aula-python-sage.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Python para prototipação em Cripto\n",
    "Ou: Uma introdução ao SageMath\n",
    "\n",
    "Thales Paiva \\\n",
    "Cripto 2021 - IME - USP\n",
    "\n",
    "***"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Pequena nota sobre implementação de Cripto\n",
    "\n",
    "Implementar Criptografia é muito difícil:\n",
    "* Não basta mirar em eliminar bugs\n",
    "\n",
    "Metas são \n",
    "* Algoritmos devem rodar em tempo independente dos segredos\n",
    "* Consumo de energia não pode depender de segredos\n",
    "* Acesso a memória deve ser uniforme\n",
    "* Implementação deve ser eficiente ou não será usada\n",
    "\n",
    "Python parece adequado para implementação criptográfica?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Implementação é quase sempre feita em C (às vezes C++ e hoje em dia em Rust, que é um alívio)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Por que estamos fazendo vocês aprenderem Python então?\n",
    "Para implementar Cripto:\n",
    "1. Primeiro prototipamos a especificação em Python\n",
    "2. Implementamos em C\n",
    "3. Testamos a implementação em C usando a implementação em Python\n",
    "4. Fazemos as otimizações e implementação em tempo constante"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Sage"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### O que é o Sage?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Sage é um sistema de computação simbólica \n",
    "- Oferece uma interface em Python para Álgebra, Grafos, Análise Numérica, e muito mais\n",
    "- No Backend, usa sistemas como LAPACK, ATLAS, PARI/GP e M4RI\n",
    "- Distribuído sob a licença GPLv3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "Formalmente ( :\n",
    "$$\n",
    "\\text{Sage} > \\left( \\text{Magma} \\cup \\text{Maple} \\cup \\text{Mathematica} \\cup \\text{MATLAB} \\right)\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Há duas semanas dei uma monitoria básica sobre Sage, que pode ser acessada aqui:\n",
    "https://drive.google.com/file/d/1Lf7AWQ92n7Scft1OoR_DRT3ufEgt7oLv/view?usp=sharing\n",
    "\n",
    "O texto auxiliar está aqui:\n",
    "https://github.com/thalespaiva/monitoria-cripto/blob/master/intro-sage-ep1.ipynb"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Mas o que é computação simbólica?**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sistemas de computação simbólica permitem trabalhar com expressões matemáticas:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "x, y = var('x, y')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(x, y) |--> x^2 + sqrt(y)\n"
     ]
    }
   ],
   "source": [
    "f(x, y) = x**2 + sqrt(y)\n",
    "print(f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(x, y) |--> (2*x, 1/2/sqrt(y))"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\n",
    "f.derivative()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
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       "\n",
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       "\n",
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       "\n",
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       "\n",
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       "    camera.up.set( 0, 0, 1 );\n",
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       "\n",
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       "\n",
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       "\n",
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       "\n",
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       "\n",
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       "\n",
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       "\n",
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       "        var nearClip = 0.01 * midToCorner;\n",
       "        var farClip = 100 * midToCorner;\n",
       "\n",
       "        if ( options.projection === 'orthographic' ) {\n",
       "            var camera = new THREE.OrthographicCamera( -1, 1, 1, -1, -farClip, farClip );\n",
       "            updateCameraAspect( camera, aspect );\n",
       "            return camera;\n",
       "        }\n",
       "\n",
       "        return new THREE.PerspectiveCamera( 45, aspect, nearClip, farClip );\n",
       "\n",
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       "\n",
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       "            camera.aspect = aspect;\n",
       "        } else if ( camera.isOrthographicCamera ) {\n",
       "            // Fit the camera frustum to the bounding box's diagonal so that the entire plot fits\n",
       "            // within at the default zoom level and camera position.\n",
       "            if ( aspect > 1 ) { // Wide window\n",
       "                camera.top = midToCorner;\n",
       "                camera.right = midToCorner * aspect;\n",
       "            } else { // Tall or square window\n",
       "                camera.top = midToCorner / aspect;\n",
       "                camera.right = midToCorner;\n",
       "            }\n",
       "            camera.bottom = -camera.top;\n",
       "            camera.left = -camera.right;\n",
       "        }\n",
       "\n",
       "        camera.updateProjectionMatrix();\n",
       "\n",
       "    }\n",
       "\n",
       "    var lights = [{&quot;x&quot;:-5, &quot;y&quot;:3, &quot;z&quot;:0, &quot;color&quot;:&quot;#7f7f7f&quot;, &quot;parent&quot;:&quot;camera&quot;}];\n",
       "    for ( var i=0 ; i < lights.length ; i++ ) {\n",
       "        var light = new THREE.DirectionalLight( lights[i].color, 1 );\n",
       "        light.position.set( a[0]*lights[i].x, a[1]*lights[i].y, a[2]*lights[i].z );\n",
       "        if ( lights[i].parent === 'camera' ) {\n",
       "            light.target.position.set( a[0]*xMid, a[1]*yMid, a[2]*zMid );\n",
       "            scene.add( light.target );\n",
       "            camera.add( light );\n",
       "        } else scene.add( light );\n",
       "    }\n",
       "    scene.add( camera );\n",
       "\n",
       "    var ambient = {&quot;color&quot;:&quot;#7f7f7f&quot;};\n",
       "    scene.add( new THREE.AmbientLight( ambient.color, 1 ) );\n",
       "\n",
       "    var controls = new THREE.OrbitControls( camera, renderer.domElement );\n",
       "    controls.target.set( a[0]*xMid, a[1]*yMid, a[2]*zMid );\n",
       "    controls.addEventListener( 'change', function() { if ( !animate ) render(); } );\n",
       "\n",
       "    window.addEventListener( 'resize', function() {\n",
       "\n",
       "        renderer.setSize( window.innerWidth, window.innerHeight );\n",
       "        updateCameraAspect( camera, window.innerWidth / window.innerHeight );\n",
       "        if ( window.rescaleFatLines ) rescaleFatLines();\n",
       "        if ( !animate ) render();\n",
       "\n",
       "    } );\n",
       "\n",
       "    var texts = [];\n",
       "    for ( var i=0 ; i < texts.length ; i++ ) addText( texts[i] );\n",
       "\n",
       "    function addText( json ) {\n",
       "        var sprite = addLabel( json.text, a[0]*json.x, a[1]*json.y, a[2]*json.z, json );\n",
       "        sprite.userData = json;\n",
       "    }\n",
       "\n",
       "    var points = [];\n",
       "    for ( var i=0 ; i < points.length ; i++ ) addPoint( points[i] );\n",
       "\n",
       "    function addPoint( json ) {\n",
       "\n",
       "        var geometry = new THREE.Geometry();\n",
       "        var v = json.point;\n",
       "        geometry.vertices.push( new THREE.Vector3( a[0]*v[0], a[1]*v[1], a[2]*v[2] ) );\n",
       "\n",
       "        var canvas = document.createElement( 'canvas' );\n",
       "        canvas.width = 128;\n",
       "        canvas.height = 128;\n",
       "\n",
       "        var context = canvas.getContext( '2d' );\n",
       "        context.arc( 64, 64, 64, 0, 2 * Math.PI );\n",
       "        context.fillStyle = json.color;\n",
       "        context.fill();\n",
       "\n",
       "        var texture = new THREE.Texture( canvas );\n",
       "        texture.needsUpdate = true;\n",
       "\n",
       "        var transparent = json.opacity < 1 ? true : false;\n",
       "        var size = camera.isOrthographicCamera ? json.size : json.size/100;\n",
       "        var material = new THREE.PointsMaterial( { size: size, map: texture,\n",
       "                                                   transparent: transparent, opacity: json.opacity,\n",
