задание 6

1 files changed, 657 insertions, 1 deletions

diff --git a/t4mk.ipynb b/t4mk.ipynb
index cae8c70..dcf2c42 100644
--- a/t4mk.ipynb
+++ b/t4mk.ipynb
@@ -596,7 +596,663 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "# Задание 6"
+    "# Задание 6\n",
+    "\n",
+    "Ввести параметры кривой $a$, $b$, размер поля `field` и индекс точки в полученном отсортированном списке точек `pt_idx`."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 157,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "a:  1\n",
+      "b:  1\n"
+     ]
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\begin{array}{|c|c|c|c|}\n",
+       "x & x^2 & x^3 + 1 \\cdot x + 1 & y \\\\\n",
+       "0 & 0 & 1 & \\\\\n",
+       "1 & 1 & 3 & \\\\\n",
+       "2 & 4 & 0 & \\\\\n",
+       "3 & -2 & -2 & \\\\\n",
+       "4 & 5 & 3 & \\\\\n",
+       "5 & 3 & -1 & \\\\\n",
+       "-5 & 3 & 3 & \\\\\n",
+       "-4 & 5 & -1 & \\\\\n",
+       "-3 & -2 & 4 & \\\\\n",
+       "-2 & 4 & 2 & \\\\\n",
+       "-1 & 1 & -1 & \\\\\n",
+       "\\end{array}$"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Порядок: 14\n",
+      "Точки:\n",
+      "(0, 1)\n",
+      "(0, -1)\n",
+      "(1, 5)\n",
+      "(1, -5)\n",
+      "(2, 0)\n",
+      "(3, 3)\n",
+      "(3, -3)\n",
+      "(4, 5)\n",
+      "(4, -5)\n",
+      "(-3, 2)\n",
+      "(-3, -2)\n",
+      "(-5, 5)\n",
+      "(-5, -5)\n",
+      "None\n",
+      "Проверяем точку (0, -1)\n"
+     ]
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle $"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\lambda = \\frac{3 \\cdot x_1^2 + a}{2 \\cdot y_1} = \\frac{3 \\cdot 0^2 + 1}{2 \\cdot 10} = \\frac{3 \\cdot 0 + 1}{20} = \\frac{1}{20}$"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\lambda = 1 \\cdot 20^{-1} = 1 \\cdot 5 = 5 \\pmod{11} = 5$"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle x_3 = \\lambda^2 - x_1 - x_2 = 5^2 - 0 - 0 = 25 - 0 - 0 = 25 \\pmod{11} = 3 = 3$"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle y_3 = \\lambda \\cdot (x_1 - x_3) - y_1 = 5 \\cdot (0 - 3) - 10 = 5 \\cdot -3 - 10 = -15 - 10 = -25 \\pmod{11} = 8 = -3$"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle 2P = P + P = (0, -1) + (0, -1) = (3, -3)$"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle $"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\lambda = \\frac{y_2 - y_1}{x_2 - x_1} = \\frac{10 - 8}{0 - 3} = \\frac{2}{-3}$"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\lambda = 2 \\cdot -3^{-1} = 2 \\cdot 7 = 14 \\pmod{11} = 3$"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle x_3 = \\lambda^2 - x_1 - x_2 = 3^2 - 3 - 0 = 9 - 3 - 0 = 6 \\pmod{11} = 6 = -5$"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle y_3 = \\lambda \\cdot (x_1 - x_3) - y_1 = 3 \\cdot (3 - 6) - 8 = 3 \\cdot -3 - 8 = -9 - 8 = -17 \\pmod{11} = 5 = 5$"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle 3P = 2P + P = (3, -3) + (0, -1) = (-5, 5)$"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle $"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\lambda = \\frac{y_2 - y_1}{x_2 - x_1} = \\frac{10 - 5}{0 - 