Start RSA with bezout
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2 changed files with 139 additions and 6 deletions
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@ -4,7 +4,7 @@
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\author{}
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\date{Dernière compilation~: \today{} à \currenttime}
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\usepackage{../cours}
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\usepackage{styles}
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\usepackage{tikz}
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\usetikzlibrary{shapes.multipart}
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\DeclareMathOperator\pgcd{pgcd}
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@ -704,11 +704,9 @@
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On ne peut donc pas choisir $e$ et $d$ au hasard.
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$e$ et $d$ sont liés.
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\subsection{Indicatrice d'Euler}
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\subsection{$\varphi(n)$~: Indicatrice d'Euler}
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Dans $\frac{\mathbb{Z}}{n\mathbb{Z}}$~:
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\begin{center}
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\begin{tabular}{cc|cccccrc}
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\multicolumn{2}{c}{$n=7$} & \multicolumn{7}{c}{puissances} \\
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\toprule
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@ -723,7 +721,6 @@
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$\overline{6}$ & oui & $\overline{6}$ & $\overline{1}$ & $\overline{6}$ & $\overline{1}$ & $\overline{6}$ & $\overline{1}$ & $\overline{6}$ \\
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\bottomrule
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\end{tabular}
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\end{center}
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\begin{itemize}
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\item 6 éléments inversibles
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@ -766,7 +763,6 @@
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\end{align*}
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Ceci implique que, par exemple~:
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\begin{align*}
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\text{Dans } \frac{\mathbb{Z}}{200\mathbb{Z}}\;: \varphi(200) = 80 \implies 80 \text{ éléments inversibles} \\
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\overline{37} \text{ est inversible} \\
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@ -782,4 +778,47 @@
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\eqref{eq:rsa} est le théorême fondamental de RSA\@.
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\subsection{$\varphi(n)$ appliqué à l'identité de Bezout}
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Le fonctionnement de RSA se base sur $\varphi(n)$ appliqué à l'identité de Bezout.
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Pour rappel~:
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\begin{equation*}
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\forall \; (a,b) \in \mathbb{N}^2 : \exists\, (u,v) \in \mathbb{Z}^2 \; / \; a \times u + b \times v = \pgcd(a,b)
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\end{equation*}
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Soit $n=p \times q$ \quad on a $\varphi(n)$ \quad on choisit $e \; / \pgcd(e, \varphi(n)) = 1$
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En appliquant Bezout à $e$ et $\varphi(n)$~:
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\begin{equation*}
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\exists\, (d,k) \; / \; e \cdot d + k \cdot \varphi(n) = 1
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\end{equation*}
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\begin{align*}
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\text{Soit } a \in [1,n-1] \quad
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&a^{(ed + k\varphi(n))} = a^1 \\
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\iff & a^{ed} \times a^{k\varphi(n)} = a \\
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\iff & a^{ed} \times (a^{\varphi(n)})^k = a \\
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\iff & a^{ed}[n] \times (a^{\varphi(n)}[n])^k = a
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\quad\text{ or d'après RSA~: } a^{\varphi(n)}[n] = 1 \\
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\text{donc } & a^{ed}[n] \times 1^k = a \\
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\iff & a^{ed}[n] = a
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\end{align*}
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\begin{equation*}
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\text{alors } \exists\, d\; /\; \forall\; a \in [1,n-1] \quad a^{ed}[n] = a
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\end{equation*}
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\hfill
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où $d$ est le coefficient de Bezout associé à $e$.
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\begin{center}
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\begin{tikzpicture}
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\node (a1) at (0,0) {$a$};
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\node (chiff) at (2,0) {$a^e[n]$};
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\draw [-latex] (a1) -- (chiff);
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\node (dechiff) at (6,0) {$(a^e[n])^d[n] = a^{ed}[n]$};
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\draw [-latex] (chiff) -- (dechiff);
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\node (a2) at (10,0) {$a$};
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\draw [-latex] (dechiff) -- (a2);
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\end{tikzpicture}
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\end{center}
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\end{document}
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94
algebre-non-lineaire/styles.sty
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94
algebre-non-lineaire/styles.sty
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@ -0,0 +1,94 @@
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\ProvidesPackage{styles}
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\usepackage[
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%showframe,
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a4paper,includeheadfoot,margin=2cm,top=1cm,bottom=1cm
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]{geometry}
