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@@ -36,7 +36,17 @@ On montre qu’un PRNG est qualité par des preuves théoriques et validations e
 
 \section{Travaux en relation}
 
+Random number generators (RNGs) are essential in statistical studies in several fields. They may be based on physical noise sources or on mathematical algorithms, but in both cases truly random numbers may not be obtained because of data acquisition
+systems in the first case or because machine precision in the second case~\cite{Gonzalez2005281,Guler2011}. In spite of the fact that the existence (or not) of truly random number generators (RNG) remains an open question, the random number is replaced by value, pseudo-random numbers, provided by deterministic algorithms whose properties mimic those of a truly random sequence~\cite{DeMicco20083373,Knuth1998}.
 
+Pseudo Random Number Generators (PRNGs) are widely used in science and technology, it is a critical component in modern cryptographic systems, communication systems, statistical simulation systems and any scientific area incorporating Monte Carlo methods~\cite{Vadim2011692,Marchi20093328,citeulike:867581,thecolourblue:1046} and many others. By the way, the Monte Carlo method appeared on the scientific scene in the late 1940s, for problems involving particle scattering in nuclear physics~\cite{Dyadkin1997258}. In the present era, there are few scientific fields that do not use random number. One of the most important applications of PRNGs is in cryptography to generate cryptographic keys, and to randomly initialize certain variables in cryptographic protocols. 
+
+Moreover, the idea o f applying chaos theory to randomness has produced important works very recently ~\cite{james1995,Gonzalez1999109,Ergun2007235,Zhou20093442,Gonzalez2002259,Behnia20113455}. Chaos theory has been established since 1970s by many different research areas, such as physics, mathematics, engineering, and biology, etc. ~\cite{Hao1993}. Since 1990s, many researchers have noticed that there exists the close relationship between chaos and cryptography ~\cite{Brown1996,Fridrich98symmetricciphers,Zaher20113721,Wong200367,Roland2001429,MS199850}; many properties of chaotic systems have their corresponding counterparts in traditional cryptosystems. Chaotic systems have several significant features favorable to secure communications, such as ergodicity, sensitivity to initial condition, control parameters and random like behaviour, which can be connected with some
+conventional cryptographic properties of good ciphers, such as confusion/diffusion. With all these advantages scientists
+expected to introduce new and powerful tools of chaotic cryptography. Cryptography is the art of achieving security by
+encoding messages to make them non-readable. Nowaday, some efficient techniques for encoding based on discrete time chaotic systems are presented in ~\cite{Djema2009,Belmouhoub2005}. A good random number generation improves the cryptographic security  ~\cite{Behnia2008408}.
+
+Several test suites are readily available to researchers in academia and industry who wish to analyze their newly developed PRNG. Some general purpose test suites are Diehard by George Marsaglia ~\cite{Marsaglia1996}, the National Institute of Standards and Technology Statistical Test Suite ~\cite{ANDREW2008}, and Comparative test parameters~\cite{Menezes1997}. Systematic tests of pseudorandom number generators have received much attention. Finally, The TestU01 testsuite by L'Ecuyer and Simard ~\cite{Lecuyer2009} appears to define the current state of the art in the field. 
 \section{Résume de nos contributions}
 Il a été étable dans ~\cite{guyeux09, guyeux10} que les itérations chaotiques (IC), un outil servant à l’obtention d’algorithmes itératifs rapide, satisfait la propriété de chaos topologique telle qu'elle a été définie par Devaney ~\cite{ Dev89}.
 Nous avons alors construit notre PRNG en combinant ces itérations chaotiques avec deux générateurs basés sur la suite de logistique~\cite{wang2009}.
@@ -61,10 +71,33 @@ Enfin, nous avons utilisé le négation vectorielle comme un prototype, et expli
 
 
 \section{Objectifs de le thèse}
-
+During my research studies, the quantization effects on the dynamics of the chaotic maps have been investigated. It is noticed that most of the maps will experience the problems such as short periodic cycle length, high correlation of data, and the problem of the existence of fixed points will become significant. Therefore, a new scheme based on chaotic iterations is proposed. It is found that the statistical properties of a sequence can be greatly improved by applying this newly designed chaotic iterations function. This hence can also serve as the major building block for the design of chaos-based random number generators.
 
 \section{Organisation du manuscrit}
 
+The organization of this thesis is as follows.
+
+In Chapter 2, serving as the background of this thesis, the fundamental views and definitions of random number are summarized, and an overview of the classification and explanation of different types of RNGs are provided. We present the definitions that will be used in this thesis. The use of chaos for random number generators will be summarized. Some techniques to generate random number, based on chaos, will be revisited. 
+
+In Chapter 3, beginning with an brief review of the combinatorial objects used in the design of CIs PRNG and a theoretical proof of chaotic iterations, it outlines two novel approaches for generating random number sequences based on chaotic iterations techniques. The design of CIs for random number generators will be summarized. Some typical examples of chaos-based random number generators are described.
+
+
+ 
+In Chapter 4, a detailed study is undergone, how a PRNG can be tested is described and the details of Diehard, NIST, comparative test parameters and TestU01 statistical test suites are presented. A comparative study on the quality of two novel approaches for CIs PRNGs and the associated random number generators are reported. The approach for combining two generators based on chaotic iterations would give better properties than the individual components alone. The comparison of the speed and  statistical tests results allow us to consider that our new CIs generator has better pseudo-random characteristics
+
+
+in Chapter 5. The performance of these newly designed PRNG based on chaotic iterations will be analyzed and compared with some common PRNGs. Finally, the criteria, such as uniformity, correlations, sequence patterns and complexity, and so on, which are commonly used to verify the randomness of a sequence, are also introduced.
+
+In Chapter 6, In prior literature, the iterate function is just the vectorial boolean negation.  
+In this Chapter, we propose a method using Graph with strongly connected components as a selection criterion for chaotic iterate function. 
+In order to face the challenge of using the proposed chaotic iterate functions in PRNG, these PRNGs are subjected to the NIST statistical batteries of tests.
+
+
+In Chapter 7, a potential use of above PRNGs in some Internet security field is presented, namely in information hiding. Such generators can strongly improve the confidence put in any information hiding scheme and in cryptography in general: due to their properties of unpredictability, the possibilities offered to an attacker to achieve his goal are drastically reduced in that context. 
+
+
+We conclude this thesis in Chapter 8 by summarizing the significance of our work  and discussing the possible future work in this area.
+
 
 \section{Abréviations}
 \begin{tabular}{ll}\toprule