W05_KiK_podstawy_algebraiczne.ipynb

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    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
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    "language_info": {
      "name": "python"
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  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
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      "source": [
        "<a href=\"https://colab.research.google.com/github/mrospond/kik/blob/main/W05_KiK_podstawy_algebraiczne.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "#@title Potrzebne biblioteki...\n",
        "\n",
        "# Rysowanie standardowych wykresów\n",
        "import matplotlib.pyplot as plt\n",
        "# Obróbka statystyczna przy rysowaniu wykresów i formatowaniu ich wyglądu\n",
        "import seaborn as sns\n",
        "sns.set()\n",
        "\n",
        "# Obliczenia matematyczne, dla nas głównie logarytm\n",
        "import math\n",
        "\n",
        "# Praca z wektorami i macierzami\n",
        "import numpy as np\n",
        "\n",
        "# Eleganckie formatowanie wyświetlanych danych\n",
        "import pprint\n",
        "\n",
        "# Obliczenia symboliczne\n",
        "## Faktoryzacja liczb\n",
        "from sympy import factorint\n",
        "## Sprawdzanie pierwszości\n",
        "from sympy import isprime\n",
        "## Obliczanie funkcji φ Eulera (tocjent)\n",
        "from sympy import totient\n",
        "from sympy import sieve\n",
        "## Pierwiastek pierwotny\n",
        "from sympy import primitive_root\n",
        "\n",
        "# Obliczanie entropii\n",
        "from scipy.stats import entropy\n",
        "\n",
        "# Zliczanie znaków w ciągu\n",
        "from collections import Counter\n",
        "\n",
        "# Zliczanie czasu trwania\n",
        "import time\n",
        "\n",
        "# Generacja liczb losowych\n",
        "import random"
      ],
      "metadata": {
        "id": "e9_HTJHJfmD2"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Klasy reszt $\\mathbb{Z}_n$"
      ],
      "metadata": {
        "id": "uvZYmn7Rg8b7"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "#@title { run: \"auto\" }\n",
        "#@markdown Grupa addytywna $\\mathbb{Z}_n$\n",
        "\n",
        "#@markdown Rząd grupy $n$:\n",
        "n = 5 #@param {type:\"integer\"}\n",
        "\n",
        "assert n >= 2, \"Wartość minimalna n to dwa!\"\n",
        "\n",
        "print(end='+|\\t')\n",
        "for i in range(n):\n",
        "  print(i, end='\\t')\n",
        "print()\n",
        "print(\"-\"*(n*9))\n",
        "\n",
        "for i in range(n):\n",
        "  print(i, end='|\\t')\n",
        "  for j in range(n):\n",
        "    print((i+j)%n, end='\\t')\n",
        "  print()"
      ],
      "metadata": {
        "id": "a-60Gi6ChBXp",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "623fb1f8-5ec4-4a45-a01c-f3cf03a2a27a"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "+|\t0\t1\t2\t3\t4\t\n",
            "---------------------------------------------\n",
            "0|\t0\t1\t2\t3\t4\t\n",
            "1|\t1\t2\t3\t4\t0\t\n",
            "2|\t2\t3\t4\t0\t1\t\n",
            "3|\t3\t4\t0\t1\t2\t\n",
            "4|\t4\t0\t1\t2\t3\t\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "#@title { run: \"auto\" }\n",
        "\n",
        "#@markdown Element przeciwny w $\\mathbb{Z}_n$\n",
        "\n",
        "#@markdown Element $k$:\n",
        "k = 2 #@param {type:\"integer\"}\n",
        "\n",
        "k1 = int(k) % n\n",
        "\n",
        "if k1 != k:\n",
        "  print(f\"Element przeciwny do {k} w Z_{n} (czyli do: {k1}) to {-k % n}.\")\n",
        "else:\n",
        "  print(f\"Element przeciwny w Z_{n}: -{k} = {-k % n}.\")"
      ],
      "metadata": {
        "id": "26ykOqNel3Rh",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "f7e709ed-ffc2-4c36-fc51-a4eff1e6ec53"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Element przeciwny w Z_5: -2 = 3.\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Jak to jest liczone? Po prostu:\n",
        "\n",
        "$$-a = n-a$$"
      ],
      "metadata": {
        "id": "DfkpOJ8TkyWc"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "#@title { run: \"auto\" }\n",
        "#@markdown Grupa multiplikatywna $\\mathbb{Z}^{\\ast}_p$\n",
        "\n",
        "#@markdown Rząd grupy $n$ (moduł):\n",
        "p = 7 #@param {type:\"integer\"}\n",
        "\n",
        "assert p >= 2 and isprime(p), \"Wartość p musi być liczbą pierwszą\"\n",
        "\n",
        "print(end='*|\\t')\n",
        "for i in range(1,p):\n",
        "  print(i, end='\\t')\n",
        "print()\n",
        "print(\"-\"*(p*9))\n",
        "\n",
        "for i in range(1,p):\n",
        "  print(i, end='|\\t')\n",
        "  for j in range(1,p):\n",
        "    print((i*j)%p, end='\\t')\n",
        "  print()"
      ],
      "metadata": {
        "id": "SMecYXu9i5IK",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "e6806331-df64-4280-c19a-9b506c19011f"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "*|\t1\t2\t3\t4\t5\t6\t\n",
            "---------------------------------------------------------------\n",
            "1|\t1\t2\t3\t4\t5\t6\t\n",
            "2|\t2\t4\t6\t1\t3\t5\t\n",
            "3|\t3\t6\t2\t5\t1\t4\t\n",
            "4|\t4\t1\t5\t2\t6\t3\t\n",
            "5|\t5\t3\t1\t6\t4\t2\t\n",
            "6|\t6\t5\t4\t3\t2\t1\t\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "#@title { run: \"auto\" }\n",
        "#@markdown Element odwrotny w $\\mathbb{Z}^{\\ast}_p$\n",
        "\n",
        "#@markdown Element $k$:\n",
        "k = 3 #@param {type:\"integer\"}\n",
        "\n",
        "k1 = int(k) % p\n",
        "assert k1 != 0, \"Nie ma elementu odwrotnego do 0\"\n",
        "\n",
        "if k1 != k:\n",
        "  print(f\"Element odwrotny do {k} w Z_{p} (czyli do: {k1}) to {pow(k,-1,p)}.\")\n",
        "else:\n",
        "  print(f\"Element odwrotny do {k} w Z_{p} to: {k}^-1 = {pow(k,-1,p)}.\")"
      ],
      "metadata": {
        "id": "GWWQlmxCkDjg",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "8fde3772-563b-4460-fe20-bb40b38aff03"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Element odwrotny do 3 w Z_7 to: 3^-1 = 5.\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Ale jak to jest liczone? A to już nie jest tak trywialne jak w grupie addytywnej..."