6} = \\frac{5}{-6}$"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\lambda = 5 \\cdot -6^{-1} = 5 \\cdot 9 = 45 \\pmod{11} = 1$"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle x_3 = \\lambda^2 - x_1 - x_2 = 1^2 - 6 - 0 = 1 - 6 - 0 = -5 \\pmod{11} = 6 = -5$"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle y_3 = \\lambda \\cdot (x_1 - x_3) - y_1 = 1 \\cdot (6 - 6) - 5 = 1 \\cdot 0 - 5 = 0 - 5 = -5 \\pmod{11} = 6 = -5$"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle 4P = 3P + P = (-5, 5) + (0, -1) = (-5, -5)$"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle $"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\lambda = \\frac{y_2 - y_1}{x_2 - x_1} = \\frac{10 - 6}{0 - 6} = \\frac{4}{-6}$"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\lambda = 4 \\cdot -6^{-1} = 4 \\cdot 9 = 36 \\pmod{11} = 3$"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle x_3 = \\lambda^2 - x_1 - x_2 = 3^2 - 6 - 0 = 9 - 6 - 0 = 3 \\pmod{11} = 3 = 3$"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle y_3 = \\lambda \\cdot (x_1 - x_3) - y_1 = 3 \\cdot (6 - 3) - 6 = 3 \\cdot 3 - 6 = 9 - 6 = 3 \\pmod{11} = 3 = 3$"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle 5P = 4P + P = (-5, -5) + (0, -1) = (3, 3)$"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle $"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\lambda = \\frac{y_2 - y_1}{x_2 - x_1} = \\frac{10 - 3}{0 - 3} = \\frac{7}{-3}$"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\lambda = 7 \\cdot -3^{-1} = 7 \\cdot 7 = 49 \\pmod{11} = 5$"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle x_3 = \\lambda^2 - x_1 - x_2 = 5^2 - 3 - 0 = 25 - 3 - 0 = 22 \\pmod{11} = 0 = 0$"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle y_3 = \\lambda \\cdot (x_1 - x_3) - y_1 = 5 \\cdot (3 - 0) - 3 = 5 \\cdot 3 - 3 = 15 - 3 = 12 \\pmod{11} = 1 = 1$"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle 6P = 5P + P = (3, 3) + (0, -1) = (0, 1)$"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle $"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\text{Координаты } x \\text{ данных точек одинаковы} \\implies \\text{ сумма равна } \\mathcal{O}$"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle 7P = 6P + P = (0, 1) + (0, -1) = \\mathcal{O}$"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\text{Порядок точки } (0, -1) \\text{ равен } 7$"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "a, b, field, pt_idx = 1, 1, 11, 1\n",
+    "\n",
+    "from math import gcd\n",
+    "\n",
+    "def point_neg(point, A, B, p):\n",
+    "    if not is_on_curve(point, A, B, p):\n",
+    "        raise RuntimeError\n",
+    "    if point is None:\n",
+    "        return point\n",
+    "    \n",
+    "    x, y = point\n",
+    "\n",
+    "    result = x, -y % p\n",
+    "    if not is_on_curve(result, A, B, p):\n",
+    "        raise RuntimeError\n",
+    "    return result\n",
+    "\n",
+    "def extended_gcd(a, b):\n",
+    "    if a == 0:\n",
+    "        return b, 0, 1\n",
+    "    else:\n",
+    "        gcd, x, y = extended_gcd(b % a, a)\n",
+    "        return gcd, y - (b // a) * x, x\n",
+    "\n",
