6_more_state_of_the_art_research_questions/Chapter_6_Homomorphic_Encryption_NN.ipynb

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      "source": [
        "# Chapter 6: Homomorphic Encryption NN\n",
        "\n",
        "| **Platform**           | **Link**                                                                                                                                                                                                                                                                                               | **Notes**                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |\n",
        "|------------------------|--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|\n",
        "| _Local Folder_         | [Deep Dive Example: Homomorphic Encryption NN](6_more_state_of_the_art_research_questions/Chapter_6_Homomorphic_Encryption_NN.ipynb)                                                                                                                                                                                                                                |  `6_more_state_of_the_art_research_questions/`<br>`Chapter_6_Homomorphic_Encryption_NN.ipynb`                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |\n",
        "| _Book Chapter_         | [Chapter 6: More State-of-the-Art Research Questions](https://learning.oreilly.com/library/view/practicing-trustworthy-machine/9781098120269/ch03.html)                                                                                                                                                                                 | Ebook available from [O'Reilly](https://www.oreilly.com/library/view/practicing-trustworthy-machine/9781098120269/).<br>Physical book available from [Amazon](https://www.amazon.com/Practicing-Trustworthy-Machine-Learning-Transparent/dp/1098120272)                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |\n",
        "| _GitHub_               | [![GitHub](https://img.shields.io/badge/-View%20on%20GitHub-181717?logo=github&logoColor=ffffff)](https://github.com/matthew-mcateer/practicing_trustworthy_machine_learning/blob/main/6_more_state_of_the_art_research_questions/Chapter_6_Homomorphic_Encryption_NN.ipynb)                                                                          | Please star on GitHub so others can<br>find this repo and get help with implenting<br>their trustworthy AI pipelines<br> [![GitHub forks](https://img.shields.io/github/forks/matthew-mcateer/practicing_trustworthy_machine_learning.svg?style=social&label=Fork&maxAge=2592000)](https://github.com/matthew-mcateer/practicing_trustworthy_machine_learning/network/) <br> [![GitHub stars](https://img.shields.io/github/stars/matthew-mcateer/practicing_trustworthy_machine_learning.svg?style=social&label=Star&maxAge=2592000)](https://github.com/matthew-mcateer/practicing_trustworthy_machine_learning/stargazers/) <br> [![GitHub watchers](https://img.shields.io/github/watchers/matthew-mcateer/practicing_trustworthy_machine_learning.svg?style=social&label=Watch&maxAge=2592000)](https://github.com/matthew-mcateer/practicing_trustworthy_machine_learning/watchers/) <br>[`@matthew-mcateer`](https://github.com/matthew-mcateer) [![GitHub followers](https://img.shields.io/github/followers/matthew-mcateer?style=social&label=Follow&maxAge=2592000)](https://github.com/matthew-mcateer?tab=followers)<br>[`@pruksmhc`](https://github.com/pruksmhc) [![GitHub followers](https://img.shields.io/github/followers/pruksmhc?style=social&label=Follow&maxAge=2592000)](https://github.com/pruksmhc?tab=followers)<br>[`@shubhobm`](https://github.com/shubhobm) [![GitHub followers](https://img.shields.io/github/followers/shubhobm?style=social&label=Follow&maxAge=2592000)](https://github.com/shubhobm?tab=followers)  |\n",
        "| _Colab_                | [![Open In Colab](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/matthew-mcateer/practicing_trustworthy_machine_learning/blob/main/6_more_state_of_the_art_research_questions/Chapter_6_Homomorphic_Encryption_NN.ipynb)                                                                         | Enable GPU<br> Colab Pro recommended                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |\n",
        "| _Kaggle_               | [ ![ Kaggle ]( https://kaggle.com/static/images/open-in-kaggle.svg ) ]( https://kaggle.com/kernels/welcome?src=https://github.com/matthew-mcateer/practicing_trustworthy_machine_learning/blob/main/6_more_state_of_the_art_research_questions/Chapter_6_Homomorphic_Encryption_NN.ipynb )                                                            | Enable GPU<br> Enable Internet                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |\n",
        "| _Gradient_             | [ ![ Gradient ]( https://assets.paperspace.io/img/gradient-badge.svg ) ]( https://console.paperspace.com/github/matthew-mcateer/practicing_trustworthy_machine_learning/blob/main/6_more_state_of_the_art_research_questions/Chapter_6_Homomorphic_Encryption_NN.ipynb )                                                                              | Enable GPU                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |\n",
        "| _SageMaker Studio Lab_ | [![ Open In SageMaker Studio Lab ]( https://studiolab.sagemaker.aws/studiolab.svg ) ]( https://studiolab.sagemaker.aws/import/github/matthew-mcateer/practicing_trustworthy_machine_learning/blob/main/6_more_state_of_the_art_research_questions/Chapter_6_Homomorphic_Encryption_NN.ipynb )                                                         | Recommend `ml.g4dn.xlarge` instance at minimum<br>(alternatively, `g4dn.2xlarge` if using EC2 instance)                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |\n",
        "| _Binder_               | [ ![ Binder ]( https://mybinder.org/badge_logo.svg ) ]( https://mybinder.org/v2/gh/matthew-mcateer/practicing_trustworthy_machine_learning/HEAD?urlpath=https%3A%2F%2Fgithub.com%2Fmatthew-mcateer%2Fpracticing_trustworthy_machine_learning%2Fblob%2Fmain%2F6_more_state_of_the_art_research_questions%2FChapter_6_Homomorphic_Encryption_NN.ipynb ) | May run slowly without GPU                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |\n",
        "\n",
        "\n",
        "The contents of this notebook combine materials from the book [\"Practicing Trustworthy Machine Learning\"](https://www.oreilly.com/library/view/practicing-trustworthy-machine/9781098120269/), and the [ODSC West 2022](https://odsc.com/california/) Presentation promoting said book. Much of the plain text follows that presentation.\n",
        "\n",
        "<!--\n",
        "Originally found on GitHub at https://github.com/matthew-mcateer/practicing_trustworthy_machine_learning/blob/main/6_more_state_of_the_art_research_questions/Chapter_6_Homomorphic_Encryption_NN.ipynb\n",
        "-->\n",
        "\n",
        "Acknowledgements:\n",
        "\n",
        "* Ayoub Benaissa - Twitter: [@y0uben11](https://twitter.com/y0uben11)\n",
        "* Bogdan Cebere - Twitter: [@bcebere](https://twitter.com/bcebere)\n",
        "* Bilal Retiat - Twitter: [@philomath213](https://twitter.com/philomath213)\n",
        "\n",
        "Oh, and Alexander Ziller,\n",
        "Andrew Trask,\n",
        "Antonio Lopardo,\n",
        "Benjamin Szymkow,\n",
        "Bobby Wagner,\n",
        "Emma Bluemke,\n",
        "Jean-Mickael Nounahon,\n",
        "Jonathan Passerat-Palmbach,\n",
        "Kritika Prakash,\n",
        "Nick Rose,\n",
        "Théo Ryffel,\n",
        "Zarreen Naowal Reza,\n",
        "and Georgios Kaissis, **along with the rest of the OpenMined community**\n"
      ]
    },
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      "cell_type": "markdown",
      "metadata": {
        "id": "h-U4n1O7a-BD"
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      "source": [
        "So, privacy is a big concern in machine learning.\n",
        "\n",
        "You usually need a lot of data.\n",
        "\n",
        "While there are plenty of ways to lock up and secure your data to restrict access, the data itself usually needs to be exposed in some way to the model in order for it to learn.\n",
        "\n",
        "Now, what if you could keep the data locked up or encrypted, while still letting the ML model learn the patterns it needs to.\n",
        "\n",
        "This is the promise behind **homomorphically encrypted machine learning**.\n",
        "\n",
        "Now, there is a lot in this phrase, so let's break it down."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "jkV9v7yo7W9a"
      },
      "source": [
        "## Part 1: What is a homomorphism?\n",
        "\n",
        "A homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). ([Wikipedia](https://en.wikipedia.org/wiki/Homomorphism))\n",
        "\n",
        "A homomorphism is a map between two algebraic structures of the same type (that is of the same name), that preserves the operations of the structures. This means a map $f:A\\to B$ between two sets $A$, $B$ equipped with the same structure such that, if the operation $🔴$ is an operation of the structure (supposed here, for simplification, to be a binary operation), then\n",
        "\n",
        "$$f(x 🔴 y)=f(x)🔴 f(y)$$\n",
        "\n",
        "for every pair $ x$, $ y$ of elements of $A$.[note 1] One says often that $f$ preserves the operation or is compatible with the operation.\n",
        "\n",
        "The operations that must be preserved by a homomorphism include 0-ary operations, that is the constants. In particular, when an identity element is required by the type of structure, the identity element of the first structure must be mapped to the corresponding identity element of the second structure.\n",
        "\n",
        "So, if the operation is preserved, including the identity operation (e.g., multiplying by 1 for multiplication, adding 0 for addition), this is a homomorphism.\n",
        "\n",
        "https://graphik-team.github.io/graal/doc/homomorphism-module"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
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        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 319
        },
        "id": "_ktDd5a77WnO",
        "outputId": "bc02ef1c-a7a4-46d3-ce40-253e66d0c3cf"
      },
      "outputs": [
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",
            "text/plain": [
              "<Figure size 720x360 with 2 Axes>"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "#@title homomorphism visualization { display-mode: \"form\" }\n",
        "\n",
        "# https://github.com/marcin-serwin/cographs/blob/9ba0fd53ec39550049e778da32028861ec932a01/cographs/utilities.py\n",
        "# https://github.com/RCSimpson/Thesis/blob/239c1f2abe2577d9934aece7dddc20b53d58a225/Motifs/motifVisualization.py\n",
        "# https://networkx.org/documentation/stable/reference/algorithms/isomorphism.html\n",
        "\n",
        "import networkx as nx\n",
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "\n",
        "# A collection of Python functions to generate examples of definitions pertinent to the Thesis.\n",
        "\n",
        "# ======================================================================================================================\n",
        "# Homomorphism Example\n",
        "# ======================================================================================================================\n",
        "\n",
        "def homomorphism():\n",
        "    labels = {}\n",
        "    labels[0] = 'A'\n",
        "    labels[1] = 'B'\n",
        "    labels[2] = 'C'\n",
        "    labels[3] = 'D'\n",
        "    A = np.ones((4,4)) - np.diag(np.diag(np.ones((4,4))))\n",
        "    A[0,1] = 0\n",
        "    A[1,0] = 0\n",
        "\n",
        "    B = np.ones((3,3)) - np.diag(np.diag(np.ones((3,3))))\n",
        "\n",
        "    fig = plt.figure(figsize=(10,5))\n",
        "    plt.subplot(121)\n",
        "    G = nx.from_numpy_matrix(A)\n",
        "    pos = nx.spring_layout(G)\n",
        "    nx.draw_networkx_edges(G, pos=pos)\n",
        "    nx.draw_networkx_nodes(G, pos=pos, node_color='c')\n",
        "    nx.draw_networkx_labels(G, pos=pos)\n",
        "    plt.title('$G$')\n",
        "\n",
        "    plt.subplot(122)\n",
        "    H = nx.from_numpy_matrix(B)\n",
        "    pos = nx.spring_layout(H)\n",
        "    nx.draw_networkx_edges(H, pos=pos)\n",
        "    nx.draw_networkx_nodes(H,pos=pos, node_color='c')\n",
        "    nx.draw_networkx_labels(H, pos=pos, labels={0:'A', 1:'B', 2:'C'})\n",
        "    plt.title('$H$')\n",
        "    plt.savefig('graph_homomorphism.png')\n",
        "\n",
        "homomorphism()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "DByBXXut7ZKI"
      },
      "source": [
        "## Part 2: What is homomorphic encryption?\n",
        "\n",
        "https://blog.openmined.org/build-an-homomorphic-encryption-scheme-from-scratch-with-python/\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Y7--t8yedQSe"
      },
      "source": [
        "A basic homomorphic encryption scheme inspired from BFV https://eprint.iacr.org/2012/144.pdf\n",
        "\n",
        "You can read my blog post explaining the implementation details here: https://www.ayoub-benaissa.com/blog/build-he-scheme-from-scratch-python/\n",
        "\n",
        "Disclaimer: This implementation doesn’t neither claim to be secure nor does it follow software engineering best practices, it is designed as simple as possible for the reader to understand the concepts behind homomorphic encryption schemes.\n",
        "\n",
        "The full implementation can be found on [this github gist from the OpenMined team](https://gist.github.com/youben11/f00bc95c5dde5e11218f14f7110ad289)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "TmuX6tKrc53x"
      },
      "outputs": [],
      "source": [
        "#@title Functions for random polynomial generation { display-mode: \"form\" }\n",
        "\n",
        "import numpy as np\n",
        "from numpy.polynomial import polynomial as poly\n",
        "\n",
        "def gen_binary_poly(size):\n",
        "    \"\"\"Generates a polynomial with coeffecients in [0, 1]\n",
        "    Args:\n",
        "        size: number of coeffcients, size-1 being the degree of the\n",
        "            polynomial.\n",
        "    Returns:\n",
        "        array of coefficients with the coeff[i] being \n",
        "        the coeff of x ^ i.\n",
        "    \"\"\"\n",
        "    return np.random.randint(0, 2, size, dtype=np.int64)\n",
        "\n",
        "\n",
        "def gen_uniform_poly(size, modulus):\n",
        "    \"\"\"Generates a polynomial with coeffecients being integers in Z_modulus\n",
        "    Args:\n",
        "        size: number of coeffcients, size-1 being the degree of the\n",
        "            polynomial.\n",
        "    Returns:\n",
        "        array of coefficients with the coeff[i] being \n",
        "        the coeff of x ^ i.\n",
        "    \"\"\"\n",
        "    return np.random.randint(0, modulus, size, dtype=np.int64)\n",
        "\n",
        "\n",
        "def gen_normal_poly(size):\n",
        "    \"\"\"Generates a polynomial with coeffecients in a normal distribution\n",
        "    of mean 0 and a standard deviation of 2, then discretize it.\n",
        "    Args:\n",
        "        size: number of coeffcients, size-1 being the degree of the\n",
        "            polynomial.\n",
        "    Returns:\n",
        "        array of coefficients with the coeff[i] being \n",
        "        the coeff of x ^ i.\n",
        "    \"\"\"\n",
        "    return np.int64(np.random.normal(0, 2, size=size))"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "OJeL4Qfkc56Z"
      },
      "outputs": [],
      "source": [
        "#@title Functions for polynomial evaluation in $Z_q[X]/(X^N + 1)$ { display-mode: \"form\" }\n",
        "\n",
        "def polymul(x, y, modulus, poly_mod):\n",
        "    \"\"\"Add two polynoms\n",
        "    Args:\n",
        "        x, y: two polynoms to be added.\n",
        "        modulus: coefficient modulus.\n",
        "        poly_mod: polynomial modulus.\n",
        "    Returns:\n",
        "        A polynomial in Z_modulus[X]/(poly_mod).\n",
        "    \"\"\"\n",
        "    return np.int64(\n",
        "        np.round(poly.polydiv(poly.polymul(x, y) % modulus, poly_mod)[1] % modulus)\n",
        "    )\n",
        "\n",
        "\n",
        "def polyadd(x, y, modulus, poly_mod):\n",
        "    \"\"\"Multiply two polynoms\n",
        "    Args:\n",
        "        x, y: two polynoms to be multiplied.\n",
        "        modulus: coefficient modulus.\n",
        "        poly_mod: polynomial modulus.\n",
        "    Returns:\n",
        "        A polynomial in Z_modulus[X]/(poly_mod).\n",
        "    \"\"\"\n",
        "    return np.int64(\n",
        "        np.round(poly.polydiv(poly.polyadd(x, y) % modulus, poly_mod)[1] % modulus)\n",
        "    )"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "byhmCOsRc5za"
      },
      "outputs": [],
      "source": [
        "#@title Functions for keygen, encryption and decryption { display-mode: \"form\" }\n",
        "\n",
        "def keygen(size, modulus, poly_mod):\n",
        "    \"\"\"Generate a public and secret keys\n",
        "    Args:\n",
        "        size: size of the polynoms for the public and secret keys.\n",
        "        modulus: coefficient modulus.\n",
        "        poly_mod: polynomial modulus.\n",
        "    Returns:\n",
        "        Public and secret key.\n",
        "    \"\"\"\n",
        "    s = gen_binary_poly(size)\n",
        "    a = gen_uniform_poly(size, modulus)\n",
        "    e = gen_normal_poly(size)\n",
        "    b = polyadd(polymul(-a, s, modulus, poly_mod), -e, modulus, poly_mod)\n",
        "\n",
        "    return (b, a), s\n",
        "\n",
        "\n",
        "def encrypt(pk, size, q, t, poly_mod, pt):\n",
        "    \"\"\"Encrypt an integer.\n",
        "    Args:\n",
        "        pk: public-key.\n",
        "        size: size of polynomials.\n",
        "        q: ciphertext modulus.\n",
        "        t: plaintext modulus.\n",
        "        poly_mod: polynomial modulus.\n",
        "        pt: integer to be encrypted.\n",
        "    Returns:\n",
        "        Tuple representing a ciphertext.      \n",
        "    \"\"\"\n",
        "    # encode the integer into a plaintext polynomial\n",
        "    m = np.array([pt] + [0] * (size - 1), dtype=np.int64) % t\n",
        "    delta = q // t\n",
        "    scaled_m = delta * m\n",
        "    e1 = gen_normal_poly(size)\n",
        "    e2 = gen_normal_poly(size)\n",