      ],
      "metadata": {
        "id": "uEkbsSOLkvIh"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Rozszerzony algorytm Euklidesa (EEA)\n",
        "\n",
        "Tożsamość Bezouta:\n",
        "\n",
        "$$ax + by = \\mathit{gcd}(a,b)$$\n",
        "\n",
        "Jeśli $a$ i $b$ są względnie pierwsze, tj. $\\mathit{gcd}(a,b) = 1$, to z tej tożsamości możemy odczytywać odwrotności: $x = a^{-1} \\bmod{b}$ oraz $y = b^{-1} \\bmod{a}$.\n",
        "\n",
        "Oczywiście z liczbą pierwszą $p$ każda liczba od niej niemniejsza (tj. $1,\\dots,p-1$) jest względnie pierwsza, dlatego moduł będący liczbą pierwszą gwarantuje, że będziemy mieli odwrotność $\\bmod{p}$."
      ],
      "metadata": {
        "id": "6aQsnSHcfLoo"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "#@title { run: \"auto\" }\n",
        "#@markdown Najpierw zobaczmy, dlaczego algorytm Euklidesa w ogóle jest sprytny...\n",
        "\n",
        "pier = 1255254 #@param {type:\"integer\"}\n",
        "drug = 20900 #@param {type:\"integer\"}\n",
        "\n",
        "#@markdown Używany jest dosyć efektywny (ale ponadwielomianowy) algorytm faktoryzacji $\\rho$ Pollarda\n",
        "\n",
        "start = time.time()\n",
        "s_pier = factorint(pier)\n",
        "s_drug = factorint(drug)\n",
        "stop = time.time()\n",
        "\n",
        "print(\"Metoda z podstawówki:\")\n",
        "print(\"*\"*100)\n",
        "print(f\"Faktoryzacja {pier} = {s_pier}.\")\n",
        "print(f\"Faktoryzacja {drug} = {s_drug}.\")\n",
        "\n",
        "gcd = 1\n",
        "for k in s_pier:\n",
        "  if k in s_drug:\n",
        "    gcd *= k**min(s_pier[k],s_drug[k])\n",
        "\n",
        "print(f\"gcd({pier},{drug}) = {gcd}.\")\n",
        "print(\"*\"*100)\n",
        "print(f\"Faktoryzacja trwała {stop-start}.\")"
      ],
      "metadata": {
        "id": "tH2_Vcyp3QTx",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "a017f3ac-1259-4dd9-e060-4c410fed1744"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Metoda z podstawówki:\n",
            "****************************************************************************************************\n",
            "Faktoryzacja 1255254 = {2: 1, 3: 1, 7: 1, 11: 2, 13: 1, 19: 1}.\n",
            "Faktoryzacja 20900 = {2: 2, 5: 2, 11: 1, 19: 1}.\n",
            "gcd(1255254,20900) = 418.\n",
            "****************************************************************************************************\n",
            "Faktoryzacja trwała 0.00014281272888183594.\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "#@markdown * Rozszerzony algorytm Euklidesa (EEA: _Extended Euclidean Algorithm_)\n",
        "#@markdown * Można pokazać, że ma złożoność wielomianową\n",
        "\n",
        "def extended_euclidean(a, b):\n",
        "\n",
        "    if b == 0:\n",
        "        return a, 1, 0\n",
        "\n",
        "    else:\n",
        "        gcd, x1, y1 = extended_euclidean(b, a % b)\n",
        "        x = y1\n",
        "        y = x1 - (a // b) * y1\n",
        "        return gcd, x, y"
      ],
      "metadata": {
        "id": "5hqL5lmQ3kQN"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "#@markdown Ile to trwa?\n",
        "\n",
        "start = time.time()\n",
        "gcd, x, y = extended_euclidean(pier, drug)\n",
        "stop = time.time()\n",
        "\n",
        "print(f\"Obliczenie z EEA trwało: {stop-start}.\")\n",
        "print(f\"gcd({pier},{drug}) = {gcd}.\")"
      ],
      "metadata": {
        "id": "IGhX7noW5_RU",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "0b1f5b9a-8c7d-446b-8ba4-cbc3617018e9"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Obliczenie z EEA trwało: 6.365776062011719e-05.\n",
            "gcd(1255254,20900) = 418.\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "#@markdown To oczywiście żaden dowód, zróbmy sobie eksperyment\n",
        "\n",
        "ile = 20\n",
        "min_l = 10e25\n",
        "max_l = 10e30\n",
        "\n",
        "wyn_fact = []\n",
        "wyn_eea = []\n",
        "for _ in range(ile):\n",
        "  pier = random.randint(min_l, max_l)\n",
        "  drug = random.randint(min_l, max_l)\n",
        "  print(pier,drug)\n",
        "\n",
        "  start = time.time()\n",
