From 8cbe6d4faae325510cbbb002936afe1c4e19202b Mon Sep 17 00:00:00 2001 From: guyeux Date: Sat, 21 Jan 2012 12:46:39 +0100 Subject: [PATCH 01/16] Relecture --- prng_gpu.tex | 54 ++++++++++++++++++++++------------------------------ 1 file changed, 23 insertions(+), 31 deletions(-) diff --git a/prng_gpu.tex b/prng_gpu.tex index 55fc756..7eb93d1 100644 --- a/prng_gpu.tex +++ b/prng_gpu.tex @@ -331,17 +331,15 @@ Let us now recall how to define a suitable metric space where chaotic iterations are continuous. For further explanations, see, e.g., \cite{guyeux10}. Let $\delta $ be the \emph{discrete Boolean metric}, $\delta -(x,y)=0\Leftrightarrow x=y.$ Given a function $f$, define the function: -%%RAPH : ici j'ai coupé la dernière ligne en 2, c'est moche mais bon -\begin{equation} +(x,y)=0\Leftrightarrow x=y.$ Given a function $f$, define the function +$F_{f}: \llbracket1;\mathsf{N}\rrbracket\times \mathds{B}^{\mathsf{N}} +\longrightarrow \mathds{B}^{\mathsf{N}}$ +\begin{equation*} \begin{array}{lrll} -F_{f}: & \llbracket1;\mathsf{N}\rrbracket\times \mathds{B}^{\mathsf{N}} & -\longrightarrow & \mathds{B}^{\mathsf{N}} \\ -& (k,E) & \longmapsto & \left( E_{j}.\delta (k,j)+ \right.\\ -& & & \left. f(E)_{k}.\overline{\delta -(k,j)}\right) _{j\in \llbracket1;\mathsf{N}\rrbracket},% +& (k,E) & \longmapsto & \left( E_{j}.\delta (k,j)+ f(E)_{k}.\overline{\delta +(k,j)}\right) _{j\in \llbracket1;\mathsf{N}\rrbracket}% \end{array}% -\end{equation}% +\end{equation*}% \noindent where + and . are the Boolean addition and product operations. Consider the phase space: \begin{equation} @@ -594,11 +592,11 @@ faster, does not deflate their topological chaos properties. \subsection{Proofs of Chaos of the General Formulation of the Chaotic Iterations} \label{deuxième def} Let us consider the discrete dynamical systems in chaotic iterations having -the general form: +the general form: $\forall n\in \mathds{N}^{\ast }$, $ \forall i\in +\llbracket1;\mathsf{N}\rrbracket $, \begin{equation} -\forall n\in \mathds{N}^{\ast }, \forall i\in -\llbracket1;\mathsf{N}\rrbracket ,x_i^n=\left\{ + x_i^n=\left\{ \begin{array}{ll} x_i^{n-1} & \text{ if } i \notin \mathcal{S}^n \\ \left(f(x^{n-1})\right)_{S^n} & \text{ if }i \in \mathcal{S}^n. @@ -623,15 +621,13 @@ Let us introduce the following function: where $\mathcal{P}\left(X\right)$ is for the powerset of the set $X$, that is, $Y \in \mathcal{P}\left(X\right) \Longleftrightarrow Y \subset X$. Given a function $f:\mathds{B}^\mathsf{N} \longrightarrow \mathds{B}^\mathsf{N} $, define the function: -%%RAPH : j'ai coupé la dernière ligne en 2, c'est moche -\begin{equation} -\begin{array}{lrll} -F_{f}: & \mathcal{P}\left(\llbracket1;\mathsf{N}\rrbracket \right) \times \mathds{B}^{\mathsf{N}} & -\longrightarrow & \mathds{B}^{\mathsf{N}} \\ -& (P,E) & \longmapsto & \left( E_{j}.\chi (j,P)+\right.\\ -& & &\left.f(E)_{j}.\overline{\chi(j,P)}\right) _{j\in \llbracket1;\mathsf{N}\rrbracket},% +$F_{f}: \mathcal{P}\left(\llbracket1;\mathsf{N}\rrbracket \right) \times \mathds{B}^{\mathsf{N}} +\longrightarrow \mathds{B}^{\mathsf{N}}$ +\begin{equation*} +\begin{array}{rll} + (P,E) & \longmapsto & \left( E_{j}.\chi (j,P)+f(E)_{j}.\overline{\chi(j,P)}\right) _{j\in \llbracket1;\mathsf{N}\rrbracket}% \end{array}% -\end{equation}% +\end{equation*}% where + and . are the Boolean addition and product operations, and $\overline{x}$ is the negation of the Boolean $x$. Consider the phase space: @@ -761,16 +757,16 @@ thus after $n_{2}$, the $k+2$ first terms of $S^n$ and $S$ are equal. \noindent As a consequence, the $k+1$ first entries of the strategies of $% G_{f}(S^n,E^n)$ and $G_{f}(S,E)$ are the same ($G_{f}$ is a shift of strategies) and due to the definition of $d_{s}$, the floating part of the distance between $(S^n,E^n)$ and $(S,E)$ is strictly less than $% -10^{-(k+1)}\leqslant \varepsilon $.\bigskip \newline +10^{-(k+1)}\leqslant \varepsilon $. + In conclusion, %%RAPH : ici j'ai rajouté une ligne -\begin{flushleft}$$ -\forall \varepsilon >0,\exists N_{0}=max(n_{0},n_{1},n_{2})\in \mathds{N}% -,\forall n\geqslant N_{0},$$ -$$ d\left( G_{f}(S^n,E^n);G_{f}(S,E)\right) +$ +\forall \varepsilon >0,$ $\exists N_{0}=max(n_{0},n_{1},n_{2})\in \mathds{N} +,$ $\forall n\geqslant N_{0},$ +$ d\left( G_{f}(S^n,E^n);G_{f}(S,E)\right) \leqslant \varepsilon . -$$ -\end{flushleft} +$ $G_{f}$ is consequently continuous. \end{proof} @@ -810,11 +806,7 @@ where $(s^0,s^1, \hdots)$ is the strategy of $Y$, satisfies the properties claimed in the lemma. \end{proof} -<<<<<<< HEAD We can now prove the Theorem~\ref{t:chaos des general}. -======= -We can now prove Theorem~\ref{t:chaos des general}... ->>>>>>> e55d237aba022a66cc2d7650d295b29169878f45 \begin{proof}[Theorem~\ref{t:chaos des general}] Firstly, strong transitivity implies transitivity. -- 2.39.5 From 7c9a1a3c4f4b214a0b8075ed65fa73f25512eddb Mon Sep 17 00:00:00 2001 From: guyeux Date: Sun, 10 Jun 2012 14:18:39 +0200 Subject: [PATCH 02/16] =?utf8?q?D=C3=A9but=20de=20la=20r=C3=A9ponse=20et?= =?utf8?q?=20des=20corrections.?= MIME-Version: 1.0 Content-Type: text/plain; charset=utf8 Content-Transfer-Encoding: 8bit --- prng_gpu.tex | 6 +++--- reponse.tex | 48 ++++++++++++++++++++++++++++++++++++++++++++++++ 2 files changed, 51 insertions(+), 3 deletions(-) create mode 100644 reponse.tex diff --git a/prng_gpu.tex b/prng_gpu.tex index 7eb93d1..81f5209 100644 --- a/prng_gpu.tex +++ b/prng_gpu.tex @@ -161,7 +161,7 @@ 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. Such a proof leads to the proposition of a cryptographically secure and -chaotic generator on GPU based on the famous Blum Blum Shum +chaotic generator on GPU based on the famous Blum Blum Shub 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 @@ -1270,7 +1270,7 @@ It is possible to build a cryptographically secure PRNG based on the previous algorithm (Algorithm~\ref{algo:gpu_kernel2}). Due to Proposition~\ref{cryptopreuve}, it simply consists in replacing the {\it xor-like} PRNG by a cryptographically secure one. -We have chosen the Blum Blum Shum generator~\cite{BBS} (usually denoted by BBS) having the form: +We have chosen the Blum Blum Shub generator~\cite{BBS} (usually denoted by BBS) having the form: $$x_{n+1}=x_n^2~ mod~ M$$ where $M$ is the product of two prime numbers (these prime numbers need to be congruent to 3 modulus 4). BBS is known to be very slow and only usable for cryptographic applications. @@ -1474,7 +1474,7 @@ the possibility to develop fast and secure PRNGs using the GPU architecture. Thoughts about an improvement of the Blum-Goldwasser cryptosystem, using the proposed method, has been finally proposed. -In future work we plan to extend these researches, building a parallel PRNG for clusters or +In future work we plan to extend this research, building a parallel PRNG for clusters or grid computing. Topological properties of the various proposed generators will be investigated, and the use of other categories of PRNGs as input will be studied too. The improvement of Blum-Goldwasser will be deepened. Finally, we diff --git a/reponse.tex b/reponse.tex new file mode 100644 index 0000000..3ec5213 --- /dev/null +++ b/reponse.tex @@ -0,0 +1,48 @@ +\documentclass{article} + + +\begin{document} +\section{Editor} + +As the reviewers point out, the paper is well written, is interesting, but there are some major concerns about both the practical aspects of the paper, as well as more theoretical aspects. While the paper has only been reviewed by two reviewers, their concerns are enough to recommend that the author consider them carefully and then resubmit this paper as a new paper. + +Most of the issues raised are related to cryptography, and not to the acceleration work on a GPU. The issue may be that during their preparation of this paper the authors were too focused on the acceleration work, and did not spend enough time being precise about the cryptography discussion. The two reviewers are experts on cryptography, as well as acceleration techniques, and the review indicate that the analysis needs to be strengthened. + + + +\section{Reviewer: 1} + + +Comments: +The authors should include a summary of test measurements showing their method passes the test sets mentioned (NIST, Diehard, TestU01) instead of the one sentence saying it passed that is in section 1. + + +Section 9: +The authors say they replace the xor-like PRNG with a cryptographically secure one, BBS, but then proceed to use extremely small values, as far as a cryptographer is concerned (modulus of $2^16$), in the computation due to the need to use 32 bit integers in the GPU and combine bits from multiple BBS generated values, but they never prove (or even discuss) how this can be considered cryptographically secure due to the small individual values. At the end of 9.1, the authors say $S^n$ is secure because it is formed from bits from the BBS generator, but do not consider if the use of such small values will lead to exhaust searches to determine individual bits. The authors either need to remove all of section 9 and or prove the resulting PRNG is cryptographically secure. + +\textit{In the conclusion: +Reword last sentence of 1st paragraph +In the 2nd paragraph, change "these researches" to "this research" in "we plan to extend ..."} + +Done. + + +\section{Reviewer: 2} + + +Comments: +The paper is, overall, well written and clear, with appropriate references to the relevant concepts and prior work. The motivation of the work, however, is not quite clear: the authors present (provable) chaotic properties of a PRNG as a security improvement, but provide no convincing argument beyond opinion (or hope). There seems to have been no effort in showing how the new PRNG improves on a single (say) xorshift generator, considering the slowdown of calling 3 of them per iteration (cf. Listing 1). This could be done, if not with the mathematical rigor of chaos theory, then with simpler bit diffusion metrics, often used in cryptography to evaluate building blocks of ciphers. + +The generator of Listing 1, despite being proved chaotic, has several problems. First, it doesn't seem to be new; using xor to mix the states of several independent generators is standard procedure (e.g., [1]). Secondly, the periods of the 3 xorshift generators are not coprime --- this reduces the useful period of combining the sequences. Thirdly, by combining 3 linear generators with xor, another linear operation, you still get a linear generator, potentially vulnerable to stringent high-dimensional spectral tests. + +The BBS-based generator of section 9 is anything but cryptographically secure. A 16-bit modulus (trivially factorable) gives out a period of at most $2^16$, which is neither useful nor secure. Its speed is irrelevant, as this generator as no practical applications whatsoever (a larger modulus, at least 1024-bit long, might be useful in some situations, but it will be a terrible GPU performer, of course). + +To sum it up, while the theoretical part of the paper is interesting, the practical results leave much to be desired, and do not back the thesis that chaos improves some quality metric of the generators. On the theoretical side, you may be interested in Vladimir Anashin's work on ergodic theory on p-adic (specifically, 2-adic) numbers to prove uniform distribution and maximal period of generators. The $d_s(S, \check{S})$ distance loosely resembles the p-adic norm. + +Typos and other nitpicks: + - Blub Blum Shub is misspelled in a few places as "Blum Blum Shum"; + - Page 12, right column, line 54: In "t<<=4", the << operation is using the « character instead. + + [1] Howes, L., and Thomas, D. "Efficient random number generation and application using CUDA." In GPU Gems 3, H. Nguyen, Ed. NVIDIA, 2007, Ch. 37. + +\end{document} -- 2.39.5 From 4ad2ccae91afa1f83fad9be3c87213a9b8d81734 Mon Sep 17 00:00:00 2001 From: guyeux Date: Mon, 11 Jun 2012 09:12:01 +0200 Subject: [PATCH 03/16] =?utf8?q?Avanc=C3=A9es=20dansla=20r=C3=A9ponse?= MIME-Version: 1.0 Content-Type: text/plain; charset=utf8 Content-Transfer-Encoding: 8bit --- prng_gpu.tex | 4 +-- reponse.tex | 78 +++++++++++++++++++++++++++++++++++++++++----------- 2 files changed, 64 insertions(+), 18 deletions(-) diff --git a/prng_gpu.tex b/prng_gpu.tex index 81f5209..a57e2a0 100644 --- a/prng_gpu.tex +++ b/prng_gpu.tex @@ -1471,8 +1471,8 @@ namely the BigCrush. Furthermore, we have shown that when the inputted generator is cryptographically secure, then it is the case too for the PRNG we propose, thus leading to the possibility to develop fast and secure PRNGs using the GPU architecture. -Thoughts about an improvement of the Blum-Goldwasser cryptosystem, using the -proposed method, has been finally proposed. +An improvement of the Blum-Goldwasser cryptosystem, making it +behaves chaotically, has finally been proposed. In future work we plan to extend this research, building a parallel PRNG for clusters or grid computing. Topological properties of the various proposed generators will be investigated, diff --git a/reponse.tex b/reponse.tex index 3ec5213..3b3e986 100644 --- a/reponse.tex +++ b/reponse.tex @@ -1,25 +1,31 @@ \documentclass{article} - +\usepackage{color} \begin{document} \section{Editor} -As the reviewers point out, the paper is well written, is interesting, but there are some major concerns about both the practical aspects of the paper, as well as more theoretical aspects. While the paper has only been reviewed by two reviewers, their concerns are enough to recommend that the author consider them carefully and then resubmit this paper as a new paper. +\bigskip +\textit{As the reviewers point out, the paper is well written, is interesting, but there are some major concerns about both the practical aspects of the paper, as well as more theoretical aspects. While the paper has only been reviewed by two reviewers, their concerns are enough to recommend that the author consider them carefully and then resubmit this paper as a new paper.