        "    u = gen_binary_poly(size)\n",
        "    ct0 = polyadd(\n",
        "        polyadd(\n",
        "            polymul(pk[0], u, q, poly_mod),\n",
        "            e1, q, poly_mod),\n",
        "        scaled_m, q, poly_mod\n",
        "    )\n",
        "    ct1 = polyadd(\n",
        "        polymul(pk[1], u, q, poly_mod),\n",
        "        e2, q, poly_mod\n",
        "    )\n",
        "    return (ct0, ct1)\n",
        "\n",
        "\n",
        "def decrypt(sk, size, q, t, poly_mod, ct):\n",
        "    \"\"\"Decrypt a ciphertext\n",
        "    Args:\n",
        "        sk: secret-key.\n",
        "        size: size of polynomials.\n",
        "        q: ciphertext modulus.\n",
        "        t: plaintext modulus.\n",
        "        poly_mod: polynomial modulus.\n",
        "        ct: ciphertext.\n",
        "    Returns:\n",
        "        Integer representing the plaintext.\n",
        "    \"\"\"\n",
        "    scaled_pt = polyadd(\n",
        "        polymul(ct[1], sk, q, poly_mod),\n",
        "        ct[0], q, poly_mod\n",
        "    )\n",
        "    delta = q // t\n",
        "    decrypted_poly = np.round(scaled_pt / delta) % t\n",
        "    return int(decrypted_poly[0])"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "IVk-5rZrc5u9"
      },
      "outputs": [],
      "source": [
        "#@title Function for adding and multiplying encrypted values { display-mode: \"form\" }\n",
        "\n",
        "def add_plain(ct, pt, q, t, poly_mod):\n",
        "    \"\"\"Add a ciphertext and a plaintext.\n",
        "    Args:\n",
        "        ct: ciphertext.\n",
        "        pt: integer to add.\n",
        "        q: ciphertext modulus.\n",
        "        t: plaintext modulus.\n",
        "        poly_mod: polynomial modulus.\n",
        "    Returns:\n",
        "        Tuple representing a ciphertext.\n",
        "    \"\"\"\n",
        "    size = len(poly_mod) - 1\n",
        "    # encode the integer into a plaintext polynomial\n",
        "    m = np.array([pt] + [0] * (size - 1), dtype=np.int64) % t\n",
        "    delta = q // t\n",
        "    scaled_m = delta * m\n",
        "    new_ct0 = polyadd(ct[0], scaled_m, q, poly_mod)\n",
        "    return (new_ct0, ct[1])\n",
        "\n",
        "\n",
        "def add_cipher(ct1, ct2, q, poly_mod):\n",
        "    \"\"\"Add a ciphertext and a ciphertext.\n",
        "    Args:\n",
        "        ct1, ct2: ciphertexts.\n",
        "        q: ciphertext modulus.\n",
        "        poly_mod: polynomial modulus.\n",
        "    Returns:\n",
        "        Tuple representing a ciphertext.\n",
        "    \"\"\"\n",
        "    new_ct0 = polyadd(ct1[0], ct2[0], q, poly_mod)\n",
        "    new_ct1 = polyadd(ct1[1], ct2[1], q, poly_mod)\n",
        "    return (new_ct0, new_ct1)\n",
        "\n",
        "\n",
        "def mul_plain(ct, pt, q, t, poly_mod):\n",
        "    \"\"\"Multiply a ciphertext and a plaintext.\n",
        "    Args:\n",
        "        ct: ciphertext.\n",
        "        pt: integer to multiply.\n",
        "        q: ciphertext modulus.\n",
        "        t: plaintext modulus.\n",
        "        poly_mod: polynomial modulus.\n",
        "    Returns:\n",
        "        Tuple representing a ciphertext.\n",
        "    \"\"\"\n",
        "    size = len(poly_mod) - 1\n",
        "    # encode the integer into a plaintext polynomial\n",
        "    m = np.array([pt] + [0] * (size - 1), dtype=np.int64) % t\n",
        "    new_c0 = polymul(ct[0], m, q, poly_mod)\n",
        "    new_c1 = polymul(ct[1], m, q, poly_mod)\n",
        "    return (new_c0, new_c1)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "J3SS6sODc5oO",
        "outputId": "eafabd44-910d-48f7-ba08-28203497e8f7"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "[+] Ciphertext ct1(73):\n",
            "\n",
            "\t ct1_0: [ 8177 17334 11008 19020  2896   769  3188 28856 25614 12125   693 27369\n",
            " 15305  6553 19151  8878]\n",
            "\t ct1_1: [ 2716 27725 14329  2040 17208 11308  7756  5462  1024 14866  6512 32337\n",
            "  3631  7668 32354 21934]\n",
            "\n",
            "[+] Ciphertext ct2(20):\n",
            "\n",
            "\t ct1_0: [26038  5448 24707 15722 30352 24668 14796 15622  9313  1331 24746 12055\n",
            " 16515  9300    77 19295]\n",
            "\t ct1_1: [15894 31964 28508 17732 23778 14131 21079 20051 30172 16576  7995   516\n",
            "   665  2276  6832  2056]\n",
            "\n",
            "[+] Decrypted ct3(ct1 + 7): 80\n",
            "[+] Decrypted ct4(ct2 * 5): 99\n"
          ]
        }
      ],
      "source": [
        "#@title Demo of the toy encryption scheme\n",
        "\n",
        "# Scheme's parameters\n",
        "# polynomial modulus degree\n",
        "n = 2**4\n",
        "# ciphertext modulus\n",
        "q = 2**15\n",
        "# plaintext modulus\n",
        "t = 2**8\n",
        "# polynomial modulus\n",
        "poly_mod = np.array([1] + [0] * (n - 1) + [1])\n",
        "# Keygen\n",
        "pk, sk = keygen(n, q, poly_mod)\n",
        "# Encryption\n",
        "pt1, pt2 = 73, 20\n",
        "cst1, cst2 = 7, 5\n",
        "ct1 = encrypt(pk, n, q, t, poly_mod, pt1)\n",
        "ct2 = encrypt(pk, n, q, t, poly_mod, pt2)\n",
        "\n",
        "print(\"[+] Ciphertext ct1({}):\".format(pt1))\n",
        "print(\"\")\n",
        "print(\"\\t ct1_0:\", ct1[0])\n",
        "print(\"\\t ct1_1:\", ct1[1])\n",
        "print(\"\")\n",
        "print(\"[+] Ciphertext ct2({}):\".format(pt2))\n",
        "print(\"\")\n",
        "print(\"\\t ct1_0:\", ct2[0])\n",
        "print(\"\\t ct1_1:\", ct2[1])\n",
        "print(\"\")\n",
        "\n",
        "# Evaluation\n",
        "ct3 = add_plain(ct1, cst1, q, t, poly_mod)\n",
        "ct4 = mul_plain(ct2, cst2, q, t, poly_mod)\n",
        "\n",
        "# Decryption\n",
        "decrypted_ct3 = decrypt(sk, n, q, t, poly_mod, ct3)\n",
        "decrypted_ct4 = decrypt(sk, n, q, t, poly_mod, ct4)\n",
        "\n",
        "print(\"[+] Decrypted ct3(ct1 + {}): {}\".format(cst1, decrypted_ct3))\n",
        "print(\"[+] Decrypted ct4(ct2 * {}): {}\".format(cst2, decrypted_ct4))"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "2PLTsXO2ibM2"
      },
      "source": [
        "#### NOTE: THIS IS A TOY EXAMPLE, AND IT IS NOT SECURE"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "LEwbEJwY9en-"
      },
      "source": [
        "## Part 3: Homomorphicly Encrypted Database queries\n",
        "\n",
        "https://blintzbase.com/posts/pir-and-fhe-from-scratch/\n",
        "\n",
        "|                         | **No Secure Location**                           | **Client**                                                                                       | **Secure Server** | **Crypto Coprocessor** | **FPGA**   |\n",
        "|-------------------------|--------------------------------------------------|--------------------------------------------------------------------------------------------------|-------------------|------------------------|------------|\n",
        "| **Non Homomorphic**     |                                                  | [\"Blob\" Store](https://learn.microsoft.com/en-us/azure/storage/blobs/storage-blobs-introduction) | [AWS GovCloud](https://aws.amazon.com/govcloud-us/)      |                        |            |\n",
        "| **Partial Homomorphic** | [CryptDB](http://www.css.csail.mit.edu/cryptdb/) | [Monomi](https://people.csail.mit.edu/nickolai/papers/tu-monomi.pdf)                             |                   | [TrustedDB](https://ieeexplore.ieee.org/document/6468039/)              | [Cipherbase](https://www.microsoft.com/en-us/research/publication/transaction-processing-on-confidential-data-using-cipherbase/) |\n",
        "| **Full Homomorphic**    |                                                  |                                                                                                  |                   |                        |            |"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "0RxMKvANgndf"
      },
      "outputs": [],
      "source": [
        "#@title PIR Utilities { display-mode: \"form\" }\n",
        "\n",
        "import random\n",
        "\n",
        "\n",
        "def mat_mul(A, B):\n",
        "    C = [[0 for _ in range(len(B[0]))] for _ in range(len(A))]\n",
        "    for i in range(len(A)):\n",
        "        for j in range(len(B[0])):\n",
        "            for k in range(len(B)):\n",
        "                C[i][j] += A[i][k] * B[k][j]\n",
        "    return C\n",
        "\n",
        "\n",
        "def mat_add(A, B):\n",
        "    return [[(A[i][j] + B[i][j]) for j in range(len(A[0]))]\n",
        "            for i in range(len(A))]\n",
        "\n",
        "\n",
        "def mat_sub(A, B):\n",
        "    return [[(A[i][j] - B[i][j]) for j in range(len(A[0]))]\n",
        "            for i in range(len(A))]\n",
        "\n",
        "\n",
        "class Matrix:\n",
        "\n",
        "    def __init__(self, mod, rows, cols, mat):\n",
        "        self.mod = mod\n",
        "        self.rows = rows\n",
        "        self.cols = cols\n",
        "        self.mat = [[x % mod for x in row] for row in mat]\n",
        "\n",
        "    def fill(self, value):\n",
        "        self.mat = [[value] * self.cols for i in range(self.rows)]\n",
        "\n",
        "    def __mul__(self, other):\n",
        "        assert (self.cols == other.rows)\n",
        "        return Matrix(self.mod, self.rows, other.cols,\n",
        "                      mat_mul(self.mat, other.mat))\n",
        "\n",
        "    def __add__(self, other):\n",
        "        if type(other) == int:\n",
        "            assert (self.rows == 1 and self.cols == 1)\n",
        "            return Matrix(self.mod, 1, 1, [[self.mat[0][0] + other]])\n",
        "        assert (self.rows == other.rows and self.cols == other.cols)\n",
        "        return Matrix(self.mod, self.rows, self.cols,\n",
        "                      mat_add(self.mat, other.mat))\n",
        "\n",
        "    def __sub__(self, other):\n",
        "        if type(other) == int:\n",
        "            assert (self.rows == 1 and self.cols == 1)\n",
        "            return Matrix(self.mod, 1, 1, [[self.mat[0][0] - other]])\n",
        "        assert (self.rows == other.rows and self.cols == other.cols)\n",
        "        return Matrix(self.mod, self.rows, self.cols,\n",
        "                      mat_sub(self.mat, other.mat))\n",
        "\n",
        "    def __int__(self):\n",
        "        assert (self.rows == 1 and self.cols == 1)\n",
        "        return self.mat[0][0]\n",
        "\n",
        "    def __repr__(self):\n",
        "        return \"Matrix(\" + str(self.mat) + \")\"\n",
        "\n",
        "\n",
        "def random_bit():\n",
        "    return random.randint(0, 1)\n",
        "\n",
        "\n",
        "def zero_matrix(mod, rows, cols):\n",
        "    mat = [[0 for _ in range(cols)] for _ in range(rows)]\n",
        "    return Matrix(mod, rows, cols, mat)\n",
        "\n",
        "\n",
        "def zero_vector(mod, rows):\n",
        "    return zero_matrix(mod, rows, 1)\n",
        "\n",
        "\n",
        "def random_value(mod):\n",
        "    return random.randint(0, mod - 1)\n",
        "\n",
        "\n",
        "def random_matrix(mod, rows, cols):\n",
        "    mat = [[random_value(mod) for _ in range(cols)] for _ in range(rows)]\n",
        "    return Matrix(mod, rows, cols, mat)\n",
        "\n",
        "\n",
        "def random_vector(mod, rows):\n",
        "    return random_matrix(mod, rows, 1)\n",
        "\n",
        "\n",
        "def random_noise_vector(mod, nd, rows):\n",
        "    return Matrix(mod, rows, 1,\n",
        "                  [[random.randint(nd[0], nd[1])] for _ in range(rows)])"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "RmdD-x96gALk",
        "outputId": "c1fb24af-d2f2-4845-ea34-6a563569ff70"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "LWE working correctly.\n"
          ]
        }
      ],
      "source": [
        "import math\n",
        "\n",
        "#@title LWE  { display-mode: \"form\" }\n",
        "\n",
        "n = 512\n",
        "q = 3329\n",
        "noise_distribution = [-3, 3]\n",
        "\n",
        "\n",
        "def floor(x):\n",
        "    return int(math.floor(x))\n",
        "\n",
        "\n",
        "def get_LWE_sample(s):\n",
        "    A = random_matrix(q, 1, n)\n",
        "    e = random_noise_vector(q, noise_distribution, 1)\n",
        "    b = A * s + e\n",
        "    return (A, b)\n",
        "\n",
        "\n",
        "def keygen():\n",
        "    s = random_vector(q, n)\n",
        "    return s\n",
        "\n",
        "\n",
        "def encrypt(s, m):\n",
        "    assert m == 0 or m == 1\n",
        "    (A, b) = get_LWE_sample(s)\n",
        "    c = b + floor(q / 2) * m\n",
        "    return (A, c)\n",
        "\n",
        "\n",
        "def decrypt(s, ct):\n",
        "    (A, c) = ct\n",
        "    raw = c - A * s\n",
        "    return round(int(raw) * 2 / q) % 2\n",
        "\n",
        "\n",
        "s = keygen()\n",
        "assert (decrypt(s, encrypt(s, 1)) == 1)\n",
        "assert (decrypt(s, encrypt(s, 0)) == 0)\n",
        "print(\"LWE working correctly.\")"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "iaZouQ9QgAI_",
        "outputId": "22a19a82-d7e3-4781-f06e-708e5c1f7ed0"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Sending the query to the server...\n",
            "Got the answer back from the server...\n",
            "The item at index 24 of the database is 0\n",
            "PIR was correct!\n"
          ]
        }
      ],
      "source": [
        "#@title PIR  { display-mode: \"form\" }\n",
        "\n",
        "num_items_in_db = 50\n",
        "desired_idx = 24\n",
        "db = [random_bit() for i in range(num_items_in_db)]\n",
        "\n",
        "\n",
        "def generate_query(desired_idx):\n",
        "    v = []\n",
        "    for i in range(num_items_in_db):\n",
        "        bit = 1 if i == desired_idx else 0\n",
        "        ct = encrypt(s, bit)\n",
        "        v.append(ct)\n",
        "    return v\n",
        "\n",
        "\n",
        "def answer_query(query, db):\n",
        "    summed_A = zero_matrix(q, 1, n)\n",
        "    summed_c = zero_vector(q, 1)\n",
        "    for i in range(num_items_in_db):\n",
        "        if db[i] == 1:\n",
        "            (A, c) = query[i]\n",
        "            summed_A += A\n",
        "            summed_c += c\n",
        "    return (summed_A, summed_c)\n",
        "\n",
        "\n",
        "s = keygen()\n",
        "query = generate_query(desired_idx)\n",
        "\n",
        "print(\"Sending the query to the server...\")\n",
        "\n",
        "answer = answer_query(query, db)\n",
        "\n",
        "print(\"Got the answer back from the server...\")\n",
        "\n",
        "result = decrypt(s, answer)\n",
        "\n",
        "print(\"The item at index %d of the database is %d\" % (desired_idx, result))\n",
        "\n",
        "assert result == db[desired_idx]\n",
        "print(\"PIR was correct!\")"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "EGk9H0LY9j3-"
      },
      "source": [
        "## Part 4: Homomorphic Machine Learning\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Pett89YT7NbT"
      },
      "source": [
        "### TenSEAL Library\n",
        "\n",
        "Python wrapper built around Microsoft's SEAL library, created by the OpenMined community\n",
        "\n",
        "\n",
        "Acknowledgements\n",
        "\n",
        "* Ayoub Benaissa - Twitter: [@y0uben11](https://twitter.com/y0uben11)\n",
        "* Bogdan Cebere - Twitter: [@bcebere](https://twitter.com/bcebere)\n",
        "* Bilal Retiat - Twitter: [@philomath213](https://twitter.com/philomath213)\n",
        "\n",
        "Oh, and Alexander Ziller,\n",
        "Andrew Trask,\n",
        "Antonio Lopardo,\n",
        "Benjamin Szymkow,\n",
        "Bobby Wagner,\n",
        "Emma Bluemke,\n",
        "Jean-Mickael Nounahon,\n",
        "Jonathan Passerat-Palmbach,\n",
        "Kritika Prakash,\n",
        "Nick Rose,\n",
        "Théo Ryffel,\n",
        "Zarreen Naowal Reza,\n",
        "and Georgios Kaissis, **along with the rest of the OpenMined community**"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "-CWwqi_W75JN"
      },
      "source": [
        "### Tutorials\n",
        "\n",
        "- [Getting Started](tutorials%2FTutorial%200%20-%20Getting%20Started.ipynb)\n",
        "- [Tutorial 1 - Training and Evaluation of Logistic Regression on Encrypted Data](tutorials%2FTutorial%201%20-%20Training%20and%20Evaluation%20of%20Logistic%20Regression%20on%20Encrypted%20Data.ipynb)\n",
        "- [Tutorial 2 - Working with Approximate Numbers](tutorials%2FTutorial%202%20-%20Working%20with%20Approximate%20Numbers.ipynb)\n",
        "- [Tutorial 3 - Benchmarks](tutorials%2FTutorial%203%20-%20Benchmarks.ipynb)\n",
        "- [Tutorial 4 - Encrypted Convolution on MNIST](tutorials%2FTutorial%204%20-%20Encrypted%20Convolution%20on%20MNIST.ipynb)\n",
        "\n",
        "##### Publications\n",
        "\n",
        "A. Benaissa, B. Retiat, B. Cebere, A.E. Belfedhal, [\"TenSEAL: A Library for Encrypted Tensor Operations Using Homomorphic Encryption\"](https://arxiv.org/abs/2104.03152), ICLR 2021 Workshop on Distributed and Private Machine Learning (DPML 2021).\n",
        "\n",
        "```\n",
        "@misc{tenseal2021,\n",
        "    title={TenSEAL: A Library for Encrypted Tensor Operations Using Homomorphic Encryption}, \n",
        "    author={Ayoub Benaissa and Bilal Retiat and Bogdan Cebere and Alaa Eddine Belfedhal},\n",
        "    year={2021},\n",
        "    eprint={2104.03152},\n",
        "    archivePrefix={arXiv},\n",
        "    primaryClass={cs.CR}\n",
        "}\n",
        "```"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "O5yWHVpi68N6"
      },
      "source": [
        "### Installation"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "WLyMgEk97EHI",
        "outputId": "569660fe-ced6-4ed9-cfc6-07df27025950"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "\u001b[K     |████████████████████████████████| 4.9 MB 12.1 MB/s \n",
            "\u001b[?25h"
          ]
        }
      ],
      "source": [
        "!pip -qq install tenseal"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Yc27Edhu7EiZ"
      },
      "source": [
        "### Quick Example"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "eD1R1xRb6cuI",
        "outputId": "0cb9f978-9842-4176-f16b-1f7591d407fb"