        "  gcd, x, y = extended_euclidean(pier, drug)\n",
        "  stop = time.time()\n",
        "  wyn_eea.append(stop-start)\n",
        "\n",
        "  start = time.time()\n",
        "  s_pier = factorint(pier)\n",
        "  s_drug = factorint(drug)\n",
        "  stop = time.time()\n",
        "  wyn_fact.append(stop-start)\n",
        "  print(s_pier,s_drug)\n",
        "\n",
        "print(\"*\"*100)\n",
        "print(f\"Średnia dla faktoryzacji = {sum(wyn_fact)/ile}, średnia dla EEA = {sum(wyn_eea)/ile}.\")"
      ],
      "metadata": {
        "id": "0mRF2Ye97S46",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "8c6cb9d4-fb57-4022-c9ba-365b781f7ef7"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "201257733231392609951219212916 1963144873761505291534050875806\n",
            "{2: 2, 3889: 1, 12937627489804101951094061: 1} {2: 1, 3465677: 1, 726367: 1, 389922382273448917: 1}\n",
            "5942303304776045377303580191008 4434373297208036414360232490176\n",
            "{2: 5, 3: 3, 11: 1, 21227: 1, 488879: 1, 60250185062936269: 1} {2: 6, 3: 3, 7: 1, 35407: 1, 2040719: 1, 59981: 1, 84587157547: 1}\n",
            "1124058166015897201655510865559 4538683998280368690236369901422\n",
            "{107: 1, 60497: 1, 173648552900498688767221: 1} {2: 1, 31397: 1, 1256879261237: 1, 57506672319199: 1}\n",
            "1112353925175690232438786529162 1439483481650985889834807270652\n",
            "{2: 1, 37: 1, 16139: 1, 10403831: 1, 89524388129532757: 1} {2: 2, 11: 1, 97: 1, 5303: 1, 443011: 1, 143564194252501433: 1}\n",
            "1130861323046501696702981642359 6798509149477155723816719000618\n",
            "{744361921: 1, 1519235859791518938679: 1} {2: 1, 7: 1, 313: 1, 232090037: 1, 6684744542820144527: 1}\n",
            "8152030795509606082944125721011 5431377431022769123924483301259\n",
            "{11: 1, 179: 1, 4140188316663080793775584419: 1} {3: 1, 59: 1, 34354215293: 1, 893216390948136919: 1}\n",
            "2113051561634399166617531699899 5088874250605828714954368731503\n",
            "{1367: 1, 1545758274787417093355911997: 1} {37: 1, 1231: 1, 111727978804440000767435149: 1}\n",
            "2877425408586947262011648429779 2223594856555206902324088500646\n",
            "{491: 1, 103913: 1, 4170244459: 1, 13523564755507: 1} {2: 1, 3: 2, 52581983503879: 1, 2349341720387693: 1}\n",
            "7148030858121198401495977353772 3669427830526629053609122113402\n",
            "{2: 2, 43: 1, 23971: 1, 1733691499835847773786731: 1} {2: 1, 3: 2, 17: 2, 521: 1, 7451: 1, 527316617: 1, 344591260543: 1}\n",
            "1099655245968531992989066017618 8142746609410882414626218035040\n",
            "{2: 1, 3: 1, 11: 1, 249072561461: 1, 66893932527145093: 1} {2: 5, 5: 1, 117363011: 1, 433630373617612921429: 1}\n",
            "1277780700080619582230378691277 8927344330050912884828179856954\n",
            "{1697: 1, 42925397: 1, 17541234778875983353: 1} {2: 1, 587: 1, 1775909: 1, 1004737099: 1, 4261682021281: 1}\n",
            "7120082830549295902847772111331 3453602961377269544940289561516\n",
            "{11: 1, 64781: 1, 992513: 1, 10067195654845013357: 1} {2: 2, 8758103: 1, 8210179: 1, 12007421124760367: 1}\n",
            "3686136132368811194353670714872 5182116917359347960999240976765\n",
            "{2: 3, 359: 1, 189930281: 1, 6757603774059997121: 1} {5: 1, 41: 1, 1657039: 1, 40468399: 1, 376968093146353: 1}\n",
            "2137333576110079721630365522255 3164496178643069037918554152934\n",
            "{5: 1, 23: 1, 43: 2, 131: 1, 22303: 1, 3440352768454579841: 1} {2: 1, 521: 1, 3036944509254384873242374427: 1}\n",
            "1413232218568989167676658600092 174972935488439887886057519383\n",
            "{2: 2, 3: 1, 73: 1, 109: 1, 5527: 1, 3445737437: 1, 777161503387: 1} {71: 1, 337: 1, 5431: 1, 1346489048795437253159: 1}\n",
            "2292669362014879575447022774618 4572201380352307569657306776417\n",
            "{2: 1, 7: 2, 617: 1, 432882937: 1, 87591047501552029: 1} {141503331251: 1, 32311616552984832667: 1}\n",
            "9045960662162479108762982056799 5357862892598513506919486721152\n",
            "{43: 1, 3623: 1, 55029937: 1, 11579317: 1, 91124665279: 1} {2: 7, 13: 1, 548136261863: 1, 5874213677924411: 1}\n",
