X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/prng_gpu.git/blobdiff_plain/9879779d913285ee14baad568f69be401dfd0fb3..0be2ff183726ec026c763c2d0f9b4d3a326360fd:/prng_gpu.tex?ds=sidebyside diff --git a/prng_gpu.tex b/prng_gpu.tex index 0c9f9c7..38431e5 100644 --- a/prng_gpu.tex +++ b/prng_gpu.tex @@ -40,6 +40,9 @@ \newcommand{\alert}[1]{\begin{color}{blue}\textit{#1}\end{color}} + +\newcommand{\PCH}[1]{\begin{color}{blue}#1\end{color}} + \title{Efficient and Cryptographically Secure Generation of Chaotic Pseudorandom Numbers on GPU} \begin{document} @@ -166,6 +169,25 @@ property. Last, but not least, we propose a rewriting of the Blum-Goldwasser asymmetric key encryption protocol by using the proposed method. + +\PCH{ +{\bf Main contributions.} In this paper a new PRNG using chaotic iteration +is defined. From a theoretical point of view, it is proven that it has fine +topological chaotic properties and that it is cryptographically secured (when +the based PRNG is also cryptographically secured). From a practical point of +view, experiments point out a very good statistical behavior. Optimized +original implementation of this PRNG are also proposed and experimented. +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 based +random generator. The generation speed is significantly weaker but, as far +as we know, it is the first cryptographically secured PRNG proposed on GPU. +Note too that an original qualitative comparison between topological chaotic +properties and statistical test is also proposed. +} + + + The remainder of this paper is organized as follows. In Section~\ref{section:related works} we review some GPU implementations of PRNGs. Section~\ref{section:BASIC RECALLS} gives some basic recalls on the well-known Devaney's formulation of chaos, @@ -185,10 +207,11 @@ Section~\ref{sec:experiments}. We show in Section~\ref{sec:security analysis} that, if the inputted generator is cryptographically secure, then it is the case too for the generator provided by the post-treatment. +\begin{color}{red} A practical +security evaluation is also outlined in Section~\ref{sec:Practicak evaluation}.\end{color} 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}, \begin{color}{red} to a practical -security evaluation in Section~\ref{sec:Practicak evaluation}, \end{color} and to an improvement of the +in Section~\ref{sec:CSGPU} 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. @@ -255,7 +278,7 @@ with basic notions on topology (see for instance~\cite{Devaney}). \subsection{Devaney's Chaotic Dynamical Systems} - +\label{subsec:Devaney} In the sequel $S^{n}$ denotes the $n^{th}$ term of a sequence $S$ and $V_{i}$ denotes the $i^{th}$ component of a vector $V$. $f^{k}=f\circ ...\circ f$ is for the $k^{th}$ composition of a function $f$. Finally, the following @@ -596,7 +619,7 @@ $(m^n)_{n \in \mathds{N}} \in \mathcal{M}^\mathds{N}$ is computed from $PRNG_1$, 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 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 +This function must be chosen such that the outputs of the resulted PRNG are 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} @@ -650,8 +673,8 @@ N \text{ if }\sum_{i=0}^{N-1}{C^i_{32}}\leqslant{y^n}<1.\\ \subsection{Improving the Speed of the Former Generator} -Instead of updating only one cell at each iteration,\begin{color}{red} we now propose to choose a -subset of components and to update them together, for speed improvements. Such a proposition leads\end{color} +Instead of updating only one cell at each iteration, \begin{color}{red} we now propose to choose a +subset of components and to update them together, for speed improvements. Such a proposition leads \end{color} to a kind of merger of the two sequences used in Algorithms \ref{CI Algorithm} and \ref{Chaotic iteration1}. When the updating function is the vectorial negation, this algorithm can be rewritten as follows: @@ -878,6 +901,8 @@ the distance between $(S^n,E^n)$ and $(S,E)$ is strictly less than $% In conclusion, %%RAPH : ici j'ai rajouté une ligne +%%TOF : ici j'ai rajouté un commentaire +%%TOF : ici aussi $ \forall \varepsilon >0,$ $\exists N_{0}=max(n_{0},n_{1},n_{2})\in \mathds{N} ,$ $\forall n\geqslant N_{0},$ @@ -960,33 +985,116 @@ have $d((S,E),(\tilde S,E))<\epsilon$. Let us now explain why we are reasonable grounds to