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "[4.000000000019502,\n",
              " 4.000000002421371,\n",
              " 4.00000000433207,\n",
              " 4.000000000712195,\n",
              " 3.999999999134501]"
            ]
          },
          "execution_count": 11,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "import tenseal as ts\n",
        "\n",
        "# Setup TenSEAL context\n",
        "context = ts.context(\n",
        "            ts.SCHEME_TYPE.CKKS,\n",
        "            poly_modulus_degree=8192,\n",
        "            coeff_mod_bit_sizes=[60, 40, 40, 60]\n",
        "          )\n",
        "context.generate_galois_keys()\n",
        "context.global_scale = 2**40\n",
        "\n",
        "v1 = [0, 1, 2, 3, 4]\n",
        "v2 = [4, 3, 2, 1, 0]\n",
        "\n",
        "# encrypted vectors\n",
        "enc_v1 = ts.ckks_vector(context, v1)\n",
        "enc_v2 = ts.ckks_vector(context, v2)\n",
        "\n",
        "result = enc_v1 + enc_v2\n",
        "result.decrypt() # ~ [4, 4, 4, 4, 4]"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "Y4TT73Uy6crh",
        "outputId": "c25af8cd-f3ce-41e5-9d03-5aa853650975"
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "[10.000002505232832]"
            ]
          },
          "execution_count": 12,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "\n",
        "result = enc_v1.dot(enc_v2)\n",
        "result.decrypt() # ~ [10]"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "AZy89EOf6cpK",
        "outputId": "5a969785-8310-4c52-9a3c-078509fffd95"
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "[157.00002093913858, -90.00001229463638, 153.00002052039056]"
            ]
          },
          "execution_count": 13,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "\n",
        "matrix = [\n",
        "  [73, 0.5, 8],\n",
        "  [81, -5, 66],\n",
        "  [-100, -78, -2],\n",
        "  [0, 9, 17],\n",
        "  [69, 11 , 10],\n",
        "]\n",
        "result = enc_v1.matmul(matrix)\n",
        "result.decrypt() # ~ [157, -90, 153]"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "_x6_tAljW3Hh"
      },
      "source": [
        "## 4a. Addition and Subtraction HE\n",
        "\n",
        "- **Source:** [OpenMined TenSEAL Tutorial 1 - Training and Evaluation of Logistic Regression on Encrypted Data](https://github.com/OpenMined/TenSEAL/blob/main/tutorials/Tutorial%201%20-%20Training%20and%20Evaluation%20of%20Logistic%20Regression%20on%20Encrypted%20Data.ipynb)\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "lQwagann6cmU",
        "outputId": "9a9d5abf-603e-47f4-f03d-fd1a9ff521a9"
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "<tenseal.enc_context.Context at 0x7f028b769690>"
            ]
          },
          "execution_count": 14,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "import tenseal as ts\n",
        "\n",
        "context = ts.context(ts.SCHEME_TYPE.BFV, poly_modulus_degree=4096, plain_modulus=1032193)\n",
        "context"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "lRSeXpfP6cjc",
        "outputId": "dda18362-a03c-4ed7-93a3-24411b10d4d8"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Created ciphertext vector from plaintext vector of size:  5\n"
          ]
        },
        {
          "data": {
            "text/plain": [
              "<tenseal.tensors.bfvvector.BFVVector at 0x7f028b765450>"
            ]
          },
          "execution_count": 15,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "plain_vector = [69, 70, 71, 72, 73]\n",
        "encrypted_vector = ts.bfv_vector(context, plain_vector)\n",
        "print(\"Created ciphertext vector from plaintext vector of size: \", encrypted_vector.size())\n",
        "encrypted_vector"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "h6g7vab46chH",
        "outputId": "ce838750-e922-4304-bc75-4ac4f64c3b9d"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "c2p addition result (decrypted):  [70, 72, 74, 76, 78]\n",
            "c2p subtraction result (decrypted):  [68, 68, 68, 68, 68]\n",
            "c2p multiplication result (decrypted):  [69, 140, 213, 288, 365]\n",
            "c2c addition result (decrypted):  [138, 140, 142, 144, 146]\n",
            "c2c subtraction result (decrypted):  [69, 70, 71, 72, 73]\n",
            "c2c multiplication result (decrypted):  [9522, 9800, 10082, 10368, 10658]\n"
          ]
        }
      ],
      "source": [
        "add_result = encrypted_vector + [1, 2, 3, 4, 5]\n",
        "print(\"c2p addition result (decrypted): \", add_result.decrypt())\n",
        "sub_result = encrypted_vector - [1, 2, 3, 4, 5]\n",
        "print(\"c2p subtraction result (decrypted): \", sub_result.decrypt())\n",
        "mul_result = encrypted_vector * [1, 2, 3, 4, 5]\n",
        "print(\"c2p multiplication result (decrypted): \", mul_result.decrypt())\n",
        "encrypted_add = add_result + sub_result\n",
        "print(\"c2c addition result (decrypted): \", encrypted_add.decrypt())\n",
        "encrypted_sub = encrypted_add - encrypted_vector\n",
        "print(\"c2c subtraction result (decrypted): \", encrypted_sub.decrypt())\n",
        "encrypted_mul = encrypted_add * encrypted_sub\n",
        "print(\"c2c multiplication result (decrypted): \", encrypted_mul.decrypt())"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "i4Lotjik6ceg",
        "outputId": "3cb19744-a5aa-4ddd-ea4a-925392bcde4c"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "c2c multiply time: 14.771223068237305 ms\n",
            "c2p multiply time: 1.5020370483398438 ms\n"
          ]
        }
      ],
      "source": [
        "from time import time\n",
        "\n",
        "t_start = time()\n",
        "_ = encrypted_add * encrypted_mul\n",
        "t_end = time()\n",
        "print(\"c2c multiply time: {} ms\".format((t_end - t_start) * 1000))\n",
        "\n",
        "t_start = time()\n",
        "_ = encrypted_add * [1, 2, 3, 4, 5]\n",
        "t_end = time()\n",
        "print(\"c2p multiply time: {} ms\".format((t_end - t_start) * 1000))"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "KE5JmDuUXxxs"
      },
      "source": [
        "## 4b. Approximate numbers\n",
        "\n",
        "- Source: [OpenMined TenSEAL Tutorial 2 - Working with Approximate Numbers](https://github.com/OpenMined/TenSEAL/blob/main/tutorials/Tutorial%202%20-%20Working%20with%20Approximate%20Numbers.ipynb)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "cLshimgpYnej"
      },
      "outputs": [],
      "source": [
        "import torch\n",
        "from torchvision import transforms\n",
        "from random import randint\n",
        "import pickle\n",
        "from PIL import Image\n",
        "import numpy as np\n",
        "from matplotlib.pyplot import imshow\n",
        "from typing import Dict\n",
        "\n",
        "import tenseal as ts"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "KgzvFYcbYnb6"
      },
      "outputs": [],
      "source": [
        "def context():\n",
        "    context = ts.context(ts.SCHEME_TYPE.CKKS, 8192, coeff_mod_bit_sizes=[60, 40, 40, 60])\n",
        "    context.global_scale = pow(2, 40)\n",
        "    context.generate_galois_keys()\n",
        "    return context\n",
        "\n",
        "context = context()\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "19Fjjb3gYnZ5",
        "outputId": "5e22756e-6994-4b56-c171-fe53e18e3a32"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            " First tensor: Shape = [2, 2] Data = [[1.0, 2.0], [3.0, 4.0]]\n",
            " Second tensor: Shape = [2, 2] Data = [[5.0, 6.0], [7.0, 8.0]]\n"
          ]
        }
      ],
      "source": [
        "plain1 = ts.plain_tensor([1,2,3,4], [2,2])\n",
        "\n",
        "print(\" First tensor: Shape = {} Data = {}\".format(plain1.shape, plain1.tolist()))\n",
        "\n",
        "plain2 = ts.plain_tensor(np.array([5,6,7,8]).reshape(2,2))\n",
        "print(\" Second tensor: Shape = {} Data = {}\".format(plain2.shape, plain2.tolist()))"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "SQ1PSESKYnXj",
        "outputId": "7bfc5a6c-0204-4dd6-f5bc-4e5b8cda10fa"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            " Shape = [2, 2]\n",
            " Encrypted Data = <tenseal.tensors.ckkstensor.CKKSTensor object at 0x7f021afcce50>.\n",
            " Shape = [2, 2]\n"
          ]
        }
      ],
      "source": [
        "encrypted_tensor1 = ts.ckks_tensor(context, plain1)\n",
        "encrypted_tensor2 = ts.ckks_tensor(context, plain2)\n",
        "\n",
        "print(\" Shape = {}\".format(encrypted_tensor1.shape))\n",
        "print(\" Encrypted Data = {}.\".format(encrypted_tensor1))\n",
        "\n",
        "\n",
        "encrypted_tensor_from_np = ts.ckks_tensor(context, np.array([5,6,7,8]).reshape([2,2]))\n",
        "print(\" Shape = {}\".format(encrypted_tensor_from_np.shape))"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "75NNJ3oKYnVI"
      },
      "outputs": [],
      "source": [
        "def decrypt(enc):\n",
        "    return enc.decrypt().tolist()"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "GsppZjqfYnSz",
        "outputId": "596df8f7-712c-46e3-c513-cf0e9dcc707a"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Plain equivalent: [[1.0, 2.0], [3.0, 4.0]] + [[5.0, 6.0], [7.0, 8.0]]\n",
            "Decrypted result: [[5.999999998738882, 7.999999998947602], [10.000000000756188, 11.999999999544915]].\n"
          ]
        }
      ],
      "source": [
        "result = encrypted_tensor1 + encrypted_tensor2\n",
        "print(\"Plain equivalent: {} + {}\\nDecrypted result: {}.\".format(plain1.tolist(), plain2.tolist(), decrypt(result)))"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "cDnS5GDyZDJc",
        "outputId": "f62b2115-3af0-4982-f029-9a430145d737"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Plain equivalent: [[1.0, 2.0], [3.0, 4.0]] - [[5.0, 6.0], [7.0, 8.0]]\n",
            "Decrypted result: [[-3.9999999980184393, -4.000000000271845], [-4.000000000487303, -4.000000001325261]].\n"
          ]
        }
      ],
      "source": [
        "result = encrypted_tensor1 - encrypted_tensor2\n",
        "print(\"Plain equivalent: {} - {}\\nDecrypted result: {}.\".format(plain1.tolist(), plain2.tolist(), decrypt(result)))"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "ffrXUU8-ZDHF",
        "outputId": "b97b8f9c-f795-4bc1-ba2d-47f6ae205612"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Plain equivalent: [[1.0, 2.0], [3.0, 4.0]] * [[5.0, 6.0], [7.0, 8.0]]\n",
            "Decrypted result: [[5.00000067133901, 12.000001604693615], [21.00000281850078, 32.00000428564696]].\n"
          ]
        }
      ],
      "source": [
        "result = encrypted_tensor1 * encrypted_tensor2\n",
        "print(\"Plain equivalent: {} * {}\\nDecrypted result: {}.\".format(plain1.tolist(), plain2.tolist(), decrypt(result)))"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "gi3cEHMgZDE0",
        "outputId": "288f98f4-57da-496e-9b02-f13affbabae5"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Plain equivalent: [[1.0, 2.0], [3.0, 4.0]] * [[5.0, 6.0], [7.0, 8.0]]\n",
            "Decrypted result: [[5.000000672651283, 12.000001606041671], [21.000002817341006, 32.00000428372986]].\n"
          ]
        }
      ],
      "source": [
        "plain = ts.plain_tensor([5,6,7,8], [2,2])\n",
        "result = encrypted_tensor1 * plain\n",
        "\n",
        "print(\"Plain equivalent: {} * {}\\nDecrypted result: {}.\".format(plain1.tolist(), plain.tolist(), decrypt(result)))"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "VZN1CCCmZDCj",
        "outputId": "f16a3f19-06ea-4bf8-d5e6-5a865a274e9b"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Plain equivalent: -[[1.0, 2.0], [3.0, 4.0]]\n",
            "Decrypted result: [[-1.0000000003602225, -1.9999999993378785], [-3.0000000001344413, -3.999999999109826]].\n"
          ]
        }
      ],
      "source": [
        "result = -encrypted_tensor1 \n",
        "\n",
        "print(\"Plain equivalent: -{}\\nDecrypted result: {}.\".format(plain1.tolist(), decrypt(result)))"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "fAmYSjr6YnQP",
        "outputId": "e244b44a-f9c9-40ab-8222-91664c25ab08"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Plain equivalent: [[1.0, 2.0], [3.0, 4.0]] ^ 3\n",
            "Decrypted result: [[1.0000008060332688, 8.000006431334013], [27.00002173078232, 64.0000514588384]].\n"
          ]
        }
      ],
      "source": [
        "result = encrypted_tensor1 ** 3\n",
        "print(\"Plain equivalent: {} ^ 3\\nDecrypted result: {}.\".format(plain1.tolist(), decrypt(result)))"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "v5SrZNJ6YnOC",
        "outputId": "3cb5eec7-cc0c-44db-f937-b48eef6cf364"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "X = [[1.0, 2.0], [3.0, 4.0]]\n",
            "1 + X^2 + X^3 = [[3.0000009417362126, 13.000006965522154], [37.00002294025727, 81.00005359884193]].\n"
          ]
        }
      ],
      "source": [
        "result = encrypted_tensor1.polyval([1,0,1,1])\n",
        "\n",
        "print(\"X = {}\".format(plain1.tolist()))\n",
        "print(\"1 + X^2 + X^3 = {}.\".format(decrypt(result)))"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "VowQ9yIq6cbo",
        "outputId": "75b9951c-4204-4234-c9f5-e8bd20e069bd"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "X = [[1.0, 2.0], [3.0, 4.0]]\n",
            "0.5 + 0.197 X - 0.004 x^X = [[0.6930000230951937, 0.862000022438146], [0.9829999736643001, 1.0319998570728586]].\n"
          ]
        }
      ],
      "source": [
        "result = encrypted_tensor1.polyval([0.5, 0.197, 0, -0.004])\n",
        "\n",
        "\n",
        "print(\"X = {}\".format(plain1.tolist()))\n",
        "print(\"0.5 + 0.197 X - 0.004 x^X = {}.\".format(decrypt(result)))"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "MBld7sPCaqv_",
        "outputId": "6a0a1a47-2bf2-4697-a144-9bee59cbf32c"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "/content/data\n",
            "/content/data/mnist-samples\n",
            "/content\n"
          ]
        }
      ],
      "source": [
        "# Download MNIST samples { display-mode: \"form\" }\n",
        "\n",
        "!mkdir data\n",
        "%cd data\n",
        "!mkdir mnist-samples\n",
        "%cd mnist-samples\n",
        "!wget -qq https://github.com/OpenMined/TenSEAL/raw/main/tutorials/data/mnist-samples/img_1.jpg\n",
        "!wget -qq https://github.com/OpenMined/TenSEAL/raw/main/tutorials/data/mnist-samples/img_2.jpg\n",
        "!wget -qq https://github.com/OpenMined/TenSEAL/raw/main/tutorials/data/mnist-samples/img_3.jpg\n",
        "!wget -qq https://github.com/OpenMined/TenSEAL/raw/main/tutorials/data/mnist-samples/img_4.jpg\n",
        "!wget -qq https://github.com/OpenMined/TenSEAL/raw/main/tutorials/data/mnist-samples/img_5.jpg\n",
        "!wget -qq https://github.com/OpenMined/TenSEAL/raw/main/tutorials/data/mnist-samples/img_6.jpg\n",
        "%cd ../.."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "VpA4SlUX6cYf"
      },
      "outputs": [],
      "source": [
        "# Create the TenSEAL security context { display-mode: \"form\" }\n",
        "\n",
        "def create_ctx():\n",
        "    \"\"\"Helper for creating the CKKS context.\n",
        "    CKKS params:\n",
        "        - Polynomial degree: 8192.\n",
        "        - Coefficient modulus size: [40, 21, 21, 21, 21, 21, 21, 40].\n",
        "        - Scale: 2 ** 21.\n",
        "        - The setup requires the Galois keys for evaluating the convolutions.\n",
        "    \"\"\"\n",
        "    poly_mod_degree = 8192\n",
        "    coeff_mod_bit_sizes = [40, 21, 21, 21, 21, 21, 21, 40]\n",
        "    ctx = ts.context(ts.SCHEME_TYPE.CKKS, poly_mod_degree, -1, coeff_mod_bit_sizes)\n",
        "    ctx.global_scale = 2 ** 21\n",
        "    ctx.generate_galois_keys()\n",
        "    return ctx\n",
        "\n",
        "# Sample an image\n",
        "def load_input():\n",
        "    transform = transforms.Compose(\n",
        "        [transforms.ToTensor(), transforms.Normalize((0.1307,), (0.3081,))]\n",
        "    )\n",
        "    idx = randint(1, 6)\n",
        "    img_name = \"data/mnist-samples/img_{}.jpg\".format(idx)\n",
        "    print(img_name)\n",
        "    img = Image.open(img_name)\n",
        "    return transform(img).view(28, 28).tolist(), img\n",
        "\n",
        "# Helper for encoding the image\n",
        "def prepare_input(ctx, plain_input):\n",
        "    enc_input, windows_nb = ts.im2col_encoding(ctx, plain_input, 7, 7, 3)\n",
        "    assert windows_nb == 64\n",
        "    return enc_input"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "cbYEpK6bZihT"
      },
      "outputs": [],
      "source": [
        "# Load a pretrained model and adapt the forward call for encrypted input { display-mode: \"form\" }\n",
        "\n",
        "class ConvMNIST():\n",
        "    \"\"\"CNN for classifying MNIST data.\n",
        "    Input should be an encoded 28x28 matrix representing the image.\n",
        "    TenSEAL can be used for encoding `tenseal.im2col_encoding(ctx, input_matrix, 7, 7, 3)`\n",
        "    The input should also be normalized with a mean=0.1307 and an std=0.3081 before encryption.\n",
        "    \"\"\"\n",
        "\n",
        "    def __init__(self, parameters: Dict[str, list]):\n",
        "        self.conv1_weight = parameters[\"conv1_weight\"]\n",
        "        self.conv1_bias = parameters[\"conv1_bias\"]\n",
        "        self.fc1_weight = parameters[\"fc1_weight\"]\n",
        "        self.fc1_bias = parameters[\"fc1_bias\"]\n",
        "        self.fc2_weight = parameters[\"fc2_weight\"]\n",
        "        self.fc2_bias = parameters[\"fc2_bias\"]\n",
        "        self.windows_nb = parameters[\"windows_nb\"]\n",
        "\n",
        "    def forward(self, enc_x: ts.CKKSVector) -> ts.CKKSVector:\n",
        "        # conv layer\n",
        "        channels = []\n",
        "        for kernel, bias in zip(self.conv1_weight, self.conv1_bias):\n",
        "            y = enc_x.conv2d_im2col(kernel, self.windows_nb) + bias\n",
        "            channels.append(y)\n",
        "        out = ts.CKKSVector.pack_vectors(channels)\n",
        "        # squaring\n",
        "        out.square_()\n",
        "        # no need to flat\n",
        "        # fc1 layer\n",
        "        out = out.mm_(self.fc1_weight) + self.fc1_bias\n",
        "        # squaring\n",
        "        out.square_()\n",
        "        # output layer\n",
        "        out = out.mm_(self.fc2_weight) + self.fc2_bias\n",
        "        return out\n",
        "\n",
        "    @staticmethod\n",
        "    def prepare_input(context: bytes, ckks_vector: bytes) -> ts.CKKSVector:\n",