            "7073924008350223894803639303295 87183534209131039270901031038\n",
            "{5: 1, 19: 1, 269: 1, 1063: 1, 7717: 1, 3720375779: 1, 9070180541: 1} {2: 1, 991: 1, 2203: 1, 588154358059: 1, 33948845017: 1}\n",
            "6383322047830465648521687599331 9790094221126071011740804466768\n",
            "{3: 1, 59: 1, 36063966371923534737410664403: 1} {2: 4, 7: 1, 41: 1, 107: 1, 143527: 1, 2962621: 1, 46858837502291: 1}\n",
            "616357787050703450567880667399 6037276747924773743959768870681\n",
            "{7: 1, 37: 1, 2379759795562561585204172461: 1} {977: 1, 1093: 1, 12347: 1, 2772542897: 1, 165153064319: 1}\n",
            "****************************************************************************************************\n",
            "Średnia dla faktoryzacji = 0.5501025557518006, średnia dla EEA = 4.4965744018554686e-05.\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "To oczywiście dalej nie jest dowód :) ale mamy odpowiednie analizy złożoności..."
      ],
      "metadata": {
        "id": "EG5_CNMkmiJj"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "#@title { run: \"auto\" }\n",
        "#@markdown Zobaczmy EEA w działaniu\n",
        "\n",
        "a = 12345 #@param {type:\"integer\"}\n",
        "b = 11111 #@param {type:\"integer\"}\n",
        "\n",
        "assert a>0 and b>0, \"Wartości mają być nieujemne!\"\n",
        "\n",
        "gcd, x, y = extended_euclidean(a, b)\n",
        "\n",
        "print(f\"Tożsamość Bezout: {a}*{x} + {b}*{y} = {gcd}\")\n",
        "\n",
        "if a>b:\n",
        "  a, x, b, y = b, y, a, x\n",
        "\n",
        "if gcd == 1:\n",
        "  x1 = x % b\n",
        "  dodatek = f\", czyli {x} mod {b}\" if x1 != x else \"\"\n",
        "  print(f\"Zatem odwrotność {a} w Z_{b} równa się: {x1}{dodatek}.\")\n",
        "  print(f\"Jest tak ponieważ {a}*{x1} = {a*x1}, natomiast {a*x1} mod {b} = {(a*x1)%b}.\")"
      ],
      "metadata": {
        "id": "dA73aaHa3wPF",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "257833c9-8e56-45c1-888b-39ad0a43e370"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Tożsamość Bezout: 12345*-2224 + 11111*2471 = 1\n",
            "Zatem odwrotność 11111 w Z_12345 równa się: 2471.\n",
            "Jest tak ponieważ 11111*2471 = 27455281, natomiast 27455281 mod 12345 = 1.\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Algorytm EEA uogólnia się do całkiem egzotycznych tworów, np. wielomianów, elementów ciał Galois $\\mathit{G}(p^m)$... One nas będą interesować w kontekście kodowania i kryptografii."
      ],
      "metadata": {
        "id": "aSlEJVUS32fh"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "#@title { run: \"auto\" }\n",
        "#@markdown Zobaczmy odwrotności w $\\mathbb{Z}^{\\ast}_n$\n",
        "\n",
        "#@markdown Rząd grupy $n$ (moduł):\n",
        "n = 19 #@param {type:\"integer\"}\n",
        "\n",
        "assert n >= 2 and isprime(n), \"Wartość n musi być liczbą pierwszą\"\n",
        "\n",
        "wartosci = []\n",
        "for i in range(1,n):\n",
        "  gcd, x, y = extended_euclidean(i, n)\n",
        "  odwr = x % n\n",
        "  print(f\"{i}^-1 mod {n} = {odwr}\")\n",
        "  wartosci.append(odwr)"
      ],
      "metadata": {
        "id": "uKfp9MbI9y4_",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "0082424c-ecc0-427b-deb5-db0fb379dc92"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "1^-1 mod 19 = 1\n",
            "2^-1 mod 19 = 10\n",
            "3^-1 mod 19 = 13\n",
            "4^-1 mod 19 = 5\n",
            "5^-1 mod 19 = 4\n",
            "6^-1 mod 19 = 16\n",
            "7^-1 mod 19 = 11\n",
            "8^-1 mod 19 = 12\n",
            "9^-1 mod 19 = 17\n",
            "10^-1 mod 19 = 2\n",
            "11^-1 mod 19 = 7\n",
            "12^-1 mod 19 = 8\n",
            "13^-1 mod 19 = 3\n",
            "14^-1 mod 19 = 15\n",
            "15^-1 mod 19 = 14\n",
            "16^-1 mod 19 = 6\n",
            "17^-1 mod 19 = 9\n",
            "18^-1 mod 19 = 18\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Widać, że jako ciąg wartości występuje tu pewna nieprzewidywalność (jak ją skwantyfikować?). Ten efekt jest jeszcze mocniejszy w przypadku odwrotności w ciele $\\mathit{G}(p^m)$, co ma zastosowanie w algorytmie AES."