} -Most of the issues raised are related to cryptography, and not to the acceleration work on a GPU. The issue may be that during their preparation of this paper the authors were too focused on the acceleration work, and did not spend enough time being precise about the cryptography discussion. The two reviewers are experts on cryptography, as well as acceleration techniques, and the review indicate that the analysis needs to be strengthened. +\bigskip +\textit{Most of the issues raised are related to cryptography, and not to the acceleration work on a GPU. The issue may be that during their preparation of this paper the authors were too focused on the acceleration work, and did not spend enough time being precise about the cryptography discussion. The two reviewers are experts on cryptography, as well as acceleration techniques, and the review indicate that the analysis needs to be strengthened.} \section{Reviewer: 1} -Comments: -The authors should include a summary of test measurements showing their method passes the test sets mentioned (NIST, Diehard, TestU01) instead of the one sentence saying it passed that is in section 1. +\bigskip +\textit{The authors should include a summary of test measurements showing their method passes the test sets mentioned (NIST, Diehard, TestU01) instead of the one sentence saying it passed that is in section 1.} + +\begin{color}{red} Raph, c'est pour toi ça.\end{color} -Section 9: -The authors say they replace the xor-like PRNG with a cryptographically secure one, BBS, but then proceed to use extremely small values, as far as a cryptographer is concerned (modulus of $2^16$), in the computation due to the need to use 32 bit integers in the GPU and combine bits from multiple BBS generated values, but they never prove (or even discuss) how this can be considered cryptographically secure due to the small individual values. At the end of 9.1, the authors say $S^n$ is secure because it is formed from bits from the BBS generator, but do not consider if the use of such small values will lead to exhaust searches to determine individual bits. The authors either need to remove all of section 9 and or prove the resulting PRNG is cryptographically secure. +\bigskip +\textit{Section 9: +The authors say they replace the xor-like PRNG with a cryptographically secure one, BBS, but then proceed to use extremely small values, as far as a cryptographer is concerned (modulus of $2^{16}$), in the computation due to the need to use 32 bit integers in the GPU and combine bits from multiple BBS generated values, but they never prove (or even discuss) how this can be considered cryptographically secure due to the small individual values. At the end of 9.1, the authors say $S^n$ is secure because it is formed from bits from the BBS generator, but do not consider if the use of such small values will lead to exhaust searches to determine individual bits. The authors either need to remove all of section 9 and or prove the resulting PRNG is cryptographically secure.} +\bigskip \textit{In the conclusion: Reword last sentence of 1st paragraph In the 2nd paragraph, change "these researches" to "this research" in "we plan to extend ..."} @@ -30,19 +36,59 @@ Done. \section{Reviewer: 2} -Comments: -The paper is, overall, well written and clear, with appropriate references to the relevant concepts and prior work. The motivation of the work, however, is not quite clear: the authors present (provable) chaotic properties of a PRNG as a security improvement, but provide no convincing argument beyond opinion (or hope). There seems to have been no effort in showing how the new PRNG improves on a single (say) xorshift generator, considering the slowdown of calling 3 of them per iteration (cf. Listing 1). This could be done, if not with the mathematical rigor of chaos theory, then with simpler bit diffusion metrics, often used in cryptography to evaluate building blocks of ciphers. +\bigskip +\textit{The paper is, overall, well written and clear, with appropriate references to the relevant concepts and prior work. The motivation of the work, however, is not quite clear: the authors present (provable) chaotic properties of a PRNG as a security improvement, but provide no convincing argument beyond opinion (or hope).} + + +\bigskip +\textit{There seems to have been no effort in showing how the new PRNG improves on a single (say) xorshift generator, considering the slowdown of calling 3 of them per iteration (cf. Listing 1). This could be done, if not with the mathematical rigor of chaos theory, then with simpler bit diffusion metrics, often used in cryptography to evaluate building blocks of ciphers.} + +\bigskip +\textit{The generator of Listing 1, despite being proved chaotic, has several problems. First, it doesn't seem to be new; using xor to mix the states of several independent generators is standard procedure (e.g., [1]).} + +\bigskip +\textit{Secondly, the periods of the 3 xorshift generators are not coprime --- this reduces the useful period of combining the sequences.} + +\bigskip +\textit{Thirdly, by combining 3 linear generators with xor, another linear operation, you still get a linear generator, potentially vulnerable to stringent high-dimensional spectral tests.} + +\bigskip +\textit{The BBS-based generator of section 9 is anything but cryptographically secure.} + +This claim is surprising, as this result is mathematically proven in the article: +either there is something wrong in the proof, or the generator is cryptographically +secure. Indeed, there is probably a misunderstanding of this notion, which does +not deal with the practical aspects of security. For instance, BBS is +cryptographically secure, but whatever the size of the keys, a brute force attack always +achieve to break it. It is only a question of time: with sufficiently large primes, +the time required to break it is astronomically large, making this attack completely +impracticable in practice. To sum up, being cryptographically secure is not a +question of key size, + + + +\bigskip +\textit{A 16-bit modulus (trivially factorable) gives out a period of at most $2^{16}$, which is neither useful nor secure. Its speed is irrelevant, as this generator as no practical applications whatsoever (a larger modulus, at least 1024-bit long, might be useful in some situations, but it will be a terrible GPU performer, of course).} + +\bigskip +\textit{To sum it up, while the theoretical part of the paper is interesting, the practical results leave much to be desired, and do not back the thesis that chaos improves some quality metric of the generators.} -The generator of Listing 1, despite being proved chaotic, has several problems. First, it doesn't seem to be new; using xor to mix the states of several independent generators is standard procedure (e.g., [1]). Secondly, the periods of the 3 xorshift generators are not coprime --- this reduces the useful period of combining the sequences. Thirdly, by combining 3 linear generators with xor, another linear operation, you still get a linear generator, potentially vulnerable to stringent high-dimensional spectral tests. -The BBS-based generator of section 9 is anything but cryptographically secure. A 16-bit modulus (trivially factorable) gives out a period of at most $2^16$, which is neither useful nor secure. Its speed is irrelevant, as this generator as no practical applications whatsoever (a larger modulus, at least 1024-bit long, might be useful in some situations, but it will be a terrible GPU performer, of course). +\bigskip +\textit{On the theoretical side, you may be interested in Vladimir Anashin's work on ergodic theory on p-adic (specifically, 2-adic) numbers to prove uniform distribution and maximal period of generators. The $d_s(S, \check{S})$ distance loosely resembles the p-adic norm.} -To sum it up, while the theoretical part of the paper is interesting, the practical results leave much to be desired, and do not back the thesis that chaos improves some quality metric of the generators. On the theoretical side, you may be interested in Vladimir Anashin's work on ergodic theory on p-adic (specifically, 2-adic) numbers to prove uniform distribution and maximal period of generators. The $d_s(S, \check{S})$ distance loosely resembles the p-adic norm. +We have already established the uniform distribution in \cite{FCt}. -Typos and other nitpicks: - - Blub Blum Shub is misspelled in a few places as "Blum Blum Shum"; - - Page 12, right column, line 54: In "t<<=4", the << operation is using the « character instead. +\bigskip +\textit{Typos and other nitpicks:\\ + - Blub Blum Shub is misspelled in a few places as "Blum Blum Shum";} + +These misspells have been corrected (sorry for that). + +\bigskip +\textit{ - Page 12, right column, line 54: In "$t<<=4$", the $<<$ operation is using the `` character instead.} - [1] Howes, L., and Thomas, D. "Efficient random number generation and application using CUDA." In GPU Gems 3, H. Nguyen, Ed. NVIDIA, 2007, Ch. 37. +\bigskip +\textit{ [1] Howes, L., and Thomas, D. "Efficient random number generation and application using CUDA." In GPU Gems 3, H. Nguyen, Ed. NVIDIA, 2007, Ch. 37. } \end{document} -- 2.39.5 From 860ecbe3a673a4ac258e24e6c0284a56e3427b6e Mon Sep 17 00:00:00 2001 From: guyeux Date: Mon, 11 Jun 2012 10:40:49 +0200 Subject: [PATCH 04/16] =?utf8?q?Ajout=20d'une=20partie=20=C3=A9valuation,?= =?utf8?q?=20=C3=A0=20revoir.?= MIME-Version: 1.0 Content-Type: text/plain; charset=utf8 Content-Transfer-Encoding: 8bit --- mabase.bib | 17 +++++++++++++++++ prng_gpu.tex | 38 ++++++++++++++++++++++++++++++++++++-- 2 files changed, 53 insertions(+), 2 deletions(-) diff --git a/mabase.bib b/mabase.bib index 21fb105..15c282d 100644 --- a/mabase.bib +++ b/mabase.bib @@ -14,6 +14,23 @@ timestamp = {2009.06.29} } +@INPROCEEDINGS{Fischlin, + author = {Fischlin, R. and Schnorr, C. P.}, + title = {Stronger security proofs for RSA and rabin bits}, + booktitle = {Proceedings of the 16th annual international conference on Theory + and application of cryptographic techniques}, + year = {1997}, + series = {EUROCRYPT'97}, + pages = {267--279}, + address = {Berlin, Heidelberg}, + publisher = {Springer-Verlag}, + acmid = {1754569}, + isbn = {3-540-62975-0}, + location = {Konstanz, Germany}, + numpages = {13}, + url = {http://dl.acm.org/citation.cfm?id=1754542.1754569} +} + @INPROCEEDINGS{BattiatoCGG99, author = {Sebastiano Battiato and Dario Catalano and Giovanni Gallo and Rosario Gennaro}, diff --git a/prng_gpu.tex b/prng_gpu.tex index a57e2a0..34ec700 100644 --- a/prng_gpu.tex +++ b/prng_gpu.tex @@ -1389,6 +1389,40 @@ secure. +\begin{color}{red} +\subsection{Practical Security 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. + +\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 @@ -1471,8 +1505,8 @@ namely the BigCrush. Furthermore, we have shown that when the inputted generator is cryptographically secure, then it is the case too for the PRNG we propose, thus leading to the possibility to develop fast and secure PRNGs using the GPU architecture. -An improvement of the Blum-Goldwasser cryptosystem, making it -behaves chaotically, has finally been proposed. +\begin{color}{red} An improvement of the Blum-Goldwasser cryptosystem, making it +behaves chaotically, has finally been proposed. \end{color} In future work we plan to extend this research, building a parallel PRNG for clusters or grid computing. Topological properties of the various proposed generators will be investigated, -- 2.39.5 From 9e057cd5768916849c2767ef4bd0f54dd9adc3b4 Mon Sep 17 00:00:00 2001 From: guyeux Date: Wed, 13 Jun 2012 14:41:45 +0200 Subject: [PATCH 05/16] ajout du chaos --- prng_gpu.tex | 430 +++++++++++++++++++++++++++++++++++++++++++++++++++ reponse.tex | 2 + 2 files changed, 432 insertions(+) diff --git a/prng_gpu.tex b/prng_gpu.tex index 34ec700..90f00f8 100644 --- a/prng_gpu.tex +++ b/prng_gpu.tex @@ -11,6 +11,8 @@ \usepackage[ruled,vlined]{algorithm2e} \usepackage{listings} \usepackage[standard]{ntheorem} +\usepackage{algorithmic} +\usepackage{slashbox} % Pour mathds : les ensembles IR, IN, etc. \usepackage{dsfont} @@ -835,6 +837,434 @@ have $d((S,E),(\tilde S,E))<\epsilon$. \end{proof} +\begin{color}{red} +\section{Improving Statistical Properties Using Chaotic Iterations} + + +\subsection{The CIPRNG family} + +Three categories of PRNGs have been derived from chaotic iterations. They are +recalled in what follows. + +\subsubsection{Old CIPRNG} + +Let $\mathsf{N} = 4$. Some chaotic iterations are fulfilled to generate a sequence $\left(x^n\right)_{n\in\mathds{N}} \in \left(\mathds{B}^4\right)^\mathds{N}$ of Boolean vectors: the successive states of the iterated system. Some of these vectors are randomly extracted and their components constitute our pseudorandom bit flow~\cite{bgw09:ip}. +Chaotic iterations are realized as follows. Initial state $x^0 \in \mathds{B}^4$ is a Boolean vector taken as a seed and chaotic strategy $\left(S^n\right)_{n\in\mathds{N}}\in \llbracket 1, 4 \rrbracket^\mathds{N}$ is constructed with $PRNG_2$. Lastly, iterate function $f$ is the vectorial Boolean negation. +At each iteration, only the $S^n$-th component of state $x^n$ is updated. Finally, some $x^n$ are selected by a sequence $m^n$, provided by a second generator $PRNG_1$, as the pseudorandom bit sequence of our generator. + +The basic design procedure of the Old CI generator is summed up in Algorithm~\ref{Chaotic iteration}. +The internal state is $x$, the output array is $r$. $a$ and $b$ are those computed by $PRNG_1$ and $PRNG_2$. + + +\begin{algorithm} +\textbf{Input:} the internal state $x$ (an array of 4-bit words)\\ +\textbf{Output:} an array $r$ of 4-bit words +\begin{algorithmic}[1] + +\STATE$a\leftarrow{PRNG_1()}$; +\STATE$m\leftarrow{a~mod~2+13}$; +\WHILE{$i=0,\dots,m$} +\STATE$b\leftarrow{PRNG_2()}$; +\STATE$S\leftarrow{b~mod~4}$; +\STATE$x_S\leftarrow{ \overline{x_S}}$; +\ENDWHILE +\STATE$r\leftarrow{x}$; +\STATE return $r$; +\medskip +\caption{An arbitrary round of the old CI generator} +\label{Chaotic iteration} +\end{algorithmic} +\end{algorithm} + +\subsubsection{New CIPRNG} + +The New CI generator is designed by the following process~\cite{bg10:ip}. First of all, some chaotic iterations have to be done to generate a sequence $\left(x^n\right)_{n\in\mathds{N}} \in \left(\mathds{B}^{32}\right)^\mathds{N}$ of Boolean vectors, which are the successive states of the iterated system. Some of these vectors will be randomly extracted and our pseudo-random bit flow will be constituted by their components. Such chaotic iterations are realized as follows. Initial state $x^0 \in \mathds{B}^{32}$ is a Boolean vector taken as a seed and chaotic strategy $\left(S^n\right)_{n\in\mathds{N}}\in \llbracket 1, 32 \rrbracket^\mathds{N}$ is +an \emph{irregular decimation} of $PRNG_2$ sequence, as described in Algorithm~\ref{Chaotic iteration1}. + +Another time, at each iteration, only the $S^n$-th component of state $x^n$ is updated, as follows: $x_i^n = x_i^{n-1}$ if $i \neq S^n$, else $x_i^n = \overline{x_i^{n-1}}$. +Finally, some $x^n$ are selected +by a sequence $m^n$ as the pseudo-random bit sequence of our generator. +$(m^n)_{n \in \mathds{N}} \in \mathcal{M}^\mathds{N}$ is computed from $PRNG_1$, where $\mathcal{M}\subset \mathds{N}^*$ is a finite nonempty set of integers. + +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 $g_1(a)$ is an integer defined as in Eq.