       "                                                   alphaTest: .1 } );\n",
       "\n",
       "        var c = new THREE.Vector3();\n",
       "        geometry.computeBoundingBox();\n",
       "        geometry.boundingBox.getCenter( c );\n",
       "        geometry.translate( -c.x, -c.y, -c.z );\n",
       "\n",
       "        var mesh = new THREE.Points( geometry, material );\n",
       "        mesh.position.set( c.x, c.y, c.z );\n",
       "        mesh.userData = json;\n",
       "        scene.add( mesh );\n",
       "\n",
       "    }\n",
       "\n",
       "    var lines = [];\n",
       "    for ( var i=0 ; i < lines.length ; i++ ) addLine( lines[i] );\n",
       "\n",
       "    function addLine( json ) {\n",
       "\n",
       "        var geometry = new THREE.Geometry();\n",
       "        for ( var i=0 ; i < json.points.length ; i++ ) {\n",
       "            var v = json.points[i];\n",
       "            geometry.vertices.push( new THREE.Vector3( a[0]*v[0], a[1]*v[1], a[2]*v[2] ) );\n",
       "        }\n",
       "\n",
       "        var c = new THREE.Vector3();\n",
       "        geometry.computeBoundingBox();\n",
       "        geometry.boundingBox.getCenter( c );\n",
       "        geometry.translate( -c.x, -c.y, -c.z );\n",
       "\n",
       "        var transparent = json.opacity < 1 ? true : false;\n",
       "        var materialOptions = { color: json.color, linewidth: json.linewidth,\n",
       "                                transparent: transparent, opacity: json.opacity };\n",
       "\n",
       "        var mesh;\n",
       "        if ( json.linewidth > 1 && window.createFatLineStrip ) {\n",
       "            mesh = createFatLineStrip( geometry, materialOptions );\n",
       "        } else {\n",
       "            var material = new THREE.LineBasicMaterial( materialOptions );\n",
       "            mesh = new THREE.Line( geometry, material );\n",
       "        }\n",
       "\n",
       "        mesh.position.set( c.x, c.y, c.z );\n",
       "        mesh.userData = json;\n",
       "        scene.add( mesh );\n",
       "\n",
       "    }\n",
       "\n",
       "    var surfaces = [{&quot;vertices&quot;: [{&quot;x&quot;: -1.0, &quot;y&quot;: 0.02564102564102587, &quot;z&quot;: 1.1601281538050878}, {&quot;x&quot;: -0.9487179487179487, &quot;y&quot;: 0.02564102564102587, &quot;z&quot;: 1.0601939000246803}, {&quot;x&quot;: -1.0, &quot;y&quot;: 0.07692307692307715, &quot;z&quot;: 1.277350098112615}, {&quot;x&quot;: -0.9487179487179487, &quot;y&quot;: 0.07692307692307715, &quot;z&quot;: 1.1774158443322074}, {&quot;x&quot;: -1.0, &quot;y&quot;: 0.12820512820512842, &quot;z&quot;: 1.3580574370197167}, {&quot;x&quot;: -0.9487179487179487, &quot;y&quot;: 0.12820512820512842, &quot;z&quot;: 1.258123183239309}, {&quot;x&quot;: -1.0, &quot;y&quot;: 0.17948717948717968, &quot;z&quot;: 1.4236592728681619}, {&quot;x&quot;: -0.9487179487179487, &quot;y&quot;: 0.17948717948717968, &quot;z&quot;: 1.3237250190877543}, {&quot;x&quot;: -1.0, &quot;y&quot;: 0.23076923076923095, &quot;z&quot;: 1.4803844614152615}, {&quot;x&quot;: -0.9487179487179487, 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572, 571], [552, 553, 573, 572], [553, 554, 574, 573], [554, 555, 575, 574], [555, 556, 576, 575], [556, 557, 577, 576], [557, 558, 578, 577], [558, 559, 579, 578], [560, 561, 581, 580], [561, 562, 582, 581], [562, 563, 583, 582], [563, 564, 584, 583], [564, 565, 585, 584], [565, 566, 586, 585], [566, 567, 587, 586], [567, 568, 588, 587], [568, 569, 589, 588], [569, 570, 590, 589], [570, 571, 591, 590], [571, 572, 592, 591], [572, 573, 593, 592], [573, 574, 594, 593], [574, 575, 595, 594], [575, 576, 596, 595], [576, 577, 597, 596], [577, 578, 598, 597], [578, 579, 599, 598], [580, 581, 601, 600], [581, 582, 602, 601], [582, 583, 603, 602], [583, 584, 604, 603], [584, 585, 605, 604], [585, 586, 606, 605], [586, 587, 607, 606], [587, 588, 608, 607], [588, 589, 609, 608], [589, 590, 610, 609], [590, 591, 611, 610], [591, 592, 612, 611], [592, 593, 613, 612], [593, 594, 614, 613], [594, 595, 615, 614], [595, 596, 616, 615], [596, 597, 617, 616], [597, 598, 618, 617], [598, 599, 619, 618], [600, 601, 621, 620], [601, 602, 622, 621], [602, 603, 623, 622], [603, 604, 624, 623], [604, 605, 625, 624], [605, 606, 626, 625], [606, 607, 627, 626], [607, 608, 628, 627], [608, 609, 629, 628], [609, 610, 630, 629], [610, 611, 631, 630], [611, 612, 632, 631], [612, 613, 633, 632], [613, 614, 634, 633], [614, 615, 635, 634], [615, 616, 636, 635], [616, 617, 637, 636], [617, 618, 638, 637], [618, 619, 639, 638], [620, 621, 641, 640], [621, 622, 642, 641], [622, 623, 643, 642], [623, 624, 644, 643], [624, 625, 645, 644], [625, 626, 646, 645], [626, 627, 647, 646], [627, 628, 648, 647], [628, 629, 649, 648], [629, 630, 650, 649], [630, 631, 651, 650], [631, 632, 652, 651], [632, 633, 653, 652], [633, 634, 654, 653], [634, 635, 655, 654], [635, 636, 656, 655], [636, 637, 657, 656], [637, 638, 658, 657], [638, 639, 659, 658], [640, 641, 661, 660], [641, 642, 662, 661], [642, 643, 663, 662], [643, 644, 664, 663], [644, 645, 665, 664], [645, 646, 666, 665], [646, 647, 667, 666], [647, 648, 668, 667], [648, 649, 669, 668], [649, 650, 670, 669], [650, 651, 671, 670], [651, 652, 672, 671], [652, 653, 673, 672], [653, 654, 674, 673], [654, 655, 675, 674], [655, 656, 676, 675], [656, 657, 677, 676], [657, 658, 678, 677], [658, 659, 679, 678], [660, 661, 681, 680], [661, 662, 682, 681], [662, 663, 683, 682], [663, 664, 684, 683], [664, 665, 685, 684], [665, 666, 686, 685], [666, 667, 687, 686], [667, 668, 688, 687], [668, 669, 689, 688], [669, 670, 690, 689], [670, 671, 691, 690], [671, 672, 692, 691], [672, 673, 693, 692], [673, 674, 694, 693], [674, 675, 695, 694], [675, 676, 696, 695], [676, 677, 697, 696], [677, 678, 698, 697], [678, 679, 699, 698], [680, 681, 701, 700], [681, 682, 702, 701], [682, 683, 703, 702], [683, 684, 704, 703], [684, 685, 705, 704], [685, 686, 706, 705], [686, 687, 707, 706], [687, 688, 708, 707], [688, 689, 709, 708], [689, 690, 710, 709], [690, 691, 711, 710], [691, 692, 712, 711], [692, 693, 713, 712], [693, 694, 714, 713], [694, 695, 715, 714], [695, 696, 716, 715], [696, 697, 717, 716], [697, 698, 718, 717], [698, 699, 719, 718], [700, 701, 721, 720], [701, 702, 722, 721], [702, 703, 723, 722], [703, 704, 724, 723], [704, 705, 725, 724], [705, 706, 726, 725], [706, 707, 727, 726], [707, 708, 728, 727], [708, 709, 729, 728], [709, 710, 730, 729], [710, 711, 731, 730], [711, 712, 732, 731], [712, 713, 733, 732], [713, 714, 734, 733], [714, 715, 735, 734], [715, 716, 736, 735], [716, 717, 737, 736], [717, 718, 738, 737], [718, 719, 739, 738], [720, 721, 741, 740], [721, 722, 742, 741], [722, 723, 743, 742], [723, 724, 744, 743], [724, 725, 745, 744], [725, 726, 746, 745], [726, 727, 747, 746], [727, 728, 748, 747], [728, 729, 749, 748], [729, 730, 750, 749], [730, 731, 751, 750], [731, 732, 752, 751], [732, 733, 753, 752], [733, 734, 754, 753], [734, 735, 755, 754], [735, 736, 756, 755], [736, 737, 757, 756], [737, 738, 758, 757], [738, 739, 759, 758], [740, 741, 761, 760], [741, 742, 762, 761], [742, 743, 763, 762], [743, 744, 764, 763], [744, 745, 765, 764], [745, 746, 766, 765], [746, 747, 767, 766], [747, 748, 768, 767], [748, 749, 769, 768], [749, 750, 770, 769], [750, 751, 771, 770], [751, 752, 772, 771], [752, 753, 773, 772], [753, 754, 774, 773], [754, 755, 775, 774], [755, 756, 776, 775], [756, 757, 777, 776], [757, 758, 778, 777], [758, 759, 779, 778], [760, 761, 781, 780], [761, 762, 782, 781], [762, 763, 783, 782], [763, 764, 784, 783], [764, 765, 785, 784], [765, 766, 786, 785], [766, 767, 787, 786], [767, 768, 788, 787], [768, 769, 789, 788], [769, 770, 790, 789], [770, 771, 791, 790], [771, 772, 792, 791], [772, 773, 793, 792], [773, 774, 794, 793], [774, 775, 795, 794], [775, 776, 796, 795], [776, 777, 797, 796], [777, 778, 798, 797], [778, 779, 799, 798]], &quot;color&quot;: &quot;#6666ff&quot;, &quot;opacity&quot;: 1.0}];\n",
       "    for ( var i=0 ; i < surfaces.length ; i++ ) addSurface( surfaces[i] );\n",
       "\n",
       "    function addSurface( json ) {\n",
       "\n",
       "        var useFaceColors = 'faceColors' in json ? true : false;\n",
       "\n",
       "        var geometry = new THREE.Geometry();\n",
       "        for ( var i=0 ; i < json.vertices.length ; i++ ) {\n",
       "            var v = json.vertices[i];\n",
       "            geometry.vertices.push( new THREE.Vector3( a[0]*v.x, a[1]*v.y, a[2]*v.z ) );\n",
       "        }\n",
