fdkjlqsjflqs

[prng_gpu.git] / prng_gpu.tex

diff --git a/prng_gpu.tex b/prng_gpu.tex
index 085ce1e4d83cfa91529ca4974ade1af78d5154f4..807f6dfa2c56eecdeb0a522ca445b8db83f5398c 100644 (file)
--- a/prng_gpu.tex
+++ b/prng_gpu.tex
@@ -46,7 +46,7 @@
 \begin{document}
 
 \author{Jacques M. Bahi, Rapha\"{e}l Couturier,  Christophe
 \begin{document}
 
 \author{Jacques M. Bahi, Rapha\"{e}l Couturier,  Christophe

-Guyeux, and Pierre-Cyrille Héam\thanks{Authors in alphabetic order}}
+Guyeux, and Pierre-Cyrille Héam*\\ FEMTO-ST Institute, UMR  6174 CNRS,\\ University of Franche-Comt\'{e}, Besan\c con, France\\ * Authors in alphabetic order}

    
 
 %\IEEEcompsoctitleabstractindextext{
    
 
 %\IEEEcompsoctitleabstractindextext{


@@ -92,7 +92,7 @@ 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
 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
 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 +141,7 @@ 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
   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
 
 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


@@ -178,7 +178,7 @@ 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
 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.
+properties and statistical tests is also proposed.

 
 
 
 
 
 


@@ -189,14 +189,14 @@ 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{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.
-
- Section~\ref{sec:efficient PRNG
-  gpu}   describes and evaluates theoretically  the  GPU   implementation. 
+%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.
+ 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}.
 We show in Section~\ref{sec:security analysis} that, if the inputted
 Such generators are experimented in 
 Section~\ref{sec:experiments}.
 We show in Section~\ref{sec:security analysis} that, if the inputted


@@ -520,7 +520,7 @@ 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. 
 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.

 
 
 
 
 
 


@@ -662,6 +662,22 @@ N \text{ if }\sum_{i=0}^{N-1}{C^i_{32}}\leqslant{y^n}<1.\\
 \end{algorithmic}
 \end{algorithm}
 
 \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
 \subsection{Improving the Speed of the Former Generator}
 
 Instead of updating only one cell at each iteration, we now propose to choose a


@@ -1293,9 +1309,9 @@ have $d((S,E),(\tilde S,E))<\epsilon$.
 
 
 \section{Toward Efficiency and Improvement for CI PRNG}
 
 
 \section{Toward Efficiency and Improvement for CI PRNG}

+\label{sec:efficient PRNG}

 
 \subsection{First Efficient Implementation of a PRNG based on Chaotic Iterations}
 
 \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}. 
 %
 %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}. 


@@ -2052,10 +2068,10 @@ To encrypt his message, Bob will compute
 c = \left(m_0 \oplus (b_0 \oplus S^0), m_1 \oplus (b_0 \oplus b_1 \oplus S^0), \hdots, \right.
  \left. m_{L-1} \oplus (b_0 \oplus b_1 \hdots \oplus b_{L-1} \oplus S^0) \right)
 \end{equation*}
 c = \left(m_0 \oplus (b_0 \oplus S^0), m_1 \oplus (b_0 \oplus b_1 \oplus S^0), \hdots, \right.
  \left. m_{L-1} \oplus (b_0 \oplus b_1 \hdots \oplus b_{L-1} \oplus S^0) \right)
 \end{equation*}

-instead of $$\left(m_0 \oplus b_0, m_1 \oplus b_1, \hdots, m_{L-1} \oplus b_{L-1} \right)$$. 
+instead of $$\left(m_0 \oplus b_0, m_1 \oplus b_1, \hdots, m_{L-1} \oplus b_{L-1} \right).$$ 

 
 The same decryption stage as in Blum-Goldwasser leads to the sequence 
 
 The same decryption stage as in Blum-Goldwasser leads to the sequence 

-$$\left(m_0 \oplus S^0, m_1 \oplus S^0, \hdots, m_{L-1} \oplus S^0 \right)$$.
+$$\left(m_0 \oplus S^0, m_1 \oplus S^0, \hdots, m_{L-1} \oplus S^0 \right).$$

 Thus, with a simple use of $S^0$, Alice can obtain the plaintext.
 By doing so, the proposed generator is used in place of BBS, leading to
 the inheritance of all the properties presented in this paper.
 Thus, with a simple use of $S^0$, Alice can obtain the plaintext.
 By doing so, the proposed generator is used in place of BBS, leading to
 the inheritance of all the properties presented in this paper.