~\ref{Formula}. + +\begin{equation} +\label{Formula} +m^n = g_1(y^n)= +\left\{ +\begin{array}{l} +0 \text{ if }0 \leqslant{y^n}<{C^0_{32}},\\ +1 \text{ if }{C^0_{32}} \leqslant{y^n}<\sum_{i=0}^1{C^i_{32}},\\ +2 \text{ if }\sum_{i=0}^1{C^i_{32}} \leqslant{y^n}<\sum_{i=0}^2{C^i_{32}},\\ +\vdots~~~~~ ~~\vdots~~~ ~~~~\\ +N \text{ if }\sum_{i=0}^{N-1}{C^i_{32}}\leqslant{y^n}<1.\\ +\end{array} +\right. +\end{equation} + +\begin{algorithm} +\textbf{Input:} the internal state $x$ (32 bits)\\ +\textbf{Output:} a state $r$ of 32 bits +\begin{algorithmic}[1] +\FOR{$i=0,\dots,N$} +{ +\STATE$d_i\leftarrow{0}$\; +} +\ENDFOR +\STATE$a\leftarrow{PRNG_1()}$\; +\STATE$m\leftarrow{f(a)}$\; +\STATE$k\leftarrow{m}$\; +\WHILE{$i=0,\dots,k$} + +\STATE$b\leftarrow{PRNG_2()~mod~\mathsf{N}}$\; +\STATE$S\leftarrow{b}$\; + \IF{$d_S=0$} + { +\STATE $x_S\leftarrow{ \overline{x_S}}$\; +\STATE $d_S\leftarrow{1}$\; + + } + \ELSIF{$d_S=1$} + { +\STATE $k\leftarrow{ k+1}$\; + }\ENDIF +\ENDWHILE\\ +\STATE $r\leftarrow{x}$\; +\STATE return $r$\; +\medskip +\caption{An arbitrary round of the new CI generator} +\label{Chaotic iteration1} +\end{algorithmic} +\end{algorithm} + + +\subsubsection{Xor CIPRNG} + +Instead of updating only one cell at each iteration as Old CI and New CI, we can try to choose a +subset of components and to update them together. Such an attempt leads +to a kind of merger of the two random sequences. When the updating function is the vectorial negation, +this algorithm can be rewritten as follows~\cite{arxivRCCGPCH}: + +\begin{equation} +\left\{ +\begin{array}{l} +x^0 \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket, S \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket^\mathds{N} \\ +\forall n \in \mathds{N}^*, x^n = x^{n-1} \oplus S^n, +\end{array} +\right. +\label{equation Oplus} +\end{equation} +%This rewriting can be understood as follows. The $n-$th term $S^n$ of the +%sequence $S$, which is an integer of $\mathsf{N}$ binary digits, presents +%the list of cells to update in the state $x^n$ of the system (represented +%as an integer having $\mathsf{N}$ bits too). More precisely, the $k-$th +%component of this state (a binary digit) changes if and only if the $k-$th +%digit in the binary decomposition of $S^n$ is 1. + +The single basic component presented in Eq.~\ref{equation Oplus} is of +ordinary use as a good elementary brick in various PRNGs. It corresponds +to the discrete dynamical system in chaotic iterations. + +\subsection{About some Well-known PRNGs} +\label{The generation of pseudo-random sequence} + + + + +Let us now give illustration on the fact that chaos appears to improve statistical properties. + +\subsection{Details of some Existing Generators} + +Here are the modules of PRNGs we have chosen to experiment. + +\subsubsection{LCG} +This PRNG implements either the simple or the combined linear congruency generator (LCGs). The simple LCG is defined by the recurrence: +\begin{equation} +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{testU01}. In what follows, 2LCGs and 3LCGs refer as two (resp. three) combinations of such LCGs. +For further details, see~\cite{combined_lcg}. + +\subsubsection{MRG} +This module implements multiple recursive generators (MRGs), based on a linear recurrence of order $k$, modulo $m$~\cite{testU01}: +\begin{equation} +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 be used in this paper. + +\subsubsection{UCARRY} +Generators based on linear recurrences with carry are implemented in this module. This includes the add-with-carry (AWC) generator, based on the recurrence: +\begin{equation} +\label{AWC} +\begin{array}{l} +x^n = (x^{n-r} + x^{n-s} + c^{n-1})~mod~m, \\ +c^n= (x^{n-r} + x^{n-s} + c^{n-1}) / m, \end{array}\end{equation} +the SWB generator, having the recurrence: +\begin{equation} +\label{SWB} +\begin{array}{l} +x^n = (x^{n-r} - x^{n-s} - c^{n-1})~mod~m, \\ +c^n=\left\{ +\begin{array}{l} +1 ~~~~~\text{if}~ (x^{i-r} - x^{i-s} - c^{i-1})<0\\ +0 ~~~~~\text{else},\end{array} \right. \end{array}\end{equation} +and the SWC generator designed by R. Couture, which is based on the following recurrence: +\begin{equation} +\label{SWC} +\begin{array}{l} +x^n = (a^1x^{n-1} \oplus ~...~ \oplus a^rx^{n-r} \oplus c^{n-1}) ~ mod ~ 2^w, \\ +c^n = (a^1x^{n-1} \oplus ~...~ \oplus a^rx^{n-r} \oplus c^{n-1}) ~ / ~ 2^w. \end{array}\end{equation} + +\subsubsection{GFSR} +This module implements the generalized feedback shift register (GFSR) generator, that is: +\begin{equation} +x^n = x^{n-r} \oplus x^{n-k} +\label{GFSR} +\end{equation} + + +\subsubsection{INV} +Finally, this module implements the nonlinear inversive generator, as defined in~\cite{testU01}, which is: + +\begin{equation} +\label{INV} +\begin{array}{l} +x^n=\left\{ +\begin{array}{ll} +(a^1 + a^2 / z^{n-1})~mod~m & \text{if}~ z^{n-1} \neq 0 \\ +a^1 & \text{if}~ z^{n-1} = 0 .\end{array} \right. \end{array}\end{equation} + + + + + +\subsection{Statistical tests} +\label{Security analysis} + +%A theoretical proof for the randomness of a generator is impossible to give, therefore statistical inference based on observed sample sequences produced by the generator seems to be the best option. +Considering the properties of binary random sequences, various statistical tests can be designed to evaluate the assertion that the sequence is generated by a perfectly random source. We have performed some statistical tests for the CIPRNGs proposed here. These tests include NIST suite~\cite{ANDREW2008} and DieHARD battery of tests~\cite{DieHARD}. For completeness and for reference, we give in the following subsection a brief description of each of the aforementioned tests. + + + +\subsubsection{NIST statistical tests suite} + +Among the numerous standard tests for pseudo-randomness, a convincing way to show the randomness of the produced sequences is to confront them to the NIST (National Institute of Standards and Technology) statistical tests, being an up-to-date tests suite proposed by the Information Technology Laboratory (ITL). A new version of the Statistical tests suite has been released in August 11, 2010. + +The NIST tests suite SP 800-22 is a statistical package consisting of 15 tests. They were developed to test the randomness of binary sequences produced by hardware or software based cryptographic pseudorandom number generators. These tests focus on a variety of different types of non-randomness that could exist in a sequence. + +For each statistical test, a set of $P-values$ (corresponding to the set of sequences) is produced. +The interpretation of empirical results can be conducted in various ways. +In this paper, the examination of the distribution of P-values to check for uniformity ($ P-value_{T}$) is used. +The distribution of $P-values$ is examined to ensure uniformity. +If $P-value_{T} \geqslant 0.0001$, then the sequences can be considered to be uniformly distributed. + +In our experiments, 100 sequences (s = 100), each with 1,000,000-bit long, are generated and tested. If the $P-value_{T}$ of any test is smaller than 0.0001, the sequences are considered to be not good enough and the generating algorithm is not suitable for usage. + + + + + +\subsubsection{DieHARD battery of tests} +The DieHARD battery of tests has been the most sophisticated standard for over a decade. Because of the stringent requirements in the DieHARD tests suite, a generator passing this battery of +tests can be considered good as a rule of thumb. + +The DieHARD battery of tests consists of 18 different independent statistical tests. This collection + of tests is based on assessing the randomness of bits comprising 32-bit integers obtained from +a random number generator. Each test requires $2^{23}$ 32-bit integers in order to run the full set +of tests. Most of the tests in DieHARD return a $P-value$, which should be uniform on $[0,1)$ if the input file +contains truly independent random bits. These $P-values$ are obtained by +$P=F(X)$, where $F$ is the assumed distribution of the sample random variable $X$ (often normal). +But that assumed $F$ is just an asymptotic approximation, for which the fit will be worst +in the tails. Thus occasional $P-values$ near 0 or 1, such as 0.0012 or 0.9983, can occur. +An individual test is considered to be failed if the $P-value$ approaches 1 closely, for example $P>0.9999$. + + +\subsection{Results and discussion} +\label{Results and discussion} +\begin{table*} +\renewcommand{\arraystretch}{1.3} +\caption{NIST and DieHARD tests suite passing rates for PRNGs without CI} +\label{NIST and DieHARD tests suite passing rate the for PRNGs without CI} +\centering + \begin{tabular}{|l||c|c|c|c|c|c|c|c|c|c|} + \hline\hline +Types of PRNGs & \multicolumn{2}{c|}{Linear PRNGs} & \multicolumn{4}{c|}{Lagged PRNGs} & \multicolumn{1}{c|}{ICG PRNGs} & \multicolumn{3}{c|}{Mixed PRNGs}\\ \hline +\backslashbox{\textbf{$Tests$}} {\textbf{$PRNG$}} & LCG& MRG& AWC & SWB & SWC & GFSR & INV & LCG2& LCG3& MRG2 \\ \hline +NIST & 11/15 & 14/15 &\textbf{15/15} & \textbf{15/15} & 14/15 & 14/15 & 14/15 & 14/15& 14/15& 14/15 \\ \hline +DieHARD & 16/18 & 16/18 & 15/18 & 16/18 & \textbf{18/18} & 16/18 & 16/18 & 16/18& 16/18& 16/18\\ \hline +\end{tabular} +\end{table*} + +Table~\ref{NIST and DieHARD tests suite passing rate the for PRNGs without CI} shows the results on the batteries recalled above, indicating that almost all the PRNGs cannot pass all their tests. In other words, the statistical quality of these PRNGs cannot fulfill the up-to-date standards presented previously. We will show that the CIPRNG can solve this issue. + +To illustrate the effects of this CIPRNG in detail, experiments will be divided in three parts: +\begin{enumerate} + \item \textbf{Single CIPRNG}: The PRNGs involved in CI computing are of the same category. + \item \textbf{Mixed CIPRNG}: Two different types of PRNGs are mixed during the chaotic iterations process. + \item \textbf{Multiple CIPRNG}: The generator is obtained by repeating the composition of the iteration function as follows: $x^0\in \mathds{B}^{\mathsf{N}}$, and $\forall n\in \mathds{N}^{\ast },\forall i\in \llbracket1;\mathsf{N}\rrbracket,$ +\begin{equation} +\begin{array}{l} +x_i^n=\left\{ +\begin{array}{l} +x_i^{n-1}~~~~~\text{if}~S^n\neq i \\ +\forall j\in \llbracket1;\mathsf{m}\rrbracket,f^m(x^{n-1})_{S^{nm+j}}~\text{if}~S^{nm+j}=i.