       "        for ( var i=0 ; i < json.faces.length ; i++ ) {\n",
       "            var f = json.faces[i];\n",
       "            for ( var j=0 ; j < f.length - 2 ; j++ ) {\n",
       "                var face = new THREE.Face3( f[0], f[j+1], f[j+2] );\n",
       "                if ( useFaceColors ) face.color.set( json.faceColors[i] );\n",
       "                geometry.faces.push( face );\n",
       "            }\n",
       "        }\n",
       "        geometry.computeVertexNormals();\n",
       "\n",
       "        var side = json.singleSide ? THREE.FrontSide : THREE.DoubleSide;\n",
       "        var transparent = json.opacity < 1 ? true : false;\n",
       "        var depthWrite = 'depthWrite' in json ? json.depthWrite : !transparent;\n",
       "        var flatShading = json.useFlatShading ? json.useFlatShading : false;\n",
       "\n",
       "        var material = new THREE.MeshPhongMaterial( { side: side,\n",
       "                                     color: useFaceColors ? 'white' : json.color,\n",
       "                                     vertexColors: useFaceColors ? THREE.FaceColors : THREE.NoColors,\n",
       "                                     transparent: transparent, opacity: json.opacity,\n",
       "                                     shininess: 20, flatShading: flatShading,\n",
       "                                     depthWrite: depthWrite } );\n",
       "\n",
       "        var c = new THREE.Vector3();\n",
       "        geometry.computeBoundingBox();\n",
       "        geometry.boundingBox.getCenter( c );\n",
       "        geometry.translate( -c.x, -c.y, -c.z );\n",
       "\n",
       "        var mesh = new THREE.Mesh( geometry, material );\n",
       "        mesh.position.set( c.x, c.y, c.z );\n",
       "        if ( transparent && json.renderOrder ) mesh.renderOrder = json.renderOrder;\n",
       "        mesh.userData = json;\n",
       "        scene.add( mesh );\n",
       "\n",
       "        if ( json.showMeshGrid ) addSurfaceMeshGrid( json );\n",
       "\n",
       "    }\n",
       "\n",
       "    function addSurfaceMeshGrid( json ) {\n",
       "\n",
       "        var geometry = new THREE.Geometry();\n",
       "\n",
       "        for ( var i=0 ; i < json.faces.length ; i++ ) {\n",
       "            var f = json.faces[i];\n",
       "            for ( var j=0 ; j < f.length ; j++ ) {\n",
       "                var k = j === f.length-1 ? 0 : j+1;\n",
       "                var v1 = json.vertices[f[j]];\n",
       "                var v2 = json.vertices[f[k]];\n",
       "                // vertices in opposite directions on neighboring faces\n",
       "                var nudge = f[j] < f[k] ? .0005*zRange : -.0005*zRange;\n",
       "                geometry.vertices.push( new THREE.Vector3( a[0]*v1.x, a[1]*v1.y, a[2]*(v1.z+nudge) ) );\n",
       "                geometry.vertices.push( new THREE.Vector3( a[0]*v2.x, a[1]*v2.y, a[2]*(v2.z+nudge) ) );\n",
       "            }\n",
       "        }\n",
       "\n",
       "        var c = new THREE.Vector3();\n",
       "        geometry.computeBoundingBox();\n",
       "        geometry.boundingBox.getCenter( c );\n",
       "        geometry.translate( -c.x, -c.y, -c.z );\n",
       "\n",
       "        var gridColor = options.theme === 'dark' ? 'white' : 'black';\n",
       "        var linewidth = json.linewidth || 1;\n",
       "        var materialOptions = { color: gridColor, linewidth: linewidth };\n",
       "\n",
       "        var mesh;\n",
       "        if ( linewidth > 1 && window.createFatLineSegments ) {\n",
       "            mesh = createFatLineSegments( geometry, materialOptions );\n",
       "        } else {\n",
       "            var material = new THREE.LineBasicMaterial( materialOptions );\n",
       "            mesh = new THREE.LineSegments( geometry, material );\n",
       "        }\n",
       "\n",
       "        mesh.position.set( c.x, c.y, c.z );\n",
       "        mesh.userData = json;\n",
       "        scene.add( mesh );\n",
       "\n",
       "    }\n",
       "\n",
       "    function render() {\n",
       "\n",
       "        if ( window.updateAnimation ) animate = updateAnimation();\n",
       "        if ( animate ) requestAnimationFrame( render );\n",
       "\n",
       "        renderer.render( scene, camera );\n",
       "\n",
       "    }\n",
       "\n",
       "    render();\n",
       "    controls.update();\n",
       "    if ( !animate ) render();\n",
       "\n",
       "\n",
       "    // menu functions\n",
       "\n",
       "    function toggleMenu() {\n",
       "\n",
       "        var m = document.getElementById( 'menu-content' );\n",
       "        if ( m.style.display === 'block' ) m.style.display = 'none'\n",
       "        else m.style.display = 'block';\n",
       "\n",
       "    }\n",
       "\n",
       "\n",
       "    function saveAsPNG() {\n",
       "\n",
       "        var a = document.body.appendChild( document.createElement( 'a' ) );\n",
       "        a.href = renderer.domElement.toDataURL( 'image/png' );\n",
       "        a.download = 'screenshot';\n",
       "        a.click();\n",
       "\n",
       "    }\n",
       "\n",
       "    function saveAsHTML() {\n",
       "\n",
       "        toggleMenu(); // otherwise visible in output\n",
       "        event.stopPropagation();\n",
       "\n",
       "        var blob = new Blob( [ '<!DOCTYPE html>\\n' + document.documentElement.outerHTML ] );\n",
       "        var a = document.body.appendChild( document.createElement( 'a' ) );\n",
       "        a.href = window.URL.createObjectURL( blob );\n",
       "        a.download = suggestFilename();\n",
       "        a.click();\n",
       "\n",
       "        function suggestFilename() {\n",
       "            if ( !document.title ) {\n",
       "                return 'graphic.html';\n",
       "            } else if ( /\\.html?$/i.test( document.title ) ) {\n",
       "                return document.title; // already ends in .htm or .html\n",
       "            } else {\n",
       "                return document.title + '.html';\n",
       "            }\n",
       "        }\n",
       "\n",
       "    }\n",
       "\n",
       "    function getViewpoint() {\n",
       "\n",
       "        function roundTo( x, n ) { return +x.toFixed(n); }\n",
       "\n",
       "        var v = camera.quaternion.inverse();\n",
       "        var r = Math.sqrt( v.x*v.x + v.y*v.y + v.z*v.z );\n",
       "        var axis = [ roundTo( v.x / r, 4 ), roundTo( v.y / r, 4 ), roundTo( v.z / r, 4 ) ];\n",
       "        var angle = roundTo( 2 * Math.atan2( r, v.w ) * 180 / Math.PI, 2 );\n",
       "\n",
       "        var textArea = document.createElement( 'textarea' );\n",
       "        textArea.textContent = JSON.stringify( axis ) + ',' + angle;\n",
       "        textArea.style.csstext = 'position: absolute; top: -100%';\n",
       "        document.body.append( textArea );\n",
       "        textArea.select();\n",
       "        document.execCommand( 'copy' );\n",
       "\n",
       "        var m = document.getElementById( 'menu-message' );\n",
       "        m.innerHTML = 'Viewpoint copied to clipboard';\n",
       "        m.style.display = 'block';\n",
       "        setTimeout( function() { m.style.display = 'none'; }, 2000 );\n",
       "\n",
       "    }\n",
       "\n",
       "</script>\n",
       "\n",
       "<div id=&quot;menu-container&quot; onclick=&quot;toggleMenu()&quot;>&#x24d8;\n",
       "<div id=&quot;menu-message&quot;></div>\n",
       "<div id=&quot;menu-content&quot;>\n",
       "<div onclick=&quot;saveAsPNG()&quot;>Save as PNG</div>\n",
       "<div onclick=&quot;saveAsHTML()&quot;>Save as HTML</div>\n",
       "<div onclick=&quot;getViewpoint()&quot;>Get Viewpoint</div>\n",
       "<div>Close Menu</div>\n",
       "</div></div>\n",
       "\n",
       "\n",
       "</body>\n",
       "</html>\n",
       "\"\n",
       "        width=\"100%\"\n",
       "        height=\"400\"\n",
       "        style=\"border: 0;\">\n",
       "</iframe>\n"
      ],
      "text/plain": [
