HE-Complexity-Speed.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Complexity of Paillier Algorithms "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[A Comparison of El Gamal and Paillier\n",
    "Cryptosystems](http://koclab.cs.ucsb.edu/teaching/cren/project/2018/Chen.pdf]) mentioned the time complexity of Paillier Algorithms:\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](figs/quote.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](figs/table.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From the sencond column of table above, we see number of operations required inrease linearly with the increase of the size of plaintext."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Experiments"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the setting of federated learning, there are few basic operations: \n",
    "* Encryption: $c = enc(x)$\n",
    "* Decryption: $x =  dec(c)$\n",
    "* Adding ciphertext with plaintext: $x + c$\n",
    "* Adding ciphertext with ciphertext: $c_1 + c_2$\n",
    "* Multiply a ciphertext with a plaintext: $cx$\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "model name\t: Intel(R) Xeon(R) CPU E5-2698 v4 @ 2.20GHz\n"
     ]
    }
   ],
   "source": [
    "!cat /proc/cpuinfo | grep 'model name' | uniq"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Implementation: [python-paillier](https://github.com/data61/python-paillier)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [],
   "source": [
    "import phe\n",
    "import numpy as np\n",
    "\n",
    "def encrypt_array(public_key:phe.PaillierPublicKey, x:np.array):\n",
    "    h, w = x.shape[0], x.shape[1]\n",
    "    c = [public_key.encrypt(float(item)) for item in x.flatten()]\n",
    "    c = np.array(c).reshape(h, w)\n",
    "    return c \n",
    "\n",
    "def decrypt_array(private_key:phe.PaillierPrivateKey, c:np.array):\n",
    "    h, w = c.shape[0], c.shape[1]\n",
    "    x = [private_key.decrypt(item) for item in c.flatten()]\n",
    "    x = np.array(x).reshape(h, w)\n",
    "    return x    \n",
    "\n",
    "def cipher_add(c1, c2):\n",
    "    return c1 + c2\n",
    "    \n",
    "def cipher_plain_add(c, x):\n",
    "    return c+x\n",
    "\n",
    "def cipher_plain_mul(c, a):\n",
    "    return a * c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1 1 1]\n",
      " [1 1 1]\n",
      " [1 1 1]]\n",
      "[[6. 4. 4.]\n",
      " [4. 6. 4.]\n",
      " [4. 4. 6.]]\n"
     ]
    }
   ],
   "source": [
    "from phe import paillier\n",
    "KEY_SIZE = 128\n",
    "pubkey, prikey = paillier.generate_paillier_keypair(n_length=KEY_SIZE)\n",
    "x = np.array([1]*9).reshape(3, 3)\n",
    "print(x)\n",
    "c = encrypt_array(pubkey, x)\n",
    "c = cipher_add(c, c)\n",
    "c = cipher_plain_add(c, np.eye(3, 3))\n",
    "c = cipher_plain_mul(c, 2)\n",
    "decrypted_x = decrypt_array(prikey, c)\n",
    "print(decrypted_x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We fix the key dimension and see how the time required of 5 basic operations varies when we increase the size of plaintext"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [],
   "source": [
    "import time\n",
    "def timeit(method, num_trails, *args, **kwargs):\n",
    "    start = time.time()\n",
    "    for _ in range(num_trails):\n",
    "        res = method(*args, **kwargs)\n",
    "    used_time = time.time() - start\n",
    "    return used_time, res"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Samples #100 | Feature Size 10\n",
      "Samples #100 | Feature Size 50\n",
      "Samples #100 | Feature Size 100\n",
      "Samples #100 | Feature Size 250\n",
      "Samples #100 | Feature Size 500\n",
      "Samples #100 | Feature Size 1000\n"
     ]
    }
   ],
   "source": [
    "import time\n",
    "feat_sizes = [10, 50, 100, 250, 500, 1000]\n",
    "NUM_SAMPLES = 100\n",
    "NUM_TRAILS = 5 \n",
    "record = []\n",
    "for feat_size in feat_sizes:\n",
    "    print(f\"Samples #{NUM_SAMPLES} | Feature Size {feat_size}\")\n",
    "    x = np.random.randn(NUM_SAMPLES, feat_size)\n",
    "    enc_time, c = timeit(encrypt_array, NUM_TRAILS, pubkey, x)\n",
    "    c_add_time, _ = timeit(cipher_add, NUM_TRAILS, c, c)\n",
    "    p_add_time, _ = timeit(cipher_plain_add, NUM_TRAILS, c, x)\n",
