X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/prng_gpu.git/blobdiff_plain/75b9bc464cc8a1276712fa012f5ed62c3d4b9f64..baf2ff7c5bd7a44b68600f37eb61b76e67a8be17:/prng_gpu.tex

diff --git a/prng_gpu.tex b/prng_gpu.tex
index 55b834d..4880738 100644
--- a/prng_gpu.tex
+++ b/prng_gpu.tex
@@ -40,14 +40,25 @@
 
 \newcommand{\alert}[1]{\begin{color}{blue}\textit{#1}\end{color}}
 
+\begin{document}
+\begin{frontmatter}
+\title{Efficient and Cryptographically Secure Generation of Chaotic Pseudorandom Numbers on GPU}
+
+
+\author{Jacques M. Bahi}
+\ead{jacques.bahi@univ-fcomte.fr}
+\author{ Rapha\"{e}l Couturier \corref{cor1}}
+\ead{raphael.couturier@univ-fcomte.fr}
+\cortext[cor1]{Corresponding author}
+\author{  Christophe Guyeux}
+\ead{christophe.guyeux@univ-fcomte.fr}
+\author{ Pierre-Cyrille HÃ©am }
+\ead{pierre-cyrille.heam@univ-fcomte.fr}
+
+\address{FEMTO-ST Institute, UMR  6174 CNRS,\\ University of Franche Comte, Belfort, France\\  Authors in alphabetic order}
 
 
-\title{Efficient and Cryptographically Secure Generation of Chaotic Pseudorandom Numbers on GPU}
-\begin{document}
 
-\author{Jacques M. Bahi, Rapha\"{e}l Couturier,  Christophe
-Guyeux, and Pierre-Cyrille HÃ©am*\\ FEMTO-ST Institute, UMR  6174 CNRS,\\ University of Franche-Comt\'{e}, Besan\c con, France\\ * Authors in alphabetic order}
-   
 
 %\IEEEcompsoctitleabstractindextext{
 \begin{abstract}
@@ -65,8 +76,13 @@ A chaotic version of the Blum-Goldwasser asymmetric key encryption scheme is fin
 
 \end{abstract}
 %}
+\begin{keyword}
+   pseudo random number\sep parallelization\sep GPU\sep cryptography\sep chaos
 
-\maketitle
+\end{keyword}
+\end{frontmatter}
+
+%\maketitle
 
 %\IEEEdisplaynotcompsoctitleabstractindextext
 %\IEEEpeerreviewmaketitle
@@ -92,7 +108,7 @@ 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.  However, in an equivalent formulation, he or she should not be
+sequence.  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
@@ -141,7 +157,7 @@ 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].
 Chaos, for its part, refers to the well-established definition of a
-chaotic dynamical system proposed by Devaney~\cite{Devaney}.
+chaotic dynamical system defined 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
@@ -177,8 +193,8 @@ Pseudorandom numbers are generated at a rate of 20GSamples/s, which is faster
 than in~\cite{conf/fpga/ThomasHL09,Marsaglia2003} (and with a better
 statistical behavior). Experiments are also provided using BBS as the initial
 random generator. The generation speed is significantly weaker.
-Note also that an original qualitative comparison between topological chaotic
-properties and statistical test is also proposed.
+%Note also that an original qualitative comparison between topological chaotic
+%properties and statistical tests is also proposed.
 
 
 
@@ -193,8 +209,9 @@ The proposed PRNG and its proof of chaos are given in  Section~\ref{sec:pseudora
 %improvement related to the chaotic iteration based post-treatment, for
 %our previously released PRNGs and a new efficient 
 %implementation on CPU.
- Section~\ref{sec:efficient PRNG
-  gpu}   describes and evaluates theoretically new effective versions of
+ Section~\ref{sec:efficient PRNG} %{sec:efficient PRNG
+%  gpu}  
+ describes and evaluates theoretically new effective versions of
 our pseudorandom generators,  in particular with a  GPU   implementation. 
 Such generators are experimented in 
 Section~\ref{sec:experiments}.
@@ -519,7 +536,7 @@ two PRNGs as inputs. These two generators are mixed with chaotic iterations,
 leading thus to a new PRNG that 
 should improve the statistical properties of each
 generator taken alone. 
-Furthermore, the generator obtained in this way possesses various chaos properties that none of the generators used as present input.
+Furthermore, the generator obtained in this way possesses various chaos properties that none of the generators used as  input present.
 
 
 
@@ -661,6 +678,22 @@ N \text{ if }\sum_{i=0}^{N-1}{C^i_{32}}\leqslant{y^n}<1.\\
 \end{algorithmic}
 \end{algorithm}
 
+
+We have shown in~\cite{bfg12a:ip} that the use of chaotic iterations 
+implies an improvement of the statistical properties for all the
+inputted defective generators we have investigated.
+For instance, when considering the TestU01 battery with its 588 tests, we obtained 261 
+failures for a PRNG based on the logistic map alone, and 
+this number of failures falls below 138 in the Old CI(Logistic,Logistic) generator.
+In the XORshift case (146 failures when considering it alone), the results are more amazing,
+as the chaotic iterations post-treatment makes it fails only 8 tests. 
+Further investigations have been systematically realized in \cite{bfg12a:ip}
+using a large set of inputted defective PRNGs, the three most used batteries of
+tests (DieHARD, NIST, and TestU01), and for all the versions of generators we have proposed.
+In all situations, an obvious improvement of the statistical behavior has
+been obtained, reinforcing the impression that chaos leads to statistical 
+enhancement~\cite{bfg12a:ip}.
+
 \subsection{Improving the Speed of the Former Generator}
 
 Instead of updating only one cell at each iteration, we now propose to choose a
@@ -1292,9 +1325,9 @@ have $d((S,E),(\tilde S,E))<\epsilon$.
 
 
 \section{Toward Efficiency and Improvement for CI PRNG}
+\label{sec:efficient PRNG}
 
 \subsection{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 
 %improve the speed of the generator formerly presented in~\cite{bgw09:ip,guyeux10}. 
@@ -1769,14 +1802,7 @@ Let $\varepsilon > 0$.
 $\mathcal{D}$ is called a $(T,\varepsilon)-$distinguishing attack on pseudorandom
 generator $G$ if
 
-\begin{flushleft}
-$\left| Pr[\mathcal{D}(G(k)) = 1 \mid k \in_R \{0,1\}^\ell ]\right.$
-\end{flushleft}
-
-\begin{flushright}
-$ - \left. Pr[\mathcal{D}(s) = 1 \mid s \in_R \mathds{B}^M ]\right| \geqslant \varepsilon,$
-\end{flushright}
-
+$$\left| Pr[\mathcal{D}(G(k)) = 1 \mid k \in_R \{0,1\}^\ell ]\right. - \left. Pr[\mathcal{D}(s) = 1 \mid s \in_R \mathds{B}^M ]\right| \geqslant \varepsilon,$$
 \noindent where the probability is taken over the internal coin flips of $\mathcal{D}$, and the notation
 ``$\in_R$'' indicates the process of selecting an element at random and uniformly over the
 corresponding set.