       "Graphics3d Object"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot3d(f, (x, -1, 1), (y, -1, 1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Importante.** Quando queremos respostas numéricas, computação simbólica mais atrapalha do que ajuda."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "93/13"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "93/13"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "log(1201)/log(2)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "log(1201, 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para obtermos a aproximação numérica de uma expressão, usamos a função `n` que toma como parâmetros uma expressão  e, opcionalmente, o número de dígitos da aproximação."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "7.15384615384615"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n(93/13)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10.2300204357056340945901830909"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n(log(1201, 2), digits=30)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10.2300204357056"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "numerical_approx(log(1201, 2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Por que usar Sage?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Vamos comparar código em C e Sage para fazer multiplicações de matrizes binárias."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Em C, a biblioteca mais rápida para fazer multiplicação binária eficiente é chamada M4RI. \n",
    "\n",
    "Considere a implementação abaixo:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "```C\n",
    "mzd_t* multiply_two_random_matrices(rci_t n) {\n",
    "  mzd_t* A = mzd_init(n, n);\n",
    "  mzd_t* B = mzd_init(n, n);\n",
    "  mzd_randomize(A);\n",
    "  mzd_randomize(B);\n",
    "  mzd_t* C = mzd_mul(NULL, A, B, NULL);\n",
    "  \n",
    "  mzd_free(A);\n",
    "  mzd_free(B);\n",
    "    \n",
    "  return C;\n",
    "}\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Agora o código em Sage que realiza exatemente a mesma computação:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "def multiply_two_random_matrices(n):\n",
    "    A = random_matrix(GF(2), n, n)\n",
    "    B = random_matrix(GF(2), n, n)\n",
    "    return A*B"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0 1 1 1 0 0 1 1]\n",
      "[1 0 1 1 1 0 1 0]\n",
      "[0 1 0 1 0 0 1 1]\n",
      "[0 0 0 1 0 1 0 1]\n",
      "[0 1 1 1 0 1 1 1]\n",
      "[0 1 1 0 0 1 1 1]\n",
      "[0 0 1 0 0 1 0 1]\n",
      "[1 0 0 0 0 0 1 0]\n"
     ]
    }
   ],
   "source": [
    "C = multiply_two_random_matrices(8)\n",
    "print(C)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As duas implementações rodam em tempo similar **pois Sage chama a rotina da M4RI para fazer a multiplicação!**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Além disso, temos a vantagem de ter a mesma interface para vários sistemas bem diferentes entre si. Por exemplo: podemos usar $\\mathbf{C}$ diretamente como matriz de adjacência para construir um grafo"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "G = sage.graphs.digraph.DiGraph(C)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Podemos obter informações importantes sobre o grafo gerado. Por exemplo, se o grafo é conexo ou não."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "G.is_connected()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Por onde você começaria a implementar em C isso que acabamos de fazer tão facilmente?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "### Como usar o Sage?\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "O SageMath pode ser usado de 3 formas:\n",
    "- Usando o Jupyter Notebook (essa é a forma sendo usada neste documento)\n",
    "- Com um interpretador interativo\n",
    "- Por meio de scripts\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Um dos melhores editores para editar scripts em é o Atom (https://atom.io/), porém qualquer editor de texto pode ser usado.\n",
    "\n",
    "Para que seu script possa usar todas as funções do sage, ele deve conter `from sage.all import *` no topo."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`script.py`\n",
    "```python\n",
    "from sage.all import *\n",
    "\n",
    "def func1():\n",
    "    print('Esta é a função 1')\n",
    "\n",
    "def func2():\n",
    "    print('Esta é a função 2')\n",
    "\n",
    "if __name__ == \"__main__\":\n",
    "    func1()  # Chama a função 1\n",
    "    func2()  # Chama a função 2\n",
    "\n",
    "\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para rodar o arquivo `script.py`, use:\n",
    "\n",
    "`sage script.py`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para a depuração do seu script, é muito útil rodar o script dentro do interpretador interativo do Sage. Da mesma forma que fazemos isso no iPython, podemos rodar `%run script.py` dentro do Sage.\n",
    "\n",
    "```\n",
    "/home/user/$ sage\n",
    "┌────────────────────────────────────────────────────────────────────┐\n",
    "  SageMath version 9.0, Release Date: 2020-01-01                                  \n",
    "  Using Python 3.8.2. Type \"help()\" for help.                                     \n",
    "└────────────────────────────────────────────────────────────────────┘\n",
    "sage: %run script.py \n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Aritmética modular"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Nesta seção, trataremos de $Z_n$, que é o anel de inteiros módulo $n$. \n",
    "Primeiro relembremos a definição de Anel."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Definição.** Um anel $R$ é um conjunto munido de duas operações $+$ (adição) e $.$ (multiplicação), que, para todos $a, b, c \\in R$, satisfazem:\n",
    "1. Associatividade de $+$\n",
    "$$\n",
    "(a + b) + c = a + ( b + c)\n",
    "$$\n",
    "2. Comutatividade de $+$\n",
    "$$\n",
    "a + b = b + a\n",
    "$$\n",
    "3. Existência do identidade de $+$. Existe um elemento $0 \\in R$ tal que \n",
    "    $$\n",
    "    a + 0 = a\n",
    "    $$\n",
    "\n",
    "4. Existência da inversa aditiva. Existe um elemento $(-a) \\in R$ tal que \n",
    "    $$\n",
    "    a + (-a) = 0\n",
    "    $$\n",
    "\n",
    "5. Associatividade de $.$\n",
    "$$\n",
    "(a b) c = a ( b c)\n",
    "$$\n",
    "6. Existência da identidade de $.$\n",
    "\n",
    "    Existe um elemento $1 \\in R$ tal que \n",
    "    $$ \n",
    "    a1 = 1a = a\n",
    "    $$\n",
    "7. Distributividade de $.$ sobre $+$:\n",
    "$$\n",
    "    (a + b)c = ac + bc \\\\\n",
    "    c(a + b) = ca + cb\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Importante**. Note que num anel não é necessário que $.$ seja comutativa, e nem que exista a inversa multiplicativa de todos os elementos."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "Em Sage, é muito simples trabalhar com $Z_n$, o anel de inteiros $\\bmod n$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Ring of integers modulo 13\n"
     ]
    }
   ],
   "source": [
    "n = 13\n",
    "Zn = Integers(n)\n",
    "print(Zn)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       " +   0  1  2  3  4  5  6  7  8  9 10 11 12\n",
       "  +---------------------------------------\n",
       " 0|  0  1  2  3  4  5  6  7  8  9 10 11 12\n",
       " 1|  1  2  3  4  5  6  7  8  9 10 11 12  0\n",
       " 2|  2  3  4  5  6  7  8  9 10 11 12  0  1\n",
       " 3|  3  4  5  6  7  8  9 10 11 12  0  1  2\n",
       " 4|  4  5  6  7  8  9 10 11 12  0  1  2  3\n",
       " 5|  5  6  7  8  9 10 11 12  0  1  2  3  4\n",
       " 6|  6  7  8  9 10 11 12  0  1  2  3  4  5\n",
       " 7|  7  8  9 10 11 12  0  1  2  3  4  5  6\n",
       " 8|  8  9 10 11 12  0  1  2  3  4  5  6  7\n",
       " 9|  9 10 11 12  0  1  2  3  4  5  6  7  8\n",
       "10| 10 11 12  0  1  2  3  4  5  6  7  8  9\n",
       "11| 11 12  0  1  2  3  4  5  6  7  8  9 10\n",
       "12| 12  0  1  2  3  4  5  6  7  8  9 10 11\n"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Zn.addition_table(names='elements')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       " *   0  1  2  3  4  5  6  7  8  9 10 11 12\n",
       "  +---------------------------------------\n",
       " 0|  0  0  0  0  0  0  0  0  0  0  0  0  0\n",
       " 1|  0  1  2  3  4  5  6  7  8  9 10 11 12\n",
       " 2|  0  2  4  6  8 10 12  1  3  5  7  9 11\n",
       " 3|  0  3  6  9 12  2  5  8 11  1  4  7 10\n",
       " 4|  0  4  8 12  3  7 11  2  6 10  1  5  9\n",
       " 5|  0  5 10  2  7 12  4  9  1  6 11  3  8\n",
       " 6|  0  6 12  5 11  4 10  3  9  2  8  1  7\n",
       " 7|  0  7  1  8  2  9  3 10  4 11  5 12  6\n",
       " 8|  0  8  3 11  6  1  9  4 12  7  2 10  5\n",
       " 9|  0  9  5  1 10  6  2 11  7  3 12  8  4\n",
       "10|  0 10  7  4  1 11  8  5  2 12  9  6  3\n",
       "11|  0 11  9  7  5  3  1 12 10  8  6  4  2\n",