believe that chaos can improve statistical properties. -We will show in this section that, when mixing defective PRNGs with -chaotic iterations, the result presents better statistical properties -(this section summarizes the work of~\cite{bfg12a:ip}). +We will show in this section that chaotic properties as defined in the +mathematical theory of chaos are related to some statistical tests that can be found +in the NIST battery. Furthermore, we will check that, when mixing defective PRNGs with +chaotic iterations, the new generator presents better statistical properties +(this section summarizes and extends the work of~\cite{bfg12a:ip}). + + + +\subsection{Qualitative relations between topological properties and statistical tests} + + +There are various relations between topological properties that describe an unpredictable behavior for a discrete +dynamical system on the one +hand, and statistical tests to check the randomness of a numerical sequence +on the other hand. These two mathematical disciplines follow a similar +objective in case of a recurrent sequence (to characterize an intrinsically complicated behavior for a +recurrent sequence), with two different but complementary approaches. +It is true that the following illustrative links give only qualitative arguments, +and proofs should be provided later to make such arguments irrefutable. However +they give a first understanding of the reason why we think that chaotic properties should tend +to improve the statistical quality of PRNGs. +% +Let us now list some of these relations between topological properties defined in the mathematical +theory of chaos and tests embedded into the NIST battery. %Such relations need to be further +%investigated, but they presently give a first illustration of a trend to search similar properties in the +%two following fields: mathematical chaos and statistics. + + +\begin{itemize} + \item \textbf{Regularity}. As stated in Section~\ref{subsec:Devaney}, a chaotic dynamical system must +have an element of regularity. Depending on the chosen definition of chaos, this element can be the existence of +a dense orbit, the density of periodic points, etc. The key idea is that a dynamical system with no periodicity +is not as chaotic as a system having periodic orbits: in the first situation, we can predict something and gain a +knowledge about the behavior of the system, that is, it never enters into a loop. A similar importance for periodicity is emphasized in +the two following NIST tests~\cite{Nist10}: + \begin{itemize} + \item \textbf{Non-overlapping Template Matching Test}. Detect generators that produce too many occurrences of a given non-periodic (aperiodic) pattern. + \item \textbf{Discrete Fourier Transform (Spectral) Test}. Detect periodic features (i.e., repetitive patterns that are near each other) in the tested sequence that would indicate a deviation from the assumption of randomness. + \end{itemize} + +\item \textbf{Transitivity}. This topological property introduced previously states that the dynamical system is intrinsically complicated: it cannot be simplified into +two subsystems that do not interact, as we can find in any neighborhood of any point another point whose orbit visits the whole phase space. +This focus on the places visited by orbits of the dynamical system takes various nonequivalent formulations in the mathematical theory +of chaos, namely: transitivity, strong transitivity, total transitivity, topological mixing, and so on~\cite{bg10:ij}. A similar attention +is brought on states visited during a random walk in the two tests below~\cite{Nist10}: + \begin{itemize} + \item \textbf{Random Excursions Variant Test}. Detect deviations from the expected number of visits to various states in the random walk. + \item \textbf{Random Excursions Test}. Determine if the number of visits to a particular state within a cycle deviates from what one would expect for a random sequence. + \end{itemize} + +\item \textbf{Chaos according to Li and Yorke}. Two points of the phase space $(x,y)$ define a couple of Li-Yorke when $\limsup_{n \rightarrow +\infty} d(f^{(n)}(x), f^{(n)}(y))>0$ et $\liminf_{n \rightarrow +\infty} d(f^{(n)}(x), f^{(n)}(y))=0$, meaning that their orbits always oscillates as the iterations pass. When a system is compact and contains an uncountable set of