\end{array} \right. \end{array} +\end{equation} +$m$ is called the \emph{functional power}. +\end{enumerate} + + +We have performed statistical analysis of each of the aforementioned CIPRNGs. +The results are reproduced in Tables~\ref{NIST and DieHARD tests suite passing rate the for PRNGs without CI} and \ref{NIST and DieHARD tests suite passing rate the for single CIPRNGs}. +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. + +\subsubsection{Tests based on the Single CIPRNG} + +\begin{table*} +\renewcommand{\arraystretch}{1.3} +\caption{NIST and DieHARD tests suite passing rates for PRNGs with CI} +\label{NIST and DieHARD tests suite passing rate the for single CIPRNGs} +\centering + \begin{tabular}{|l||c|c|c|c|c|c|c|c|c|c|c|c|} + \hline +Types of PRNGs & \multicolumn{2}{c|}{Linear PRNGs} & \multicolumn{4}{c|}{Lagged PRNGs} & \multicolumn{1}{c|}{ICG PRNGs} & \multicolumn{3}{c|}{Mixed PRNGs}\\ \hline +\backslashbox{\textbf{$Tests$}} {\textbf{$Single~CIPRNG$}} & LCG & MRG & AWC & SWB & SWC & GFSR & INV& LCG2 & LCG3& MRG2 \\ \hline\hline +Old CIPRNG\\ \hline \hline +NIST & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} & \textbf{15/15} & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} *& \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} \\ \hline +DieHARD & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} & \textbf{18/18} * & \textbf{18/18} *& \textbf{18/18} * & \textbf{18/18} *& \textbf{18/18} * \\ \hline +New CIPRNG\\ \hline \hline +NIST & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} & \textbf{15/15} & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} *& \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} \\ \hline +DieHARD & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} *& \textbf{18/18} *\\ \hline +Xor CIPRNG\\ \hline\hline +NIST & 14/15*& \textbf{15/15} * & \textbf{15/15} & \textbf{15/15} & 14/15 & \textbf{15/15} * & 14/15& \textbf{15/15} * & \textbf{15/15} *& \textbf{15/15} \\ \hline +DieHARD & 16/18 & 16/18 & 17/18* & \textbf{18/18} * & \textbf{18/18} & \textbf{18/18} * & 16/18 & 16/18 & 16/18& 16/18\\ \hline +\end{tabular} +\end{table*} + +The statistical tests results of the PRNGs using the single CIPRNG method are given in Table~\ref{NIST and DieHARD tests suite passing rate the for single CIPRNGs}. +We can observe that, except for the Xor CIPRNG, all of the CIPRNGs have passed the 15 tests of the NIST battery and the 18 tests of the DieHARD one. +Moreover, considering these scores, we can deduce that both the single Old CIPRNG and the single New CIPRNG are relatively steadier than the single Xor CIPRNG approach, when applying them to different PRNGs. +However, the Xor CIPRNG is obviously the fastest approach to generate a CI random sequence, and it still improves the statistical properties relative to each generator taken alone, although the test values are not as good as desired. + +Therefore, all of these three ways are interesting, for different reasons, in the production of pseudorandom numbers and, +on the whole, the single CIPRNG method can be considered to adapt to or improve all kinds of PRNGs. + +To have a realization of the Xor CIPRNG that can pass all the tests embedded into the NIST battery, the Xor CIPRNG with multiple functional powers are investigated in Section~\ref{Tests based on Multiple CIPRNG}. + + +\subsubsection{Tests based on the Mixed CIPRNG} + +To compare the previous approach with the CIPRNG design that uses a Mixed CIPRNG, we have taken into account the same inputted generators than in the previous section. +These inputted couples $(PRNG_1,PRNG_2)$ of PRNGs are used in the Mixed approach as follows: +\begin{equation} +\left\{ +\begin{array}{l} +x^0 \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket, S \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket^\mathds{N} \\ +\forall n \in \mathds{N}^*, x^n = x^{n-1} \oplus PRNG_1\oplus PRNG_2, +\end{array} +\right. +\label{equation Oplus} +\end{equation} + +With this Mixed CIPRNG approach, both the Old CIPRNG and New CIPRNG continue to pass all the NIST and DieHARD suites. +In addition, we can see that the PRNGs using a Xor CIPRNG approach can pass more tests than previously. +The main reason of this success is that the Mixed Xor CIPRNG has a longer period. +Indeed, let $n_{P}$ be the period of a PRNG $P$, then the period deduced from the single Xor CIPRNG approach is obviously equal to: +\begin{equation} +n_{SXORCI}= +\left\{ +\begin{array}{ll} +n_{P}&\text{if~}x^0=x^{n_{P}}\\ +2n_{P}&\text{if~}x^0\neq x^{n_{P}}.\\ +\end{array} +\right. +\label{equation Oplus} +\end{equation} + +Let us now denote by $n_{P1}$ and $n_{P2}$ the periods of respectively the $PRNG_1$ and $PRNG_2$ generators, then the period of the Mixed Xor CIPRNG will be: +\begin{equation} +n_{XXORCI}= +\left\{ +\begin{array}{ll} +LCM(n_{P1},n_{P2})&\text{if~}x^0=x^{LCM(n_{P1},n_{P2})}\\ +2LCM(n_{P1},n_{P2})&\text{if~}x^0\neq x^{LCM(n_{P1},n_{P2})}.\\ +\end{array} +\right. +\label{equation Oplus} +\end{equation} + +In Table~\ref{DieHARD fail mixex CIPRNG}, we only show the results for the Mixed CIPRNGs that cannot pass all DieHARD suites (the NIST tests are all passed). It demonstrates that Mixed Xor CIPRNG involving LCG, MRG, LCG2, LCG3, MRG2, or INV cannot pass the two following tests, namely the ``Matrix Rank 32x32'' and the ``COUNT-THE-1's'' tests contained into the DieHARD battery. Let us recall their definitions: + +\begin{itemize} + \item \textbf{Matrix Rank 32x32.} A random 32x32 binary matrix is formed, each row having a 32-bit random vector. Its rank is an integer that ranges from 0 to 32. Ranks less than 29 must be rare, and their occurences must be pooled with those of rank 29. To achieve the test, ranks of 40,000 such random matrices are obtained, and a chisquare test is performed on counts for ranks 32,31,30 and for ranks $\leq29$. + + \item \textbf{COUNT-THE-1's TEST} Consider the file under test as a stream of bytes (four per 2 bit integer). Each byte can contain from 0 to 8 1's, with probabilities 1,8,28,56,70,56,28,8,1 over 256. Now let the stream of bytes provide a string of overlapping 5-letter words, each ``letter'' taking values A,B,C,D,E. The letters are determined by the number of 1's in a byte: 0,1, or 2 yield A, 3 yields B, 4 yields C, 5 yields D and 6,7, or 8 yield E. Thus we have a monkey at a typewriter hitting five keys with various probabilities (37,56,70,56,37 over 256). There are $5^5$ possible 5-letter words, and from a string of 256,000 (over-lapping) 5-letter words, counts are made on the frequencies for each word. The quadratic form in the weak inverse of the covariance matrix of the cell counts provides a chisquare test: Q5-Q4, the difference of the naive Pearson sums of $(OBS-EXP)^2/EXP$ on counts for 5- and 4-letter cell counts. +\end{itemize} + +The reason of these fails is that the output of LCG, LCG2, LCG3, MRG, and MRG2 under the experiments are in 31-bit. Compare with the Single CIPRNG, using different PRNGs to build CIPRNG seems more efficient in improving random number quality (mixed Xor CI can 100\% pass NIST, but single cannot). + +\begin{table*} +\renewcommand{\arraystretch}{1.3} +\caption{Scores of mixed Xor CIPRNGs when considering the DieHARD battery} +\label{DieHARD fail mixex CIPRNG} +\centering + \begin{tabular}{|l||c|c|c|c|c|c|} + \hline +\backslashbox{\textbf{$PRNG_1$}} {\textbf{$PRNG_0$}} & LCG & MRG & INV & LCG2 & LCG3 & MRG2 \\ \hline\hline +LCG &\backslashbox{} {} &16/18&16/18 &16/18 &16/18 &16/18\\ \hline +MRG &16/18 &\backslashbox{} {} &16/18&16/18 &16/18 &16/18\\ \hline +INV &16/18 &16/18&\backslashbox{} {} &16/18 &16/18&16/18 \\ \hline +LCG2 &16/18 &16/18 &16/18 &\backslashbox{} {} &16/18&16/18\\ \hline +LCG3 &16/18 &16/18 &16/18&16/18&\backslashbox{} {} &16/18\\ \hline +MRG2 &16/18 &16/18 &16/18&16/18 &16/18 &\backslashbox{} {} \\ \hline +\end{tabular} +\end{table*} + +\subsubsection{Tests based on the Multiple CIPRNG} +\label{Tests based on Multiple CIPRNG} + +Until now, the combination of at most two input PRNGs has been investigated. +We now regard the possibility to use a larger number of generators to improve the statistics of the generated pseudorandom numbers, leading to the multiple functional power approach. +For the CIPRNGs which have already pass both the NIST and DieHARD suites with 2 inputted PRNGs (all the Old and New CIPRNGs, and some of the Xor CIPRNGs), it is not meaningful to consider their adaption of this multiple CIPRNG method, hence only the Multiple Xor CIPRNGs, having the following form, will be investigated. +\begin{equation} +\left\{ +\begin{array}{l} +x^0 \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket, S \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket^\mathds{N} \\ +\forall n \in \mathds{N}^*, x^n = x^{n-1} \oplus S^{nm}\oplus S^{nm+1}\ldots \oplus S^{nm+m-1} , +\end{array} +\right. +\label{equation Oplus} +\end{equation} + +The question is now to determine the value of the threshold $m$ (the functional power) making the multiple CIPRNG being able to pass the whole NIST battery. +Such a question is answered in Table~\ref{threshold}. + + +\begin{table*} +\renewcommand{\arraystretch}{1.3} +\caption{Functional power $m$ making it possible to pass the whole NIST battery} +\label{threshold} +\centering + \begin{tabular}{|l||c|c|c|c|c|c|c|c|} + \hline +Inputted $PRNG$ & LCG & MRG & SWC & GFSR & INV& LCG2 & LCG3 & MRG2 \\ \hline\hline +Threshold value $m$& 19 & 7 & 2& 1 & 11& 9& 3& 4\\ \hline\hline +\end{tabular} +\end{table*} + +\subsubsection{Results Summary} + +We can summarize the obtained results as follows. +\begin{enumerate} +\item The CIPRNG method is able to improve the statistical properties of a large variety of PRNGs. +\item Using different PRNGs in the CIPRNG approach is better than considering several instances of one unique PRNG. +\item The statistical quality of the outputs increases with the functional power $m$. +\end{enumerate} + +\end{color} \section{Efficient PRNG based on Chaotic Iterations} \label{sec:efficient PRNG} diff --git a/reponse.tex b/reponse.tex index 3b3e986..0ffb36a 100644 --- a/reponse.tex +++ b/reponse.tex @@ -25,6 +25,8 @@ \textit{Section 9: The authors say they replace the xor-like PRNG with a cryptographically secure one, BBS, but then proceed to use extremely small values, as far as a cryptographer is concerned (modulus of $2^{16}$), in the computation due to the need to use 32 bit integers in the GPU and combine bits from multiple BBS generated values, but they never prove (or even discuss) how this can be considered cryptographically secure due to the small individual values. At the end of 9.1, the authors say $S^n$ is secure because it is formed from bits from the BBS generator, but do not consider if the use of such small values will lead to exhaust searches to determine individual bits. The authors either need to remove all of section 9 and or prove the resulting PRNG is cryptographically secure.} +A new section has been added to measure practically the security of the generator. + \bigskip \textit{In the conclusion: Reword last sentence of 1st paragraph -- 2.39.5 From 0f430d9a654120c023030faedfd501aa3f6195e9 Mon Sep 17 00:00:00 2001 From: guyeux Date: Thu, 14 Jun 2012 10:21:16 +0200 Subject: [PATCH 06/16] Plein de trucs --- prng_gpu.tex | 276 ++++++++++++++++++++++----------------------------- 1 file changed, 119 insertions(+), 157 deletions(-) diff --git a/prng_gpu.tex b/prng_gpu.tex index 90f00f8..ba76fe2 100644 --- a/prng_gpu.tex +++ b/prng_gpu.tex @@ -418,7 +418,7 @@ the metric space $(\mathcal{X},d)$. \end{proposition} The chaotic property of $G_f$ has been firstly established for the vectorial -Boolean negation $f(x_1,\hdots, x_\mathsf{N}) = (\overline{x_1},\hdots, \overline{x_\mathsf{N}})$ \cite{guyeux10}. To obtain a characterization, we have secondly +Boolean negation $f_0(x_1,\hdots, x_\mathsf{N}) = (\overline{x_1},\hdots, \overline{x_\mathsf{N}})$ \cite{guyeux10}. To obtain a characterization, we have secondly introduced the notion of asynchronous iteration graph recalled bellow. Let $f$ be a map from $\mathds{B}^\mathsf{N}$ to itself. The @@ -475,33 +475,58 @@ Let us finally remark that the vectorial negation satisfies the hypotheses of bo We have proposed in~\cite{bgw09:ip} a new family of generators that receives two PRNGs as inputs. These two generators are mixed with chaotic iterations, -leading thus to a new PRNG that improves the statistical properties of each -generator taken alone. Furthermore, our generator -possesses various chaos properties that none of the generators used as input +leading thus to a new PRNG that +\begin{color}{red} +should improves the statistical properties of each +generator taken alone. +Furthermore, the generator obtained by this way possesses various chaos properties that none of the generators used as input present. + \begin{algorithm}[h!] \begin{small} \KwIn{a function $f$, an iteration number $b$, an initial configuration $x^0$ ($n$ bits)} \KwOut{a configuration $x$ ($n$ bits)} $x\leftarrow x^0$\; -$k\leftarrow b + \textit{XORshift}(b)$\; +$k\leftarrow b + PRNG_1(b)$\; \For{$i=0,\dots,k$} { -$s\leftarrow{\textit{XORshift}(n)}$\; +$s\leftarrow{PRNG_2(n)}$\; $x\leftarrow{F_f(s,x)}$\; } return $x$\; \end{small} -\caption{PRNG with chaotic functions} +\caption{An arbitrary round of $Old~ CI~ PRNG_f(PRNG_1,PRNG_2)$} \label{CI Algorithm} \end{algorithm} +This generator is synthesized in Algorithm~\ref{CI Algorithm}. +It takes as input: a Boolean function $f$ satisfying Theorem~\ref{Th:Caractérisation des IC chaotiques}; +an integer $b$, ensuring that the number of executed iterations +between two outputs is at least $b$ +and at most $2b+1$; and an initial configuration $x^0$. +It returns the new generated configuration $x$. Internally, it embeds two +inputted generators $PRNG_i(k), i=1,2$, + which must return integers +uniformly distributed +into $\llbracket 1 ; k \rrbracket$. +For instance, these PRNGs can be the \textit{XORshift}~\cite{Marsaglia2003}, +being a category of very fast PRNGs designed by George Marsaglia +that repeatedly uses the transform of exclusive or (XOR, $\oplus$) on a number +with a bit shifted version of it. Such a PRNG, which has a period of +$2^{32}-1=4.29\times10^9$, is summed up in Algorithm~\ref{XORshift}. +This XORshift, or any other reasonable PRNG, is used +in our own generator to compute both the number of iterations between two +outputs (provided by $PRNG_1$) and the strategy elements ($PRNG_2$). + +%This former generator has successively passed various batteries of statistical tests, as the NIST~\cite{bcgr11:ip}, DieHARD~\cite{Marsaglia1996}, and TestU01~\cite{LEcuyerS07} ones. + + \begin{algorithm}[h!] \begin{small} \KwIn{the internal configuration $z$ (a 32-bit word)} @@ -517,31 +542,95 @@ return $y$\; \end{algorithm} +\subsection{A ``New CI PRNG''} + +In order to make the Old CI PRNG usable in practice, we have proposed +an adapted version of the chaotic iteration based generator in~\cite{bg10:ip}. +In this ``New CI PRNG'', we prevent from changing twice a given +bit between two outputs. +This new generator is designed by the following process. + +First of all, some chaotic iterations have to be done to generate a sequence +$\left(x^n\right)_{n\in\mathds{N}} \in \left(\mathds{B}^{32}\right)^\mathds{N}$ +of Boolean vectors, which are the successive states of the iterated system. +Some of these vectors will be randomly extracted and our pseudo-random bit +flow will be constituted by their components. Such chaotic iterations are +realized as follows. Initial state $x^0 \in \mathds{B}^{32}$ is a Boolean +vector taken as a seed and chaotic strategy $\left(S^n\right)_{n\in\mathds{N}}\in +\llbracket 1, 32 \rrbracket^\mathds{N}$ is +an \emph{irregular decimation} of $PRNG_2$ sequence, as described in +Algorithm~\ref{Chaotic iteration1}. + +Then, at each iteration, only the $S^n$-th component of state $x^n$ is +updated, as follows: $x_i^n = x_i^{n-1}$ if $i \neq S^n$, else $x_i^n = \overline{x_i^{n-1}}$. +Such a procedure is equivalent to achieve chaotic iterations with +the Boolean vectorial negation $f_0$ and some well-chosen strategies. +Finally, some $x^n$ are selected +by a sequence $m^n$ as the pseudo-random bit sequence of our generator. +$(m^n)_{n \in \mathds{N}} \in \mathcal{M}^\mathds{N}$ is computed from $PRNG_1$, where $\mathcal{M}\subset \mathds{N}^*$ is a finite nonempty set of integers. +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 is required to make the outputs uniform in $\llbracket 0, 2^\mathsf{N}-1 \rrbracket$ +(the reader is referred to~\cite{bg10:ip} for more information). +\begin{equation} +\label{Formula} +m^n = g(y^n)= +\left\{ +\begin{array}{l} +0 \text{ if }0 \leqslant{y^n}<{C^0_{32}},\\ +1 \text{ if }{C^0_{32}} \leqslant{y^n}<\sum_{i=0}^1{C^i_{32}},\\ +2 \text{ if }\sum_{i=0}^1{C^i_{32}} \leqslant{y^n}<\sum_{i=0}^2{C^i_{32}},\\ +\vdots~~~~~ ~~\vdots~~~ ~~~~\\ +N \text{ if }\sum_{i=0}^{N-1}{C^i_{32}}\leqslant{y^n}<1.