\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}
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. \begin{color}{red} Or, in an equivalent formulation, he or she should not be
+sequence. \begin{color}{red} However, 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
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 initial 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 initial
+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 also 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,
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.
\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:
\end{itemize}
-We have proven in our previous works~\cite{} that chaotic iterations satisfying Theorem~\ref{Th:Caractérisation des IC chaotiques} are, among other
+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.
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
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}
\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
\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 $uv$.
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}.
\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
\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}
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
