More latex please

TTM4135.ipynb

@@ -10,7 +10,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 1,
+   "execution_count": 14,
    "metadata": {},
    "outputs": [
     {
@@ -19,7 +19,7 @@
        "5"
       ]
      },
-     "execution_count": 1,
+     "execution_count": 14,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -46,7 +46,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 2,
+   "execution_count": 17,
    "metadata": {},
    "outputs": [
     {
@@ -72,7 +72,7 @@
     "    x = y1 - (b//a) * x1  \n",
     "    y = x1  \n",
     "\n",
-    "    return gcd,x,y \n",
+    "    return gcd, x, y \n",
     "\n",
     "a, b = 35,10\n",
     "g, x, y = gcdExtended(a, b)\n",
@@ -92,7 +92,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 3,
+   "execution_count": 16,
    "metadata": {},
    "outputs": [
     {
@@ -130,7 +130,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 16,
+   "execution_count": 4,
    "metadata": {},
    "outputs": [
     {
@@ -167,6 +167,57 @@
     "print(crt(n, a))"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Integer factorisation\n",
+    "Given an integer, find its prime factors"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "The factors of 641361 are [3, 7, 7, 4363], verified\n",
+      "Manual test: True\n"
+     ]
+    }
+   ],
+   "source": [
+    "# Brute force solution\n",
+    "def prime_factors(n):\n",
+    "    i = 2\n",
+    "    factors = []\n",
+    "    while i * i <= n:\n",
+    "        if n % i:\n",
+    "            i += 1\n",
+    "        else:\n",
+    "            n //= i\n",
+    "            factors.append(i)\n",
+    "    if n > 1:\n",
+    "        factors.append(n)\n",
+    "    return factors\n",
+    "\n",
+    "# Check if the numbers in array a is the factors of n\n",
+    "def factor_test(a, n):\n",
+    "    k = 1\n",
+    "    for i in factors:\n",
+    "        k=k*i\n",
+    "    return k==n\n",
+    "\n",
+    "n = 641361\n",
+    "factors = prime_factors(n)\n",
+    "print(f\"The factors of {n} are {factors}, {'verified' if factor_test(factors, n) else 'ERROR'}\")\n",
+    "\n",
+    "print(\"Manual test:\", factor_test([3, 7, 7, 4363], 641361))"
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},
@@ -179,7 +230,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 14,
+   "execution_count": 6,
    "metadata": {},
    "outputs": [
     {
@@ -233,7 +284,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 13,
+   "execution_count": 7,
    "metadata": {},
    "outputs": [
     {
@@ -286,16 +337,16 @@
     "- 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",
+    "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",
+    "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",
+    "1. compute all the distinct prime factors of $p − 1$ and call them $f_1, f_2, ..., f_r$\n",
-    "2. then g is a generator as long as $g^{\\frac{p−1}{fi}} \\neq 1 \\mod(p)$ for $i = 1,2,,...,r$"
+    "2. then $g$ is a generator as long as $g^{\\frac{p−1}{f_i}} \\neq 1 \\mod(p)$ for $i = 1,2,,...,r$"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 52,
+   "execution_count": 8,
    "metadata": {},
    "outputs": [
     {
@@ -331,7 +382,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 7,
+   "execution_count": 9,
    "metadata": {},
    "outputs": [
     {
@@ -355,7 +406,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 8,
+   "execution_count": 10,
    "metadata": {},
    "outputs": [
     {
@@ -409,7 +460,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 9,
+   "execution_count": 11,
    "metadata": {},
    "outputs": [
     {
@@ -482,57 +533,6 @@
     "    print(f\"{value} is {miller_rabin_prime(value)}.\")\n"
    ]
   },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "# Integer factorisation\n",
-    "Given an integer, find its prime factors"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 10,
-   "metadata": {},
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "The factors of 641361 are [3, 7, 7, 4363], verified\n",
-      "Manual test: True\n"
-     ]
-    }
-   ],
-   "source": [
-    "# Brute force solution\n",
-    "def prime_factors(n):\n",
-    "    i = 2\n",
-    "    factors = []\n",
-    "    while i * i <= n:\n",
-    "        if n % i:\n",
-    "            i += 1\n",
-    "        else:\n",
-    "            n //= i\n",
-    "            factors.append(i)\n",
-    "    if n > 1:\n",
-    "        factors.append(n)\n",
-    "    return factors\n",
-    "\n",
-    "# Check if the numbers in array a is the factors of n\n",
-    "def factor_test(a, n):\n",
-    "    k = 1\n",
-    "    for i in factors:\n",
-    "        k=k*i\n",
-    "    return k==n\n",
-    "\n",
-    "n = 641361\n",
-    "factors = prime_factors(n)\n",
-    "print(f\"The factors of {n} are {factors}, {'verified' if factor_test(factors, n) else 'ERROR'}\")\n",
-    "\n",
-    "print(\"Manual test:\", factor_test([3, 7, 7, 4363], 641361))"
-   ]
-  },
   {
    "cell_type": "markdown",
    "metadata": {},
@@ -552,7 +552,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 11,
+   "execution_count": 12,
    "metadata": {},
    "outputs": [
     {
@@ -629,7 +629,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 12,
+   "execution_count": 13,
    "metadata": {},
    "outputs": [
     {