\\ +\end{array} +\right. +\end{equation} -This generator is synthesized in Algorithm~\ref{CI Algorithm}. -It takes as input: a Boolean function $f$ satisfying Theorem~\ref{Th:Caractérisation des IC chaotiques}; -an integer $b$, ensuring that the number of executed iterations is at least $b$ -and at most $2b+1$; and an initial configuration $x^0$. -It returns the new generated configuration $x$. Internally, it embeds two -\textit{XORshift}$(k)$ PRNGs~\cite{Marsaglia2003} that return integers -uniformly distributed -into $\llbracket 1 ; k \rrbracket$. -\textit{XORshift} is a category of very fast PRNGs designed by George Marsaglia, -which repeatedly uses the transform of exclusive or (XOR, $\oplus$) on a number -with a bit shifted version of it. This PRNG, which has a period of -$2^{32}-1=4.29\times10^9$, is summed up in Algorithm~\ref{XORshift}. It is used -in our PRNG to compute the strategy length and the strategy elements. +\begin{algorithm} +\textbf{Input:} the internal state $x$ (32 bits)\\ +\textbf{Output:} a state $r$ of 32 bits +\begin{algorithmic}[1] +\FOR{$i=0,\dots,N$} +{ +\STATE$d_i\leftarrow{0}$\; +} +\ENDFOR +\STATE$a\leftarrow{PRNG_1()}$\; +\STATE$m\leftarrow{g(a)}$\; +\STATE$k\leftarrow{m}$\; +\WHILE{$i=0,\dots,k$} -This former generator has successively passed various batteries of statistical tests, as the NIST~\cite{bcgr11:ip}, DieHARD~\cite{Marsaglia1996}, and TestU01~\cite{LEcuyerS07} ones. +\STATE$b\leftarrow{PRNG_2()~mod~\mathsf{N}}$\; +\STATE$S\leftarrow{b}$\; + \IF{$d_S=0$} + { +\STATE $x_S\leftarrow{ \overline{x_S}}$\; +\STATE $d_S\leftarrow{1}$\; + + } + \ELSIF{$d_S=1$} + { +\STATE $k\leftarrow{ k+1}$\; + }\ENDIF +\ENDWHILE\\ +\STATE $r\leftarrow{x}$\; +\STATE return $r$\; +\medskip +\caption{An arbitrary round of the new CI generator} +\label{Chaotic iteration1} +\end{algorithmic} +\end{algorithm} +\end{color} \subsection{Improving the Speed of the Former Generator} -Instead of updating only one cell at each iteration, we can try to choose a -subset of components and to update them together. Such an attempt leads -to a kind of merger of the two sequences used in Algorithm -\ref{CI Algorithm}. When the updating function is the vectorial negation, +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: \begin{equation} @@ -551,7 +640,7 @@ x^0 \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket, S \in \llbracket 0, 2^\mathsf{N \forall n \in \mathds{N}^*, x^n = x^{n-1} \oplus S^n, \end{array} \right. -\label{equation Oplus} +\label{equation Oplus0} \end{equation} where $\oplus$ is for the bitwise exclusive or between two integers. This rewriting can be understood as follows. The $n-$th term $S^n$ of the @@ -561,7 +650,7 @@ as an integer having $\mathsf{N}$ bits too). More precisely, the $k-$th component of this state (a binary digit) changes if and only if the $k-$th digit in the binary decomposition of $S^n$ is 1. -The single basic component presented in Eq.~\ref{equation Oplus} is of +The single basic component presented in Eq.~\ref{equation Oplus0} is of ordinary use as a good elementary brick in various PRNGs. It corresponds to the following discrete dynamical system in chaotic iterations: @@ -582,8 +671,8 @@ than the ones presented in Definition \ref{Def:chaotic iterations} because, inst we select a subset of components to change. -Obviously, replacing Algorithm~\ref{CI Algorithm} by -Equation~\ref{equation Oplus}, which is possible when the iteration function is +Obviously, replacing the previous CI PRNG Algorithms by +Equation~\ref{equation Oplus0}, which is possible when the iteration function is the vectorial negation, leads to a speed improvement. However, proofs of chaos obtained in~\cite{bg10:ij} have been established only for chaotic iterations of the form presented in Definition @@ -838,134 +927,7 @@ have $d((S,E),(\tilde S,E))<\epsilon$. \begin{color}{red} -\section{Improving Statistical Properties Using Chaotic Iterations} - - -\subsection{The CIPRNG family} - -Three categories of PRNGs have been derived from chaotic iterations. They are -recalled in what follows. - -\subsubsection{Old CIPRNG} - -Let $\mathsf{N} = 4$. Some chaotic iterations are fulfilled to generate a sequence $\left(x^n\right)_{n\in\mathds{N}} \in \left(\mathds{B}^4\right)^\mathds{N}$ of Boolean vectors: the successive states of the iterated system. Some of these vectors are randomly extracted and their components constitute our pseudorandom bit flow~\cite{bgw09:ip}. -Chaotic iterations are realized as follows. Initial state $x^0 \in \mathds{B}^4$ is a Boolean vector taken as a seed and chaotic strategy $\left(S^n\right)_{n\in\mathds{N}}\in \llbracket 1, 4 \rrbracket^\mathds{N}$ is constructed with $PRNG_2$. Lastly, iterate function $f$ is the vectorial Boolean negation. -At each iteration, only the $S^n$-th component of state $x^n$ is updated. Finally, some $x^n$ are selected by a sequence $m^n$, provided by a second generator $PRNG_1$, as the pseudorandom bit sequence of our generator. - -The basic design procedure of the Old CI generator is summed up in Algorithm~\ref{Chaotic iteration}. -The internal state is $x$, the output array is $r$. $a$ and $b$ are those computed by $PRNG_1$ and $PRNG_2$. - - -\begin{algorithm} -\textbf{Input:} the internal state $x$ (an array of 4-bit words)\\ -\textbf{Output:} an array $r$ of 4-bit words -\begin{algorithmic}[1] - -\STATE$a\leftarrow{PRNG_1()}$; -\STATE$m\leftarrow{a~mod~2+13}$; -\WHILE{$i=0,\dots,m$} -\STATE$b\leftarrow{PRNG_2()}$; -\STATE$S\leftarrow{b~mod~4}$; -\STATE$x_S\leftarrow{ \overline{x_S}}$; -\ENDWHILE -\STATE$r\leftarrow{x}$; -\STATE return $r$; -\medskip -\caption{An arbitrary round of the old CI generator} -\label{Chaotic iteration} -\end{algorithmic} -\end{algorithm} - -\subsubsection{New CIPRNG} - -The New CI generator is designed by the following process~\cite{bg10:ip}. First of all, some chaotic iterations have to be done to generate a sequence $\left(x^n\right)_{n\in\mathds{N}} \in \left(\mathds{B}^{32}\right)^\mathds{N}$ of Boolean vectors, which are the successive states of the iterated system. Some of these vectors will be randomly extracted and our pseudo-random bit flow will be constituted by their components. Such chaotic iterations are realized as follows. Initial state $x^0 \in \mathds{B}^{32}$ is a Boolean vector taken as a seed and chaotic strategy $\left(S^n\right)_{n\in\mathds{N}}\in \llbracket 1, 32 \rrbracket^\mathds{N}$ is -an \emph{irregular decimation} of $PRNG_2$ sequence, as described in Algorithm~\ref{Chaotic iteration1}. - -Another time, at each iteration, only the $S^n$-th component of state $x^n$ is updated, as follows: $x_i^n = x_i^{n-1}$ if $i \neq S^n$, else $x_i^n = \overline{x_i^{n-1}}$. -Finally, some $x^n$ are selected -by a sequence $m^n$ as the pseudo-random bit sequence of our generator. -$(m^n)_{n \in \mathds{N}} \in \mathcal{M}^\mathds{N}$ is computed from $PRNG_1$, where $\mathcal{M}\subset \mathds{N}^*$ is a finite nonempty set of integers. - -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 $g_1(a)$ is an integer defined as in Eq.~\ref{Formula}. - -\begin{equation} -\label{Formula} -m^n = g_1(y^n)= -\left\{ -\begin{array}{l} -0 \text{ if }0 \leqslant{y^n}<{C^0_{32}},\\ -1 \text{ if }{C^0_{32}} \leqslant{y^n}<\sum_{i=0}^1{C^i_{32}},\\ -2 \text{ if }\sum_{i=0}^1{C^i_{32}} \leqslant{y^n}<\sum_{i=0}^2{C^i_{32}},\\ -\vdots~~~~~ ~~\vdots~~~ ~~~~\\ -N \text{ if }\sum_{i=0}^{N-1}{C^i_{32}}\leqslant{y^n}<1.\\ -\end{array} -\right. -\end{equation} - -\begin{algorithm} -\textbf{Input:} the internal state $x$ (32 bits)\\ -\textbf{Output:} a state $r$ of 32 bits -\begin{algorithmic}[1] -\FOR{$i=0,\dots,N$} -{ -\STATE$d_i\leftarrow{0}$\; -} -\ENDFOR -\STATE$a\leftarrow{PRNG_1()}$\; -\STATE$m\leftarrow{f(a)}$\; -\STATE$k\leftarrow{m}$\; -\WHILE{$i=0,\dots,k$} - -\STATE$b\leftarrow{PRNG_2()~mod~\mathsf{N}}$\; -\STATE$S\leftarrow{b}$\; - \IF{$d_S=0$} - { -\STATE $x_S\leftarrow{ \overline{x_S}}$\; -\STATE $d_S\leftarrow{1}$\; - - } - \ELSIF{$d_S=1$} - { -\STATE $k\leftarrow{ k+1}$\; - }\ENDIF -\ENDWHILE\\ -\STATE $r\leftarrow{x}$\; -\STATE return $r$\; -\medskip -\caption{An arbitrary round of the new CI generator} -\label{Chaotic iteration1} -\end{algorithmic} -\end{algorithm} - - -\subsubsection{Xor CIPRNG} - -Instead of updating only one cell at each iteration as Old CI and New CI, we can try to choose a -subset of components and to update them together. Such an attempt leads -to a kind of merger of the two random sequences. When the updating function is the vectorial negation, -this algorithm can be rewritten as follows~\cite{arxivRCCGPCH}: - -\begin{equation} -\left\{ -\begin{array}{l} -x^0 \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket, S \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket^\mathds{N} \\ -\forall n \in \mathds{N}^*, x^n = x^{n-1} \oplus S^n, -\end{array} -\right. -\label{equation Oplus} -\end{equation} -%This rewriting can be understood as follows. The $n-$th term $S^n$ of the -%sequence $S$, which is an integer of $\mathsf{N}$ binary digits, presents -%the list of cells to update in the state $x^n$ of the system (represented -%as an integer having $\mathsf{N}$ bits too). More precisely, the $k-$th -%component of this state (a binary digit) changes if and only if the $k-$th -%digit in the binary decomposition of $S^n$ is 1. - -The single basic component presented in Eq.~\ref{equation Oplus} is of -ordinary use as a good elementary brick in various PRNGs. It corresponds -to the discrete dynamical system in chaotic iterations. +\section{Statistical Improvements Using Chaotic Iterations} \subsection{About some Well-known PRNGs} \label{The generation of pseudo-random sequence} -- 2.39.5 From 11f1ec59d1b84c34db2d61d773d6fe2c9a480938 Mon Sep 17 00:00:00 2001 From: guyeux Date: Thu, 14 Jun 2012 10:56:47 +0200 Subject: [PATCH 07/16] pouet --- mabase.bib | 26 ++++++++++++++++++++++++++ prng_gpu.tex | 42 +++++++++++++++++++++++------------------- 2 files changed, 49 insertions(+), 19 deletions(-) diff --git a/mabase.bib b/mabase.bib index 15c282d..6b41f33 100644 --- a/mabase.bib +++ b/mabase.bib @@ -14,6 +14,32 @@ timestamp = {2009.06.29} } +@inproceedings{bfg12a:ip, +inhal = {no}, +domainehal = {INFO:INFO_DC, INFO:INFO_CR, INFO:INFO_MO}, +equipe = {and}, +classement = {ACTI}, +author = {Bahi, Jacques and Fang, Xiaole and Guyeux, Christophe}, +title = {An optimization technique on pseudorandom generators based on chaotic iterations}, +booktitle = {INTERNET'2012, 4-th Int. Conf. on Evolving Internet}, +pages = {***--***}, +address = {Venice, Italy}, +month = jun, +year = 2012, +note = {To appear}, + +} + +@Article{combined_lcg, + title = "Efficient and portable combined random number generators", + journal = "Communications of the ACM", + volume = "31", + number = "6", + pages = "742--749", + year = "1988", +} + + @INPROCEEDINGS{Fischlin, author = {Fischlin, R. and Schnorr, C. P.}, title = {Stronger security proofs for RSA and rabin bits}, diff --git a/prng_gpu.tex b/prng_gpu.tex index ba76fe2..468b218 100644 --- a/prng_gpu.tex +++ b/prng_gpu.tex @@ -929,37 +929,39 @@ have $d((S,E),(\tilde S,E))<\epsilon$. \begin{color}{red} \section{Statistical Improvements Using Chaotic Iterations} -\subsection{About some Well-known PRNGs} \label{The generation of pseudo-random sequence} - - -Let us now give illustration on the fact that chaos appears to improve statistical properties. +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}). \subsection{Details of some Existing Generators} -Here are the modules of PRNGs we have chosen to experiment. +The list of defective PRNGs we will use +as inputs for the statistical tests to come is introduced here. -\subsubsection{LCG} -This PRNG implements either the simple or the combined linear congruency generator (LCGs). The simple LCG is defined by the recurrence: +Firstly, the simple linear congruency generator (LCGs) is defined by the following recurrence: \begin{equation} 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{testU01}. In what follows, 2LCGs and 3LCGs refer as two (resp. three) combinations of such LCGs. -For further details, see~\cite{combined_lcg}. +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}. -\subsubsection{MRG} -This module implements multiple recursive generators (MRGs), based on a linear recurrence of order $k$, modulo $m$~\cite{testU01}: +Secondly, the multiple recursive generators (MRGs) is 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 \label{MRG} \end{equation} Combination of two MRGs (referred as 2MRGs) is also be used in this paper. -\subsubsection{UCARRY} -Generators based on linear recurrences with carry are implemented in this module. This includes the add-with-carry (AWC) generator, based on the recurrence: +Thirdly, generators based on linear recurrences with carry will be regarded too in experimentations. +This includes the add-with-carry (AWC) generator, based on the recurrence: \begin{equation} \label{AWC} \begin{array}{l} @@ -981,16 +983,14 @@ and the SWC generator designed by R. Couture, which is based on the following re x^n = (a^1x^{n-1} \oplus ~...