       "12|  0 12 11 10  9  8  7  6  5  4  3  2  1\n"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Zn.multiplication_table(names='elements')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "O Sage cuida para que as operações sejam feitas dentro deste anel."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "19"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "10 + 9  # Operação fora de Zn"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(10 + 9) % 13"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "100000000000000000000"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "10**20  # Operação fora de Zn"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "9"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "10**20 % 13"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Zn(10) + Zn(9)  # Operação em Zn"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "9"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Zn(10)**20  # Operação em Zn"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Comparação de tempo**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.013619533739984035\n",
      "7.692199869779869e-07\n"
     ]
    }
   ],
   "source": [
    "import timeit\n",
    "\n",
    "def power_mod_python(x, e):\n",
    "    return lambda: x**e % n\n",
    "def power_mod_Zn(x, e):\n",
    "    return lambda: Zn(x)**e\n",
    "\n",
    "N = 100\n",
    "print(timeit.timeit(power_mod_python(10, 2000000), number=N)/N)\n",
    "print(timeit.timeit(power_mod_Zn(10, 2000000), number=N)/N)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note que, quando $\\gcd(a, n) = 1$, existe inversa de $a \\bmod n$, que pode é computada diretamente:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Zn(4)^-1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Zn(10)*Zn(4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Além disso, para casos em que precisamos apenas da exponenciação ou inversão modular, sem que seja necessário fazer outras operações em $Z_n$, podemos usar as funções `power_mod` ou `inverse_mod`. Isso evita a sobrecarga de criar o $Z_n$, como fizemos anteriormente."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "9"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "power_mod(10, 20, 13)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "inverse_mod(4, 13)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compare as respotas acima com as calculadas sobre elementos de $Z_n$ diretamente."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "***"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Funções importantes como o Máximo Divisor Comum, Algoritmo de Euclides estendido, e Teorema Chinês do Resto são implementados em Sage."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "O MDC (Greatest Common Divisor - GCD) pode ser computado usando a função `gcd`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gcd(133, 48)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gcd(12, 18)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "O algoritmo de Euclides estendido é feito pela função `xgcd`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "res, u, v = xgcd(133, 48)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "res"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u*133 + v*48"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Considere os seguintes resíduos para os quais queremos aplicar o Teorema Chinês do Resto (TCR).\n",
    "$$\n",
    "\\begin{cases}\n",
    "    x \\equiv 17 \\mod 23 \\\\\n",
    "    x \\equiv 91 \\mod 111 \\\\\n",
    "    x \\equiv 40 \\mod 57 \\\\\n",
    "\\end{cases}\n",
    "$$\n",
    "\n",
    "O Teorema Chinês do Resto (Chinese Remainder Theorem - CRT) pode ser usado para encontrar uma solução para as congruências usando a função `crt`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = crt([17, 91, 40], [23, 111, 57])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "31504"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(17, 91, 40)"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x % 23, x % 111, x % 57"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## As funções do Sage podem ser usadas para cripto segura?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 119,
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "import timeit\n",
    "\n",
    "MAX_INT  = 10000\n",
    "NSAMPLES = 1000\n",
    "\n",
    "def compare_modular_exponentiation_efficiency(nsamples=NSAMPLES, max_int=MAX_INT):\n",
    "    \n",
    "    data = {\n",
    "        'n': [],\n",
    "        'p': [],\n",
    "        'exponent': [],\n",
    "        'time_power_mod': [],\n",
    "        'time_regular_exponentiation': []\n",
    "    }\n",
    "    \n",
    "    for _ in range(nsamples):\n",
    "        n = randint(1, max_int)\n",
    "        p = random_prime(max_int)\n",
    "        exponent = randint(1, max_int)\n",
    "\n",
    "        def compute_with_power_mod():\n",
    "            v = power_mod(n, exponent, p)\n",
    "\n",
    "        time_power_mod = timeit.timeit(compute_with_power_mod, number=10)\n",
    "\n",
    "        def compute_with_regular_exponentiation():\n",
    "            v = n**exponent % p\n",
    "\n",
    "        time_regular_exponentiation = timeit.timeit(compute_with_regular_exponentiation, number=10)\n",
    "\n",
    "        data['n'].append(n)\n",
    "        data['p'].append(p)\n",
    "        data['exponent'].append(exponent)\n",
    "        data['time_power_mod'].append(time_power_mod)\n",
    "        data['time_regular_exponentiation'].append(time_regular_exponentiation)\n",
    "\n",
    "        \n",
    "    return data\n",
    "\n",
    "data = compare_modular_exponentiation_efficiency()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 128,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend object at 0x7f8cd2520280>"
      ]
     },
     "execution_count": 128,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "import pandas as pd\n",
    "\n",
    "df = pd.DataFrame.from_dict(data)\n",
    "\n",
    "ax = plt.gca()\n",
    "\n",
    "df.plot.scatter(x='exponent', y='time_power_mod', label='Usando power_mod', c='red', ax=ax, s=0.4)\n",
    "df.plot.scatter(x='exponent', y='time_regular_exponentiation', c='blue', label='Sem usar power_mod', ax=ax, s=0.4)\n",
    "\n",
    "plt.ylabel('Tempo de execução')\n",
    "plt.legend()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 127,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<AxesSubplot:xlabel='exponent', ylabel='time_power_mod'>"
      ]
     },
     "execution_count": 127,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "df.plot.scatter(x='exponent', y='time_power_mod', label='Usando power_mod', c='red', s=0.4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 166,
   "metadata": {},
   "outputs": [],
   "source": [
    "def compare_modular_exponentiation_efficiency_larg_exp(a=6000, b=7000):\n",
    "    \n",
    "    data = {\n",
    "        'n': [],\n",
    "        'p': [],\n",
    "        'exponent': [],\n",
    "        'time_power_mod': [],\n",
    "    }\n",
    "#     n = 203123\n",
    "#     p = random_prime(max_int)\n",
    "    for exponent in range(a, b):\n",
    "        n = randint(2, 10000)\n",
    "        p = random_prime(10000)\n",
    "\n",
    "        def compute_with_power_mod():\n",
    "            v = power_mod(n, exponent, p)\n",
    "\n",
    "        time_power_mod = timeit.timeit(compute_with_power_mod, number=10)\n",
    "\n",
    "        data['n'].append(n)\n",
    "        data['p'].append(p)\n",
    "        data['exponent'].append(exponent)\n",
    "        data['time_power_mod'].append(time_power_mod)\n",
    "\n",
    "        \n",
    "    return data\n",
    "\n",
    "data_large = compare_modular_exponentiation_efficiency_larg_exp()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 167,
   "metadata": {},
   "outputs": [],
   "source": [
    "df_large_exp = pd.DataFrame.from_dict(data_large)\n",
    "\n",
    "def hamming_weight(x):\n",
    "    return bin(x)[2:].count('1')\n",
    "\n",
    "df_large_exp['exp_hamming_weight'] = df_large_exp.exponent.apply(hamming_weight, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 155,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'0b1001100111001'"
      ]
     },
     "execution_count": 155,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bin(4921)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 168,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>n</th>\n",
       "      <th>p</th>\n",
       "      <th>exponent</th>\n",
       "      <th>time_power_mod</th>\n",