such points, it is claimed as chaotic according +to Li-Yorke~\cite{Li75,Ruette2001}. A similar property is regarded in the following NIST test~\cite{Nist10}. + \begin{itemize} + \item \textbf{Runs Test}. To determine whether the number of runs of ones and zeros of various lengths is as expected for a random sequence. In particular, this test determines whether the oscillation between such zeros and ones is too fast or too slow. + \end{itemize} + \item \textbf{Topological entropy}. The desire to formulate an equivalency of the thermodynamics entropy +has emerged both in the topological and statistical fields. Another time, a similar objective has led to two different +rewritten of an entropy based disorder: the famous Shannon definition of entropy is approximated in the statistical approach, +whereas topological entropy is defined as follows. +$x,y \in \mathcal{X}$ are $\varepsilon-$\emph{separated in time $n$} if there exists $k \leqslant n$ such that $d\left(f^{(k)}(x),f^{(k)}(y)\right)>\varepsilon$. Then $(n,\varepsilon)-$separated sets are sets of points that are all $\varepsilon-$separated in time $n$, which +leads to the definition of $s_n(\varepsilon,Y)$, being the maximal cardinality of all $(n,\varepsilon)-$separated sets. Using these notations, +the topological entropy is defined as follows: $$h_{top}(\mathcal{X},f) = \displaystyle{\lim_{\varepsilon \rightarrow 0} \Big[ \limsup_{n \rightarrow +\infty} \dfrac{1}{n} \log s_n(\varepsilon,\mathcal{X})\Big]}.$$ +This value measures the average exponential growth of the number of distinguishable orbit segments. +In this sense, it measures complexity of the topological dynamical system, whereas +the Shannon approach is in mind when defining the following test~\cite{Nist10}: + \begin{itemize} +\item \textbf{Approximate Entropy Test}. Compare the frequency of overlapping blocks of two consecutive/adjacent lengths ($m$ and $m+1$) against the expected result for a random sequence. + \end{itemize} + + \item \textbf{Non-linearity, complexity}. Finally, let us remark that non-linearity and complexity are +not only sought in general to obtain chaos, but they are also required for randomness, as illustrated by the two tests below~\cite{Nist10}. + \begin{itemize} +\item \textbf{Binary Matrix Rank Test}. Check for linear dependence among fixed length substrings of the original sequence. +\item \textbf{Linear Complexity Test}. Determine whether or not the sequence is complex enough to be considered random. + \end{itemize} +\end{itemize} + + +We have proven in our previous works~\cite{guyeux12:bc} that chaotic iterations satisfying Theorem~\ref{Th:Caractérisation des IC chaotiques} are, among other +things, strongly transitive, topologically mixing, chaotic as defined by Li and Yorke, +and that they have a topological entropy and an exponent of Lyapunov both equal to $ln(\mathsf{N})$, +where $\mathsf{N}$ is the size of the iterated vector. +These topological properties make that we are ground to believe that a generator based on chaotic +iterations will probably be able to pass all the existing statistical batteries for pseudorandomness like +the NIST one. The following subsections, in which we prove that defective generators have their +statistical properties improved by chaotic iterations, show that such an assumption is true. \subsection{Details of some Existing Generators} The list of defective PRNGs we will use as inputs for the statistical tests to come is introduced here. -Firstly, the simple linear congruency generator (LCGs) will be used. -It is defined by the following recurrence: +Firstly, the simple linear congruency generators (LCGs) will be used. +They are defined by the following recurrence: \begin{equation} -x^n = (ax^{n-1} + c)~mod~m +x^n = (ax^{n-1} + c)~mod~m, \label{LCG} \end{equation} where $a$, $c$, and $x^0$ must be, among other things, non-negative and less than $m$~\cite{LEcuyerS07}. In what follows, 2LCGs and 3LCGs refer as two (resp. three) combinations of such LCGs. For further details, see~\cite{bfg12a:ip,combined_lcg}. -Secondly, the multiple recursive generators (MRGs) will be used too, which +Secondly, the multiple recursive generators (MRGs) will be used, which are based on a linear recurrence of order $k$, modulo $m$~\cite{LEcuyerS07}: \begin{equation} -x^n = (a^1x^{n-1}+~...