~ \oplus a^rx^{n-r} \oplus c^{n-1}) ~ mod ~ 2^w, \\ c^n = (a^1x^{n-1} \oplus ~...~ \oplus a^rx^{n-r} \oplus c^{n-1}) ~ / ~ 2^w. \end{array}\end{equation} -\subsubsection{GFSR} -This module implements the generalized feedback shift register (GFSR) generator, that is: +Then the generalized feedback shift register (GFSR) generator has been implemented, that is: \begin{equation} x^n = x^{n-r} \oplus x^{n-k} \label{GFSR} \end{equation} -\subsubsection{INV} -Finally, this module implements the nonlinear inversive generator, as defined in~\cite{testU01}, which is: +Finally, the nonlinear inversive generator~\cite{LEcuyerS07} has been regarded too, which is: \begin{equation} \label{INV} @@ -1007,8 +1007,12 @@ a^1 & \text{if}~ z^{n-1} = 0 .\end{array} \right. \end{array}\end{equation} \subsection{Statistical tests} \label{Security analysis} -%A theoretical proof for the randomness of a generator is impossible to give, therefore statistical inference based on observed sample sequences produced by the generator seems to be the best option. -Considering the properties of binary random sequences, various statistical tests can be designed to evaluate the assertion that the sequence is generated by a perfectly random source. We have performed some statistical tests for the CIPRNGs proposed here. These tests include NIST suite~\cite{ANDREW2008} and DieHARD battery of tests~\cite{DieHARD}. For completeness and for reference, we give in the following subsection a brief description of each of the aforementioned tests. +Considering the properties of binary random sequences, various statistical tests can be designed +to evaluate the assertion that the sequence is generated by a perfectly random source. We have +performed some statistical tests for the CIPRNGs proposed here. These tests include NIST +suite~\cite{ANDREW2008} and DieHARD battery of tests~\cite{DieHARD}. For completeness and +for reference, we give in the following subsection a brief description of each of the +aforementioned tests. -- 2.39.5 From ddb01e4b5bfe53afe6dba0b77f3d5322ac38c81f Mon Sep 17 00:00:00 2001 From: guyeux Date: Thu, 14 Jun 2012 11:27:03 +0200 Subject: [PATCH 08/16] blabla --- mabase.bib | 9 +++++++ prng_gpu.tex | 72 ++++++++++++++++------------------------------------ 2 files changed, 31 insertions(+), 50 deletions(-) diff --git a/mabase.bib b/mabase.bib index 6b41f33..30ad80e 100644 --- a/mabase.bib +++ b/mabase.bib @@ -30,6 +30,15 @@ note = {To appear}, } +@UNPUBLISHED{ANDREW2008, + author = {NIST Special Publication 800-22 rev. 1}, + title = {A Statistical Test Suite for Random and Pseudorandom Number Generators + for Cryptographic Applications}, + year = {August 2008}, + owner = {qianxue}, + timestamp = {2009.01.22} +} + @Article{combined_lcg, title = "Efficient and portable combined random number generators", journal = "Communications of the ACM", diff --git a/prng_gpu.tex b/prng_gpu.tex index 468b218..3228083 100644 --- a/prng_gpu.tex +++ b/prng_gpu.tex @@ -943,7 +943,8 @@ chaotic iterations, the result presents better statistical properties 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) is defined by the following recurrence: +Firstly, the simple linear congruency generator (LCGs) will be used. +It is defined by the following recurrence: \begin{equation} x^n = (ax^{n-1} + c)~mod~m \label{LCG} @@ -952,16 +953,17 @@ where $a$, $c$, and $x^0$ must be, among other things, non-negative and less tha $m$~\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) is based on a linear recurrence of order +Secondly, the multiple recursive generators (MRGs) will be used too, 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 \label{MRG} \end{equation} -Combination of two MRGs (referred as 2MRGs) is also be used in this paper. +Combination of two MRGs (referred as 2MRGs) is also used in these experimentations. -Thirdly, generators based on linear recurrences with carry will be regarded too in experimentations. -This includes the add-with-carry (AWC) generator, based on the recurrence: +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: \begin{equation} \label{AWC} \begin{array}{l} @@ -1007,49 +1009,14 @@ a^1 & \text{if}~ z^{n-1} = 0 .\end{array} \right. \end{array}\end{equation} \subsection{Statistical tests} \label{Security analysis} -Considering the properties of binary random sequences, various statistical tests can be designed -to evaluate the assertion that the sequence is generated by a perfectly random source. We have -performed some statistical tests for the CIPRNGs proposed here. These tests include NIST -suite~\cite{ANDREW2008} and DieHARD battery of tests~\cite{DieHARD}. For completeness and -for reference, we give in the following subsection a brief description of each of the -aforementioned tests. +Three batteries of tests are reputed and usually used +to evaluate the statistical properties of newly designed pseudorandom +number generators. These batteries are named DieHard~\cite{Marsaglia1996}, +the NIST suite~\cite{ANDREW2008}, and the most stringent one called +TestU01~\cite{LEcuyerS07}, which encompasses the two other batteries. -\subsubsection{NIST statistical tests suite} - -Among the numerous standard tests for pseudo-randomness, a convincing way to show the randomness of the produced sequences is to confront them to the NIST (National Institute of Standards and Technology) statistical tests, being an up-to-date tests suite proposed by the Information Technology Laboratory (ITL). A new version of the Statistical tests suite has been released in August 11, 2010. - -The NIST tests suite SP 800-22 is a statistical package consisting of 15 tests. They were developed to test the randomness of binary sequences produced by hardware or software based cryptographic pseudorandom number generators. These tests focus on a variety of different types of non-randomness that could exist in a sequence. - -For each statistical test, a set of $P-values$ (corresponding to the set of sequences) is produced. -The interpretation of empirical results can be conducted in various ways. -In this paper, the examination of the distribution of P-values to check for uniformity ($ P-value_{T}$) is used. -The distribution of $P-values$ is examined to ensure uniformity. -If $P-value_{T} \geqslant 0.0001$, then the sequences can be considered to be uniformly distributed. - -In our experiments, 100 sequences (s = 100), each with 1,000,000-bit long, are generated and tested. If the $P-value_{T}$ of any test is smaller than 0.0001, the sequences are considered to be not good enough and the generating algorithm is not suitable for usage. - - - - - -\subsubsection{DieHARD battery of tests} -The DieHARD battery of tests has been the most sophisticated standard for over a decade. Because of the stringent requirements in the DieHARD tests suite, a generator passing this battery of -tests can be considered good as a rule of thumb. - -The DieHARD battery of tests consists of 18 different independent statistical tests. This collection - of tests is based on assessing the randomness of bits comprising 32-bit integers obtained from -a random number generator. Each test requires $2^{23}$ 32-bit integers in order to run the full set -of tests. Most of the tests in DieHARD return a $P-value$, which should be uniform on $[0,1)$ if the input file -contains truly independent random bits. These $P-values$ are obtained by -$P=F(X)$, where $F$ is the assumed distribution of the sample random variable $X$ (often normal). -But that assumed $F$ is just an asymptotic approximation, for which the fit will be worst -in the tails. Thus occasional $P-values$ near 0 or 1, such as 0.0012 or 0.9983, can occur. -An individual test is considered to be failed if the $P-value$ approaches 1 closely, for example $P>0.9999$. - - -\subsection{Results and discussion} \label{Results and discussion} \begin{table*} \renewcommand{\arraystretch}{1.3} @@ -1065,16 +1032,21 @@ DieHARD & 16/18 & 16/18 & 15/18 & 16/18 & \textbf{18/18} & 16/18 & 16/18 & 16/18 \end{tabular} \end{table*} -Table~\ref{NIST and DieHARD tests suite passing rate the for PRNGs without CI} shows the results on the batteries recalled above, indicating that almost all the PRNGs cannot pass all their tests. In other words, the statistical quality of these PRNGs cannot fulfill the up-to-date standards presented previously. We will show that the CIPRNG can solve this issue. - -To illustrate the effects of this CIPRNG in detail, experiments will be divided in three parts: +Table~\ref{NIST and DieHARD tests suite passing rate the for PRNGs without CI} shows the +results on the two firsts batteries recalled above, indicating that all the PRNGs presented +in the previous section +cannot pass all these tests. In other words, the statistical quality of these PRNGs cannot +fulfill the up-to-date standards presented previously. We have shown in~\cite{bfg12a:ip} that the use of chaotic +iterations can solve this issue. +More precisely, to +illustrate the effects of chaotic iterations on these defective PRNGs, experiments have been divided in three parts~\cite{bfg12a:ip}: \begin{enumerate} \item \textbf{Single CIPRNG}: The PRNGs involved in CI computing are of the same category. \item \textbf{Mixed CIPRNG}: Two different types of PRNGs are mixed during the chaotic iterations process. - \item \textbf{Multiple CIPRNG}: The generator is obtained by repeating the composition of the iteration function as follows: $x^0\in \mathds{B}^{\mathsf{N}}$, and $\forall n\in \mathds{N}^{\ast },\forall i\in \llbracket1;\mathsf{N}\rrbracket,$ + \item \textbf{Multiple CIPRNG}: The generator is obtained by repeating the composition of the iteration function as follows: $x^0\in \mathds{B}^{\mathsf{N}}$, and $\forall n\in \mathds{N}^{\ast },\forall i\in \llbracket1;\mathsf{N}\rrbracket, x_i^n=$ \begin{equation} \begin{array}{l} -x_i^n=\left\{ +\left\{ \begin{array}{l} x_i^{n-1}~~~~~\text{if}~S^n\neq i \\ \forall j\in \llbracket1;\mathsf{m}\rrbracket,f^m(x^{n-1})_{S^{nm+j}}~\text{if}~S^{nm+j}=i.\end{array} \right. \end{array} -- 2.39.5 From cc4c6aa868df1972b090ffcce36c865b8bf53644 Mon Sep 17 00:00:00 2001 From: guyeux Date: Thu, 14 Jun 2012 12:16:02 +0200 Subject: [PATCH 09/16] Nettoyage --- prng_gpu.tex | 167 ++++++++++++--------------------------------------- 1 file changed, 37 insertions(+), 130 deletions(-) diff --git a/prng_gpu.tex b/prng_gpu.tex index 3228083..bc40b2d 100644 --- a/prng_gpu.tex +++ b/prng_gpu.tex @@ -673,7 +673,10 @@ we select a subset of components to change. Obviously, replacing the previous CI PRNG Algorithms by Equation~\ref{equation Oplus0}, which is possible when the iteration function is -the vectorial negation, leads to a speed improvement. However, proofs +the vectorial negation, leads to a speed improvement +(the resulting generator will be referred as ``Xor CI PRNG'' +in what follows). +However, proofs of chaos obtained in~\cite{bg10:ij} have been established only for chaotic iterations of the form presented in Definition \ref{Def:chaotic iterations}. The question is now to determine whether the @@ -1038,28 +1041,30 @@ in the previous section cannot pass all these tests. In other words, the statistical quality of these PRNGs cannot fulfill the up-to-date standards presented previously. We have shown in~\cite{bfg12a:ip} that the use of chaotic iterations can solve this issue. -More precisely, to -illustrate the effects of chaotic iterations on these defective PRNGs, experiments have been divided in three parts~\cite{bfg12a:ip}: -\begin{enumerate} - \item \textbf{Single CIPRNG}: The PRNGs involved in CI computing are of the same category. - \item \textbf{Mixed CIPRNG}: Two different types of PRNGs are mixed during the chaotic iterations process. - \item \textbf{Multiple CIPRNG}: The generator is obtained by repeating the composition of the iteration function as follows: $x^0\in \mathds{B}^{\mathsf{N}}$, and $\forall n\in \mathds{N}^{\ast },\forall i\in \llbracket1;\mathsf{N}\rrbracket, x_i^n=$ -\begin{equation} -\begin{array}{l} -\left\{ -\begin{array}{l} -x_i^{n-1}~~~~~\text{if}~S^n\neq i \\ -\forall j\in \llbracket1;\mathsf{m}\rrbracket,f^m(x^{n-1})_{S^{nm+j}}~\text{if}~S^{nm+j}=i.\end{array} \right. \end{array} -\end{equation} -$m$ is called the \emph{functional power}. -\end{enumerate} - - -We have performed statistical analysis of each of the aforementioned CIPRNGs. -The results are reproduced in Tables~\ref{NIST and DieHARD tests suite passing rate the for PRNGs without CI} and \ref{NIST and DieHARD tests suite passing rate the for single CIPRNGs}. -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. +%More precisely, to +%illustrate the effects of chaotic iterations on these defective PRNGs, experiments have been divided in three parts~\cite{bfg12a:ip}: +%\begin{enumerate} +% \item \textbf{Single CIPRNG}: The PRNGs involved in CI computing are of the same category. +% \item \textbf{Mixed CIPRNG}: Two different types of PRNGs are mixed during the chaotic iterations process. +% \item \textbf{Multiple CIPRNG}: The generator is obtained by repeating the composition of the iteration function as follows: $x^0\in \mathds{B}^{\mathsf{N}}$, and $\forall n\in \mathds{N}^{\ast },\forall i\in \llbracket1;\mathsf{N}\rrbracket, x_i^n=$ +%\begin{equation} +%\begin{array}{l} +%\left\{ +%\begin{array}{l} +%x_i^{n-1}~~~~~\text{if}~S^n\neq i \\ +%\forall j\in \llbracket1;\mathsf{m}\rrbracket,f^m(x^{n-1})_{S^{nm+j}}~\text{if}~S^{nm+j}=i.