        "        try:\n",
        "            ctx = ts.context_from(context)\n",
        "            enc_x = ts.ckks_vector_from(ctx, ckks_vector)\n",
        "        except:\n",
        "            raise DeserializationError(\"cannot deserialize context or ckks_vector\")\n",
        "        try:\n",
        "            _ = ctx.galois_keys()\n",
        "        except:\n",
        "            raise InvalidContext(\"the context doesn't hold galois keys\")\n",
        "\n",
        "        return enc_x\n",
        "    def __call__(self, *args, **kwargs):\n",
        "        return self.forward(*args, **kwargs)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "2izmc341Zie9",
        "outputId": "f986ac00-001c-4bb3-8fee-adcd64460090"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "/content/parameters\n",
            "--2022-11-03 03:34:21--  https://github.com/OpenMined/TenSEAL/raw/main/tutorials/parameters/ConvMNIST-0.1.pickle\n",
            "Resolving github.com (github.com)... 140.82.113.4\n",
            "Connecting to github.com (github.com)|140.82.113.4|:443... connected.\n",
            "HTTP request sent, awaiting response... 302 Found\n",
            "Location: https://raw.githubusercontent.com/OpenMined/TenSEAL/main/tutorials/parameters/ConvMNIST-0.1.pickle [following]\n",
            "--2022-11-03 03:34:21--  https://raw.githubusercontent.com/OpenMined/TenSEAL/main/tutorials/parameters/ConvMNIST-0.1.pickle\n",
            "Resolving raw.githubusercontent.com (raw.githubusercontent.com)... 185.199.111.133, 185.199.108.133, 185.199.109.133, ...\n",
            "Connecting to raw.githubusercontent.com (raw.githubusercontent.com)|185.199.111.133|:443... connected.\n",
            "HTTP request sent, awaiting response... 200 OK\n",
            "Length: 157239 (154K) [application/octet-stream]\n",
            "Saving to: ‘ConvMNIST-0.1.pickle’\n",
            "\n",
            "ConvMNIST-0.1.pickl 100%[===================>] 153.55K  --.-KB/s    in 0.002s  \n",
            "\n",
            "2022-11-03 03:34:21 (77.2 MB/s) - ‘ConvMNIST-0.1.pickle’ saved [157239/157239]\n",
            "\n",
            "/content\n"
          ]
        }
      ],
      "source": [
        "#@title Load saved parameters for model { display-mode: \"form\" }\n",
        "\n",
        "!mkdir parameters\n",
        "%cd parameters\n",
        "!wget -qq https://github.com/OpenMined/TenSEAL/raw/main/tutorials/parameters/ConvMNIST-0.1.pickle\n",
        "%cd .."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "A8Gi1cCIZick",
        "outputId": "d33ad4ff-3684-4963-d1f3-1ac9da3d153d"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Model loaded from 'parameters/ConvMNIST-0.1.pickle'\n"
          ]
        }
      ],
      "source": [
        "import pickle\n",
        "import os\n",
        "\n",
        "def load_parameters(file_path: str) -> dict:\n",
        "    try:\n",
        "        parameters = pickle.load(open(file_path, \"rb\"))\n",
        "        print(f\"Model loaded from '{file_path}'\")\n",
        "    except OSError as ose:\n",
        "        print(\"error\", ose)\n",
        "        raise ose\n",
        "    return parameters\n",
        "\n",
        "\n",
        "\n",
        "parameters = load_parameters(\"parameters/ConvMNIST-0.1.pickle\")\n",
        "model = ConvMNIST(parameters)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 319
        },
        "id": "y9yBII5CZiaO",
        "outputId": "d287d352-a5d4-4e1d-b861-927555bde766"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "data/mnist-samples/img_3.jpg\n",
            "Encrypted image  <tenseal.tensors.ckksvector.CKKSVector object at 0x7f028c8d6c10>\n",
            "Original image \n"
          ]
        },
        {
          "data": {
            "image/png": 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",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          },
          "output_type": "display_data"
        }
      ],
      "source": [
        "# CKKS context generation.\n",
        "context = create_ctx()\n",
        "\n",
        "# Random image sampling\n",
        "image, orig = load_input()\n",
        "\n",
        "# Image encoding\n",
        "encrypted_image = prepare_input(context, image)\n",
        "\n",
        "print(\"Encrypted image \", encrypted_image)\n",
        "print(\"Original image \")\n",
        "imshow(np.asarray(orig))\n",
        "\n",
        "# We prepare the context for the server, by making it public(we drop the secret key)\n",
        "server_context = context.copy()\n",
        "server_context.make_context_public()\n",
        "\n",
        "# Context and ciphertext serialization\n",
        "server_context = server_context.serialize()\n",
        "encrypted_image = encrypted_image.serialize()\n",
        "\n",
        "\n",
        "client_query = {\n",
        "    \"data\" : encrypted_image,\n",
        "    \"context\" : server_context,\n",
        "}"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "Ps1zb2RZZiXa"
      },
      "outputs": [],
      "source": [
        "encrypted_query = model.prepare_input(client_query[\"context\"], client_query[\"data\"])\n",
        "encrypted_result = model(encrypted_query).serialize()\n",
        "\n",
        "server_response = {\n",
        "    \"data\" : encrypted_result\n",
        "}"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "Wu_hQ4FUZiU-",
        "outputId": "d5a3645d-3a7d-449f-9618-6ec286fa30a6"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Maximum probability for label 9\n"
          ]
        }
      ],
      "source": [
        "result = ts.ckks_vector_from(context, server_response[\"data\"]).decrypt()\n",
        "\n",
        "probs = torch.softmax(torch.tensor(result), 0)\n",
        "label_max = torch.argmax(probs)\n",
        "print(\"Maximum probability for label {}\".format(label_max))"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "0s-CUHffX4aI"
      },
      "source": [
        "## 4c. Benchmarking\n",
        "\n",
        "- Source: [OpenMined TenSEAL Tutorial 3 - Benchmarks](https://github.com/OpenMined/TenSEAL/blob/main/tutorials/Tutorial%203%20-%20Benchmarks.ipynb)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "offEKrAjdSQi"
      },
      "outputs": [],
      "source": [
        "!pip -qq install tabulate"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "hOe82pVuxsl1"
      },
      "source": [
        "## Homomorphic Encryption Parameters Review\n",
        "\n",
        "#### The scaling factor(CKKS)\n",
        "The first step of the CKKS scheme is encoding a vector of real numbers into a plaintext polynomial.\n",
        "\n",
        "\n",
        "The scaling factor defines the encoding precision for the binary representation of the number. Intuitively, we are talking about binary precision as pictured below:\n",
        "\n",
        "\n",
        "<img src=\"assets/floating_point.png\" alt=\"ckks-high-level\" width=\"400\"/>\n",
        "\n",
        "#### The polynomial modulus degree (poly_modulus_degree)\n",
        "\n",
        "The polynomial modulus($N$ in the diagram) directly affects:\n",
        " - The number of coefficients in plaintext polynomials\n",
        " - The size of ciphertext elements\n",
        " - The computational performance of the scheme (bigger is worse)\n",
        " - The security level (bigger is better).\n",
        "\n",
        "In TenSEAL, as in Microsoft SEAL, the degree of the polynomial modulus must be a power of 2 (e.g. $1024$, $2048$, $4096$, $8192$, $16384$, or $32768$).\n",
        "\n",
        "#### The coefficient modulus sizes\n",
        "\n",
        "The last parameter required for the scheme is a list of binary sizes.\n",
        "Using this list, SEAL will generate a list of primes of those binary sizes, called the coefficient modulus($q$ in the diagram).\n",
        "\n",
        "The coefficient modulus directly affects:\n",
        " - The size of ciphertext elements\n",
        " - The length of the list indicates the level of the scheme(or the number of encrypted multiplications supported).\n",
        " - The security level (bigger is worse).\n",
        " \n",
        "In TenSEAL, as in Microsoft SEAL, each of the prime numbers in the coefficient modulus must be at most 60 bits and must be congruent to 1 modulo 2*poly_modulus_degree."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "pdaa4g4n6cWK"
      },
      "outputs": [],
      "source": [
        "import tenseal as ts\n",
        "import tenseal.sealapi as seal\n",
        "import random\n",
        "import pickle\n",
        "import numpy as np\n",
        "import math\n",
        "from IPython.display import HTML, display\n",
        "import tabulate\n",
        "import pytest\n",
        " \n",
        "def convert_size(size_bytes):\n",
        "    if size_bytes == 0:\n",
        "        return \"0B\"\n",
        "    size_name = (\"B\", \"KB\", \"MB\", \"GB\", \"TB\", \"PB\", \"EB\", \"ZB\", \"YB\")\n",
        "    i = int(math.floor(math.log(size_bytes, 1024)))\n",
        "    p = math.pow(1024, i)\n",
        "    s = round(size_bytes / p, 2)\n",
        "    return \"%s %s\" % (s, size_name[i])\n",
        "\n",
        "enc_type_str = {\n",
        "    ts.ENCRYPTION_TYPE.SYMMETRIC : \"symmetric\", \n",
        "    ts.ENCRYPTION_TYPE.ASYMMETRIC : \"asymmetric\",\n",
        "}\n",
        "\n",
        "scheme_str = {\n",
        "    ts.SCHEME_TYPE.CKKS : \"ckks\", \n",
        "    ts.SCHEME_TYPE.BFV : \"bfv\",\n",
        "}\n",
        "\n",
        "def decrypt(enc):\n",
        "    return enc.decrypt()"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 1000
        },
        "id": "9fLbf7ls6cTk",
        "outputId": "a39456eb-cc59-45c0-8e1d-e9daa580133f"
      },
      "outputs": [
        {
          "data": {
            "text/html": [
              "<table>\n",
              "<tbody>\n",
              "<tr><td>Encryption Type</td><td>Scheme Type</td><td>Polynomial modulus</td><td>Coefficient modulus sizes       </td><td>Saved keys </td><td>Context serialized size</td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>all        </td><td>263.1 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>Secret key </td><td>263.09 KB              </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>Galois keys</td><td>86.88 MB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>Relin keys </td><td>3.63 MB                </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>none       </td><td>112.0 B                </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>8192              </td><td>[40, 20, 40]                    </td><td>all        </td><td>124.94 KB              </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>8192              </td><td>[40, 20, 40]                    </td><td>Secret key </td><td>124.93 KB              </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>8192              </td><td>[40, 20, 40]                    </td><td>Galois keys</td><td>11.89 MB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>8192              </td><td>[40, 20, 40]                    </td><td>Relin keys </td><td>505.11 KB              </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>8192              </td><td>[40, 20, 40]                    </td><td>none       </td><td>92.0 B                 </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>8192              </td><td>[20, 20, 20]                    </td><td>all        </td><td>79.35 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>8192              </td><td>[20, 20, 20]                    </td><td>Secret key </td><td>79.34 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>8192              </td><td>[20, 20, 20]                    </td><td>Galois keys</td><td>7.48 MB                </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>8192              </td><td>[20, 20, 20]                    </td><td>Relin keys </td><td>317.41 KB              </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>8192              </td><td>[20, 20, 20]                    </td><td>none       </td><td>83.0 B                 </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>8192              </td><td>[17, 17]                        </td><td>all        </td><td>43.74 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>8192              </td><td>[17, 17]                        </td><td>Secret key </td><td>43.73 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>8192              </td><td>[17, 17]                        </td><td>Galois keys</td><td>2.15 MB                </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>8192              </td><td>[17, 17]                        </td><td>Relin keys </td><td>86.98 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>8192              </td><td>[17, 17]                        </td><td>none       </td><td>78.0 B                 </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>4096              </td><td>[40, 20, 40]                    </td><td>all        </td><td>62.55 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>4096              </td><td>[40, 20, 40]                    </td><td>Secret key </td><td>62.55 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>4096              </td><td>[40, 20, 40]                    </td><td>Galois keys</td><td>5.48 MB                </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>4096              </td><td>[40, 20, 40]                    </td><td>Relin keys </td><td>253.1 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>4096              </td><td>[40, 20, 40]                    </td><td>none       </td><td>92.0 B                 </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>4096              </td><td>[30, 20, 30]                    </td><td>all        </td><td>51.09 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>4096              </td><td>[30, 20, 30]                    </td><td>Secret key </td><td>51.08 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>4096              </td><td>[30, 20, 30]                    </td><td>Galois keys</td><td>4.53 MB                </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>4096              </td><td>[30, 20, 30]                    </td><td>Relin keys </td><td>208.33 KB              </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>4096              </td><td>[30, 20, 30]                    </td><td>none       </td><td>95.0 B                 </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>4096              </td><td>[20, 20, 20]                    </td><td>all        </td><td>40.06 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>4096              </td><td>[20, 20, 20]                    </td><td>Secret key </td><td>40.06 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>4096              </td><td>[20, 20, 20]                    </td><td>Galois keys</td><td>3.44 MB                </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>4096              </td><td>[20, 20, 20]                    </td><td>Relin keys </td><td>159.16 KB              </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>4096              </td><td>[20, 20, 20]                    </td><td>none       </td><td>82.0 B                 </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>4096              </td><td>[19, 19, 19]                    </td><td>all        </td><td>37.72 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>4096              </td><td>[19, 19, 19]                    </td><td>Secret key </td><td>37.72 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>4096              </td><td>[19, 19, 19]                    </td><td>Galois keys</td><td>3.3 MB                 </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>4096              </td><td>[19, 19, 19]                    </td><td>Relin keys </td><td>150.85 KB              </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>4096              </td><td>[19, 19, 19]                    </td><td>none       </td><td>82.0 B                 </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>4096              </td><td>[18, 18, 18]                    </td><td>all        </td><td>35.51 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>4096              </td><td>[18, 18, 18]                    </td><td>Secret key </td><td>35.51 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>4096              </td><td>[18, 18, 18]                    </td><td>Galois keys</td><td>3.15 MB                </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>4096              </td><td>[18, 18, 18]                    </td><td>Relin keys </td><td>143.18 KB              </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>4096              </td><td>[18, 18, 18]                    </td><td>none       </td><td>81.0 B                 </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>4096              </td><td>[18, 18]                        </td><td>all        </td><td>24.02 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>4096              </td><td>[18, 18]                        </td><td>Secret key </td><td>24.01 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>4096              </td><td>[18, 18]                        </td><td>Galois keys</td><td>1.05 MB                </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>4096              </td><td>[18, 18]                        </td><td>Relin keys </td><td>47.55 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>4096              </td><td>[18, 18]                        </td><td>none       </td><td>78.0 B                 </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>4096              </td><td>[17, 17]                        </td><td>all        </td><td>21.68 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>4096              </td><td>[17, 17]                        </td><td>Secret key </td><td>21.68 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>4096              </td><td>[17, 17]                        </td><td>Galois keys</td><td>1004.6 KB              </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>4096              </td><td>[17, 17]                        </td><td>Relin keys </td><td>44.38 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>4096              </td><td>[17, 17]                        </td><td>none       </td><td>78.0 B                 </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>2048              </td><td>[20, 20]                        </td><td>all        </td><td>13.09 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>2048              </td><td>[20, 20]                        </td><td>Secret key </td><td>13.08 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>2048              </td><td>[20, 20]                        </td><td>Galois keys</td><td>532.8 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>2048              </td><td>[20, 20]                        </td><td>Relin keys </td><td>26.77 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>2048              </td><td>[20, 20]                        </td><td>none       </td><td>78.0 B                 </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>2048              </td><td>[18, 18]                        </td><td>all        </td><td>12.28 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>2048              </td><td>[18, 18]                        </td><td>Secret key </td><td>12.28 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>2048              </td><td>[18, 18]                        </td><td>Galois keys</td><td>495.29 KB              </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>2048              </td><td>[18, 18]                        </td><td>Relin keys </td><td>24.56 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>2048              </td><td>[18, 18]                        </td><td>none       </td><td>79.0 B                 </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>2048              </td><td>[16, 16]                        </td><td>all        </td><td>9.7 KB                 </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>2048              </td><td>[16, 16]                        </td><td>Secret key </td><td>9.7 KB                 </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>2048              </td><td>[16, 16]                        </td><td>Galois keys</td><td>417.21 KB              </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>2048              </td><td>[16, 16]                        </td><td>Relin keys </td><td>18.99 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>ckks       </td><td>2048              </td><td>[16, 16]                        </td><td>none       </td><td>77.0 B                 </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>all        </td><td>262.14 KB              </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>Secret key </td><td>262.13 KB              </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>Galois keys</td><td>86.88 MB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>Relin keys </td><td>3.63 MB                </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>none       </td><td>111.0 B                </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 40]    </td><td>all        </td><td>234.43 KB              </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 40]    </td><td>Secret key </td><td>234.42 KB              </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 40]    </td><td>Galois keys</td><td>66.86 MB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 40]    </td><td>Relin keys </td><td>2.78 MB                </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 40]    </td><td>none       </td><td>106.0 B                </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 21, 40]        </td><td>all        </td><td>207.9 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 21, 40]        </td><td>Secret key </td><td>207.89 KB              </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 21, 40]        </td><td>Galois keys</td><td>49.34 MB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 21, 40]        </td><td>Relin keys </td><td>2.06 MB                </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 21, 40]        </td><td>none       </td><td>104.0 B                </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 40]            </td><td>all        </td><td>179.47 KB              </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 40]            </td><td>Secret key </td><td>179.47 KB              </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 40]            </td><td>Galois keys</td><td>34.35 MB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 40]            </td><td>Relin keys </td><td>1.42 MB                </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 40]            </td><td>none       </td><td>100.0 B                </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 40]                </td><td>all        </td><td>153.54 KB              </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 40]                </td><td>Secret key </td><td>153.54 KB              </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 40]                </td><td>Galois keys</td><td>21.89 MB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 40]                </td><td>Relin keys </td><td>935.94 KB              </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 40]                </td><td>none       </td><td>92.0 B                 </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>8192              </td><td>[40, 20, 40]                    </td><td>all        </td><td>124.72 KB              </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>8192              </td><td>[40, 20, 40]                    </td><td>Secret key </td><td>124.72 KB              </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>8192              </td><td>[40, 20, 40]                    </td><td>Galois keys</td><td>11.89 MB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>8192              </td><td>[40, 20, 40]                    </td><td>Relin keys </td><td>504.31 KB              </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>8192              </td><td>[40, 20, 40]                    </td><td>none       </td><td>91.0 B                 </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>4096              </td><td>[40, 20, 40]                    </td><td>all        </td><td>62.7 KB                </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>4096              </td><td>[40, 20, 40]                    </td><td>Secret key </td><td>62.69 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>4096              </td><td>[40, 20, 40]                    </td><td>Galois keys</td><td>5.48 MB                </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>4096              </td><td>[40, 20, 40]                    </td><td>Relin keys </td><td>253.45 KB              </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>4096              </td><td>[40, 20, 40]                    </td><td>none       </td><td>91.0 B                 </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>4096              </td><td>[30, 20, 30]                    </td><td>all        </td><td>51.11 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>4096              </td><td>[30, 20, 30]                    </td><td>Secret key </td><td>51.11 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>4096              </td><td>[30, 20, 30]                    </td><td>Galois keys</td><td>4.53 MB                </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>4096              </td><td>[30, 20, 30]                    </td><td>Relin keys </td><td>208.37 KB              </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>4096              </td><td>[30, 20, 30]                    </td><td>none       </td><td>93.0 B                 </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>4096              </td><td>[20, 20, 20]                    </td><td>all        </td><td>40.18 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>4096              </td><td>[20, 20, 20]                    </td><td>Secret key </td><td>40.17 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>4096              </td><td>[20, 20, 20]                    </td><td>Galois keys</td><td>3.44 MB                </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>4096              </td><td>[20, 20, 20]                    </td><td>Relin keys </td><td>159.44 KB              </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>4096              </td><td>[20, 20, 20]                    </td><td>none       </td><td>87.0 B                 </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>4096              </td><td>[19, 19, 19]                    </td><td>all        </td><td>37.83 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>4096              </td><td>[19, 19, 19]                    </td><td>Secret key </td><td>37.83 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>4096              </td><td>[19, 19, 19]                    </td><td>Galois keys</td><td>3.3 MB                 </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>4096              </td><td>[19, 19, 19]                    </td><td>Relin keys </td><td>151.05 KB              </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>4096              </td><td>[19, 19, 19]                    </td><td>none       </td><td>87.0 B                 </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>4096              </td><td>[18, 18, 18]                    </td><td>all        </td><td>35.6 KB                </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>4096              </td><td>[18, 18, 18]                    </td><td>Secret key </td><td>35.59 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>4096              </td><td>[18, 18, 18]                    </td><td>Galois keys</td><td>3.15 MB                </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>4096              </td><td>[18, 18, 18]                    </td><td>Relin keys </td><td>143.31 KB              </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>4096              </td><td>[18, 18, 18]                    </td><td>none       </td><td>86.0 B                 </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>2048              </td><td>[20, 20]                        </td><td>all        </td><td>13.12 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>2048              </td><td>[20, 20]                        </td><td>Secret key </td><td>13.11 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>2048              </td><td>[20, 20]                        </td><td>Galois keys</td><td>533.05 KB              </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>2048              </td><td>[20, 20]                        </td><td>Relin keys </td><td>26.65 KB               </td></tr>\n",
              "<tr><td>symmetric      </td><td>bfv        </td><td>2048              </td><td>[20, 20]                        </td><td>none       </td><td>83.0 B                 </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>all        </td><td>791.27 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>Public key </td><td>529.31 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>Secret key </td><td>262.06 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>Galois keys</td><td>86.88 MB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>Relin keys </td><td>3.63 MB                </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>none       </td><td>110.0 B                </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>8192              </td><td>[40, 20, 40]                    </td><td>all        </td><td>375.34 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>8192              </td><td>[40, 20, 40]                    </td><td>Public key </td><td>250.57 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>8192              </td><td>[40, 20, 40]                    </td><td>Secret key </td><td>124.86 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>8192              </td><td>[40, 20, 40]                    </td><td>Galois keys</td><td>11.89 MB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>8192              </td><td>[40, 20, 40]                    </td><td>Relin keys </td><td>504.5 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>8192              </td><td>[40, 20, 40]                    </td><td>none       </td><td>90.0 B                 </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>8192              </td><td>[20, 20, 20]                    </td><td>all        </td><td>237.93 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>8192              </td><td>[20, 20, 20]                    </td><td>Public key </td><td>158.62 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>8192              </td><td>[20, 20, 20]                    </td><td>Secret key </td><td>79.39 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>8192              </td><td>[20, 20, 20]                    </td><td>Galois keys</td><td>7.48 MB                </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>8192              </td><td>[20, 20, 20]                    </td><td>Relin keys </td><td>317.54 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>8192              </td><td>[20, 20, 20]                    </td><td>none       </td><td>81.0 B                 </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>8192              </td><td>[17, 17]                        </td><td>all        </td><td>130.28 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>8192              </td><td>[17, 17]                        </td><td>Public key </td><td>87.04 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>8192              </td><td>[17, 17]                        </td><td>Secret key </td><td>43.3 KB                </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>8192              </td><td>[17, 17]                        </td><td>Galois keys</td><td>2.15 MB                </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>8192              </td><td>[17, 17]                        </td><td>Relin keys </td><td>86.98 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>8192              </td><td>[17, 17]                        </td><td>none       </td><td>76.0 B                 </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[40, 20, 40]                    </td><td>all        </td><td>188.83 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[40, 20, 40]                    </td><td>Public key </td><td>126.17 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[40, 20, 40]                    </td><td>Secret key </td><td>62.74 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[40, 20, 40]                    </td><td>Galois keys</td><td>5.48 MB                </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[40, 20, 40]                    </td><td>Relin keys </td><td>253.2 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[40, 20, 40]                    </td><td>none       </td><td>90.0 B                 </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[30, 20, 30]                    </td><td>all        </td><td>154.41 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[30, 20, 30]                    </td><td>Public key </td><td>103.41 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[30, 20, 30]                    </td><td>Secret key </td><td>51.09 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[30, 20, 30]                    </td><td>Galois keys</td><td>4.53 MB                </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[30, 20, 30]                    </td><td>Relin keys </td><td>208.48 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[30, 20, 30]                    </td><td>none       </td><td>93.0 B                 </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[20, 20, 20]                    </td><td>all        </td><td>119.68 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[20, 20, 20]                    </td><td>Public key </td><td>79.79 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[20, 20, 20]                    </td><td>Secret key </td><td>39.97 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[20, 20, 20]                    </td><td>Galois keys</td><td>3.44 MB                </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[20, 20, 20]                    </td><td>Relin keys </td><td>159.02 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[20, 20, 20]                    </td><td>none       </td><td>80.0 B                 </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[19, 19, 19]                    </td><td>all        </td><td>112.94 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[19, 19, 19]                    </td><td>Public key </td><td>75.25 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[19, 19, 19]                    </td><td>Secret key </td><td>37.76 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[19, 19, 19]                    </td><td>Galois keys</td><td>3.3 MB                 </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[19, 19, 19]                    </td><td>Relin keys </td><td>151.02 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[19, 19, 19]                    </td><td>none       </td><td>80.0 B                 </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[18, 18, 18]                    </td><td>all        </td><td>106.75 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[18, 18, 18]                    </td><td>Public key </td><td>71.18 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[18, 18, 18]                    </td><td>Secret key </td><td>35.65 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[18, 18, 18]                    </td><td>Galois keys</td><td>3.16 MB                </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[18, 18, 18]                    </td><td>Relin keys </td><td>143.03 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[18, 18, 18]                    </td><td>none       </td><td>79.0 B                 </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[18, 18]                        </td><td>all        </td><td>71.5 KB                </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[18, 18]                        </td><td>Public key </td><td>47.55 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[18, 18]                        </td><td>Secret key </td><td>24.01 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[18, 18]                        </td><td>Galois keys</td><td>1.05 MB                </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[18, 18]                        </td><td>Relin keys </td><td>47.61 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[18, 18]                        </td><td>none       </td><td>76.0 B                 </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[17, 17]                        </td><td>all        </td><td>65.77 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[17, 17]                        </td><td>Public key </td><td>44.24 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[17, 17]                        </td><td>Secret key </td><td>21.6 KB                </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[17, 17]                        </td><td>Galois keys</td><td>1005.28 KB             </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[17, 17]                        </td><td>Relin keys </td><td>43.72 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[17, 17]                        </td><td>none       </td><td>76.0 B                 </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>2048              </td><td>[20, 20]                        </td><td>all        </td><td>39.93 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>2048              </td><td>[20, 20]                        </td><td>Public key </td><td>26.83 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>2048              </td><td>[20, 20]                        </td><td>Secret key </td><td>13.17 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>2048              </td><td>[20, 20]                        </td><td>Galois keys</td><td>532.91 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>2048              </td><td>[20, 20]                        </td><td>Relin keys </td><td>26.65 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>2048              </td><td>[20, 20]                        </td><td>none       </td><td>76.0 B                 </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>2048              </td><td>[18, 18]                        </td><td>all        </td><td>36.84 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>2048              </td><td>[18, 18]                        </td><td>Public key </td><td>24.59 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>2048              </td><td>[18, 18]                        </td><td>Secret key </td><td>12.32 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>2048              </td><td>[18, 18]                        </td><td>Galois keys</td><td>495.42 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>2048              </td><td>[18, 18]                        </td><td>Relin keys </td><td>24.73 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>2048              </td><td>[18, 18]                        </td><td>none       </td><td>77.0 B                 </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>2048              </td><td>[16, 16]                        </td><td>all        </td><td>28.51 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>2048              </td><td>[16, 16]                        </td><td>Public key </td><td>19.0 KB                </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>2048              </td><td>[16, 16]                        </td><td>Secret key </td><td>9.59 KB                </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>2048              </td><td>[16, 16]                        </td><td>Galois keys</td><td>417.3 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>2048              </td><td>[16, 16]                        </td><td>Relin keys </td><td>18.51 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>2048              </td><td>[16, 16]                        </td><td>none       </td><td>75.0 B                 </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>all        </td><td>790.31 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>Public key </td><td>527.71 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>Secret key </td><td>262.7 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>Galois keys</td><td>86.88 MB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>Relin keys </td><td>3.63 MB                </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>none       </td><td>109.0 B                </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 40]    </td><td>all        </td><td>707.17 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 40]    </td><td>Public key </td><td>472.09 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 40]    </td><td>Secret key </td><td>235.18 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 40]    </td><td>Galois keys</td><td>66.86 MB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 40]    </td><td>Relin keys </td><td>2.78 MB                </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 40]    </td><td>none       </td><td>104.0 B                </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 21, 40]        </td><td>all        </td><td>629.19 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 21, 40]        </td><td>Public key </td><td>421.31 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 21, 40]        </td><td>Secret key </td><td>207.98 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 21, 40]        </td><td>Galois keys</td><td>49.34 MB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 21, 40]        </td><td>Relin keys </td><td>2.06 MB                </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 21, 40]        </td><td>none       </td><td>102.0 B                </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 40]            </td><td>all        </td><td>542.17 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 40]            </td><td>Public key </td><td>361.56 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 40]            </td><td>Secret key </td><td>180.7 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 40]            </td><td>Galois keys</td><td>34.35 MB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 40]            </td><td>Relin keys </td><td>1.42 MB                </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 40]            </td><td>none       </td><td>98.0 B                 </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 40]                </td><td>all        </td><td>464.25 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 40]                </td><td>Public key </td><td>309.61 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 40]                </td><td>Secret key </td><td>154.72 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 40]                </td><td>Galois keys</td><td>21.89 MB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 40]                </td><td>Relin keys </td><td>935.89 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 40]                </td><td>none       </td><td>90.0 B                 </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>8192              </td><td>[40, 20, 40]                    </td><td>all        </td><td>376.67 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>8192              </td><td>[40, 20, 40]                    </td><td>Public key </td><td>251.82 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>8192              </td><td>[40, 20, 40]                    </td><td>Secret key </td><td>124.94 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>8192              </td><td>[40, 20, 40]                    </td><td>Galois keys</td><td>11.89 MB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>8192              </td><td>[40, 20, 40]                    </td><td>Relin keys </td><td>504.9 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>8192              </td><td>[40, 20, 40]                    </td><td>none       </td><td>89.0 B                 </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>4096              </td><td>[40, 20, 40]                    </td><td>all        </td><td>188.65 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>4096              </td><td>[40, 20, 40]                    </td><td>Public key </td><td>126.16 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>4096              </td><td>[40, 20, 40]                    </td><td>Secret key </td><td>62.57 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>4096              </td><td>[40, 20, 40]                    </td><td>Galois keys</td><td>5.48 MB                </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>4096              </td><td>[40, 20, 40]                    </td><td>Relin keys </td><td>253.61 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>4096              </td><td>[40, 20, 40]                    </td><td>none       </td><td>89.0 B                 </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>4096              </td><td>[30, 20, 30]                    </td><td>all        </td><td>154.32 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>4096              </td><td>[30, 20, 30]                    </td><td>Public key </td><td>103.4 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>4096              </td><td>[30, 20, 30]                    </td><td>Secret key </td><td>51.01 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>4096              </td><td>[30, 20, 30]                    </td><td>Galois keys</td><td>4.53 MB                </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>4096              </td><td>[30, 20, 30]                    </td><td>Relin keys </td><td>208.46 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>4096              </td><td>[30, 20, 30]                    </td><td>none       </td><td>91.0 B                 </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>4096              </td><td>[20, 20, 20]                    </td><td>all        </td><td>119.96 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>4096              </td><td>[20, 20, 20]                    </td><td>Public key </td><td>79.89 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>4096              </td><td>[20, 20, 20]                    </td><td>Secret key </td><td>40.15 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>4096              </td><td>[20, 20, 20]                    </td><td>Galois keys</td><td>3.44 MB                </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>4096              </td><td>[20, 20, 20]                    </td><td>Relin keys </td><td>159.2 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>4096              </td><td>[20, 20, 20]                    </td><td>none       </td><td>85.0 B                 </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>4096              </td><td>[19, 19, 19]                    </td><td>all        </td><td>113.01 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>4096              </td><td>[19, 19, 19]                    </td><td>Public key </td><td>75.29 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>4096              </td><td>[19, 19, 19]                    </td><td>Secret key </td><td>37.81 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>4096              </td><td>[19, 19, 19]                    </td><td>Galois keys</td><td>3.3 MB                 </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>4096              </td><td>[19, 19, 19]                    </td><td>Relin keys </td><td>150.98 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>4096              </td><td>[19, 19, 19]                    </td><td>none       </td><td>85.0 B                 </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>4096              </td><td>[18, 18, 18]                    </td><td>all        </td><td>106.66 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>4096              </td><td>[18, 18, 18]                    </td><td>Public key </td><td>71.23 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>4096              </td><td>[18, 18, 18]                    </td><td>Secret key </td><td>35.51 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>4096              </td><td>[18, 18, 18]                    </td><td>Galois keys</td><td>3.15 MB                </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>4096              </td><td>[18, 18, 18]                    </td><td>Relin keys </td><td>143.01 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>4096              </td><td>[18, 18, 18]                    </td><td>none       </td><td>84.0 B                 </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>2048              </td><td>[20, 20]                        </td><td>all        </td><td>39.68 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>2048              </td><td>[20, 20]                        </td><td>Public key </td><td>26.55 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>2048              </td><td>[20, 20]                        </td><td>Secret key </td><td>13.2 KB                </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>2048              </td><td>[20, 20]                        </td><td>Galois keys</td><td>533.02 KB              </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>2048              </td><td>[20, 20]                        </td><td>Relin keys </td><td>26.71 KB               </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>2048              </td><td>[20, 20]                        </td><td>none       </td><td>81.0 B                 </td></tr>\n",
              "</tbody>\n",
              "</table>"
            ],
            "text/plain": [
              "<IPython.core.display.HTML object>"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "#@title Context serialization { display-mode: \"form\" }\n",
        "#@markdown The TenSEAL context is required for defining the perfomance and security of your application.\n",
        "#@markdown \n",
        "#@markdown In a client-server setup, it is required only on the initial handshake, as the ciphertexts can be linked with an existing context on deserialization.\n",
        "#@markdown \n",
        "#@markdown Here we are benchmarking the size of the context, depending on the input parameters.\n",
        "\n",
        "ctx_size_benchmarks = [[\"Encryption Type\", \"Scheme Type\", \"Polynomial modulus\", \"Coefficient modulus sizes\", \"Saved keys\", \"Context serialized size\", ]]\n",
        "\n",
        "for enc_type in [ts.ENCRYPTION_TYPE.SYMMETRIC, ts.ENCRYPTION_TYPE.ASYMMETRIC]:\n",
        "    for (poly_mod, coeff_mod_bit_sizes) in [\n",
        "        (8192, [40, 21, 21, 21, 21, 21, 21, 40]),\n",
        "        (8192, [40, 20, 40]),\n",
        "        (8192, [20, 20, 20]),\n",
        "        (8192, [17, 17]),\n",
        "        (4096, [40, 20, 40]),\n",
        "        (4096, [30, 20, 30]),\n",
        "        (4096, [20, 20, 20]),\n",
        "        (4096, [19, 19, 19]),\n",
        "        (4096, [18, 18, 18]),\n",
        "        (4096, [18, 18]),\n",
        "        (4096, [17, 17]),\n",
        "        (2048, [20, 20]),\n",
        "        (2048, [18, 18]),\n",
        "        (2048, [16, 16]),\n",
        "    ]:\n",
        "        context = ts.context(\n",
        "            scheme=ts.SCHEME_TYPE.CKKS,\n",
        "            poly_modulus_degree=poly_mod,\n",
        "            coeff_mod_bit_sizes=coeff_mod_bit_sizes,\n",
        "            encryption_type=enc_type,\n",
        "        )\n",
        "        context.generate_galois_keys()\n",
        "        context.generate_relin_keys()\n",
        "        \n",
        "        ser = context.serialize(save_public_key=True, save_secret_key=True, save_galois_keys=True, save_relin_keys=True)\n",
        "        ctx_size_benchmarks.append([enc_type_str[enc_type], scheme_str[ts.SCHEME_TYPE.CKKS], poly_mod, coeff_mod_bit_sizes, \"all\", convert_size(len(ser))])\n",
        "        \n",
        "        if enc_type is ts.ENCRYPTION_TYPE.ASYMMETRIC:\n",
        "            ser = context.serialize(save_public_key=True, save_secret_key=False, save_galois_keys=False, save_relin_keys=False)\n",
        "            ctx_size_benchmarks.append([enc_type_str[enc_type], scheme_str[ts.SCHEME_TYPE.CKKS], poly_mod, coeff_mod_bit_sizes, \"Public key\", convert_size(len(ser))])\n",
        "  \n",
        "        ser = context.serialize(save_public_key=False, save_secret_key=True, save_galois_keys=False, save_relin_keys=False)\n",
        "        ctx_size_benchmarks.append([enc_type_str[enc_type], scheme_str[ts.SCHEME_TYPE.CKKS], poly_mod, coeff_mod_bit_sizes, \"Secret key\",  convert_size(len(ser))])\n",
        "                                    \n",
        "        ser = context.serialize(save_public_key=False, save_secret_key=False, save_galois_keys=True, save_relin_keys=False)\n",
        "        ctx_size_benchmarks.append([enc_type_str[enc_type], scheme_str[ts.SCHEME_TYPE.CKKS], poly_mod, coeff_mod_bit_sizes, \"Galois keys\",  convert_size(len(ser))])\n",
        "                                                                      \n",
        "        ser = context.serialize(save_public_key=False, save_secret_key=False, save_galois_keys=False, save_relin_keys=True)\n",
        "        ctx_size_benchmarks.append([enc_type_str[enc_type], scheme_str[ts.SCHEME_TYPE.CKKS], poly_mod, coeff_mod_bit_sizes, \"Relin keys\",  convert_size(len(ser))])\n",
        "  \n",
        "        ser = context.serialize(save_public_key=False, save_secret_key=False, save_galois_keys=False, save_relin_keys=False)\n",
        "        ctx_size_benchmarks.append([enc_type_str[enc_type], scheme_str[ts.SCHEME_TYPE.CKKS], poly_mod, coeff_mod_bit_sizes, \"none\", convert_size(len(ser))])\n",
        "        \n",
        "    for (poly_mod, coeff_mod_bit_sizes) in [\n",
        "        (8192, [40, 21, 21, 21, 21, 21, 21, 40]),\n",
        "        (8192, [40, 21, 21, 21, 21, 21, 40]),\n",
        "        (8192, [40, 21, 21, 21, 21, 40]),\n",
        "        (8192, [40, 21, 21, 21, 40]),\n",
        "        (8192, [40, 21, 21, 40]),\n",
        "        (8192, [40, 20, 40]),\n",
        "        (4096, [40, 20, 40]),\n",
        "        (4096, [30, 20, 30]),\n",
        "        (4096, [20, 20, 20]),\n",
        "        (4096, [19, 19, 19]),\n",
        "        (4096, [18, 18, 18]),\n",
        "        (2048, [20, 20]),\n",
        "    ]:\n",
        "        context = ts.context(\n",
        "            scheme=ts.SCHEME_TYPE.BFV,\n",
        "            poly_modulus_degree=poly_mod,\n",
        "            plain_modulus=786433,\n",
        "            coeff_mod_bit_sizes=coeff_mod_bit_sizes,\n",
        "            encryption_type=enc_type,\n",
        "        )\n",
        "        \n",
        "        context.generate_galois_keys()\n",
        "        context.generate_relin_keys()\n",
        "        \n",
        "        ser = context.serialize(save_public_key=True, save_secret_key=True, save_galois_keys=True, save_relin_keys=True)\n",
        "        ctx_size_benchmarks.append([enc_type_str[enc_type], scheme_str[ts.SCHEME_TYPE.BFV], poly_mod, coeff_mod_bit_sizes, \"all\", convert_size(len(ser))])\n",
        "        \n",
        "        if enc_type is ts.ENCRYPTION_TYPE.ASYMMETRIC:\n",
        "            ser = context.serialize(save_public_key=True, save_secret_key=False, save_galois_keys=False, save_relin_keys=False)\n",
        "            ctx_size_benchmarks.append([enc_type_str[enc_type], scheme_str[ts.SCHEME_TYPE.BFV], poly_mod, coeff_mod_bit_sizes, \"Public key\", convert_size(len(ser))])\n",
        "  \n",
        "        ser = context.serialize(save_public_key=False, save_secret_key=True, save_galois_keys=False, save_relin_keys=False)\n",