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\setcounter{tocdepth}{2}
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\usepackage[utf8]{inputenc}
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\usepackage[T1]{fontenc}
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\usepackage{babel}
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\usepackage{datetime}
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\usepackage{color}
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\usepackage{amsmath}
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\usepackage{amsfonts}
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\everymath{\displaystyle}
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\newcommand*\dif{\mathop{}\!\mathrm{d}}
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\usepackage{booktabs}
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\usepackage{tabularx}
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\newcolumntype{Y}{>{\centering\arraybackslash}X}
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\usepackage{multicol}
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\usepackage{multirow}
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\usepackage{makecell}
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\usepackage{parskip}
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\usepackage{microtype}
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\usepackage{listingsutf8}
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\lstset{%
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inputencoding=utf8/latin1,
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basicstyle=\small\ttfamily,
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columns=flexible,
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breaklines=true,
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extendedchars=true,
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literate=
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{á}{{\'a}}1 {é}{{\'e}}1 {í}{{\'\i}}1 {ó}{{\'o}}1 {ú}{{\'u}}1
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{Á}{{\'A}}1 {É}{{\'E}}1 {Í}{{\'I}}1 {Ó}{{\'O}}1 {Ú}{{\'U}}1
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{à}{{\`a}}1 {è}{{\`e}}1 {ì}{{\`\i}}1 {ò}{{\`o}}1 {ù}{{\`u}}1
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{À}{{\`A}}1 {È}{{\`E}}1 {Ì}{{\`I}}1 {Ò}{{\`O}}1 {Ù}{{\`U}}1
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{ä}{{\"a}}1 {ë}{{\"e}}1 {ï}{{\"\i}}1 {ö}{{\"o}}1 {ü}{{\"u}}1
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{Ä}{{\"A}}1 {Ë}{{\"E}}1 {Ï}{{\"I}}1 {Ö}{{\"O}}1 {Ü}{{\"U}}1
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{â}{{\^a}}1 {ê}{{\^e}}1 {î}{{\^\i}}1 {ô}{{\^o}}1 {û}{{\^u}}1
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{Â}{{\^A}}1 {Ê}{{\^E}}1 {Î}{{\^I}}1 {Ô}{{\^O}}1 {Û}{{\^U}}1
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{Ã}{{\~A}}1 {ã}{{\~a}}1 {Õ}{{\~O}}1 {õ}{{\~o}}1 {œ}{{\oe}}1
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{Œ}{{\OE}}1 {æ}{{\ae}}1 {Æ}{{\AE}}1 {ß}{{\ss}}1 {ű}{{\H{u}}}1
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{Ű}{{\H{U}}}1 {ő}{{\H{o}}}1 {Ő}{{\H{O}}}1 {ç}{{\c c}}1
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{Ç}{{\c C}}1 {ø}{{\o}}1 {å}{{\r a}}1 {Å}{{\r A}}1
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{€}{{\euro}}1 {£}{{\pounds}}1 {«}{{\guillemotleft}}1
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{»}{{\guillemotright}}1 {ñ}{{\~n}}1 {Ñ}{{\~N}}1 {¿}{{?`}}1
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}
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\let\oldlstinputlisting\lstinputlisting
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% make \lstinputlisting always have frame
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\renewcommand{\lstinputlisting}[2][]{\oldlstinputlisting[frame=single,#1]{#2}}
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\lstnewenvironment{console}[1][]{%
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\lstset{%
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inputencoding=utf8/latin1,
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basicstyle=\small\ttfamily\color{white},
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columns=flexible,
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breaklines=true,
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extendedchars=true,
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literate=
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{á}{{\'a}}1 {é}{{\'e}}1 {í}{{\'\i}}1 {ó}{{\'o}}1 {ú}{{\'u}}1
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{Á}{{\'A}}1 {É}{{\'E}}1 {Í}{{\'I}}1 {Ó}{{\'O}}1 {Ú}{{\'U}}1
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{à}{{\`a}}1 {è}{{\`e}}1 {ì}{{\`\i}}1 {ò}{{\`o}}1 {ù}{{\`u}}1
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{À}{{\`A}}1 {È}{{\`E}}1 {Ì}{{\`I}}1 {Ò}{{\`O}}1 {Ù}{{\`U}}1
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{ä}{{\"a}}1 {ë}{{\"e}}1 {ï}{{\"\i}}1 {ö}{{\"o}}1 {ü}{{\"u}}1
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{Ä}{{\"A}}1 {Ë}{{\"E}}1 {Ï}{{\"I}}1 {Ö}{{\"O}}1 {Ü}{{\"U}}1
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{â}{{\^a}}1 {ê}{{\^e}}1 {î}{{\^\i}}1 {ô}{{\^o}}1 {û}{{\^u}}1
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{Â}{{\^A}}1 {Ê}{{\^E}}1 {Î}{{\^I}}1 {Ô}{{\^O}}1 {Û}{{\^U}}1
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{Ã}{{\~A}}1 {ã}{{\~a}}1 {Õ}{{\~O}}1 {õ}{{\~o}}1 {œ}{{\oe}}1
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{Œ}{{\OE}}1 {æ}{{\ae}}1 {Æ}{{\AE}}1 {ß}{{\ss}}1 {ű}{{\H{u}}}1
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{Ű}{{\H{U}}}1 {ő}{{\H{o}}}1 {Ő}{{\H{O}}}1 {ç}{{\c c}}1
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{Ç}{{\c C}}1 {ø}{{\o}}1 {å}{{\r a}}1 {Å}{{\r A}}1
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{€}{{\euro}}1 {£}{{\pounds}}1 {«}{{\guillemotleft}}1
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{»}{{\guillemotright}}1 {ñ}{{\~n}}1 {Ñ}{{\~N}}1 {¿}{{?`}}1,
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backgroundcolor=\color{black}, #1
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}%
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}{}
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\usepackage{graphicx}
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\usepackage[hyphens]{url}
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\usepackage[colorlinks,hidelinks]{hyperref}
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\usepackage{letltxmacro}
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\LetLtxMacro{\normalparagraph}{\paragraph}
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\renewcommand{\paragraph}[1]{\normalparagraph{#1}\mbox{}}
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\sloppy
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