TCG-Block-Cipher/research.html at master · CharCoding/TCG-Block-Cipher · GitHub

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<!DOCTYPE html>
<html>
<head>
  <meta charset="utf-8">
  <title>Polynomial Congruential Generators</title>
  <style type="text/css">
  body {
    position: absolute;
    background-color: #111;
    left: 50%;
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    width: 1080px;
    margin-left: -540px;
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  span.g { color: #888; }
  a { text-decoration: none; }
  a:hover { text-decoration: underline; }
  #latticeGraph {
    width: 1080px;
    height: 640px;
  }
  #cycleChart {
    margin: 0 28px;
    width: 1024px;
    height: 512px;
    image-rendering: pixelated;
    image-rendering: optimizeSpeed;
    image-rendering: crisp-edges;
    image-rendering: -moz-crisp-edges;
    image-rendering: -o-crisp-edges;
    -ms-interpolation-mode: nearest-neighbor;
  }
  #cycleGraph {
    margin: 0 140px;
    width: 800px;
    height: 800px;
  }
  #data {
    font-family: "Consolas", monospace;
    text-align: center;
  }
  input {
    color: #000;
    font-size: 1em;
  }
  input[type="number"] { width: 60px; }
  pre { line-height: 120%; }
  </style>
  <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
  <script type="text/javascript">
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    tex: {inlineMath: [['$', '$']]}
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</head>
<body>
  <h1>Polynomial Congruential Generators</h1>
  $$x_{n+1} = a\frac{x_n(x_n+1)}{2}+c \mod m$$
  <p>
    where $a, 1 \le a < m$ is the <i>multiplier</i>,<br>
    $c, 1 \le c < m$ is the <i>adder</i>,<br>
    $m = 2^k$ is the <i>modulus</i>, and<br>
    $x, 0 \le x_n < m$ is the sequence of pseudorandom numbers.<br>
    The polynomial isn't limited to 2 degrees. It can be any degree that's a power of 2 in the following form:<br>
    $y_{n+1} = a(y_n(y_n+1)-1)\frac{y_n(y_n+1)}{2}+c \mod m$<br>
    $z_{n+1} = a(z_n(z_n+1)-1)(z_n(z_n+1)(z_n(z_n+1)-1)-1)\frac{z_n(z_n+1)}{2}+c \mod m$ , etc.
  </p>
  <h3>Advantages over the Linear Congruential Generator:</h3>
  <li>
    <ul><a href="#period">- Every bit of the output has equal period. 🔗</a></ul>
    <ul><a href="#lattice">- n-consecutive values do not fall within a hyperplane lattice in $\mathbb{Z}^n$. 🔗</a></ul>
    <ul>- Cannot reverse engineer parameters $a$ and $c$ from a small sample of consecutive output<sup>[<i style="color:#0ff;">citation needed</i>]</sup>.</ul>
    <ul>- Difficult to jump forward or backward - although undesirable for some applications.</ul>
  </li>
  <h3>Disadvantages:</h3>
  <li>
    <ul><a href="#parameters">- Difficult to find suitable parameters to achieve maximal period. 🔗</a>
      <li>
        <ul>- Bruteforcing maximal period takes exponential time.</ul>
      </li>
    </ul>
    <ul>- Multiplication is slower than other operations.</ul>
    <ul><a href="#overflow">- Special case needed when implementing generator with $m - 1$ equal to <code style="color:#abc;">UINT_MAX</code>. 🔗</a></ul>
    <ul>- Possibly harder to analyze.</ul>
  </li>
  <h3 id="period">Lower bit periods</h3>
  $x_0 = 0$ in both generators. They have the same period $2^{24}$.<input type="button" id="genNext" value="generate" />
  <table>
    <thead>
      <tr>
        <th>$n$</th><th>LCG: $\small{x_{n+1}=137x_n+9531381 \mod 2^{24}}$</th><th>PCG: $\small{x_{n+1}=137\frac{x_n(x_n+1)}{2}+9531382 \mod 2^{24}}$</th>
      </tr>
    </thead>
    <tbody id="data">
    </tbody>
    <tfoot>
      <tr><td colspan="3">By observation, the $n$-th lowest bit of the LCG output has period $2^n$, while the PCG output does not show this pattern.</td></tr>
    </tfoot>
  </table>
  <h3 id="lattice">Consecutive output as points in 3-space</h3>
  <input type="button" value="LCG" id="genlcg" />
  <input type="button" value="QCG" id="genqcg" />
  <input type="button" value="PCG²" id="genpcg2" />
  <input type="button" value="PCG⁴" id="genpcg4" />
  $PCG^n$ denotes a PCG of degree $n$.<br>
  <canvas id="latticeGraph" width="1080" height="640"></canvas>
  The QCG (Quadratic Congruential Generator) by Donald E. Knuth is of the form $x_{n+1}=ax^2+bx+c \mod m$, $a, b, c \in \mathbb{Z}$.<br>
  In LCG, consecutive outputs fall into a regular lattice. In QCG, the points fall into lines. In PCG² and PCG⁴, most points appear to be randomly placed while only a few small sets of points fall into lines.