~+a^kx^{n-k})~mod~m +x^n = (a^1x^{n-1}+~...~+a^kx^{n-k})~mod~m . \label{MRG} \end{equation} -Combination of two MRGs (referred as 2MRGs) is also used in these experimentations. +Combination of two MRGs (referred as 2MRGs) is also used in these experiments. Generators based on linear recurrences with carry will be regarded too. This family of generators includes the add-with-carry (AWC) generator, based on the recurrence: @@ -1013,12 +1121,12 @@ c^n = (a^1x^{n-1} \oplus ~...~ \oplus a^rx^{n-r} \oplus c^{n-1}) ~ / ~ 2^w. \end Then the generalized feedback shift register (GFSR) generator has been implemented, that is: \begin{equation} -x^n = x^{n-r} \oplus x^{n-k} +x^n = x^{n-r} \oplus x^{n-k} . \label{GFSR} \end{equation} -Finally, the nonlinear inversive generator~\cite{LEcuyerS07} has been regarded too, which is: +Finally, the nonlinear inversive (INV) generator~\cite{LEcuyerS07} has been studied, which is: \begin{equation} \label{INV} @@ -1132,8 +1240,9 @@ The obtained results are reproduced in Table The scores written in boldface indicate that all the tests have been passed successfully, whereas an asterisk ``*'' means that the considered passing rate has been improved. The improvements are obvious for both the ``Old CI'' and ``New CI'' generators. -Concerning the ``Xor CI PRNG'', the speed improvement makes that statistical -results are not as good as for the two other versions of these CIPRNGs. +Concerning the ``Xor CI PRNG'', the score is less spectacular: a large speed improvement makes that statistics + are not as good as for the two other versions of these CIPRNGs. +However 8 tests have been improved (with no deflation for the other results). \begin{table*} @@ -1160,8 +1269,9 @@ DieHARD & 16/18 & 16/18 & 17/18* & \textbf{18/18} * & \textbf{18/18} & \textbf{ We have then investigate in~\cite{bfg12a:ip} if it is possible to improve the statistical behavior of the Xor CI version by combining more than one -$\oplus$ operation. Results are summarized in~\ref{threshold}, showing -that rapid and perfect PRNGs, regarding the NIST and DieHARD batteries, can be obtained +$\oplus$ operation. Results are summarized in Table~\ref{threshold}, illustrating +the progressive increasing effects of chaotic iterations, when giving time to chaos to get settled in. +Thus rapid and perfect PRNGs, regarding the NIST and DieHARD batteries, can be obtained using chaotic iterations on defective generators. \begin{table*} @@ -1176,19 +1286,23 @@ Threshold value $m$& 19 & 7 & 2& 1 & 11& 9& 3& 4\\ \hline\hline \end{tabular} \end{table*} -Finally, the TestU01 battery as been launched on three well-known generators +Finally, the TestU01 battery has 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. +The obvious improvement speaks for itself, and together with the other +results recalled in this section, it reinforces the opinion that a strong +correlation between topological properties and statistical behavior exists. -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 +Next subsection will now give a concrete original implementation of the Xor CI PRNG, the +fastest generator in the chaotic iteration based family. In the remainder, +this generator will be simply referred as CIPRNG, or ``the proposed PRNG'', if this statement does not raise ambiguity. \end{color} -\subsection{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 @@ -1267,7 +1381,13 @@ works with 32-bits, we use the command \texttt{(unsigned int)}, that selects the Thus producing a pseudorandom number needs 6 xor operations with 6 32-bits numbers that are provided by 3 64-bits PRNGs. This version successfully passes the -stringent BigCrush battery of tests~\cite{LEcuyerS07}. +stringent BigCrush battery of tests~\cite{LEcuyerS07}. +\begin{color}{red}At this point, we thus +have defined an efficient and statistically unbiased generator. Its speed is +directly related to the use of linear operations, but for the same reason, +this fast generator cannot be proven as secure. +\end{color} + \section{Efficient PRNGs based on Chaotic Iterations on GPU} \label{sec:efficient PRNG gpu} @@ -1403,7 +1523,9 @@ version\label{IR}} \label{algo:gpu_kernel2} \end{algorithm} -\subsection{Theoretical Evaluation of the Improved Version} +\begin{color}{red} +\subsection{Chaos Evaluation of the Improved Version} +\end{color} A run of Algorithm~\ref{algo:gpu_kernel2} consists