       "      <th>exp_hamming_weight</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>4782</td>\n",
       "      <td>6263</td>\n",
       "      <td>6000</td>\n",
       "      <td>0.000045</td>\n",
       "      <td>7</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>5395</td>\n",
       "      <td>6221</td>\n",
       "      <td>6001</td>\n",
       "      <td>0.000040</td>\n",
       "      <td>8</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>6328</td>\n",
       "      <td>2399</td>\n",
       "      <td>6002</td>\n",
       "      <td>0.000040</td>\n",
       "      <td>8</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>5280</td>\n",
       "      <td>9029</td>\n",
       "      <td>6003</td>\n",
       "      <td>0.000040</td>\n",
       "      <td>9</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>7639</td>\n",
       "      <td>3719</td>\n",
       "      <td>6004</td>\n",
       "      <td>0.000041</td>\n",
       "      <td>8</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>995</th>\n",
       "      <td>1075</td>\n",
       "      <td>9043</td>\n",
       "      <td>6995</td>\n",
       "      <td>0.000040</td>\n",
       "      <td>8</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>996</th>\n",
       "      <td>2732</td>\n",
       "      <td>1979</td>\n",
       "      <td>6996</td>\n",
       "      <td>0.000038</td>\n",
       "      <td>7</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>997</th>\n",
       "      <td>6527</td>\n",
       "      <td>5659</td>\n",
       "      <td>6997</td>\n",
       "      <td>0.000039</td>\n",
       "      <td>8</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>998</th>\n",
       "      <td>8110</td>\n",
       "      <td>8447</td>\n",
       "      <td>6998</td>\n",
       "      <td>0.000040</td>\n",
       "      <td>8</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>999</th>\n",
       "      <td>945</td>\n",
       "      <td>4721</td>\n",
       "      <td>6999</td>\n",
       "      <td>0.000053</td>\n",
       "      <td>9</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>1000 rows × 5 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "        n     p  exponent  time_power_mod  exp_hamming_weight\n",
       "0    4782  6263      6000        0.000045                   7\n",
       "1    5395  6221      6001        0.000040                   8\n",
       "2    6328  2399      6002        0.000040                   8\n",
       "3    5280  9029      6003        0.000040                   9\n",
       "4    7639  3719      6004        0.000041                   8\n",
       "..    ...   ...       ...             ...                 ...\n",
       "995  1075  9043      6995        0.000040                   8\n",
       "996  2732  1979      6996        0.000038                   7\n",
       "997  6527  5659      6997        0.000039                   8\n",
       "998  8110  8447      6998        0.000040                   8\n",
       "999   945  4721      6999        0.000053                   9\n",
       "\n",
       "[1000 rows x 5 columns]"
      ]
     },
     "execution_count": 168,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df_large_exp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 175,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<AxesSubplot:xlabel='exp_hamming_weight', ylabel='time_power_mod'>"
      ]
     },
     "execution_count": 175,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import seaborn as sns\n",
    "sns.boxplot(data=df_large_exp, x='exp_hamming_weight', y='time_power_mod', showfliers=False)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**É possível obter informação parcial sobre o peso de Hamming do expoente pelo tempo de execução do algoritmo!!!**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Matrizes"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Como vimos no começo da aula, Sage é capaz de trabalhar com álgebra de matrizes."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Criando matrizes e vetores"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para criar uma matriz usando uma lista de listas, usamos a função `matrix`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1 3 2]\n",
      "[2 5 1]\n",
      "[7 8 3]\n"
     ]
    }
   ],
   "source": [
    "A = matrix(\n",
    "    [\n",
    "        [1, 3, 2],\n",
    "        [2, 5, 1], \n",
    "        [7, 8, 3],\n",
    "    ]\n",
    ")\n",
    "print(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Como não foi explicitado a que anel pertencem os elementos de `A`, o Sage supõe que são inteiros. Portanto, ao calcular `A*A`, a resposta é dada em inteiros."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[ 955297791 1820777299  600591356]\n",
       "[1021112620 1946219871  641969205]\n",
       "[2391178115 4557536902 1503323989]"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A**10"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Em criptografia, é muito comum trabalharmos com matrizes de inteiros $\\bmod n$. Para isso, podemos fazer a inicialização da seguinte forma."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1 3 2]\n",
      "[2 5 1]\n",
      "[7 8 3]\n"
     ]
    }
   ],
   "source": [
    "Zn = Integers(13)  # Apenas para lembrar do que é Zn\n",
    "\n",
    "An = matrix(Zn,\n",
    "    [\n",
    "        [1, 3, 2],\n",
    "        [2, 5, 1], \n",
    "        [7, 8, 3],\n",
    "    ]\n",
    ")\n",
    "print(An)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Agora, o produto `An*An` e soma `An + An` são calculado em `Zn`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[ 6  3  1]\n",
       "[ 8 11  7]\n",
       "[ 1  9 11]"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "An**10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[ 2  6  4]\n",
       "[ 4 10  2]\n",
       "[ 1  3  6]"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "An + An"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Muitas vezes, em criptografia, precisamos gerar matrizes aleatórias. Isto pode ser feito usando a função `random_matrix`, que recebe como parâmetro um Anel de base, o número de linhas e o número de colunas."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[10  6  1]\n",
       "[ 8  4 12]\n",
       "[10  2  2]"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "random_matrix(Zn, 3, 3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Operações sobre matrizes"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "O método `inverse` calcula inversa de uma matriz quadrada, quando esta é não singular."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 3  3 10]\n",
      "[ 6 12  5]\n",
      "[ 3  0  7]\n"
     ]
    }
   ],
   "source": [
    "An_inv = An.inverse()\n",
    "print(An_inv)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1 0 0]\n",
       "[0 1 0]\n",
       "[0 0 1]"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "An_inv*An"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[ 3  3 10]\n",
       "[ 6 12  5]\n",
       "[ 3  0  7]"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "An**(-1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "O determinante pode ser calculado usando o método `determinant`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-28"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.determinant()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "O método `echelon_form` devolve a matriz escalonada. Não confunda com o método `echelonize`, que escalona a matriz, alterando-a."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Matriz B\n",
      "[ 4  2 11 12  1  8  2  6  4  0]\n",
      "[11  6  4  7 10  5  7 10  3  9]\n",
      "[ 4  8  1  5  9  9  9  9  6  6]\n",
      "[ 5  5 12  8  3  1  9  0  6  2]\n",
      "[11  8  4 11  6  1  7  5  4 11]\n",
      "Matriz B escalonada\n",
      "[ 1  0  0  0  0  1 11  0 12  3]\n",
      "[ 0  1  0  0  0 11 11  9 12 10]\n",
      "[ 0  0  1  0  0  9 12  4  6  2]\n",
      "[ 0  0  0  1  0 12  4  8  9  8]\n",
      "[ 0  0  0  0  1 12  3  4  5  6]\n"
     ]
    }
   ],
   "source": [
    "B = random_matrix(Zn, 5, 10)\n",
    "print('Matriz B')\n",
    "print(B)\n",
    " \n",
    "print('Matriz B escalonada')\n",
    "print(B.echelon_form())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Posto e nulidade:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B.rank()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B.nullity()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Resolvendo sistemas lineares"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para resolver um sistema da forma $\\mathbf{D} \\mathbf{x} = \\mathbf{b}$, usamos o método `solve_right`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "D = \n",
