Update galois_field.ipynb

Author: BBEK-Anand

algorithms/Graph.ipynb

@@ -0,0 +1,215 @@
+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": 8,
+   "id": "8561e0a7",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/html": [
+       "<style>.container{width:100%}</style>\n"
+      ],
+      "text/plain": [
+       "<IPython.core.display.HTML object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "%%HTML\n",
+    "<style>.container{width:100%}</style>"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "id": "12f36b2f",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "21"
+      ]
+     },
+     "execution_count": 5,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "def eucld_gcd(a, b):\n",
+    "    \"\"\"\n",
+    "    Computes the greatest common divisor (GCD) of \n",
+    "    two integers using the Euclidean algorithm.\n",
+    "\n",
+    "    Parameters:\n",
+    "    a (int): The first integer.\n",
+    "    b (int): The second integer.\n",
+    "\n",
+    "    Returns:\n",
+    "    int: The greatest common divisor of a and b.\n",
+    "    \"\"\"\n",
+    "    if a < b:\n",
+    "        a, b = b, a\n",
+    "    if b == 0:\n",
+    "        return a\n",
+    "    r = a % b\n",
+    "    if r == 0:\n",
+    "        return b\n",
+    "    return eucld_gcd(b, r)\n",
+    "\n",
+    "\n",
+    "eucld_gcd(252, 105)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "id": "c060ee17",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[1, -48]"
+      ]
+     },
+     "execution_count": 6,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "import numpy as np\n",
+    "\n",
+    "\n",
+    "def ext_eucld(a, b):\n",
+    "    \"\"\"\n",
+    "    Computes the extended Euclidean algorithm to find the \n",
+    "    greatest common divisor (GCD)of two integers, and \n",
+    "    also the coefficients (x, y) of the equation:\n",
+    "    a*x + b*y = GCD(a, b)\n",
+    "\n",
+    "    This method returns the coefficients (x, y) \n",
+    "    such that a*x + b*y = GCD(a, b).\n",
+    "\n",
+    "    Parameters:\n",
+    "    a (int): The first integer.\n",
+    "    b (int): The second integer.\n",
+    "\n",
+    "    Returns:\n",
+    "    list: A list of two integers [x, y] where x and y are the \n",
+    "    coefficients for the linear combination of a and b \n",
+    "    that equals their GCD.\n",
+    "    \"\"\"\n",
+    "    swap = False\n",
+    "    if a < b:\n",
+    "        a, b = b, a\n",
+    "        swap = True\n",
+    "\n",
+    "    def eucld(a, b):\n",
+    "        if b in {0,1}:\n",
+    "            return []\n",
+    "        ls = []\n",
+    "        while b != 1:\n",
+    "            r = a % b\n",
+    "            if r == 0:\n",
+    "                return ls\n",
+    "            idx = (a - r) // b\n",
+    "            ls.append(idx)\n",
+    "            a = b\n",
+    "            b = r\n",
+    "        return ls\n",
+    "\n",
+    "    row = np.array([[1, 0], [0, 1]])\n",
+    "    ls = eucld(a, b)\n",
+    "    for i in ls:\n",
+    "        row = np.append(row, [row[-2] - i * row[-1]], axis=0)\n",
+    "\n",
+    "    if swap:\n",
+    "        return list(row[-1])[::-1]\n",
+    "\n",
+    "    return list(row[-1])\n",
+    "\n",
+    "\n",
+    "ext_eucld(97, 2)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "id": "d3edd1f6",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "a=5, b=7, gcd=1, m=3, n=-2, ma+nb=1\n"
+     ]
+    }
+   ],
+   "source": [
+<<<<<<< HEAD
+    "a, b= 5, 7\n",
+=======
+    "a = 5\n",
+    "b = 7\n",
+>>>>>>> ffbe3bbd8623bff453d16b942b7cf035c20d808c
+    "gcd = eucld_gcd(a, b)\n",
+    "m, n = ext_eucld(a, b)\n",
+    "print(f\"a={a}, b={b}, gcd={gcd}, m={m}, n={n}, ma+nb={m*a+n*b}\")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "782bb8d3",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "e224063e",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "3da0aaac-23b4-4118-ae3e-aef611f0f38c",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.12.3"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}