      ],
      "metadata": {
        "id": "SVLOcazf_hdg"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "#@markdown Entropia ciągu odwrotności\n",
        "\n",
        "print(\"Odwrotności: \", wartosci)\n",
        "roznice = [wartosci[n]-wartosci[n-1] for n in range(1,len(wartosci))]\n",
        "print(\"Różnice w ciągu odwrotności: \", roznice)\n",
        "\n",
        "ile = Counter(roznice)\n",
        "razem = sum(ile.values())\n",
        "rozklad = {k: v/razem for k, v in ile.items()}\n",
        "print(f\"Rozkład: {rozklad}\")\n",
        "\n",
        "print(f\"Entropia: {entropy(list(rozklad.values()),base=2)} bit.\")"
      ],
      "metadata": {
        "id": "eSm0n32y__Hq",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "8f530563-ab1f-4a8e-8364-7e67ffffa454"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Odwrotności:  [1, 10, 13, 5, 4, 16, 11, 12, 17, 2, 7, 8, 3, 15, 14, 6, 9, 18]\n",
            "Różnice w ciągu odwrotności:  [9, 3, -8, -1, 12, -5, 1, 5, -15, 5, 1, -5, 12, -1, -8, 3, 9]\n",
            "Rozkład: {9: 0.11764705882352941, 3: 0.11764705882352941, -8: 0.11764705882352941, -1: 0.11764705882352941, 12: 0.11764705882352941, -5: 0.11764705882352941, 1: 0.11764705882352941, 5: 0.11764705882352941, -15: 0.058823529411764705}\n",
            "Entropia: 3.146286370662104 bit.\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Grupy cykliczne\n",
        "\n",
        "$\\mathbb{Z}_n$ jest grupą cykliczną (co najmniej w sposób trywialny $\\langle{1}\\rangle = \\mathbb{Z}_n$). Ale zobaczmy też na inne opcje:"
      ],
      "metadata": {
        "id": "3-u-PTtm6pi7"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "#@title { run: \"auto\" }\n",
        "#@markdown $\\mathbb{Z}_n$ jako grupa cykliczna i rząd elementu w grupie...\n",
        "\n",
        "#@markdown Rząd grupy $n$ (moduł):\n",
        "n = 5 #@param {type:\"integer\"}\n",
        "\n",
        "#@markdown Element $a$:\n",
        "a = 3 #@param {type:\"integer\"}\n",
        "\n",
        "a = a % n\n",
        "\n",
        "wyn = 0\n",
        "rzad = 0\n",
        "rzad_flaga = True\n",
        "print(f\"a = {a}\")\n",
        "print(\"-\"*2*n)\n",
        "for i in range(n+5):\n",
        "  if i>0 and wyn==0 and rzad_flaga:\n",
        "    rzad_flaga = False\n",
        "    rzad=i\n",
        "  print(f\"{i}a = {wyn}\")\n",
        "  wyn = (wyn+a) % n\n",
        "\n",
        "print(\"-\"*2*n)\n",
        "if rzad<n:\n",
        "  print(f\"Rząd elementu {a} w podgrupie <{a}> to {rzad}.\")\n",
        "else:\n",
        "  print(f\"Rząd elementu {a} w grupie Z_{n} to {rzad}.\")"
      ],
      "metadata": {
        "id": "HNfkCpnS7GGQ",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "1e6b90ed-b028-4c52-d0fa-b12839edc993"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "a = 3\n",
            "----------\n",
            "0a = 0\n",
            "1a = 3\n",
            "2a = 1\n",
            "3a = 4\n",
            "4a = 2\n",
            "5a = 0\n",
            "6a = 3\n",
            "7a = 1\n",
            "8a = 4\n",
            "9a = 2\n",
            "----------\n",
            "Rząd elementu 3 w grupie Z_5 to 5.\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "#@title { run: \"auto\" }\n",
        "#@markdown $\\mathbb{Z}^{*}_p$ jako grupa cykliczna i rząd elementu w grupie...\n",
        "\n",
        "#@markdown Rząd grupy $p-1$ ($p$: moduł):\n",
        "p = 7 #@param {type:\"integer\"}\n",
        "assert p >= 2 and isprime(p), \"Wartość p musi być liczbą pierwszą\"\n",
        "\n",
        "#@markdown Element $a$:\n",
        "a = 3 #@param {type:\"integer\"}\n",
        "a = a % p\n",
        "assert a > 0, \"0 nie należy do grupy Z*_p\"\n",
        "\n",
        "wyn = 1\n",
        "rzad = 1\n",
        "rzad_flaga = True\n",
        "print(f\"a = {a}\")\n",
        "print(\"-\"*2*p)\n",
        "for i in range(p+5):\n",
        "  if i>0 and wyn==1 and rzad_flaga:\n",
        "    rzad_flaga = False\n",
        "    rzad=i   \n",
        "  print(f\"a^{i} = {wyn}\")\n",
        "  wyn = (wyn*a) % p\n",
        "\n",
        "print(\"-\"*2*p)\n",
        "if rzad<(n-1):\n",
        "  print(f\"Rząd elementu {a} w podgrupie <{a}> to {rzad}.\")\n",
        "else:\n",
        "  print(f\"Rząd elementu {a} w grupie Z*_{n} to {rzad}.\")"
      ],
      "metadata": {
        "id": "5U7Q-sXs8DlK",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "8935b284-5277-4f60-d0e4-f3f1ec481a7b"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "a = 3\n",
