Python/cryptography/extended_eucledian.ipynb

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    "%%HTML\n",
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    "def eucld_gcd(a, b):\n",
    "    \"\"\"\n",
    "    Computes the greatest common divisor (GCD) of 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)"
   ]
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   "source": [
    "import numpy as np\n",
    "\n",
    "\n",
    "def ext_eucld(a, b):\n",
    "    \"\"\"\n",
    "    Computes the extended Euclidean algorithm to find the greatest common divisor (GCD)\n",
    "    of two integers, and 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) 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 coefficients for the linear\n",
    "          combination of a and b 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 == 0 or b == 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)"
   ]
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     "text": [
      "a=5, b=7, gcd=1, m=3, n=-2, ma+nb=1\n"
     ]
    }
   ],
   "source": [
    "a = 5\n",
    "b = 7\n",
    "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}\")"
   ]
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