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",
- "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"
+ "φ(16) = 8 since the 8 items in Z*16 are each relatively prime to 16.\n",
+ "Z*16 = [1, 3, 5, 7, 9, 11, 13, 15]\n"
  ]
  }
  ],
@@ -204,7 +206,7 @@
  " 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",
@@ -215,13 +217,78 @@
  "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": {},
+ "source": [
  "# Testing for primality\n",
  "From slides 2020-4135-l07 - Number Theory for Public Key Cryptography"
  ]
  },
  {
  "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": [
  {