            "--------------\n",
            "a^0 = 1\n",
            "a^1 = 3\n",
            "a^2 = 2\n",
            "a^3 = 6\n",
            "a^4 = 4\n",
            "a^5 = 5\n",
            "a^6 = 1\n",
            "a^7 = 3\n",
            "a^8 = 2\n",
            "a^9 = 6\n",
            "a^10 = 4\n",
            "a^11 = 5\n",
            "--------------\n",
            "Rząd elementu 3 w grupie Z*_5 to 6.\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "#@title { run: \"auto\" }\n",
        "#@markdown Grupa multiplikatywna $\\mathbb{Z}^{*}_n$ i jej pierwiastek pierwotny\n",
        "\n",
        "#@markdown Moduł grupy $n$:\n",
        "n = 7 #@param {type:\"integer\"}\n",
        "assert n >= 2, \"Wartość n musi być większa od 1\"\n",
        "\n",
        "print(f\"Grupa multiplikatywna Z*_{n}:\")\n",
        "\n",
        "grupa_mult_n = []\n",
        "for i in range(1,n):\n",
        "  if math.gcd(i,n) == 1:\n",
        "    grupa_mult_n.append(i)\n",
        "print(f\"{grupa_mult_n} o rzędzie {len(grupa_mult_n)}.\")\n",
        "phi = totient(n)\n",
        "print(f\"Tocjent: φ({n}) = {phi}.\\n\")\n",
        "\n",
        "a = primitive_root(n)\n",
        "print(f\"Pierwiastek pierwotny tej grupy to {a}:\")\n",
        "wyn = 1\n",
        "for i in range(phi):\n",
        "  print(f\"a^{i} = {wyn}\")\n",
        "  wyn = (wyn*a) % n\n",
        "\n",
        "if a>2:\n",
        "  print(f\"\\nNie ma mniejszych pierwiastków pierwotnych, np. dla 2:\")\n",
        "  wyn = 1\n",
        "  for i in range(phi):\n",
        "    print(f\"2^{i} = {wyn}\")\n",
        "    wyn = (wyn*2) % n"
      ],
      "metadata": {
        "id": "v03JTQJh-iDT",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "d86f6acc-7f3d-4c45-9790-b7f414e6fa50"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Grupa multiplikatywna Z*_7:\n",
            "[1, 2, 3, 4, 5, 6] o rzędzie 6.\n",
            "Tocjent: φ(7) = 6.\n",
            "\n",
            "Pierwiastek pierwotny tej grupy to 3:\n",
            "a^0 = 1\n",
            "a^1 = 3\n",
            "a^2 = 2\n",
            "a^3 = 6\n",
            "a^4 = 4\n",
            "a^5 = 5\n",
            "\n",
            "Nie ma mniejszych pierwiastków pierwotnych, np. dla 2:\n",
            "2^0 = 1\n",
            "2^1 = 2\n",
            "2^2 = 4\n",
            "2^3 = 1\n",
            "2^4 = 2\n",
            "2^5 = 4\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "#@title Tocjent\n",
        "\n",
        "x_min = 2\n",
        "x_max = 1000\n",
        "x = list(range(x_min,x_max))\n",
        "y = [i for i in sieve.totientrange(x_min,x_max)]\n",
        "\n",
        "plt.figure()\n",
        "plt.scatter(x,y,s=3)\n",
        "plt.title(\"Tocjent\")\n",
        "plt.xlabel(f\"$n$\")\n",
        "plt.ylabel(f\"$\\phi(n)$\")\n",
        "plt.savefig(\"tocjent.png\")\n",
        "plt.show()"
      ],
      "metadata": {
        "id": "IOLghbTyJPiQ",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 480
        },
        "outputId": "4995048c-a33d-4c09-e609-24c62532822b"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Skąd te \"paski\"?"
      ],
      "metadata": {
        "id": "R8TSEzrlNooR"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Rozkład grupy na warstwy względem podgrupy"
      ],
      "metadata": {
        "id": "g4Q4bb1H28Db"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "#@markdown Grupa addytywna $G = \\mathbb{Z}$\n",
        "\n",
        "zakres_min = -20\n",
        "zakres_max = 20\n",
        "\n",
        "grupa_Z = list(range(zakres_min,zakres_max+1))\n",
        "\n",
        "print(\"Grupa: \",end=\"..., \")\n",
        "for i in grupa_Z:\n",
        "  print(f\"{i}, \",end=\"\")\n",
        "print(\"...\")\n",
        "print(\"*\"*200)"
      ],
      "metadata": {
        "id": "Y1HfjTSz3AY-",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "9f36d4a7-9e22-48fc-bffd-6c19402ef85b"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Grupa: ..., -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, ...\n",
            "********************************************************************************************************************************************************************************************************\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Element neutralny, element przeciwny do $a$?"