+    "def inverse_mod(k, p):\n",
+    "    if k < 0:\n",
+    "        return p - inverse_mod(-k, p)\n",
+    "    gcd, x, _ = extended_gcd(k, p)\n",
+    "    if gcd == 1:\n",
+    "        if x < 0:\n",
+    "            x += p\n",
+    "        return x\n",
+    "    else:\n",
+    "        raise RuntimeError\n",
+    "\n",
+    "def is_on_curve(point, A, B, p):\n",
+    "    if point is None:\n",
+    "        return True\n",
+    "\n",
+    "    x, y = point\n",
+    "    return (y ** 2 - x ** 3 - A * x - B) % p == 0\n",
+    "\n",
+    "\n",
+    "def point_addition(point1, point2, A, B, p):\n",
+    "    display(Math(''))\n",
+    "    if not is_on_curve(point1, A, B, p) or not is_on_curve(point2, A, B, p):\n",
+    "        raise RuntimeError\n",
+    "\n",
+    "    if point1 is None:\n",
+    "        return point2\n",
+    "    elif point2 is None:\n",
+    "        return point1\n",
+    "\n",
+    "    x1, y1 = point1\n",
+    "    x2, y2 = point2\n",
+    "\n",
+    "    if x1 - x2 == 0 and y1 != y2:\n",
+    "        display(Math(r'\\text{Координаты } x \\text{ данных точек одинаковы} \\implies \\text{ сумма равна } \\mathcal{O}'))\n",
+    "        return None\n",
+    "\n",
+    "    if x1 == x2:\n",
+    "        display(Math(fr\"\\lambda = \\frac{{3 \\cdot x_1^2 + a}}{{2 \\cdot y_1}} = \\frac{{3 \\cdot {x1}^2 + {A}}}{{2 \\cdot {y1}}} = \\frac{{3 \\cdot {x1 * x1} + {A}}}{{{2 * y1}}} = \\frac{{{3 * x1 * x1 + A}}}{{{2 * y1}}}\"))\n",
+    "        # print(f\"\\talpha = (3 * x1^2 + a) / (2 * y1) = (3 * {x1}^2 + {A}) / (2 * {y1}) = (3 * {x1 * x1} + {A}) / {2 * y1} = {3 * x1 * x1 + A} / {2 * y1}\")\n",
+    "        num = (3 * x1 ** 2 + A)\n",
+    "        den = 2 * y1\n",
+    "    else:\n",
+    "        display(Math(fr\"\\lambda = \\frac{{y_2 - y_1}}{{x_2 - x_1}} = \\frac{{{y2} - {y1}}}{{{x2} - {x1}}} = \\frac{{{y2 - y1}}}{{{x2 - x1}}}\"))\n",
+    "        # print(f\"\\talpha = (y2 - y1) / (x2 - x1) = ({y2} - {y1}) / ({x2} - {x1}) = {y2 - y1} / {x2 - x1}\")\n",
+    "        num = y2 - y1\n",
+    "        den = x2 - x1\n",
+    "\n",
+    "    if gcd(den, p) != 1:\n",
+    "        return None\n",
+    "\n",
+    "    m = num * inverse_mod(den, p)\n",
+    "    display(Math(fr\"\\lambda = {num} \\cdot {den}^{{-1}} = {num} \\cdot {inverse_mod(den, p)} = {m} \\pmod{{{p}}} = {m % p}\"))\n",
+    "    # print(f\"\\talpha = {num} * {inverse_mod(den, p)} = {m} (mod {p}) = {m % p}\")\n",
+    "    m = m % p\n",
+    "\n",
+    "    x3 = m * m - x1 - x2\n",
+    "    display(Math(fr\"x_3 = \\lambda^2 - x_1 - x_2 = {m}^2 - {x1} - {x2} = {m * m} - {x1} - {x2} = {x3} \\pmod{{{p}}} = {x3 % p} = {mm(x3 % p)}\"))\n",
+    "    # print(f\"\\tx3 = alpha^2 - x1 - x2 = {m}^2 - {x1} - {x2} = {m * m} - {x1} - {x2} = {x3} (mod {p}) = {x3 % p}\")\n",
+    "    x3 = x3 % p\n",
+    "\n",
+    "    y3 = m * (x1 - x3) - y1\n",
+    "    display(Math(fr\"y_3 = \\lambda \\cdot (x_1 - x_3) - y_1 = {m} \\cdot ({x1} - {x3}) - {y1} = {m} \\cdot {x1 - x3} - {y1} = {m * (x1 - x3)} - {y1} = {y3} \\pmod{{{p}}} = {y3 % p} = {mm(y3 % p)}\"))\n",