      "[ 1 10  8 11]\n",
      "[ 0  3  6  2]\n",
      "[ 8  7  1  3]\n",
      "[ 5  7 10 10]\n",
      "b =  (10, 5, 7, 10)\n"
     ]
    }
   ],
   "source": [
    "D = random_matrix(Zn, 4, 4)\n",
    "b = random_vector(Zn, 4)\n",
    "print('D = ')\n",
    "print(D)\n",
    "print('b = ', b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(12, 10, 3, 11)\n"
     ]
    }
   ],
   "source": [
    "x = D.solve_right(b)\n",
    "print(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D*x == b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Corpos"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*Definição.* Um Corpo K é um conjunto munido de duas operações $+$ e $.$ tais que:\n",
    "1. $(K, +, .)$ é um anel\n",
    "2. A operação $.$ é comutativa. Ou seja, para todos $a, b \\in K$:\n",
    "   $$\n",
    "   a b = b a, \n",
    "   $$\n",
    "   \n",
    "3. Para todo elemento $a$ **não nulo** de $K$, existe o seu inverso multiplicativo $a^{-1} \\in K$ tal que:\n",
    "    $$\n",
    "    aa^{-1} = a^{-1}a = 1\n",
    "    $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Observação**. Corpo é uma abstração do conjunto dos Racionais, onde todos os elementos diferentes de 0 têm inversa."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Em criptografia, tipicamente estamos interessados em corpos finitos, pois os elementos devem ser representados num número finito e prefixado de bits."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Teorema**. Se $p$ é número primo, então $Z_p$ é um corpo finito.\n",
    "\n",
    "*Demontração*. Sabemos que $(Z_p, +, .)$ é um anel, então só precisamos mostrar que valem as propriedades 2 e 3. \n",
    "Sabemos também que a multiplicação modular é comutativa, o que prova que a propriedade 2 vale.\n",
    "Além disso, como $p$ é primo, então $\\gcd(a, p) = 1$ para todo $a \\in Z_p - \\{0\\}$. Portanto existe a inversa de $a \\mod p$, e vale a propriedade 3.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Em sage, para determinar se um determinado anel é também um corpo, podemos usar o método `is_field`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Integers(13).is_field()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Integers(10).is_field()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Em geral, ao programar, devemos ser o mais explícitos possível. Portanto, ao declarar um conjunto que sabemos que é um corpo, é preferível usar o construtor `GF` (de Galois Field) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [],
   "source": [
    "K = GF(13)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Este construtor quebra quando passamos um tamanho incompatível com um corpo finito"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "the order of a finite field must be a prime power\n"
     ]
    }
   ],
   "source": [
    "try:\n",
    "    GF(10)\n",
    "except Exception as e:\n",
    "    print(e)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Há corpos de tamanho não-primo. Em particular o tamanho de um corpo é sempre $q = p^k$ para algum primo $p$ e algum inteiro positivo $k \\geq 1$.\n",
    "\n",
    "Porém, pelo modo como é feita a multiplicação em corpos finitos de tamanhos não primos, seus elementos são melhor representados por polinômios e não por inteiros."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3*t + 5"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "GF(7**2, 't').random_element()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Integers(7**2).is_field()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "z2 + 6"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "GF(7**2).random_element()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "GF(7).random_element()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para entender como chegar nisso, primeiro vejamos como construir anéis polinômiais."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Anéis Polinomiais"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Um anel polinomial $K[x]$ nada mais é que o conjunto de polinômios em $x$ com coeficientes em $K$. Tipicamente, estamos interessados em $K$ sendo um corpo, mas a construção vale para $K$ sendo apenas um anel.\n",
    "Para ver que $K[x]$ é um anel, basta usar o fato de que $K$ é um corpo (e portanto um anel) e usar as regras de adição e multiplicações de polinômios."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para criar um anel polinomial `R` sobre a variavel `x` e com coeficientes em `GF(13)`, em Sage, fazemos:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [],
   "source": [
    "R.<x> = GF(13)[]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Univariate Polynomial Ring in x over Finite Field of size 13"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [],
   "source": [
    "R2 = GF(13).polynomial_ring('x2')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Univariate Polynomial Ring in x2 over Finite Field of size 13"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "5*x^4 + x^3 + 5*x^2 + 7*x + 9\n"
     ]
    }
   ],
   "source": [
    "v = R.random_element(4)  # 10 é o grau do elemento aleatório\n",
    "print(v)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note que $GF(n)[x]$ é infinito."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "+Infinity"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R.cardinality()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para contruir um anel polinomial finito, podemos fazer o análogo do que fizemos para construir $Z_n$ a partir de $Z$: pegar os resíduos quando dividimos por um elemento $p_1 \\in R$. O anel construído é denotado por $R/p_1$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [],
   "source": [
    "p1 = x^2 + x\n",
    "R_mod_p1 = R.quotient(p1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note que o número de resíduos quando dividimos por um polinômio de grau 2 é dado por\n",
    "$$\n",
    "( \\text{possibilidades para coeficiente de grau 0}) \\times (\\text{possibilidades para coeficiente de grau 1}) = n^2\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "169"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R_mod_p1.cardinality()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Porém note que o anel quociente construído não é um corpo:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R_mod_p1.is_field()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "O motivo por que $R/p_1$ não é um corpo é análogo ao motivo de $Z_{10}$ não ser um corpo: o módulo não é primo, e há elementos $a$ em $R/p_1$ tais que $\\gcd(a, p_1) \\neq 1$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "x + 1"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gcd(x + 1, x^2 + x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Definição**. Um polinômio $m \\in GF(q)[x]$ é primo (ou irredutível) quando ele **não** pode ser fatorado em dois polinômios não-constantes de $GF(q)[x]$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "No caso anterior, $p_1= x^2 + x$ claramente pode ser fatorado em $x(x + 1)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Observação**. Para mostrar que um polinômio não é primo, basta mostrar uma fatoração. Agora, para mostrar que um polinômio é primo não é tão fácil..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Por sorte, o Sage vem com um método que devolve um polinômio irredutível num anel polinomial, dado um grau."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x^2 + 12*x + 2\n"
     ]
    }
   ],
   "source": [
    "m = R.irreducible_element(2)\n",
    "print(m)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Podemos agora fazer a mesma construção anterior só que com $m$ irredutível no lugar de $p_1$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [],
   "source": [
    "R_mod_m = R.quotient(m)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Neste caso, $R/m$ será um corpo finito de $13^2 = 169$ elementos."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R_mod_m.is_field()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "169"
      ]
     },