      ],
      "metadata": {
        "id": "pcH1Ra_B3wDU"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "#@markdown Podgrupa $H = n\\mathbb{Z}$\n",
        "\n",
        "n = 3 #@param {type: \"slider\", min:2, max:9, step:1}\n",
        "\n",
        "podgrupa_n = list(range(zakres_min*n,zakres_max*n+1,n))\n",
        "\n",
        "print(\"Podgrupa: \",end=\"..., \")\n",
        "for i in podgrupa_n:\n",
        "  print(f\"{i}, \",end=\"\")\n",
        "print(\"...\")\n",
        "print(\"*\"*200)"
      ],
      "metadata": {
        "id": "oI2ifre_30Yg",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "9a8e5e26-8365-4c08-ce2e-13168a97025c"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Podgrupa: ..., -60, -57, -54, -51, -48, -45, -42, -39, -36, -33, -30, -27, -24, -21, -18, -15, -12, -9, -6, -3, 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, ...\n",
            "********************************************************************************************************************************************************************************************************\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Jest element neutralny i element przeciwny do każdego elementu w podgrupie?"
      ],
      "metadata": {
        "id": "Z-E6ZTWR4W0o"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "#@title { run: \"auto\" }\n",
        "#@markdown Przykład podwarstwy $g+H$, gdzie $g \\in G$ (tj. $g + n\\mathbb{Z}$, $g \\in \\mathbb{Z}$)\n",
        "\n",
        "g = 7 #@param {type: \"slider\", min:-20, max:20, step:1}\n",
        "\n",
        "warstwa = [g + el for el in podgrupa_n]\n",
        "\n",
        "print(f\"Warstwa wyznaczona przez {g}: \",end=\"..., \")\n",
        "for i in warstwa:\n",
        "  print(f\"{i}, \",end=\"\")\n",
        "print(\"...\")\n",
        "print(\"*\"*200)"
      ],
      "metadata": {
        "id": "LaXGWZWD4dxY",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "31cc0d68-2e3e-4241-89b6-87c56b4dec4f"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Warstwa wyznaczona przez 7: ..., -53, -50, -47, -44, -41, -38, -35, -32, -29, -26, -23, -20, -17, -14, -11, -8, -5, -2, 1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 58, 61, 64, 67, ...\n",
            "********************************************************************************************************************************************************************************************************\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Czy każda warstwa tworzy grupę?"
      ],
      "metadata": {
        "id": "J86aDTK55XDR"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "#@markdown Różne warstwy:\n",
        "\n",
        "for g in range(n+2):\n",
        "  warstwa = [g + el for el in podgrupa_n]\n",
        "  print(f\"Warstwa wyznaczona przez {g}: \",end=\"..., \")\n",
        "  for i in warstwa:\n",
        "    print(f\"{i}, \",end=\"\")\n",
        "  print(\"...\")\n",
        "  print(\"*\"*200)"
      ],
      "metadata": {
        "id": "4_orWE9r5bmR",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "0427bcc1-ce1e-4d71-8e07-040accf2e539"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Warstwa wyznaczona przez 0: ..., -60, -57, -54, -51, -48, -45, -42, -39, -36, -33, -30, -27, -24, -21, -18, -15, -12, -9, -6, -3, 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, ...\n",
            "********************************************************************************************************************************************************************************************************\n",
            "Warstwa wyznaczona przez 1: ..., -59, -56, -53, -50, -47, -44, -41, -38, -35, -32, -29, -26, -23, -20, -17, -14, -11, -8, -5, -2, 1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 58, 61, ...\n",
            "********************************************************************************************************************************************************************************************************\n",
            "Warstwa wyznaczona przez 2: ..., -58, -55, -52, -49, -46, -43, -40, -37, -34, -31, -28, -25, -22, -19, -16, -13, -10, -7, -4, -1, 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41, 44, 47, 50, 53, 56, 59, 62, ...\n",
            "********************************************************************************************************************************************************************************************************\n",
            "Warstwa wyznaczona przez 3: ..., -57, -54, -51, -48, -45, -42, -39, -36, -33, -30, -27, -24, -21, -18, -15, -12, -9, -6, -3, 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, ...\n",
            "********************************************************************************************************************************************************************************************************\n",
            "Warstwa wyznaczona przez 4: ..., -56, -53, -50, -47, -44, -41, -38, -35, -32, -29, -26, -23, -20, -17, -14, -11, -8, -5, -2, 1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 58, 61, 64, ...\n",
            "********************************************************************************************************************************************************************************************************\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Widzimy, że $g_1 + n\\mathbb{Z} = g_2 + n\\mathbb{Z}$, o ile tylko $g_1 = g_2 \\bmod{n}$.\n",
        "\n",
        "Wszystkich różnych podwarstw jest tylko tyle, ile wynosi $n$. Warstwy są rozłączne."