+    "    # print(f\"\\ty3 = alpha * (x1 - x3) - y1 = {m} * ({x1} - {x3}) - {y1} = {m} * {x1 - x3} - {y1} = {m * (x1 - x3)} - {y1} = {y3} (mod {p}) = {y3 % p}\")\n",
+    "    y3 = y3 % p\n",
+    "    result = (x3 % p, y3 % p)\n",
+    "    if not is_on_curve(result, A, B, p):\n",
+    "        raise RuntimeError\n",
+    "    return result\n",
+    "\n",
+    "\n",
+    "def mm(x):\n",
+    "    if x <= 5:\n",
+    "        return x\n",
+    "    else:\n",
+    "        return x - field\n",
+    "\n",
+    "def mmm(p):\n",
+    "    return (mm(p[0]), mm(p[1]))\n",
+    "\n",
+    "N = 1\n",
+    "\n",
+    "print(\"a: \", a)\n",
+    "print(\"b: \", b)\n",
+    "\n",
+    "\n",
+    "points = []\n",
+    "\n",
+    "tmp = []\n",
+    "for x in range(field):\n",
+    "    ys = (x * x) % field\n",
+    "    xs = (x * x * x + a * x + b) % field\n",
+    "    tmp.append((mm(x), mm(ys), mm(xs)))\n",
+    "    # print(f\"x = {mm(x)}; x^2 = {mm(ys)}; x^3 + ax + b = {mm(xs)}\")\n",
+    "\n",
+    "ltx = [r\"\\begin{array}{|c|c|c|c|}\", fr\"x & x^2 & x^3 + {a} \\cdot x + {b} & y \\\\\"]\n",
+    "for (x, x2, x3) in tmp:\n",
+    "    ltx.append(fr\"{x} & {x2} & {x3} & \\\\\")\n",
+    "ltx.append(r\"\\end{array}\")\n",
+    "display(Math('\\n'.join(ltx)))\n",
+    "\n",
+    "for x in range(field):\n",
+    "    sqrt_y = x * x * x + a * x + b\n",
+    "    for y in range(field):\n",
+    "        if (y * y) % field == sqrt_y % field:\n",
+    "            N += 1\n",
+    "            points.append((x, y))\n",
+    "\n",
+    "def sort_key(a):\n",
+    "    m = list(mmm(a))\n",
+    "    if m[0] < 0:\n",
+    "        m[0] = abs(m[0]) + 1e9\n",
+    "    if m[1] < 0:\n",
+    "        m[1] = abs(m[1]) + 1e9\n",
+    "    return tuple(m)\n",
+    "\n",
+    "points.sort(key=sort_key)\n",
+    "points = points + [None]\n",
+    "\n",
+    "print(\"Порядок:\", N)\n",
+    "print(\"Точки:\")\n",
+    "pts_print = []\n",
+    "for pt in points:\n",
+    "    if pt is not None:\n",
+    "        pt = mmm(pt)\n",
+    "    print(pt)\n",
+    "\n",
+    "\n",
+    "pt = points[pt_idx]\n",
+    "print(f\"Проверяем точку {mmm(pt)}\")\n",
+    "\n",
+    "kpts = set([pt])\n",
+    "last = pt\n",
+    "for k in range(2, N + 1):\n",
+    "    try:\n",
+    "        kpt = point_addition(last, pt, a, b, field)\n",
+    "\n",
+    "        last1 = mm(last[0]), mm(last[1])\n",
+    "        if kpt is not None:\n",
+    "            kpt1 = mm(kpt[0]), mm(kpt[1])\n",
+    "        else:\n",
+    "            kpt1 = r'\\mathcal{O}'\n",
+    "        pt1 = mm(pt[0]), mm(pt[1])\n",
+    "        \n",
+    "        if k != 2:\n",
+    "            display(Math(f\"{k}P = {k - 1}P + P = {last1} + {pt1} = {kpt1}\"))\n",
+    "        else:\n",
+    "            display(Math(f\"{k}P = P + P = {last1} + {pt1} = {kpt1}\"))\n",
+    "        last = kpt\n",
+    "        if kpt is not None:\n",
+    "            kpts.add(kpt)\n",
+    "        else:\n",
+    "            break\n",
+    "    except RuntimeError:\n",
+    "        break\n",
+    "\n",
+    "pt1 = mm(pt[0]), mm(pt[1])\n",
+    "display(Math(fr\"\\text{{Порядок точки }} {pt1} \\text{{ равен }} {k}\"))\n"
    ]
   }
  ],