  <h3 id="parameters">Generator period and cycle decomposition</h3>
  The recurrence function of the PCG is a permutation on the set of integers $[0, m - 1]$. For it to achieve maximal period, the permutation must have a single m-cycle (strict cyclic permutation). However, the necessary conditions for $a$ and $c$ to obtain a m-cycle is unknown.<br>
  Load cycle count chart for $m$ = 2^<input id="cycleMod" type="number" value="6" step="1" min="2" max="8">. The x and y-axis represent different $c$ and $a$ values ($a \equiv 1 \pmod 2$), respectively. Pure red pixels represent a maximal generator parameter pair.<br>
  Click a pixel to view its cycle decomposition. Alternatively, change the parameters below.
  <canvas id="cycleChart" width="256" height="128"></canvas>
  <br>a = <input id="cycleA" type="number" step="2" min="1" max="255" value="1">,
  c = <input id="cycleC" type="number" step="1" min="0" max="255" value="32"> &rarr;
  cycles = <span id="cycleCount">1</span><br>
  <canvas id="cycleGraph" width="800" height="800"></canvas>
  <h3>Special properties of parameters</h3>
  If $a \equiv 1 \pmod 4$ and $c \equiv 0 \pmod 2$, the number of cycles is odd.<br>
  If $a \equiv 3 \pmod 4$ and $c \equiv 1 \pmod 2$, the number of cycles is also odd.<br>
  Otherwise, the number of cycles is even.<br>
  Generally, the generators come in pairs:<br>Let $x_{n+1}=a\frac{x_n(x_n+1)}{2}+c \mod m$, and $y_{n+1}=a'\frac{y_n(y_n+1)}{2}+c' \mod m$,<br> s.t. $a' = m - a$ and $c' = \left(\frac{m}{2} - 1\right) \oplus c$, where $\oplus$ denotes the bitwise exclusive or.<br>If $x_0 + y_0 = m - 1$, then $x_n + y_n = m - 1$ for all $n$. $x$ and $y$ are a pair of generators whose output is the 1's complement of each other, regardless of the period of the generators.
  <h3>General implementation in C:</h3>
  <pre>    <span class="a">unsigned int</span> a = <span class="y">137</span>; <span class="g">// largest known maximal period generator</span>
    <span class="a">unsigned int</span> c = <span class="y">9531382</span>;
    <span class="a">unsigned int</span> x = <span class="y">0</span>; <span class="g">// some starting seed 0 <= x <= m</span>
    <span class="a">unsigned int</span> m = <span class="y">0xffffff</span>; <span class="g">// m = 2^24 - 1, bitmask</span>
    <span class="a">unsigned int</span> <span class="l">pcg_next</span>() {
        <span class="m">return</span> x = a * (x * (x + <span class="y">1</span>) >> <span class="y">1</span>) + c & m;
    }</pre>
  <h3 id="overflow">Avoiding integer overflow:</h3>
  <pre>    <span class="a">unsigned int</span> a = <span class="y">123456789</span>; <span class="g">// unknown period, a mod 4 == 1</span>
    <span class="a">unsigned int</span> c = <span class="y">0x5f375a86</span>; <span class="g">// guarantee odd # of cycles</span>
    <span class="a">unsigned int</span> x = <span class="y">0</span>; <span class="g">// 0 <= x <= m = 2^32 - 1</span>
    <span class="a">unsigned int</span> <span class="l">pcg_next</span>() {
        <span class="a">bool</span> bit = x & <span class="y">1</span>; <span class="g">// bitshift right before overflow (x + 1) can occur</span>
        <span class="m">return</span> x = a * ((x >> !bit) * ((x >> bit) + <span class="y">1</span>)) + c;
    }</pre>
  Created by Qiyuan (Charly) Cui, 2020.
