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)}\")"
    ]
   },
   {