \hline
+Failures                               &138    &9      &0      &0       \\
+\bottomrule
+  \end{tabular}
+\end{table}
+
+
+

 
 
 \subsection{Statistical tests}
 
 
 \subsection{Statistical tests}


@@ -1106,25 +1176,32 @@ Threshold  value $m$& 19 & 7  & 2& 1 & 11& 9& 3& 4\\ \hline\hline
 \end{tabular}
 \end{table*}
 
 \end{tabular}
 \end{table*}
 

+Finally, the TestU01 battery as been launched on three well-known generators 
+(a logistic map, a simple XORshift, and the cryptographically secure ISAAC, 
+see Table~\ref{TestU011}). These results can be compared with 
+Table~\ref{TestU01 for Old CI}, which gives the scores obtained by the
+Old CI PRNG that has received these generators.
+
+

 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}
 
 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}
 

-\subsection{Efficient PRNG based on Chaotic Iterations}
+\subsection{Efficient Implementation of a PRNG based on Chaotic Iterations}

 \label{sec:efficient PRNG}
 \label{sec:efficient PRNG}

-
-Based on the proof presented in the previous section, it is now possible to 
-improve the speed of the generator formerly presented in~\cite{bgw09:ip,guyeux10}. 
-The first idea is to consider
-that the provided strategy is a pseudorandom Boolean vector obtained by a
-given PRNG.
-An iteration of the system is simply the bitwise exclusive or between
-the last computed state and the current strategy.
-Topological properties of disorder exhibited by chaotic 
-iterations can be inherited by the inputted generator, we hope by doing so to 
-obtain some statistical improvements while preserving speed.
-
+%
+%Based on the proof presented in the previous section, it is now possible to 
+%improve the speed of the generator formerly presented in~\cite{bgw09:ip,guyeux10}. 
+%The first idea is to consider
+%that the provided strategy is a pseudorandom Boolean vector obtained by a
+%given PRNG.
+%An iteration of the system is simply the bitwise exclusive or between
+%the last computed state and the current strategy.
+%Topological properties of disorder exhibited by chaotic 
+%iterations can be inherited by the inputted generator, we hope by doing so to 
+%obtain some statistical improvements while preserving speed.
+%

 %%RAPH : j'ai viré tout ca
 %% Let us give an example using 16-bits numbers, to clearly understand how the bitwise xor operations
 %% are
 %%RAPH : j'ai viré tout ca
 %% Let us give an example using 16-bits numbers, to clearly understand how the bitwise xor operations
 %% are


@@ -1156,7 +1233,7 @@ obtain some statistical improvements while preserving speed.
 
 
 
 
 
 

-\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}
 
 \begin{small}
 \begin{lstlisting}
 


@@ -1666,6 +1743,7 @@ secure.
 
 \begin{color}{red}
 \subsection{Practical Security Evaluation}
 
 \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,
 
 Suppose now that the PRNG will work during 
 $M=100$ time units, and that during this period,