  <h3>References</h3>
  <a href="https://dspace.library.uvic.ca:8443/bitstream/handle/1828/3142/Random_Number_Generators.pdf?sequence">Random Number Generators (T.E. Hull, A.R. Dobell)</a><br>
  The Art of Programming, Volume 2 (Donald E. Knuth)<br>
  <script type="text/javascript">'use strict';
function *LCG(a, b, m, x = 0) { while(true) yield x=a*x+b&m; }
function *QCG(a, b, c, m, x = 0) { while(true) yield x=(x*(a*x+b)+c)%m; }
function *PCG2(a, b, m, x = 0) { while(true) yield x=a*(x*(x+1)>>>1)+b&m; }
function *PCG4(a, b, m, x = 0) { while(true) yield x=a*(x*(x+1)*(x*(x+1)-1)>>>1)+b&m; }
const lcg24 = LCG(137, 9531381, 0xffffff), pcg24 = PCG2(137, 9531382, 0xffffff);
let genIndex = 1;
for(; genIndex < 16; ++genIndex) {
  data.innerHTML += `<tr><td>${genIndex}</td><td>${lcg24.next().value.toString(2).padStart(24, '0')}</td><td>${pcg24.next().value.toString(2).padStart(24, '0')}</td></tr>`;
}
genNext.onclick = e => {
  data.removeChild(data.firstElementChild);
  data.innerHTML += `<tr><td>${genIndex++}</td><td>${lcg24.next().value.toString(2).padStart(24, '0')}</td><td>${pcg24.next().value.toString(2).padStart(24, '0')}</td></tr>`;
}
const chartCTX = cycleChart.getContext('2d', { alpha: false }),
  graphCTX = cycleGraph.getContext('2d', { alpha: false });
let arr;
graphCTX.textAlign = 'center';
graphCTX.textBaseline = 'middle';
graphCTX.translate(400, 400);
graphCTX.fillStyle = '#fff';
cycleMod.oninput = e => {
  const v = +cycleMod.value;
  if(Number.isInteger(v) && v > 1 && v < 9) {
    const m = 1 << v, ang = Math.PI * 2 / m;
    cycleA.max = cycleC.max = m - 1;
    graphCTX.font = 20 - v + 'px Arial';
    graphCTX.clearRect(-400, -400, 800, 800);
    for(let i = m; i--;)
      graphCTX.fillText(i, Math.cos(i * ang) * 391, Math.sin(i * ang) * 391);
    cycleChart.width = m;
    cycleChart.height = m >> 1;
    fetch(`c${v}.dat`).then(data => data.arrayBuffer()).then(buffer => {
      arr = new Uint8Array(buffer);
      let scale = 360 / arr[arr.length - 1] - 1;
      // this problem might be due to http-server and not fetch()
      //if (scale == 359) // scale value is correct on Github pages
        //scale = 35; // line break = 0x0a = 10, 360 / 10 - 1 = 35
      for(let i = m >> 1; i--;) {
        for(let j = m; j--;) {
          const val = (arr[i << v | j] - 1) * scale;
          chartCTX.fillStyle = `hsl(${val},100%,50%)`;
          chartCTX.fillRect(j, i, 1, 1);
        }
      }
    }, error => console.warn(error));
  }
}
cycleMod.oninput();
cycleChart.onclick = e => {
  drawGraph(e.offsetY * cycleChart.height >> 8 | 1, e.offsetX * cycleChart.width >> 10);
}
cycleA.oninput = cycleC.oninput = e => {
  drawGraph(+cycleA.value, +cycleC.value);
}
const drawGraph = (a, c) => {
  graphCTX.clearRect(-400, -400, 800, 800);
  const m = cycleChart.width,
    mask = m - 1,
    ang = Math.PI * 2 / m,
    scale = 360 / arr[(a >> 1) * m | c] - 1,
    bitmap = new Uint8Array(m);
  cycleA.value = a;
  cycleC.value = c;
  let index = 0, cycles = 0;
  while((index = bitmap.indexOf(0, index)) >= 0) {
    graphCTX.strokeStyle = graphCTX.fillStyle = `hsl(${cycles * scale},100%,50%)`;
    let x = index, length = 0;
    graphCTX.beginPath();
    graphCTX.moveTo(Math.cos(x * ang) * 384, Math.sin(x * ang) * 384);
    do {
      bitmap[x] = 1;
      ++length;
      x = a * (x * (x + 1) >> 1) + c & mask;
      graphCTX.fillText(x, Math.cos(x * ang) * 391, Math.sin(x * ang) * 391);
      graphCTX.lineTo(Math.cos(x * ang) * 384, Math.sin(x * ang) * 384);
    } while(x != index);
    if(length == 1) {
      graphCTX.beginPath();
      graphCTX.arc(Math.cos(x * ang) * 391, Math.sin(x * ang) * 391, 10, 0, Math.PI * 2);
    } else {
      graphCTX.closePath();
    }
    graphCTX.stroke();
    ++cycles;
    ++index;
  }
  cycleCount.textContent = cycles;
}
document.addEventListener('DOMContentLoaded', function() {
  const w = 1080, h = 640,
    scene = new Phoria.Scene(), renderer = new Phoria.CanvasRenderer(latticeGraph);