\end{array} \right. \end{array} +%\end{equation} +%$m$ is called the \emph{functional power}. +%\end{enumerate} +% +The obtained results are reproduced in Table +\ref{NIST and DieHARD tests suite passing rate the for single CIPRNGs}. +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. -\subsubsection{Tests based on the Single CIPRNG} \begin{table*} \renewcommand{\arraystretch}{1.3} @@ -1082,108 +1087,16 @@ DieHARD & 16/18 & 16/18 & 17/18* & \textbf{18/18} * & \textbf{18/18} & \textbf{ \end{tabular} \end{table*} -The statistical tests results of the PRNGs using the single CIPRNG method are given in Table~\ref{NIST and DieHARD tests suite passing rate the for single CIPRNGs}. -We can observe that, except for the Xor CIPRNG, all of the CIPRNGs have passed the 15 tests of the NIST battery and the 18 tests of the DieHARD one. -Moreover, considering these scores, we can deduce that both the single Old CIPRNG and the single New CIPRNG are relatively steadier than the single Xor CIPRNG approach, when applying them to different PRNGs. -However, the Xor CIPRNG is obviously the fastest approach to generate a CI random sequence, and it still improves the statistical properties relative to each generator taken alone, although the test values are not as good as desired. - -Therefore, all of these three ways are interesting, for different reasons, in the production of pseudorandom numbers and, -on the whole, the single CIPRNG method can be considered to adapt to or improve all kinds of PRNGs. - -To have a realization of the Xor CIPRNG that can pass all the tests embedded into the NIST battery, the Xor CIPRNG with multiple functional powers are investigated in Section~\ref{Tests based on Multiple CIPRNG}. - - -\subsubsection{Tests based on the Mixed CIPRNG} - -To compare the previous approach with the CIPRNG design that uses a Mixed CIPRNG, we have taken into account the same inputted generators than in the previous section. -These inputted couples $(PRNG_1,PRNG_2)$ of PRNGs are used in the Mixed approach as follows: -\begin{equation} -\left\{ -\begin{array}{l} -x^0 \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket, S \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket^\mathds{N} \\ -\forall n \in \mathds{N}^*, x^n = x^{n-1} \oplus PRNG_1\oplus PRNG_2, -\end{array} -\right. -\label{equation Oplus} -\end{equation} -With this Mixed CIPRNG approach, both the Old CIPRNG and New CIPRNG continue to pass all the NIST and DieHARD suites. -In addition, we can see that the PRNGs using a Xor CIPRNG approach can pass more tests than previously. -The main reason of this success is that the Mixed Xor CIPRNG has a longer period. -Indeed, let $n_{P}$ be the period of a PRNG $P$, then the period deduced from the single Xor CIPRNG approach is obviously equal to: -\begin{equation} -n_{SXORCI}= -\left\{ -\begin{array}{ll} -n_{P}&\text{if~}x^0=x^{n_{P}}\\ -2n_{P}&\text{if~}x^0\neq x^{n_{P}}.\\ -\end{array} -\right. -\label{equation Oplus} -\end{equation} - -Let us now denote by $n_{P1}$ and $n_{P2}$ the periods of respectively the $PRNG_1$ and $PRNG_2$ generators, then the period of the Mixed Xor CIPRNG will be: -\begin{equation} -n_{XXORCI}= -\left\{ -\begin{array}{ll} -LCM(n_{P1},n_{P2})&\text{if~}x^0=x^{LCM(n_{P1},n_{P2})}\\ -2LCM(n_{P1},n_{P2})&\text{if~}x^0\neq x^{LCM(n_{P1},n_{P2})}.\\ -\end{array} -\right. -\label{equation Oplus} -\end{equation} - -In Table~\ref{DieHARD fail mixex CIPRNG}, we only show the results for the Mixed CIPRNGs that cannot pass all DieHARD suites (the NIST tests are all passed). It demonstrates that Mixed Xor CIPRNG involving LCG, MRG, LCG2, LCG3, MRG2, or INV cannot pass the two following tests, namely the ``Matrix Rank 32x32'' and the ``COUNT-THE-1's'' tests contained into the DieHARD battery. Let us recall their definitions: - -\begin{itemize} - \item \textbf{Matrix Rank 32x32.} A random 32x32 binary matrix is formed, each row having a 32-bit random vector. Its rank is an integer that ranges from 0 to 32. Ranks less than 29 must be rare, and their occurences must be pooled with those of rank 29. To achieve the test, ranks of 40,000 such random matrices are obtained, and a chisquare test is performed on counts for ranks 32,31,30 and for ranks $\leq29$. - - \item \textbf{COUNT-THE-1's TEST} Consider the file under test as a stream of bytes (four per 2 bit integer). Each byte can contain from 0 to 8 1's, with probabilities 1,8,28,56,70,56,28,8,1 over 256. Now let the stream of bytes provide a string of overlapping 5-letter words, each ``letter'' taking values A,B,C,D,E. The letters are determined by the number of 1's in a byte: 0,1, or 2 yield A, 3 yields B, 4 yields C, 5 yields D and 6,7, or 8 yield E. Thus we have a monkey at a typewriter hitting five keys with various probabilities (37,56,70,56,37 over 256). There are $5^5$ possible 5-letter words, and from a string of 256,000 (over-lapping) 5-letter words, counts are made on the frequencies for each word. The quadratic form in the weak inverse of the covariance matrix of the cell counts provides a chisquare test: Q5-Q4, the difference of the naive Pearson sums of $(OBS-EXP)^2/EXP$ on counts for 5- and 4-letter cell counts. -\end{itemize} - -The reason of these fails is that the output of LCG, LCG2, LCG3, MRG, and MRG2 under the experiments are in 31-bit. Compare with the Single CIPRNG, using different PRNGs to build CIPRNG seems more efficient in improving random number quality (mixed Xor CI can 100\% pass NIST, but single cannot). +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 +using chaotic iterations on defective generators. \begin{table*} \renewcommand{\arraystretch}{1.3} -\caption{Scores of mixed Xor CIPRNGs when considering the DieHARD battery} -\label{DieHARD fail mixex CIPRNG} -\centering - \begin{tabular}{|l||c|c|c|c|c|c|} - \hline -\backslashbox{\textbf{$PRNG_1$}} {\textbf{$PRNG_0$}} & LCG & MRG & INV & LCG2 & LCG3 & MRG2 \\ \hline\hline -LCG &\backslashbox{} {} &16/18&16/18 &16/18 &16/18 &16/18\\ \hline -MRG &16/18 &\backslashbox{} {} &16/18&16/18 &16/18 &16/18\\ \hline -INV &16/18 &16/18&\backslashbox{} {} &16/18 &16/18&16/18 \\ \hline -LCG2 &16/18 &16/18 &16/18 &\backslashbox{} {} &16/18&16/18\\ \hline -LCG3 &16/18 &16/18 &16/18&16/18&\backslashbox{} {} &16/18\\ \hline -MRG2 &16/18 &16/18 &16/18&16/18 &16/18 &\backslashbox{} {} \\ \hline -\end{tabular} -\end{table*} - -\subsubsection{Tests based on the Multiple CIPRNG} -\label{Tests based on Multiple CIPRNG} - -Until now, the combination of at most two input PRNGs has been investigated. -We now regard the possibility to use a larger number of generators to improve the statistics of the generated pseudorandom numbers, leading to the multiple functional power approach. -For the CIPRNGs which have already pass both the NIST and DieHARD suites with 2 inputted PRNGs (all the Old and New CIPRNGs, and some of the Xor CIPRNGs), it is not meaningful to consider their adaption of this multiple CIPRNG method, hence only the Multiple Xor CIPRNGs, having the following form, will be investigated. -\begin{equation} -\left\{ -\begin{array}{l} -x^0 \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket, S \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket^\mathds{N} \\ -\forall n \in \mathds{N}^*, x^n = x^{n-1} \oplus S^{nm}\oplus S^{nm+1}\ldots \oplus S^{nm+m-1} , -\end{array} -\right. -\label{equation Oplus} -\end{equation} - -The question is now to determine the value of the threshold $m$ (the functional power) making the multiple CIPRNG being able to pass the whole NIST battery. -Such a question is answered in Table~\ref{threshold}. - - -\begin{table*} -\renewcommand{\arraystretch}{1.3} -\caption{Functional power $m$ making it possible to pass the whole NIST battery} +\caption{Number of $\oplus$ operations to pass the whole NIST and DieHARD batteries} \label{threshold} \centering \begin{tabular}{|l||c|c|c|c|c|c|c|c|} @@ -1193,18 +1106,12 @@ Threshold value $m$& 19 & 7 & 2& 1 & 11& 9& 3& 4\\ \hline\hline \end{tabular} \end{table*} -\subsubsection{Results Summary} - -We can summarize the obtained results as follows. -\begin{enumerate} -\item The CIPRNG method is able to improve the statistical properties of a large variety of PRNGs. -\item Using different PRNGs in the CIPRNG approach is better than considering several instances of one unique PRNG. -\item The statistical quality of the outputs increases with the functional power $m$. -\end{enumerate} - +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 +raise ambiguity. \end{color} -\section{Efficient PRNG based on Chaotic Iterations} +\subsection{Efficient PRNG based on Chaotic Iterations} \label{sec:efficient PRNG} Based on the proof presented in the previous section, it is now possible to -- 2.39.5 From 850c033d45e9af70be22cb2e0c76a9de99d23c17 Mon Sep 17 00:00:00 2001 From: guyeux Date: Thu, 14 Jun 2012 12:20:23 +0200 Subject: [PATCH 10/16] truc bidule --- prng_gpu.tex | 26 +++++++++++++------------- 1 file changed, 13 insertions(+), 13 deletions(-) diff --git a/prng_gpu.tex b/prng_gpu.tex index bc40b2d..966dbaa 100644 --- a/prng_gpu.tex +++ b/prng_gpu.tex @@ -1111,20 +1111,20 @@ new be simply called CIPRNG, or ``the proposed PRNG'', if this statement does no raise ambiguity. \end{color} -\subsection{Efficient PRNG based on Chaotic Iterations} +\subsection{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}. -The first idea is to consider -that the provided strategy is a pseudorandom Boolean vector obtained by a -given PRNG. -An iteration of the system is simply the bitwise exclusive or between -the last computed state and the current strategy. -Topological properties of disorder exhibited by chaotic -iterations can be inherited by the inputted generator, we hope by doing so to -obtain some statistical improvements while preserving speed. - +% +%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}. +%The first idea is to consider +%that the provided strategy is a pseudorandom Boolean vector obtained by a +%given PRNG. +%An iteration of the system is simply the bitwise exclusive or between +%the last computed state and the current strategy. +%Topological properties of disorder exhibited by chaotic +%iterations can be inherited by the inputted generator, we hope by doing so to +%obtain some statistical improvements while preserving speed. +% %%RAPH : j'ai viré tout ca %% Let us give an example using 16-bits numbers, to clearly understand how the bitwise xor operations %% are -- 2.39.5 From 8f1af7e18d4d59611a7b16178ac5f32cfe541056 Mon Sep 17 00:00:00 2001 From: guyeux Date: Fri, 15 Jun 2012 09:44:52 +0200 Subject: [PATCH 11/16] dfjlskdjfl --- prng_gpu.tex | 57 ++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 57 insertions(+) diff --git a/prng_gpu.tex b/prng_gpu.tex index 966dbaa..c5fbd5d 100644 --- a/prng_gpu.tex +++ b/prng_gpu.tex @@ -13,6 +13,9 @@ \usepackage[standard]{ntheorem} \usepackage{algorithmic} \usepackage{slashbox} +\usepackage{ctable} +\usepackage{tabularx} +\usepackage{multirow} % Pour mathds : les ensembles IR, IN, etc. \usepackage{dsfont} @@ -1007,6 +1010,53 @@ a^1 & \text{if}~ z^{n-1} = 0 .\end{array} \right. \end{array}\end{equation} +\begin{table} +\renewcommand{\arraystretch}{1.3} +\caption{TestU01 Statistical Test} +\label{TestU011} +\centering + \begin{tabular}{lccccc} + \toprule +Test name &Tests& Logistic & XORshift & ISAAC\\ +Rabbit & 38 &21 &14 &0 \\ +Alphabit & 17 &16 &9 &0 \\ +Pseudo DieHARD &126 &0 &2 &0 \\ +FIPS\_140\_2 &16 &0 &0 &0 \\ +SmallCrush &15 &4 &5 &0 \\ +Crush &144 &95 &57 &0 \\ +Big Crush &160 &125 &55 &0 \\ \hline +Failures & &261 &146 &0 \\ +\bottomrule + \end{tabular} +\end{table} + + + +\begin{table} +\renewcommand{\arraystretch}{1.3} +\caption{TestU01 Statistical Test for Old CI algorithms ($\mathsf{N}=4$)} +\label{TestU01 for Old CI} +\centering + \begin{tabular}{lcccc} + \toprule +\multirow{3}*{Test name} & \multicolumn{4}{c}{Old CI}\\ +&Logistic& XORshift& ISAAC&ISAAC \\ +&+& +& + & + \\ +&Logistic& XORshift& XORshift&ISAAC \\ \cmidrule(r){2-5} +Rabbit &7 &2 &0 &0 \\ +Alphabit & 3 &0 &0 &0 \\ +DieHARD &0 &0 &0 &0 \\ +FIPS\_140\_2 &0 &0 &0 &0 \\ +SmallCrush &2 &0 &0 &0 \\ +Crush &47 &4 &0 &0 \\ +Big Crush &79 &3 &0 &0 \\ \hline +Failures &138 &9 &0 &0 \\ +\bottomrule + \end{tabular} +\end{table} + + + \subsection{Statistical tests} @@ -1106,6 +1156,13 @@ 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 +(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. + + 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 raise ambiguity. -- 2.39.5 From ecb1754c4e0d138d986131429812fb32f405953f Mon Sep 17 00:00:00 2001 From: couturie Date: Wed, 18 Jul 2012 11:12:37 +0100 Subject: [PATCH 12/16] reponse sur les tests --- prng_gpu.tex | 13 +++++++++++-- reponse.tex | 4 +++- 2 files changed, 14 insertions(+), 3 deletions(-) diff --git a/prng_gpu.tex b/prng_gpu.tex index c5fbd5d..bf74539 100644 --- a/prng_gpu.tex +++ b/prng_gpu.tex @@ -126,7 +126,16 @@ stringent statistical evaluation of a sequence claimed as random. This battery can be found in the well-known TestU01 package~\cite{LEcuyerS07}. Chaos, for its part, refers to the well-established definition of a chaotic dynamical system proposed by Devaney~\cite{Devaney}. - +\begin{color}{red} +More precisely, each time we performed a test on a PRNG, we ran it +twice in order to observe if all p-values are inside [0.01, 0.99]. In +fact, we observed that few p-values (less than ten) are sometimes +outside this interval but inside [0.001, 0.999], so that is why a +second run allows us to confirm that the values outside are not for +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]. +\end{color} 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 @@ -480,7 +489,7 @@ We have proposed in~\cite{bgw09:ip} a new family of generators that receives two PRNGs as inputs. These two generators are mixed with chaotic iterations, leading thus to a new PRNG that \begin{color}{red} -should improves the statistical properties of each +should improve the statistical properties of each generator taken alone. Furthermore, the generator obtained by this way possesses various chaos properties that none of the generators used as input present. diff --git a/reponse.tex b/reponse.tex index 0ffb36a..865604e 100644 --- a/reponse.tex +++ b/reponse.tex @@ -18,7 +18,9 @@ \bigskip \textit{The authors should include a summary of test measurements showing their method passes the test sets mentioned (NIST, Diehard, TestU01) instead of the one sentence saying it passed that is in section 1.