      ],
      "metadata": {
        "id": "IV_pg0Iq5w3S"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "#@title { run: \"auto\" }\n",
        "#@markdown Można też dodawać warstwy, bo one tworzą tzw. grupę ilorazową $[G/H]$, która jest izomorficzna z grupą $\\mathbb{Z}_n$\n",
        "\n",
        "g1 = 5 #@param {type: \"slider\", min:-20, max:20, step:1}\n",
        "g2 = 7 #@param {type: \"slider\", min:-20, max:20, step:1}\n",
        "\n",
        "gg = [g1,g2,g1+g2]\n",
        "\n",
        "warstwa = {}\n",
        "for g in gg:\n",
        "  warstwa[g] = [g + el for el in podgrupa_n]\n",
        "  print(f\"Warstwa wyznaczona przez {g} (= {g%n} mod-n): \",end=\"..., \")\n",
        "  for i in warstwa[g]:\n",
        "    print(f\"{i}, \",end=\"\")\n",
        "  print(\"...\")\n",
        "  print(\"*\"*200)\n",
        "\n",
        "print(f\"Wynik dodawania warstwy wyznaczonej przez {g1} i warstwy wyznaczonej przez {g2}:\")\n",
        "print(end=\"..., \")\n",
        "for i in range(len(podgrupa_n)):\n",
        "    print(f\"{warstwa[g1][i]+warstwa[g2][i]}, \",end=\"\")\n",
        "print(\"...\")"
      ],
      "metadata": {
        "id": "f3MI22EB6Ml9",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "ff16bb1d-3ae2-42a1-f60d-19f255a70cce"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Warstwa wyznaczona przez 5 (= 2 mod-n): ..., -55, -52, -49, -46, -43, -40, -37, -34, -31, -28, -25, -22, -19, -16, -13, -10, -7, -4, -1, 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41, 44, 47, 50, 53, 56, 59, 62, 65, ...\n",
            "********************************************************************************************************************************************************************************************************\n",
            "Warstwa wyznaczona przez 7 (= 1 mod-n): ..., -53, -50, -47, -44, -41, -38, -35, -32, -29, -26, -23, -20, -17, -14, -11, -8, -5, -2, 1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 58, 61, 64, 67, ...\n",
            "********************************************************************************************************************************************************************************************************\n",
            "Warstwa wyznaczona przez 12 (= 0 mod-n): ..., -48, -45, -42, -39, -36, -33, -30, -27, -24, -21, -18, -15, -12, -9, -6, -3, 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, ...\n",
            "********************************************************************************************************************************************************************************************************\n",
            "Wynik dodawania warstwy wyznaczonej przez 5 i warstwy wyznaczonej przez 7:\n",
            "..., -108, -102, -96, -90, -84, -78, -72, -66, -60, -54, -48, -42, -36, -30, -24, -18, -12, -6, 0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126, 132, ...\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Mamy równość:\n",
        "\n",
        "$$(g_1 + H) + (g_2 + H) = (g_1 + g_2) + H$$\n",
        "\n",
        "co świadczy o izomorfizmie grupy ilorazowej $[\\mathbb{Z}/n\\mathbb{Z}]$ oraz grupy $\\mathbb{Z}_n$.\n",
        "\n",
        "Tak samo można dokonywać rozkładu przestrzeni liniowej względem jej podprzestrzeni, co ma znaczenie z punktu widzenia dekodowania."
      ],
      "metadata": {
        "id": "HGJjwpfB7pFo"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Twierdzenie Fermata-Eulera"
      ],
      "metadata": {
        "id": "NIE5oqvuArKK"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "#@title { run: \"auto\" }\n",
        "#@markdown Małe tw. Fermata\n",
        "\n",
        "p = 79 #@param {type:\"integer\"}\n",
        "a = 2 #@param {type:\"integer\"}\n",
        "\n",
        "assert p >= 2 and isprime(p) and a > 0 and a < p, \"Wartość p musi być liczbą pierwszą\"\n",
        "\n",
        "wynik = a**(p-1)\n",
        "wynik_mod = wynik % p\n",
        "\n",
        "print(f\"{a}^{p-1} = {wynik} mod-{p} = {wynik_mod}.\")"
      ],
      "metadata": {
        "id": "bNYMU89gAtVP",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "5b9172f1-8eb5-45bf-c1af-6bbe70a2174e"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "2^78 = 302231454903657293676544 mod-79 = 1.\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "#@title { run: \"auto\" }\n",
        "#@markdown Tw. Fermata-Eulera\n",
        "\n",
        "n = 72 #@param {type:\"integer\"}\n",
        "a = 35 #@param {type:\"integer\"}\n",
        "\n",
        "assert n >= 2 and a > 0 and a < n and math.gcd(a, n) == 1, \"Wartość a ma być mniejsza od n oraz musi być z nią względnie pierwsza\"\n",
        "\n",
        "phi = totient(n)\n",
        "print(f\"Tocjent: φ({n}) = {phi}.\\n\")\n",
        "\n",
        "wynik = a**phi\n",
        "wynik_mod = wynik % n\n",
        "\n",
        "print(f\"{a}^φ({n}) = {wynik} mod-{n} = {wynik_mod}.\")"
      ],
      "metadata": {
        "id": "iOPyjFchCaIG",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "bba81b10-7a44-4738-eeab-01a22c018127"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Tocjent: φ(72) = 24.\n",
            "\n",
            "35^φ(72) = 11419131242070580387175083160400390625 mod-72 = 1.\n"
          ]
        }
      ]
    }
  ]
}