    scene.viewport.width = w;
    scene.viewport.height = h;
    scene.camera.position = {x:0, y:0, z:0};
    scene.camera.lookat = {x:0, y:0, z:0};
    scene.perspective.aspect = w / h;
  const grid = {
    style: {
      drawmode: "wireframe",
      shademode: 'plain',
      linewidth: 1,
      objectsortmode: "back"
    }
  }
  let plane1 = Phoria.Entity.create(Object.assign(Phoria.Util.generateTesselatedPlane(2,2,0,1024), grid)),
  plane2 = Phoria.Entity.create(Object.assign(Phoria.Util.generateTesselatedPlane(2,2,0,1024), grid)),
  plane3 = Phoria.Entity.create(Object.assign(Phoria.Util.generateTesselatedPlane(2,2,0,1024), grid));
  plane2.rotateX(Math.PI / 2);
  plane3.rotateZ(Math.PI / 2);
  scene.graph.push(plane1, plane2, plane3);
  const lcg = LCG(761, 1013, 1023), qcg = QCG(231, 43, 97, 1029, 1), pcg2 = PCG2(763, 1013, 1023), pcg4 = PCG4(763, 1013, 1023);
  const points = Phoria.Entity.create({
    points: [],
    style: {
      color: [255, 255, 0],
      drawmode: 'point',
      shademode: 'plain',
      linewidth: 256,
      linescale: 1
    }
  });
  for(let i = 1024; i--;)
    points.points.push({ x: pcg2.next().value - 512, y: pcg2.next().value - 512, z: pcg2.next().value - 512});
  scene.graph.push(points);
  let pause = false, timer, r = 2000, theta = 0, phi = 0, moved = true;
  function draw(now) {
    if(moved) {
      moved = false;
      r = Math.sqrt(4000000 - scene.camera.position.y ** 2);
      if(r == 0) r = 1;
      scene.camera.position.x = r * Math.sin(theta);
      scene.camera.position.z = r * Math.cos(theta);
      scene.modelView();
      renderer.render(scene);
    }
    timer = requestAnimationFrame(draw);
  }
  timer = requestAnimationFrame(draw);
  genlcg.onclick = e => {
    for(let i = 1024; i--;) {
      points.points[i].x = lcg.next().value - 512;
      points.points[i].y = lcg.next().value - 512;
      points.points[i].z = lcg.next().value - 512;
    }
    points.style.color = [0, 255, 255];
    latticeGraph.requestPointerLock();
    moved = true;
  };
  genqcg.onclick = e => {
    for(let i = 1024; i--;) {
      points.points[i].x = qcg.next().value - 514.5;
      points.points[i].y = qcg.next().value - 514.5;
      points.points[i].z = qcg.next().value - 514.5;
      if(i == 683 || i == 341) qcg.next(); // 1029 is a multiple of 3; offset the generator to generate 3x more points
    }
    points.style.color = [255, 255, 255];
    latticeGraph.requestPointerLock();
    moved = true;
  }
  genpcg2.onclick = e => {
    for(let i = 1024; i--;) {
      points.points[i].x = pcg2.next().value - 512;
      points.points[i].y = pcg2.next().value - 512;
      points.points[i].z = pcg2.next().value - 512;
    }
    points.style.color = [255, 255, 0];
    latticeGraph.requestPointerLock();
    moved = true;
  };
  genpcg4.onclick = e => {
    for(let i = 1024; i--;) {
      points.points[i].x = pcg4.next().value - 512;
      points.points[i].y = pcg4.next().value - 512;
      points.points[i].z = pcg4.next().value - 512;
    }
    points.style.color = [255, 0, 255];
    latticeGraph.requestPointerLock();
    moved = true;
  }
  latticeGraph.addEventListener('click', e => document.pointerLockElement || latticeGraph.requestPointerLock());
  latticeGraph.addEventListener('mousemove', e => {
    const deg = e.movementX / 200;
    points.rotateY(-deg);
    plane1.rotateY(-deg);
    plane2.rotateZ(deg);
    plane3.rotateX(-deg);
    scene.camera.position.y += e.movementY * (Math.sqrt(100 - scene.camera.position.y ** 2 * 2.5e-05) + 1);
    if(scene.camera.position.y > 2000) scene.camera.position.y = 2000;
    else if(scene.camera.position.y < -2000) scene.camera.position.y = -2000;
    moved = true;
  });
  document.addEventListener('pointerlockchange', e => {
    if(document.pointerLockElement) timer = requestAnimationFrame(draw);
    else cancelAnimationFrame(timer);
  });
})
  </script>
</body>
</html>