Finding a generator of Z*p

TTM4135.ipynb

@@ -130,7 +130,7 @@
  },
  {
  "cell_type": "code",
- "execution_count": 4,
+ "execution_count": 16,
  "metadata": {},
  "outputs": [
  {
@@ -179,15 +179,15 @@
  },
  {
  "cell_type": "code",
- "execution_count": 5,
+ "execution_count": 14,
  "metadata": {},
  "outputs": [
  {
  "name": "stdout",
  "output_type": "stream",
  "text": [
- "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"
+ "φ(21) = 12 since the 12 items in Z*21 are each relatively prime to 21.\n",
+ "Z*21 = [1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20]\n"
  ]
  }
  ],
@@ -206,7 +206,7 @@
  " r.append(i)\n",
  " return r\n",
  "\n",
- "n = 16\n",
+ "n = 21\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",
@@ -233,7 +233,7 @@
  },
  {
  "cell_type": "code",
- "execution_count": 6,
+ "execution_count": 13,
  "metadata": {},
  "outputs": [
  {
@@ -275,7 +275,50 @@
  " return (int)(result)\n",
  "\n",
  "n = 1125\n",
- "print(f\"𝜑({n}) = {totient(1125)}\")"
+ "print(f\"𝜑({n}) = {totient(n)}\")"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Finding a generator of Z*p\n",
+ "- https://crypto.stanford.edu/pbc/notes/numbertheory/gen.html\n",
+ "- Lecture 2, page 19\n",
+ "\n",
+ "A generator of $Z^∗p$ is an element of order $p − 1$\n",
+ "\n",
+ "To find a generator of Z∗p we can choose a value g and test it as follows:\n",
+ "1. compute all the distinct prime factors of p − 1 and call them f1,f2,...,fr\n",
+ "2. then g is a generator as long as $g^{\\frac{p−1}{fi}} \\neq 1 \\mod(p)$ for $i = 1,2,,...,r$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 52,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "Is 3 a generator for Z*4? True\n"
+ ]
+ }
+ ],
+ "source": [
+ "# Finding a generator g of Z∗p\n",
+ "def is_generator(p, g):\n",
+ " pf = prime_factors(p-1)\n",
+ " for f in pf:\n",
+ " if (g**((p-1)/f))%p == 1:\n",
+ " return False\n",
+ " return True\n",
+ "\n",
+ "g = 3 # Generator\n",
+ "p = 4 # Z*p\n",
+ "\n",
+ "print(f\"Is {g} a generator for Z*{p}? {is_generator(p, g)}\")"
  ]
  },
  {