    "    p_mul_time, _ = timeit(cipher_plain_mul, NUM_TRAILS, c, 3)\n",
    "    dec_time, _ = timeit(decrypt_array, NUM_TRAILS, prikey, c)\n",
    "    record.append([feat_size, enc_time, c_add_time, p_add_time, p_mul_time, dec_time])\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "df = pd.DataFrame(record, columns=[\n",
    "    'feature size', \n",
    "    'encryption time',\n",
    "    'c + c time',\n",
    "    'c + x time',\n",
    "    'c * x time',\n",
    "    'decryption time'\n",
    "])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>feature size</th>\n",
       "      <th>encryption time</th>\n",
       "      <th>c + c time</th>\n",
       "      <th>c + x time</th>\n",
       "      <th>c * x time</th>\n",
       "      <th>decryption time</th>\n",
       "      <th>Total</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>10</td>\n",
       "      <td>0.603065</td>\n",
       "      <td>0.014551</td>\n",
       "      <td>0.118330</td>\n",
       "      <td>0.024570</td>\n",
       "      <td>0.297539</td>\n",
       "      <td>11.058055</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>50</td>\n",
       "      <td>3.018854</td>\n",
       "      <td>0.074289</td>\n",
       "      <td>0.625760</td>\n",
       "      <td>0.134737</td>\n",
       "      <td>1.639590</td>\n",
       "      <td>55.493230</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>100</td>\n",
       "      <td>6.342592</td>\n",
       "      <td>0.123240</td>\n",
       "      <td>1.143241</td>\n",
       "      <td>0.462100</td>\n",
       "      <td>2.754181</td>\n",
       "      <td>110.825354</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>250</td>\n",
       "      <td>13.700213</td>\n",
       "      <td>0.275990</td>\n",
       "      <td>2.896023</td>\n",
       "      <td>0.804470</td>\n",
       "      <td>6.682265</td>\n",
       "      <td>274.358961</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>500</td>\n",
       "      <td>26.775391</td>\n",
       "      <td>0.955021</td>\n",
       "      <td>5.485407</td>\n",
       "      <td>1.370756</td>\n",
       "      <td>13.549033</td>\n",
       "      <td>548.135608</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>1000</td>\n",
       "      <td>54.089990</td>\n",
       "      <td>1.863433</td>\n",
       "      <td>10.803079</td>\n",
       "      <td>2.892955</td>\n",
       "      <td>26.988481</td>\n",
       "      <td>1096.637938</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   feature size  encryption time  c + c time  c + x time  c * x time  \\\n",
       "0            10         0.603065    0.014551    0.118330    0.024570   \n",
       "1            50         3.018854    0.074289    0.625760    0.134737   \n",
       "2           100         6.342592    0.123240    1.143241    0.462100   \n",
       "3           250        13.700213    0.275990    2.896023    0.804470   \n",
       "4           500        26.775391    0.955021    5.485407    1.370756   \n",
       "5          1000        54.089990    1.863433   10.803079    2.892955   \n",
       "\n",
       "   decryption time        Total  \n",
       "0         0.297539    11.058055  \n",
       "1         1.639590    55.493230  \n",
       "2         2.754181   110.825354  \n",
       "3         6.682265   274.358961  \n",
       "4        13.549033   548.135608  \n",
       "5        26.988481  1096.637938  "
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df['Total'] = df.sum(axis=1)\n",
    "df"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'Time for 5 run(sec)')"
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "df.to_csv('paillier-time-complexity.csv')\n",
    "import matplotlib.pyplot as plt \n",
    "plt.plot(df['feature size'], df['Total'])\n",
    "plt.title('Time used for Paillier Encryption')\n",
    "plt.xlabel('Feature Size')\n",
    "plt.ylabel('Time for 5 run(sec)')"
   ]
  }
 ],
 "metadata": {
  "interpreter": {
   "hash": "3aeeecc898ad790c3595bc2ebae48d6a2546ab5e65cf8bb476118c27fd8febf3"
  },
  "kernelspec": {
   "display_name": "Python 3.9.7 ('phy-env')",
   "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": "3.9.7"
  },
  "orig_nbformat": 4
 },
 "nbformat": 4,
 "nbformat_minor": 2
}