Add Eulers Totient

TTM4135.ipynb

@@ -172,7 +172,9 @@
    "metadata": {},
    "source": [
     "# Euler function φ and Z*n\n",
-    "From slides 2020-4135-l07 - Number Theory for Public Key Cryptography"
+    "From slides 2020-4135-l07 - Number Theory for Public Key Cryptography\n",
+    "\n",
+    "> $\\phi(16) = \\phi(2^{4}) = 16*\\left(1-\\frac{1}{2}\\right)$"
    ]
   },
   {
@@ -184,8 +186,8 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "φ(81) = 54 since the 54 items in Z*81 are each relatively prime to 81.\n",
+      "φ(16) = 8 since the 8 items in Z*16 are each relatively prime to 16.\n",
-      "Z*81 = [1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 28, 29, 31, 32, 34, 35, 37, 38, 40, 41, 43, 44, 46, 47, 49, 50, 52, 53, 55, 56, 58, 59, 61, 62, 64, 65, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80]\n"
+      "Z*16 = [1, 3, 5, 7, 9, 11, 13, 15]\n"
      ]
     }
    ],
@@ -204,13 +206,78 @@
     "            r.append(i)\n",
     "    return r\n",
     "\n",
-    "n = 81\n",
+    "n = 16\n",
     "ar = z_star(n)\n",
     "e = len(ar)\n",
     "print(f\"φ({n}) = {e} since the {e} items in Z*{n} are each relatively prime to {n}.\")\n",
     "print(f\"Z*{n} = {ar}\")"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Euler’s Totient\n",
+    "- https://www.dcode.fr/euler-totient\n",
+    "- https://www.ibnus.io/eulers-totient-function-using-python/\n",
+    "\n",
+    "In general:\n",
+    "\n",
+    "> $\\varphi(n) = n \\prod_{p \\mid n} \\left( 1 - \\frac{1}{p} \\right)$\n",
+    "\n",
+    "Examples:\n",
+    "> $\\varphi(16) = \\varphi(2^{4}) = 16*\\left(1-\\frac{1}{2}\\right) = 8$\n",
+    "\n",
+    "> $\\varphi(1125) = \\varphi(33555) = 1125\\left(1-\\frac{1}{3}\\right)\\left(1-\\frac{1}{5}\\right) = 600$\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "𝜑(1125) = 600\n"
+     ]
+    }
+   ],
+   "source": [
+    "# Euler's Totient Function\n",
+    "# using Euler's product formula\n",
+    "def totient(n) :\n",
+    "    result = n # Initialize result as n\n",
+    "\n",
+    "    # Consider all prime factors\n",
+    "    # of n and for every prime\n",
+    "    # factor p, multiply result with (1 – 1/p)\n",
+    "    p = 2\n",
+    "    while(p * p <= n) :\n",
+    "\n",
+    "        # Check if p is a prime factor.\n",
+    "        if (n % p == 0) :\n",
+    "\n",
+    "            # If yes, then update n and result\n",
+    "            while (n % p == 0) :\n",
+    "                n = n // p\n",
+    "            result = result * (1.0 - (1.0 / (float) (p)))\n",
+    "        p = p + 1\n",
+    "\n",
+    "    # If n has a prime factor\n",
+    "    # greater than sqrt(n)\n",
+    "    # (There can be at-most one\n",
+    "    # such prime factor)\n",
+    "    if (n > 1) :\n",
+    "        result = result * (1.0 - (1.0 / (float)(n)))\n",
+    "\n",
+    "    return (int)(result)\n",
+    "\n",
+    "n = 1125\n",
+    "print(f\"𝜑({n}) = {totient(1125)}\")"
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},
@@ -221,7 +288,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 6,
+   "execution_count": 7,
    "metadata": {},
    "outputs": [
     {
@@ -245,14 +312,14 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 7,
+   "execution_count": 8,
    "metadata": {},
    "outputs": [
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "1105 is composite\n"
+      "1105 is probable prime\n"
      ]
     }
    ],
@@ -299,7 +366,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 8,
+   "execution_count": 9,
    "metadata": {},
    "outputs": [
     {
@@ -382,7 +449,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 9,
+   "execution_count": 10,
    "metadata": {},
    "outputs": [
     {
@@ -442,7 +509,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 10,
+   "execution_count": 11,
    "metadata": {},
    "outputs": [
     {
@@ -519,7 +586,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 11,
+   "execution_count": 12,
    "metadata": {},
    "outputs": [
     {