        "        ctx_size_benchmarks.append([enc_type_str[enc_type], scheme_str[ts.SCHEME_TYPE.BFV], poly_mod, coeff_mod_bit_sizes, \"Secret key\",  convert_size(len(ser))])\n",
        "                                    \n",
        "        ser = context.serialize(save_public_key=False, save_secret_key=False, save_galois_keys=True, save_relin_keys=False)\n",
        "        ctx_size_benchmarks.append([enc_type_str[enc_type], scheme_str[ts.SCHEME_TYPE.BFV], poly_mod, coeff_mod_bit_sizes, \"Galois keys\",  convert_size(len(ser))])\n",
        "                                                                      \n",
        "        ser = context.serialize(save_public_key=False, save_secret_key=False, save_galois_keys=False, save_relin_keys=True)\n",
        "        ctx_size_benchmarks.append([enc_type_str[enc_type], scheme_str[ts.SCHEME_TYPE.BFV], poly_mod, coeff_mod_bit_sizes, \"Relin keys\",  convert_size(len(ser))])\n",
        "  \n",
        "        ser = context.serialize(save_public_key=False, save_secret_key=False, save_galois_keys=False, save_relin_keys=False)\n",
        "        ctx_size_benchmarks.append([enc_type_str[enc_type], scheme_str[ts.SCHEME_TYPE.BFV], poly_mod, coeff_mod_bit_sizes, \"none\", convert_size(len(ser))])\n",
        "        \n",
        "\n",
        "display(HTML(tabulate.tabulate(ctx_size_benchmarks, tablefmt='html')))"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "--w6BcLNku4C"
      },
      "source": [
        "### Context serialization interpretation\n",
        "\n",
        "Some key points here would be:\n",
        "\n",
        "* The symmetric encryption creates smaller contexts than the public key ones.\n",
        "* Decreasing the length of the coefficient modulus decreases the size of the context but also the depth of available multiplications.\n",
        "* Decreasing the coefficient modulus sizes reduces the context size, but impacts the precision as well (for CKKS).\n",
        "* Galois keys increase the context size only for public contexts (without the secret key). Send them only when you need to perform ciphertext rotations on the other end.\n",
        "* Relinearization keys increase the context size only for public contexts. Send them only when you need to perform multiplications on ciphertexts on the other end.\n",
        "* When we send the secret key, the Relinearization/Galois key can be regenerated on the other end without sending them."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 737
        },
        "id": "QbHTMCxGcT2x",
        "outputId": "836180a4-147c-4832-ba61-434f5df4bce0"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Plain data size in bytes 8.8 KB\n"
          ]
        },
        {
          "data": {
            "text/html": [
              "<table>\n",
              "<tbody>\n",
              "<tr><td>Encryption Type</td><td>Scheme Type</td><td>Polynomial modulus</td><td>Coefficient modulus sizes       </td><td>Precision</td><td>Ciphertext serialized size</td><td>Encryption increase ratio</td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>8192              </td><td>[60, 40, 60]                    </td><td>2**40    </td><td>229.8 KB                  </td><td>26.12                    </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>2**40    </td><td>427.94 KB                 </td><td>48.65                    </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>2**21    </td><td>428.65 KB                 </td><td>48.73                    </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>8192              </td><td>[40, 20, 40]                    </td><td>2**40    </td><td>153.3 KB                  </td><td>17.43                    </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>8192              </td><td>[20, 20, 20]                    </td><td>2**38    </td><td>104.52 KB                 </td><td>11.88                    </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>8192              </td><td>[60, 60]                        </td><td>2**38    </td><td>128.14 KB                 </td><td>14.57                    </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>8192              </td><td>[40, 40]                        </td><td>2**38    </td><td>92.15 KB                  </td><td>10.48                    </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>8192              </td><td>[17, 17]                        </td><td>2**15    </td><td>38.17 KB                  </td><td>4.34                     </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[40, 20, 40]                    </td><td>2**40    </td><td>78.97 KB                  </td><td>8.98                     </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[30, 20, 30]                    </td><td>2**40    </td><td>66.25 KB                  </td><td>7.53                     </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[20, 20, 20]                    </td><td>2**38    </td><td>53.5 KB                   </td><td>6.08                     </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[19, 19, 19]                    </td><td>2**35    </td><td>49.85 KB                  </td><td>5.67                     </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[18, 18, 18]                    </td><td>2**33    </td><td>47.02 KB                  </td><td>5.35                     </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[30, 30]                        </td><td>2**25    </td><td>34.23 KB                  </td><td>3.89                     </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[25, 25]                        </td><td>2**20    </td><td>30.79 KB                  </td><td>3.5                      </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[18, 18]                        </td><td>2**16    </td><td>23.74 KB                  </td><td>2.7                      </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>4096              </td><td>[17, 17]                        </td><td>2**15    </td><td>18.47 KB                  </td><td>2.1                      </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>2048              </td><td>[20, 20]                        </td><td>2**18    </td><td>13.24 KB                  </td><td>1.5                      </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>2048              </td><td>[18, 18]                        </td><td>2**16    </td><td>12.11 KB                  </td><td>1.38                     </td></tr>\n",
              "<tr><td>asymmetric     </td><td>ckks       </td><td>2048              </td><td>[16, 16]                        </td><td>2**14    </td><td>9.57 KB                   </td><td>1.09                     </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>-        </td><td>428.19 KB                 </td><td>48.67                    </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 40]    </td><td>-        </td><td>373.15 KB                 </td><td>42.42                    </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 21, 40]        </td><td>-        </td><td>318.69 KB                 </td><td>36.23                    </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 21, 40]            </td><td>-        </td><td>265.47 KB                 </td><td>30.18                    </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>8192              </td><td>[40, 21, 21, 40]                </td><td>-        </td><td>210.73 KB                 </td><td>23.96                    </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>8192              </td><td>[40, 20, 40]                    </td><td>-        </td><td>153.26 KB                 </td><td>17.42                    </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>4096              </td><td>[40, 20, 40]                    </td><td>-        </td><td>78.93 KB                  </td><td>8.97                     </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>4096              </td><td>[30, 20, 30]                    </td><td>-        </td><td>66.28 KB                  </td><td>7.53                     </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>4096              </td><td>[20, 20, 20]                    </td><td>-        </td><td>53.46 KB                  </td><td>6.08                     </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>4096              </td><td>[19, 19, 19]                    </td><td>-        </td><td>49.88 KB                  </td><td>5.67                     </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>4096              </td><td>[18, 18, 18]                    </td><td>-        </td><td>47.06 KB                  </td><td>5.35                     </td></tr>\n",
              "<tr><td>asymmetric     </td><td>bfv        </td><td>2048              </td><td>[20, 20]                        </td><td>-        </td><td>13.21 KB                  </td><td>1.5                      </td></tr>\n",
              "</tbody>\n",
              "</table>"
            ],
            "text/plain": [
              "<IPython.core.display.HTML object>"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "#@title Ciphertext serialization { display-mode: \"form\" }\n",
        "#@markdown We next review how different parameters impact the ciphertext serialization.\n",
        "#@markdown \n",
        "#@markdown The first observation here is that the symmetric/asymmetric encryption switch doesn't actually impact the size of the ciphertext, only of the context.\n",
        "#@markdown \n",
        "#@markdown We will review the benchmarks only for the asymmetric scenario.\n",
        "\n",
        "data = [random.uniform(-10, 10) for _ in range(10 ** 3)]\n",
        "network_data = pickle.dumps(data)\n",
        "print(\"Plain data size in bytes {}\".format(convert_size(len(network_data))))\n",
        "\n",
        "enc_type = ts.ENCRYPTION_TYPE.ASYMMETRIC\n",
        "ct_size_benchmarks = [[\"Encryption Type\", \"Scheme Type\", \"Polynomial modulus\", \"Coefficient modulus sizes\", \"Precision\", \"Ciphertext serialized size\", \"Encryption increase ratio\"]]\n",
        "\n",
        "\n",
        "for (poly_mod, coeff_mod_bit_sizes, prec) in [\n",
        "    (8192, [60, 40, 60], 40),\n",
        "    (8192, [40, 21, 21, 21, 21, 21, 21, 40], 40),\n",
        "    (8192, [40, 21, 21, 21, 21, 21, 21, 40], 21),\n",
        "    (8192, [40, 20, 40], 40),\n",
        "    (8192, [20, 20, 20], 38),\n",
        "    (8192, [60, 60], 38),\n",
        "    (8192, [40, 40], 38),\n",
        "    (8192, [17, 17], 15),\n",
        "    (4096, [40, 20, 40], 40),\n",
        "    (4096, [30, 20, 30], 40),\n",
        "    (4096, [20, 20, 20], 38),\n",
        "    (4096, [19, 19, 19], 35),\n",
        "    (4096, [18, 18, 18], 33),\n",
        "    (4096, [30, 30], 25),\n",
        "    (4096, [25, 25], 20),\n",
        "    (4096, [18, 18], 16),\n",
        "    (4096, [17, 17], 15),\n",
        "    (2048, [20, 20], 18),\n",
        "    (2048, [18, 18], 16),\n",
        "    (2048, [16, 16], 14),\n",
        "]:\n",
        "    context = ts.context(\n",
        "        scheme=ts.SCHEME_TYPE.CKKS,\n",
        "        poly_modulus_degree=poly_mod,\n",
        "        coeff_mod_bit_sizes=coeff_mod_bit_sizes,\n",
        "        encryption_type=enc_type,\n",
        "    )\n",
        "    scale = 2 ** prec\n",
        "    ckks_vec = ts.ckks_vector(context, data, scale)\n",
        " \n",
        "    enc_network_data = ckks_vec.serialize()\n",
        "    ct_size_benchmarks.append([enc_type_str[enc_type], scheme_str[ts.SCHEME_TYPE.CKKS], poly_mod, coeff_mod_bit_sizes, \"2**{}\".format(prec),convert_size(len(enc_network_data)), round(len(enc_network_data) / len(network_data), 2)])\n",
        "\n",
        "for (poly_mod, coeff_mod_bit_sizes) in [\n",
        "    (8192, [40, 21, 21, 21, 21, 21, 21, 40]),\n",
        "    (8192, [40, 21, 21, 21, 21, 21, 40]),\n",
        "    (8192, [40, 21, 21, 21, 21, 40]),\n",
        "    (8192, [40, 21, 21, 21, 40]),\n",
        "    (8192, [40, 21, 21, 40]),\n",
        "    (8192, [40, 20, 40]),\n",
        "    (4096, [40, 20, 40]),\n",
        "    (4096, [30, 20, 30]),\n",
        "    (4096, [20, 20, 20]),\n",
        "    (4096, [19, 19, 19]),\n",
        "    (4096, [18, 18, 18]),\n",
        "    (2048, [20, 20]),\n",
        "]:\n",
        "    context = ts.context(\n",
        "        scheme=ts.SCHEME_TYPE.BFV,\n",
        "        poly_modulus_degree=poly_mod,\n",
        "        plain_modulus=786433,\n",
        "        coeff_mod_bit_sizes=coeff_mod_bit_sizes,\n",
        "        encryption_type=enc_type,\n",
        "    )\n",
        "    vec = ts.bfv_vector(context, data)\n",
        "    enc_network_data = vec.serialize()\n",
        "    ct_size_benchmarks.append([enc_type_str[enc_type], scheme_str[ts.SCHEME_TYPE.BFV], poly_mod, coeff_mod_bit_sizes, \"-\",convert_size(len(enc_network_data)), round(len(enc_network_data) / len(network_data), 2)])\n",
        "\n",
        "    \n",
        "display(HTML(tabulate.tabulate(ct_size_benchmarks, tablefmt='html')))"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "a-hgDaoMl52L"
      },
      "source": [
        "### Ciphertext serialization interpretation\n",
        "Some observations are:\n",
        "* The polynomial modulus increase results in a ciphertext increase.\n",
        "* The length of coefficient modulus sizes impacts the ciphertext size.\n",
        "* The values of the coefficient modulus sizes impact the ciphertext size, as well as the precision.\n",
        "* For a fixed set of polynomial modulus and coefficient modulus sizes, changing the precision doesn't affect the ciphertext size."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 1000
        },
        "id": "R1x9RR-7cT0t",
        "outputId": "15f5c163-0e07-4b81-ae7e-be8044da3102"
      },
      "outputs": [
        {
          "data": {
            "text/html": [
              "<table>\n",
              "<tbody>\n",
              "<tr><td>Value range  </td><td>Polynomial modulus</td><td>Coefficient modulus sizes       </td><td>Precision</td><td>Operation</td><td>Status                  </td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>8192              </td><td>[60, 40, 60]                    </td><td>2**40    </td><td>encrypt  </td><td>decryption prec 2 ** -30</td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>8192              </td><td>[60, 40, 60]                    </td><td>2**40    </td><td>sum      </td><td>decryption prec 2 ** -29</td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>2**40    </td><td>encrypt  </td><td>decryption prec 2 ** -29</td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>2**40    </td><td>sum      </td><td>decryption prec 2 ** -28</td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>2**21    </td><td>encrypt  </td><td>decryption prec 2 ** -15</td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>2**21    </td><td>sum      </td><td>decryption prec 2 ** -14</td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>8192              </td><td>[40, 20, 40]                    </td><td>2**40    </td><td>encrypt  </td><td>decryption prec 2 ** -31</td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>8192              </td><td>[40, 20, 40]                    </td><td>2**40    </td><td>sum      </td><td>decryption prec 2 ** -30</td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>8192              </td><td>[20, 20, 20]                    </td><td>2**38    </td><td>encrypt  </td><td>decryption prec 2 ** -27</td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>8192              </td><td>[60, 60]                        </td><td>2**38    </td><td>encrypt  </td><td>decryption prec 2 ** -26</td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>8192              </td><td>[60, 60]                        </td><td>2**38    </td><td>sum      </td><td>decryption prec 2 ** -25</td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>8192              </td><td>[40, 40]                        </td><td>2**38    </td><td>encrypt  </td><td>decryption prec 2 ** -28</td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>8192              </td><td>[40, 40]                        </td><td>2**38    </td><td>sum      </td><td>decryption prec 2 ** -27</td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>8192              </td><td>[17, 17]                        </td><td>2**15    </td><td>encrypt  </td><td>decryption prec 2 ** -7 </td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>4096              </td><td>[40, 20, 40]                    </td><td>2**40    </td><td>encrypt  </td><td>decryption prec 2 ** -32</td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>4096              </td><td>[40, 20, 40]                    </td><td>2**40    </td><td>sum      </td><td>decryption prec 2 ** -31</td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>4096              </td><td>[30, 20, 30]                    </td><td>2**40    </td><td>encrypt  </td><td>decryption prec 2 ** -31</td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>4096              </td><td>[30, 20, 30]                    </td><td>2**40    </td><td>sum      </td><td>decryption prec 2 ** -30</td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>4096              </td><td>[20, 20, 20]                    </td><td>2**38    </td><td>encrypt  </td><td>decryption prec 2 ** -28</td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>4096              </td><td>[20, 20, 20]                    </td><td>2**38    </td><td>sum      </td><td>decryption prec 2 ** -27</td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>4096              </td><td>[19, 19, 19]                    </td><td>2**35    </td><td>encrypt  </td><td>decryption prec 2 ** -28</td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>4096              </td><td>[19, 19, 19]                    </td><td>2**35    </td><td>sum      </td><td>decryption prec 2 ** -27</td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>4096              </td><td>[18, 18, 18]                    </td><td>2**33    </td><td>encrypt  </td><td>decryption prec 2 ** -25</td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>4096              </td><td>[18, 18, 18]                    </td><td>2**33    </td><td>sum      </td><td>decryption prec 2 ** -24</td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>4096              </td><td>[30, 30]                        </td><td>2**25    </td><td>encrypt  </td><td>decryption prec 2 ** -15</td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>4096              </td><td>[30, 30]                        </td><td>2**25    </td><td>sum      </td><td>decryption prec 2 ** -14</td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>4096              </td><td>[25, 25]                        </td><td>2**20    </td><td>encrypt  </td><td>decryption prec 2 ** -10</td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>4096              </td><td>[25, 25]                        </td><td>2**20    </td><td>sum      </td><td>decryption prec 2 ** -9 </td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>4096              </td><td>[18, 18]                        </td><td>2**16    </td><td>encrypt  </td><td>decryption prec 2 ** -6 </td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>4096              </td><td>[18, 18]                        </td><td>2**16    </td><td>sum      </td><td>decryption prec 2 ** -5 </td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>4096              </td><td>[17, 17]                        </td><td>2**15    </td><td>encrypt  </td><td>decryption prec 2 ** -8 </td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>2048              </td><td>[20, 20]                        </td><td>2**18    </td><td>encrypt  </td><td>decryption prec 2 ** -10</td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>2048              </td><td>[20, 20]                        </td><td>2**18    </td><td>sum      </td><td>decryption prec 2 ** -9 </td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>2048              </td><td>[18, 18]                        </td><td>2**16    </td><td>encrypt  </td><td>decryption prec 2 ** -10</td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>2048              </td><td>[18, 18]                        </td><td>2**16    </td><td>sum      </td><td>decryption prec 2 ** -9 </td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>2048              </td><td>[16, 16]                        </td><td>2**14    </td><td>encrypt  </td><td>decryption prec 2 ** -6 </td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>2048              </td><td>[16, 16]                        </td><td>2**14    </td><td>sum      </td><td>decryption prec 2 ** -5 </td></tr>\n",