} -\begin{color}{red} Raph, c'est pour toi ça.\end{color} +\begin{color}{red} In section 1, we have added a small summary of test measurements performed with BigCrush of TestU01. +As other tests (NIST, Diehard, SmallCrush and Crush of TestU01 ) are deemed less selective, in this paper we did not use them. +\end{color} \bigskip -- 2.39.5 From c471dd052c6b541bcbc3712b5c3cad2e0f0df08b Mon Sep 17 00:00:00 2001 From: cguyeux Date: Thu, 30 Aug 2012 14:01:05 +0200 Subject: [PATCH 13/16] lfkdjslkjfsd --- prng_gpu.tex | 9 ++++----- 1 file changed, 4 insertions(+), 5 deletions(-) diff --git a/prng_gpu.tex b/prng_gpu.tex index bf74539..c7853b2 100644 --- a/prng_gpu.tex +++ b/prng_gpu.tex @@ -565,7 +565,7 @@ This new generator is designed by the following process. First of all, some chaotic iterations have to be done to generate a sequence $\left(x^n\right)_{n\in\mathds{N}} \in \left(\mathds{B}^{32}\right)^\mathds{N}$ of Boolean vectors, which are the successive states of the iterated system. -Some of these vectors will be randomly extracted and our pseudo-random bit +Some of these vectors will be randomly extracted and our pseudorandom bit flow will be constituted by their components. Such chaotic iterations are realized as follows. Initial state $x^0 \in \mathds{B}^{32}$ is a Boolean vector taken as a seed and chaotic strategy $\left(S^n\right)_{n\in\mathds{N}}\in @@ -578,7 +578,7 @@ updated, as follows: $x_i^n = x_i^{n-1}$ if $i \neq S^n$, else $x_i^n = \overlin Such a procedure is equivalent to achieve chaotic iterations with the Boolean vectorial negation $f_0$ and some well-chosen strategies. Finally, some $x^n$ are selected -by a sequence $m^n$ as the pseudo-random bit sequence of our generator. +by a sequence $m^n$ as the pseudorandom bit sequence of our generator. $(m^n)_{n \in \mathds{N}} \in \mathcal{M}^\mathds{N}$ is computed from $PRNG_1$, where $\mathcal{M}\subset \mathds{N}^*$ is a finite nonempty set of integers. The basic design procedure of the New CI generator is summarized in Algorithm~\ref{Chaotic iteration1}. @@ -611,8 +611,7 @@ N \text{ if }\sum_{i=0}^{N-1}{C^i_{32}}\leqslant{y^n}<1.\\ } \ENDFOR \STATE$a\leftarrow{PRNG_1()}$\; -\STATE$m\leftarrow{g(a)}$\; -\STATE$k\leftarrow{m}$\; +\STATE$k\leftarrow{g(a)}$\; \WHILE{$i=0,\dots,k$} \STATE$b\leftarrow{PRNG_2()~mod~\mathsf{N}}$\; @@ -944,7 +943,7 @@ have $d((S,E),(\tilde S,E))<\epsilon$. \begin{color}{red} \section{Statistical Improvements Using Chaotic Iterations} -\label{The generation of pseudo-random sequence} +\label{The generation of pseudorandom sequence} Let us now explain why we are reasonable grounds to believe that chaos -- 2.39.5 From 9879779d913285ee14baad568f69be401dfd0fb3 Mon Sep 17 00:00:00 2001 From: cguyeux Date: Mon, 3 Sep 2012 09:09:09 +0200 Subject: [PATCH 14/16] =?utf8?q?avanc=C3=A9es=20dans=20la=20r=C3=A9=C3=A9c?= =?utf8?q?riture?= MIME-Version: 1.0 Content-Type: text/plain; charset=utf8 Content-Transfer-Encoding: 8bit --- prng_gpu.tex | 45 +++++++++++++++++++++++++++++---------------- 1 file changed, 29 insertions(+), 16 deletions(-) diff --git a/prng_gpu.tex b/prng_gpu.tex index c7853b2..0c9f9c7 100644 --- a/prng_gpu.tex +++ b/prng_gpu.tex @@ -90,7 +90,13 @@ 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. +sequence. \begin{color}{red} 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 +of the difficulty of this challenge when the size of the parameters of the PRNG increases. +\end{color} + Finally, a small part of the community working in this domain focuses on a third requirement, that is to define chaotic generators. The main idea is to take benefits from a chaotic dynamical system to obtain a @@ -124,18 +130,18 @@ statistical perfection refers to the ability to pass the whole {\it BigCrush} battery of tests, which is widely considered as the most stringent statistical evaluation of a sequence claimed as random. This battery can be found in the well-known TestU01 package~\cite{LEcuyerS07}. -Chaos, for its part, refers to the well-established definition of a -chaotic dynamical system proposed by Devaney~\cite{Devaney}. \begin{color}{red} More precisely, each time we performed a test on a PRNG, we ran it -twice in order to observe if all p-values are inside [0.01, 0.99]. In -fact, we observed that few p-values (less than ten) are sometimes +twice in order to observe if all $p-$values are inside [0.01, 0.99]. In +fact, we observed that few $p-$values (less than ten) are sometimes outside this interval but inside [0.001, 0.999], so that is why a second run allows us to confirm that the values outside are not for the same test. With this approach all our PRNGs pass the {\it - BigCrush} successfully and all p-values are at least once inside + BigCrush} successfully and all $p-$values are at least once inside [0.01, 0.99]. \end{color} +Chaos, for its part, refers to the well-established definition of a +chaotic dynamical system proposed 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 @@ -166,8 +172,13 @@ 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}. -Section~\ref{sec:efficient PRNG} presents an efficient -implementation of this chaotic PRNG on a CPU, whereas Section~\ref{sec:efficient PRNG +\begin{color}{red} +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. +\end{color} + Section~\ref{sec:efficient PRNG gpu} describes and evaluates theoretically the GPU implementation. Such generators are experimented in Section~\ref{sec:experiments}. @@ -176,7 +187,8 @@ generator is cryptographically secure, then it is the case too for the generator provided by the post-treatment. 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}, and to an improvement of the +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 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. @@ -184,7 +196,7 @@ summarized and intended future work is presented. -\section{Related works on GPU based PRNGs} +\section{Related work on GPU based PRNGs} \label{section:related works} Numerous research works on defining GPU based PRNGs have already been proposed in the @@ -584,8 +596,8 @@ $(m^n)_{n \in \mathds{N}} \in \mathcal{M}^\mathds{N}$ 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 is required to make the outputs uniform in $\llbracket 0, 2^\mathsf{N}-1 \rrbracket$ -(the reader is referred to~\cite{bg10:ip} for more information). +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 +goal (other candidates and more information can be found in ~\cite{bg10:ip}). \begin{equation} \label{Formula} @@ -651,7 +663,7 @@ x^0 \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket, S \in \llbracket 0, 2^\mathsf{N \forall n \in \mathds{N}^*, x^n = x^{n-1} \oplus S^n, \end{array} \right. -\label{equation Oplus0} +\label{equation Oplus} \end{equation} where $\oplus$ is for the bitwise exclusive or between two integers. This rewriting can be understood as follows. The $n-$th term $S^n$ of the @@ -661,7 +673,7 @@ as an integer having $\mathsf{N}$ bits too). More precisely, the $k-$th component of this state (a binary digit) changes if and only if the $k-$th digit in the binary decomposition of $S^n$ is 1. -The single basic component presented in Eq.~\ref{equation Oplus0} is of +The single basic component presented in Eq.~\ref{equation Oplus} is of ordinary use as a good elementary brick in various PRNGs. It corresponds to the following discrete dynamical system in chaotic iterations: @@ -683,7 +695,7 @@ we select a subset of components to change. Obviously, replacing the previous CI PRNG Algorithms by -Equation~\ref{equation Oplus0}, which is possible when the iteration function is +Equation~\ref{equation Oplus}, which is possible when the iteration function is the vectorial negation, leads to a speed improvement (the resulting generator will be referred as ``Xor CI PRNG'' in what follows). @@ -1221,7 +1233,7 @@ raise ambiguity. -\lstset{language=C,caption={C code of the sequential PRNG based on chaotic iterations},label=algo:seqCIPRNG} +\lstset{language=C,caption={C code of the sequential PRNG based on chaotic iterations},label={algo:seqCIPRNG}} \begin{small} \begin{lstlisting} @@ -1731,6 +1743,7 @@ 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, -- 2.39.5 From 26a94eb935af7804c64342d5c4718e05c9e10036 Mon Sep 17 00:00:00 2001 From: cguyeux Date: Mon, 3 Sep 2012 16:48:42 +0200 Subject: [PATCH 15/16] Ava --- mabase.bib | 10 +---- prng_gpu.tex | 113 ++++++++++++++++++++++++++++++++++++++++++--------- 2 files changed, 95 insertions(+), 28 deletions(-) diff --git a/mabase.bib b/mabase.bib index 30ad80e..de47741 100644 --- a/mabase.bib +++ b/mabase.bib @@ -41,6 +41,7 @@ note = {To appear}, @Article{combined_lcg, title = "Efficient and portable combined random number generators", + author = {}, journal = "Communications of the ACM", volume = "31", number = "6", @@ -164,13 +165,6 @@ note = {To appear}, timestamp = {2009.06.29} } -@Book{Goldreich, - author = {Oded Goldreich}, - ALTeditor = {}, - title = {Foundations of Cryptography: Basic Tools}, - publisher = {Cambridge University Press}, - year = {2007}, -} @INPROCEEDINGS{DBLP:conf/cec/HiggsSHS10, author = {Trent Higgs and Bela Stantic and Tamjidul Hoque and Abdul Sattar}, @@ -4308,7 +4302,7 @@ note = {To appear}, } @Article{ZRKB10, - author = {A. Zhmurov, K. Rybnikov, Y. Kholodov, and V. Barsegov}, + author = {A. Zhmurov and K. Rybnikov and Y. Kholodov and V. Barsegov}, title = {Generation of Random Numbers on Graphics Processors: Forced Indentation In Silico of the Bacteriophage HK97}, journal = {J. Phys. Chem. B}, year = {2011}, diff --git a/prng_gpu.tex b/prng_gpu.tex index 0c9f9c7..983aa92 100644 --- a/prng_gpu.tex +++ b/prng_gpu.tex @@ -255,7 +255,7 @@ 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 +596,7 @@ $(m^n)_{n \in \mathds{N}} \in \mathcal{M}^\mathds{N}$ 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} @@ -960,33 +960,100 @@ 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 these 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 enter into a loop. A similar importance for regularity is emphasized in +the two following 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 stated 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}. This property is related to the following 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}. Both in topological and statistical fields. + \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 (m is the length of each block). + \end{itemize} + + \item \textbf{Non-linearity, complexity}. + \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 (M is the length in bits of a block). + \end{itemize} +\end{itemize} + + + + \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 +1080,12 @@ 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 +1199,9 @@ 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 +1228,9 @@ 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*} @@ -1176,15 +1245,19 @@ 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} -- 2.39.5 From a0cdbfd7933130eaba676883427b46743a6cf7ea Mon Sep 17 00:00:00 2001 From: cguyeux Date: Mon, 3 Sep 2012 18:55:08 +0200 Subject: [PATCH 16/16] balblalab --- prng_gpu.tex | 48 +++++++++++++++++++++++++++++++++--------------- 1 file changed, 33 insertions(+), 15 deletions(-) diff --git a/prng_gpu.tex b/prng_gpu.tex index 983aa92..39cee40 100644 --- a/prng_gpu.tex +++ b/prng_gpu.tex @@ -878,6 +878,8 @@ 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,$ $\exists N_{0}=max(n_{0},n_{1},n_{2})\in \mathds{N} ,$ $\forall n\geqslant N_{0},$ @@ -977,15 +979,15 @@ 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 these illustrative links give only qualitative arguments, +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. +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} @@ -993,8 +995,8 @@ two following fields: mathematical chaos and statistics. 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 enter into a loop. A similar importance for regularity is emphasized in -the two following tests~\cite{Nist10}: +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. @@ -1004,32 +1006,48 @@ the two following tests~\cite{Nist10}: 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 stated visited during a random walk in the two tests below~\cite{Nist10}: +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}. This property is related to the following test~\cite{Nist10}. +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}. Both in topological and statistical fields. + \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 (m is the length of each block). +\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}. + \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 (M is the length in bits of a block). +\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{} 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} -- 2.39.5