     "execution_count": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R_mod_m.cardinality()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note que $R/m$ e $R/p_1$ contam com exatamente os mesmos polinômios! A diferença ocorre na multiplicação de elementos, que, quando feita $\\bmod m$, mantém a propriedade da existência do inverso."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Importante**. Se tomarmos outro polinômio irredutível $m_2$ do mesmo grau de $m$, os corpos finitos $R/m$ e $R/m_2$ são isomórficos. Há um teorema importante que diz que todos os corpos finitos de mesmos tamanhos são isomórficos."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para ver isso na prática, considere."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [],
   "source": [
    "P.<t1> = GF(2)[]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Queremos m1 != m2\n",
      "t1^3 + t1 + 1\n",
      "t1^3 + t1 + 1\n"
     ]
    }
   ],
   "source": [
    "m1 = P.irreducible_element(3, algorithm='random')\n",
    "m2 = P.irreducible_element(3, algorithm='random')\n",
    "print(\"Queremos m1 != m2\")\n",
    "print(m1)\n",
    "print(m2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "*  a b c d e f g h\n",
       " +----------------\n",
       "a| a a a a a a a a\n",
       "b| a b c d e f g h\n",
       "c| a c e g d b h f\n",
       "d| a d g f h e b c\n",
       "e| a e d h g c f b\n",
       "f| a f b e c h d g\n",
       "g| a g h b f d c e\n",
       "h| a h f c b g e d\n"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P.quotient(m1, 'i').multiplication_table()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "*  a b c d e f g h\n",
       " +----------------\n",
       "a| a a a a a a a a\n",
       "b| a b c d e f g h\n",
       "c| a c e g d b h f\n",
       "d| a d g f h e b c\n",
       "e| a e d h g c f b\n",
       "f| a f b e c h d g\n",
       "g| a g h b f d c e\n",
       "h| a h f c b g e d\n"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P.quotient(m2, 'j').multiplication_table()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Observação**. É possível achar uma bijeção entre os polinômios dos dois corpos que transforma uma tabela na outra. (Não que seja uma coisa muito fácil de fazer agora, mas acho que a ideia do isomorfismo fica bem ilustrada pelas tabelas.)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## O Corpo finito $GF(2^m)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Vimos que todos os corpos finitos são isomórficos a $GF(p^k)$ para algum primo $p$ e algum inteiro $k \\geq 1$. Vimos também que para usar qualquer um desses corpos finitos em Sage, usamos o construtor `GF`, **não** sendo necessário criar anéis polinomiais e selecionar um polinômio irredutível.\n",
    "\n",
    "Em criptografia é comum usarmos corpos finitos de característica 2, isto é, $p = 2$, pois algumas operações podem ser simplificadas usando operações de bits."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Uma aplicação interessante é que podemos definir uma operação de multiplicação sobre bytes.\n",
    "\n",
    "Considere por exemplo o $GF(2^8)$ na representação usada pelo AES."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Finite Field in t of size 2^8"
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t = GF(2).polynomial_ring().gen()\n",
    "F8 = GF(2^8, 't', modulus=t^8 + t^4 + t^3 + t + 1)\n",
    "F8"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "t^7 + t^6 + t^5 + t\n",
      "t^6 + t^3 + t^2 + t + 1\n"
     ]
    }
   ],
   "source": [
    "b1 = F8.random_element()\n",
    "b2 = F8.random_element()\n",
    "print(b1)\n",
    "print(b2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "256"
      ]
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F8.cardinality()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para obter a representação de `b1` e `b2` como sequências de bits, podemos fazer:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0, 1, 0, 0, 0, 1, 1, 1]"
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "list(b1.polynomial())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "t^7 + t^6 + t^5 + t"
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0, 1, 0, 0, 0, 1, 1, 1]\n",
      "[1, 1, 1, 1, 0, 0, 1]\n"
     ]
    }
   ],
   "source": [
    "print(list(b1.polynomial()))\n",
    "print(list(b2.polynomial()))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note como a soma de `b1` e `b2` é simplesmente o XOR bit a bit "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1, 0, 1, 1, 0, 1, 0, 1]\n"
     ]
    }
   ],
   "source": [
    "print(list((b1 + b2).polynomial()))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "E agora temos uma definição para o produto de dois bytes `b1*b2`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "t^6 + t^5 + t^4 + t^2 + t + 1"
      ]
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b1*b2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1, 1, 1, 0, 1, 1, 1]\n"
     ]
    }
   ],
   "source": [
    "print(list((b1*b2).polynomial()))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "190"
      ]
     },
     "execution_count": 87,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b1.integer_representation()*b2.integer_representation() % 256"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "119"
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(b1*b2).integer_representation()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para pegar a representação em bytes de um elemento de `F8`, basta usarmos o método `integer_representation`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "226\n"
     ]
    }
   ],
   "source": [
    "n1 = b1.integer_representation()\n",
    "print(n1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'0b1010'"
      ]
     },
     "execution_count": 90,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bin(10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para o processo inverso, não há uma função pronta. Podemos usar a função `bin` para implementar a conversão de inteiros para elementos de `F8`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [],
   "source": [
    "def int_to_F8_poly(n):\n",
    "    t = F8.gen()\n",
    "    if not (0 <= n < 2**8):\n",
    "        raise ValueError('Deve-se obedecer à condição 0 <= int1, int2 < 256')\n",
    "    coeffs = reversed(bin(n)[2:])\n",
    "    return sum(GF(2)(v)*(t**i) for i, v in enumerate(coeffs))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "t^4 + t^2 + t + 1"
      ]
     },
     "execution_count": 92,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "int_to_F8_poly(23)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'0b10111'"
      ]
     },
     "execution_count": 93,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bin(23)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Teste de sanidade:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 94,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "int_to_F8_poly(b1.integer_representation()) == b1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para explicitar as operações que queremos implementar, podemos usar as seguintes funções `bytes_sum` e `bytes_product`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {},
   "outputs": [],
   "source": [
    "def bytes_sum(n1, n2):\n",
    "    '''\n",
    "        n1: Inteiro >= 0 e < 256\n",
    "        n2: Inteiro >= 0 e < 256\n",
    "    '''\n",
    "    b1 = int_to_F8_poly(n1)\n",
    "    b2 = int_to_F8_poly(n2)\n",
    "    \n",
    "    return (b1 + b2).integer_representation()\n",
    "\n",
    "def bytes_product(n1, n2):\n",
    "    '''\n",
    "        n1: Inteiro >= 0 e < 256\n",
    "        n2: Inteiro >= 0 e < 256\n",
    "    '''\n",
    "    b1 = int_to_F8_poly(n1)\n",
    "    b2 = int_to_F8_poly(n2)\n",
    "    \n",
    "    return (b1 * b2).integer_representation()\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'0x1'"
      ]
     },
     "execution_count": 96,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "hex(bytes_product(int('53', 16), int('ca', 16)))  # Deve dar 0x01"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'0x8a'"
      ]
     },
     "execution_count": 97,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "hex(bytes_product(int('45', 16), int('02', 16)))  # Deve dar 0x8a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'0x91'"
      ]
     },
     "execution_count": 98,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "hex(bytes_product(int('c5', 16), int('02', 16)))  # Deve dar 0x91"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Importante**. Este modo de implementar a soma e o produto não é o mais eficiente. Em geral, as operações são implementadas usando operações de bits, que são muito mais rápidas."
   ]
  }
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