in an operation ($x=x\oplus t$) having the form of Equation~\ref{equation Oplus}, which is equivalent to the iterative @@ -1501,9 +1623,27 @@ as it is shown in the next sections. \section{Security Analysis} -\label{sec:security analysis} +\begin{color}{red} +This section is dedicated to the security analysis of the + proposed PRNGs, both from a theoretical and a practical points of view. + +\subsection{Theoretical Proof of Security} +\label{sec:security analysis} + +The standard definition + of {\it indistinguishability} used is the classical one as defined for + instance in~\cite[chapter~3]{Goldreich}. + This property shows that predicting the future results of the PRNG + cannot be done in a reasonable time compared to the generation time. It is important to emphasize that this + is a relative notion between breaking time and the sizes of the + keys/seeds. Of course, if small keys or seeds are chosen, the system can + be broken in practice. But it also means that if the keys/seeds are large + enough, the system is secured. +As a complement, an example of a concrete practical evaluation of security +is outlined in the next subsection. +\end{color} In this section the concatenation of two strings $u$ and $v$ is classically denoted by $uv$. @@ -1525,7 +1665,15 @@ internal coin tosses of $D$. Intuitively, it means that there is no polynomial time algorithm that can distinguish a perfect uniform random generator from $G$ with a non -negligible probability. The interested reader is referred +negligible probability. +\begin{color}{red} + An equivalent formulation of this well-known +security property means that it is possible +\emph{in practice} to predict the next bit of +the generator, knowing all the previously +produced ones. +\end{color} +The interested reader is referred to~\cite[chapter~3]{Goldreich} for more information. Note that it is quite easily possible to change the function $\ell$ into any polynomial function $\ell^\prime$ satisfying $\ell^\prime(m)>m)$~\cite[Chapter 3.3]{Goldreich}. @@ -1550,7 +1698,7 @@ PRNG too. \end{proposition} \begin{proof} -The proposition is proved by contraposition. Assume that $X$ is not +The proposition is proven by contraposition. Assume that $X$ is not secure. By Definition, there exists a polynomial time probabilistic algorithm $D$, a positive polynomial $p$, such that for all $k_0$ there exists $N\geq \frac{k_0}{2}$ satisfying @@ -1613,6 +1761,100 @@ proving that $H$ is not secure, which is a contradiction. \end{proof} + +\begin{color}{red} +\subsection{Practical Security Evaluation} +\label{sec:Practicak evaluation} + +Pseudorandom generators based on Eq.~\eqref{equation Oplus} are thus cryptographically secure when +they are XORed with an already cryptographically +secure PRNG. But, as stated previously, +such a property does not mean that, whatever the +key size, no attacker can predict the next bit +knowing all the previously released ones. +However, given a key size, it is possible to +measure in practice the minimum duration needed +for an attacker to break a cryptographically +secure PRNG, if we know the power of his/her +machines. Such a concrete security evaluation +is related to the $(T,\varepsilon)-$security +notion, which is recalled and evaluated in what +follows, for the sake of completeness. + +Let us firstly recall that, +\begin{definition} +Let $\mathcal{D} : \mathds{B}^M \longrightarrow \mathds{B}$ be a probabilistic algorithm that runs +in time $T$. +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} + +\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. +\end{definition} + +Let us recall that the running time of a probabilistic algorithm is defined to be the +maximum of the expected number of steps needed to produce an output, maximized +over all inputs; the expected number is averaged over all coin flips made by the algorithm~\cite{Knuth97}. +We are now able to define the notion of cryptographically secure PRNGs: + +\begin{definition} +A pseudorandom generator is $(T,\varepsilon)-$secure if there exists no $(T,\varepsilon)-$distinguishing attack on this pseudorandom generator. +\end{definition} + + + + + + + +Suppose now that the PRNG of Eq.~\eqref{equation Oplus} will work during +$M=100$ time units, and that during this period, +an attacker can realize $10^{12}$ clock cycles. +We thus wonder whether, during the PRNG's +lifetime, the attacker can distinguish this +sequence from truly random one, with a probability +greater than $\varepsilon = 0.2$. +We consider that $N$ has 900 bits. + +Predicting the next generated bit knowing all the +previously released ones by Eq.~\eqref{equation Oplus} is obviously equivalent to predict the +next bit in the BBS generator, which +is cryptographically secure. More precisely, it +is $(T,\varepsilon)-$secure: no +$(T,\varepsilon)-$distinguishing attack can be +successfully realized on this PRNG, if~\cite{Fischlin} +\begin{equation} +T \leqslant \dfrac{L(N)}{6 N (log_2(N))\varepsilon^{-2}M^2}-2^7 N \varepsilon^{-2} M^2 log_2 (8 N \varepsilon^{-1}M) +\label{mesureConcrete} +\end{equation} +where $M$ is the length of the output ($M=100$ in +our example), and $L(N)$ is equal to +$$ +2.8\times 10^{-3} exp \left(1.9229 \times (N ~ln(2)^\frac{1}{3}) \times ln(N~ln 2)^\frac{2}{3}\right) +$$ +is the number of clock cycles to factor a $N-$bit +integer. + + + + +A direct numerical application shows that this attacker +cannot achieve its $(10^{12},0.2)$ distinguishing +attack in that context. + +\end{color} + + \section{Cryptographical Applications} \subsection{A Cryptographically Secure PRNG for GPU} @@ -1736,46 +1978,41 @@ It should be noticed that this generator has once more the form $x^{n+1} = x^n where $S^n$ is referred in this algorithm as $t$: each iteration of this PRNG ends with $x = x \wedge t$. This $S^n$ is only constituted by secure bits produced by the BBS generator, and thus, due to -Proposition~\ref{cryptopreuve}, the resulted PRNG is cryptographically -secure. - - +Proposition~\ref{cryptopreuve}, the resulted PRNG is +cryptographically 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, -an attacker can realize $10^{12}$ clock cycles. -We thus wonder whether, during the PRNG's -lifetime, the attacker can distinguish this -sequence from truly random one, with a probability -greater than $\varepsilon = 0.2$. -We consider that $N$ has 900 bits. - -The random process is the BBS generator, which -is cryptographically secure. More precisely, it -is $(T,\varepsilon)-$secure: no -$(T,\varepsilon)-$distinguishing attack can be -successfully realized on this PRNG, if~\cite{Fischlin} -$$ -T \leqslant \dfrac{L(N)}{6 N (log_2(N))\varepsilon^{-2}M^2}-2^7 N \varepsilon^{-2} M^2 log_2 (8 N \varepsilon^{-1}M) -$$ -where $M$ is the length of the output ($M=100$ in -our example), and $L(N)$ is equal to -$$ -2.8\times 10^{-3} exp \left(1.9229 \times (N ~ln(2)^\frac{1}{3}) \times ln(N~ln 2)^\frac{2}{3}\right) -$$ -is the number of clock cycles to factor a $N-$bit -integer. - -A direct numerical application shows that this attacker -cannot achieve its $(10^{12},0.2)$ distinguishing -attack in that context. - +As stated before, even if the proposed PRNG is cryptocaphically +secure, it does not mean that such a generator +can be used as described here when attacks are +awaited. The problem is to determine the minimum +time required for an attacker, with a given +computational power, to predict under a probability +lower than 0.5 the $n+1$th bit, knowing the $n$ +previous ones. The proposed GPU generator will be +useful in a security context, at least in some +situations where a secret protected by a pseudorandom +keystream is rapidly obsolete, if this time to +predict the next bit is large enough when compared +to both the generation and transmission times. +It is true that the prime numbers used in the last +section are very small compared to up-to-date +security recommends. However the attacker has not +access to each BBS, but to the output produced +by Algorithm~\ref{algo:bbs_gpu}, which is quite +more complicated than a simple BBS. Indeed, to +determine if this cryptographically secure PRNG +on GPU can be useful in security context with the +proposed parameters, or if it is only a very fast +and statistically perfect generator on GPU, its +$(T,\varepsilon)-$security must be determined, and +a formulation similar to Eq.\eqref{mesureConcrete} +must be established. Authors +hope to achieve to realize this difficult task in a future +work. \end{color} + \subsection{Toward a Cryptographically Secure and Chaotic Asymmetric Cryptosystem} \label{Blum-Goldwasser} We finish this research work by giving some thoughts about the use of