              "<tr><td>[2^0 - 2^1]  </td><td>8192              </td><td>[60, 40, 60]                    </td><td>2**40    </td><td>encrypt  </td><td>decryption prec 2 ** -28</td></tr>\n",
              "<tr><td>[2^0 - 2^1]  </td><td>8192              </td><td>[60, 40, 60]                    </td><td>2**40    </td><td>sum      </td><td>decryption prec 2 ** -27</td></tr>\n",
              "<tr><td>[2^0 - 2^1]  </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>2**40    </td><td>encrypt  </td><td>decryption prec 2 ** -30</td></tr>\n",
              "<tr><td>[2^0 - 2^1]  </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>2**40    </td><td>sum      </td><td>decryption prec 2 ** -29</td></tr>\n",
              "<tr><td>[2^0 - 2^1]  </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>2**21    </td><td>encrypt  </td><td>decryption prec 2 ** -15</td></tr>\n",
              "<tr><td>[2^0 - 2^1]  </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>2**21    </td><td>sum      </td><td>decryption prec 2 ** -14</td></tr>\n",
              "<tr><td>[2^0 - 2^1]  </td><td>8192              </td><td>[40, 20, 40]                    </td><td>2**40    </td><td>encrypt  </td><td>decryption prec 2 ** -28</td></tr>\n",
              "<tr><td>[2^0 - 2^1]  </td><td>8192              </td><td>[40, 20, 40]                    </td><td>2**40    </td><td>sum      </td><td>decryption prec 2 ** -27</td></tr>\n",
              "<tr><td>[2^0 - 2^1]  </td><td>8192              </td><td>[20, 20, 20]                    </td><td>2**38    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^0 - 2^1]  </td><td>8192              </td><td>[60, 60]                        </td><td>2**38    </td><td>encrypt  </td><td>decryption prec 2 ** -26</td></tr>\n",
              "<tr><td>[2^0 - 2^1]  </td><td>8192              </td><td>[60, 60]                        </td><td>2**38    </td><td>sum      </td><td>decryption prec 2 ** -25</td></tr>\n",
              "<tr><td>[2^0 - 2^1]  </td><td>8192              </td><td>[40, 40]                        </td><td>2**38    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^0 - 2^1]  </td><td>8192              </td><td>[17, 17]                        </td><td>2**15    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^0 - 2^1]  </td><td>4096              </td><td>[40, 20, 40]                    </td><td>2**40    </td><td>encrypt  </td><td>decryption prec 2 ** -38</td></tr>\n",
              "<tr><td>[2^0 - 2^1]  </td><td>4096              </td><td>[40, 20, 40]                    </td><td>2**40    </td><td>sum      </td><td>decryption prec 2 ** -37</td></tr>\n",
              "<tr><td>[2^0 - 2^1]  </td><td>4096              </td><td>[30, 20, 30]                    </td><td>2**40    </td><td>encrypt  </td><td>decryption prec 2 ** -31</td></tr>\n",
              "<tr><td>[2^0 - 2^1]  </td><td>4096              </td><td>[30, 20, 30]                    </td><td>2**40    </td><td>sum      </td><td>decryption prec 2 ** -30</td></tr>\n",
              "<tr><td>[2^0 - 2^1]  </td><td>4096              </td><td>[20, 20, 20]                    </td><td>2**38    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^0 - 2^1]  </td><td>4096              </td><td>[19, 19, 19]                    </td><td>2**35    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^0 - 2^1]  </td><td>4096              </td><td>[18, 18, 18]                    </td><td>2**33    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^0 - 2^1]  </td><td>4096              </td><td>[30, 30]                        </td><td>2**25    </td><td>encrypt  </td><td>decryption prec 2 ** -15</td></tr>\n",
              "<tr><td>[2^0 - 2^1]  </td><td>4096              </td><td>[30, 30]                        </td><td>2**25    </td><td>sum      </td><td>decryption prec 2 ** -14</td></tr>\n",
              "<tr><td>[2^0 - 2^1]  </td><td>4096              </td><td>[25, 25]                        </td><td>2**20    </td><td>encrypt  </td><td>decryption prec 2 ** -12</td></tr>\n",
              "<tr><td>[2^0 - 2^1]  </td><td>4096              </td><td>[25, 25]                        </td><td>2**20    </td><td>sum      </td><td>decryption prec 2 ** -11</td></tr>\n",
              "<tr><td>[2^0 - 2^1]  </td><td>4096              </td><td>[18, 18]                        </td><td>2**16    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^0 - 2^1]  </td><td>4096              </td><td>[17, 17]                        </td><td>2**15    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^0 - 2^1]  </td><td>2048              </td><td>[20, 20]                        </td><td>2**18    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^0 - 2^1]  </td><td>2048              </td><td>[18, 18]                        </td><td>2**16    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^0 - 2^1]  </td><td>2048              </td><td>[16, 16]                        </td><td>2**14    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^1 - 2^2]  </td><td>8192              </td><td>[60, 40, 60]                    </td><td>2**40    </td><td>encrypt  </td><td>decryption prec 2 ** -31</td></tr>\n",
              "<tr><td>[2^1 - 2^2]  </td><td>8192              </td><td>[60, 40, 60]                    </td><td>2**40    </td><td>sum      </td><td>decryption prec 2 ** -30</td></tr>\n",
              "<tr><td>[2^1 - 2^2]  </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>2**40    </td><td>encrypt  </td><td>decryption prec 2 ** -28</td></tr>\n",
              "<tr><td>[2^1 - 2^2]  </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>2**40    </td><td>sum      </td><td>decryption prec 2 ** -27</td></tr>\n",
              "<tr><td>[2^1 - 2^2]  </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>2**21    </td><td>encrypt  </td><td>decryption prec 2 ** -11</td></tr>\n",
              "<tr><td>[2^1 - 2^2]  </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>2**21    </td><td>sum      </td><td>decryption prec 2 ** -10</td></tr>\n",
              "<tr><td>[2^1 - 2^2]  </td><td>8192              </td><td>[40, 20, 40]                    </td><td>2**40    </td><td>encrypt  </td><td>decryption prec 2 ** -30</td></tr>\n",
              "<tr><td>[2^1 - 2^2]  </td><td>8192              </td><td>[40, 20, 40]                    </td><td>2**40    </td><td>sum      </td><td>decryption prec 2 ** -29</td></tr>\n",
              "<tr><td>[2^1 - 2^2]  </td><td>8192              </td><td>[20, 20, 20]                    </td><td>2**38    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^1 - 2^2]  </td><td>8192              </td><td>[60, 60]                        </td><td>2**38    </td><td>encrypt  </td><td>decryption prec 2 ** -29</td></tr>\n",
              "<tr><td>[2^1 - 2^2]  </td><td>8192              </td><td>[60, 60]                        </td><td>2**38    </td><td>sum      </td><td>decryption prec 2 ** -28</td></tr>\n",
              "<tr><td>[2^1 - 2^2]  </td><td>8192              </td><td>[40, 40]                        </td><td>2**38    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^1 - 2^2]  </td><td>8192              </td><td>[17, 17]                        </td><td>2**15    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^1 - 2^2]  </td><td>4096              </td><td>[40, 20, 40]                    </td><td>2**40    </td><td>encrypt  </td><td>decryption prec 2 ** -34</td></tr>\n",
              "<tr><td>[2^1 - 2^2]  </td><td>4096              </td><td>[40, 20, 40]                    </td><td>2**40    </td><td>sum      </td><td>decryption prec 2 ** -33</td></tr>\n",
              "<tr><td>[2^1 - 2^2]  </td><td>4096              </td><td>[30, 20, 30]                    </td><td>2**40    </td><td>encrypt  </td><td>decryption prec 2 ** -34</td></tr>\n",
              "<tr><td>[2^1 - 2^2]  </td><td>4096              </td><td>[30, 20, 30]                    </td><td>2**40    </td><td>sum      </td><td>decryption prec 2 ** -33</td></tr>\n",
              "<tr><td>[2^1 - 2^2]  </td><td>4096              </td><td>[20, 20, 20]                    </td><td>2**38    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^1 - 2^2]  </td><td>4096              </td><td>[19, 19, 19]                    </td><td>2**35    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^1 - 2^2]  </td><td>4096              </td><td>[18, 18, 18]                    </td><td>2**33    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^1 - 2^2]  </td><td>4096              </td><td>[30, 30]                        </td><td>2**25    </td><td>encrypt  </td><td>decryption prec 2 ** -17</td></tr>\n",
              "<tr><td>[2^1 - 2^2]  </td><td>4096              </td><td>[30, 30]                        </td><td>2**25    </td><td>sum      </td><td>decryption prec 2 ** -16</td></tr>\n",
              "<tr><td>[2^1 - 2^2]  </td><td>4096              </td><td>[25, 25]                        </td><td>2**20    </td><td>encrypt  </td><td>decryption prec 2 ** -11</td></tr>\n",
              "<tr><td>[2^1 - 2^2]  </td><td>4096              </td><td>[25, 25]                        </td><td>2**20    </td><td>sum      </td><td>decryption prec 2 ** -10</td></tr>\n",
              "<tr><td>[2^1 - 2^2]  </td><td>4096              </td><td>[18, 18]                        </td><td>2**16    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^1 - 2^2]  </td><td>4096              </td><td>[17, 17]                        </td><td>2**15    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^1 - 2^2]  </td><td>2048              </td><td>[20, 20]                        </td><td>2**18    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^1 - 2^2]  </td><td>2048              </td><td>[18, 18]                        </td><td>2**16    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^1 - 2^2]  </td><td>2048              </td><td>[16, 16]                        </td><td>2**14    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^5 - 2^6]  </td><td>8192              </td><td>[60, 40, 60]                    </td><td>2**40    </td><td>encrypt  </td><td>decryption prec 2 ** -32</td></tr>\n",
              "<tr><td>[2^5 - 2^6]  </td><td>8192              </td><td>[60, 40, 60]                    </td><td>2**40    </td><td>sum      </td><td>decryption prec 2 ** -31</td></tr>\n",
              "<tr><td>[2^5 - 2^6]  </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>2**40    </td><td>encrypt  </td><td>decryption prec 2 ** -27</td></tr>\n",
              "<tr><td>[2^5 - 2^6]  </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>2**40    </td><td>sum      </td><td>decryption prec 2 ** -26</td></tr>\n",
              "<tr><td>[2^5 - 2^6]  </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>2**21    </td><td>encrypt  </td><td>decryption prec 2 ** -11</td></tr>\n",
              "<tr><td>[2^5 - 2^6]  </td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>2**21    </td><td>sum      </td><td>decryption prec 2 ** -10</td></tr>\n",
              "<tr><td>[2^5 - 2^6]  </td><td>8192              </td><td>[40, 20, 40]                    </td><td>2**40    </td><td>encrypt  </td><td>decryption prec 2 ** -30</td></tr>\n",
              "<tr><td>[2^5 - 2^6]  </td><td>8192              </td><td>[40, 20, 40]                    </td><td>2**40    </td><td>sum      </td><td>decryption prec 2 ** -29</td></tr>\n",
              "<tr><td>[2^5 - 2^6]  </td><td>8192              </td><td>[20, 20, 20]                    </td><td>2**38    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^5 - 2^6]  </td><td>8192              </td><td>[60, 60]                        </td><td>2**38    </td><td>encrypt  </td><td>decryption prec 2 ** -29</td></tr>\n",
              "<tr><td>[2^5 - 2^6]  </td><td>8192              </td><td>[60, 60]                        </td><td>2**38    </td><td>sum      </td><td>decryption prec 2 ** -28</td></tr>\n",
              "<tr><td>[2^5 - 2^6]  </td><td>8192              </td><td>[40, 40]                        </td><td>2**38    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^5 - 2^6]  </td><td>8192              </td><td>[17, 17]                        </td><td>2**15    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^5 - 2^6]  </td><td>4096              </td><td>[40, 20, 40]                    </td><td>2**40    </td><td>encrypt  </td><td>decryption prec 2 ** -30</td></tr>\n",
              "<tr><td>[2^5 - 2^6]  </td><td>4096              </td><td>[40, 20, 40]                    </td><td>2**40    </td><td>sum      </td><td>decryption prec 2 ** -29</td></tr>\n",
              "<tr><td>[2^5 - 2^6]  </td><td>4096              </td><td>[30, 20, 30]                    </td><td>2**40    </td><td>encrypt  </td><td>decryption prec 2 ** -31</td></tr>\n",
              "<tr><td>[2^5 - 2^6]  </td><td>4096              </td><td>[30, 20, 30]                    </td><td>2**40    </td><td>sum      </td><td>decryption prec 2 ** -30</td></tr>\n",
              "<tr><td>[2^5 - 2^6]  </td><td>4096              </td><td>[20, 20, 20]                    </td><td>2**38    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^5 - 2^6]  </td><td>4096              </td><td>[19, 19, 19]                    </td><td>2**35    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^5 - 2^6]  </td><td>4096              </td><td>[18, 18, 18]                    </td><td>2**33    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^5 - 2^6]  </td><td>4096              </td><td>[30, 30]                        </td><td>2**25    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^5 - 2^6]  </td><td>4096              </td><td>[25, 25]                        </td><td>2**20    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^5 - 2^6]  </td><td>4096              </td><td>[18, 18]                        </td><td>2**16    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^5 - 2^6]  </td><td>4096              </td><td>[17, 17]                        </td><td>2**15    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^5 - 2^6]  </td><td>2048              </td><td>[20, 20]                        </td><td>2**18    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^5 - 2^6]  </td><td>2048              </td><td>[18, 18]                        </td><td>2**16    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^5 - 2^6]  </td><td>2048              </td><td>[16, 16]                        </td><td>2**14    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^11 - 2^12]</td><td>8192              </td><td>[60, 40, 60]                    </td><td>2**40    </td><td>encrypt  </td><td>decryption prec 2 ** -30</td></tr>\n",
              "<tr><td>[2^11 - 2^12]</td><td>8192              </td><td>[60, 40, 60]                    </td><td>2**40    </td><td>sum      </td><td>decryption prec 2 ** -29</td></tr>\n",
              "<tr><td>[2^11 - 2^12]</td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>2**40    </td><td>encrypt  </td><td>decryption prec 2 ** -32</td></tr>\n",
              "<tr><td>[2^11 - 2^12]</td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>2**40    </td><td>sum      </td><td>decryption prec 2 ** -31</td></tr>\n",
              "<tr><td>[2^11 - 2^12]</td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>2**21    </td><td>encrypt  </td><td>decryption prec 2 ** -10</td></tr>\n",
              "<tr><td>[2^11 - 2^12]</td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>2**21    </td><td>sum      </td><td>decryption prec 2 ** -9 </td></tr>\n",
              "<tr><td>[2^11 - 2^12]</td><td>8192              </td><td>[40, 20, 40]                    </td><td>2**40    </td><td>encrypt  </td><td>decryption prec 2 ** -31</td></tr>\n",
              "<tr><td>[2^11 - 2^12]</td><td>8192              </td><td>[40, 20, 40]                    </td><td>2**40    </td><td>sum      </td><td>decryption prec 2 ** -30</td></tr>\n",
              "<tr><td>[2^11 - 2^12]</td><td>8192              </td><td>[20, 20, 20]                    </td><td>2**38    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^11 - 2^12]</td><td>8192              </td><td>[60, 60]                        </td><td>2**38    </td><td>encrypt  </td><td>decryption prec 2 ** -30</td></tr>\n",
              "<tr><td>[2^11 - 2^12]</td><td>8192              </td><td>[60, 60]                        </td><td>2**38    </td><td>sum      </td><td>decryption prec 2 ** -29</td></tr>\n",
              "<tr><td>[2^11 - 2^12]</td><td>8192              </td><td>[40, 40]                        </td><td>2**38    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^11 - 2^12]</td><td>8192              </td><td>[17, 17]                        </td><td>2**15    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^11 - 2^12]</td><td>4096              </td><td>[40, 20, 40]                    </td><td>2**40    </td><td>encrypt  </td><td>decryption prec 2 ** -31</td></tr>\n",
              "<tr><td>[2^11 - 2^12]</td><td>4096              </td><td>[40, 20, 40]                    </td><td>2**40    </td><td>sum      </td><td>decryption prec 2 ** -30</td></tr>\n",
              "<tr><td>[2^11 - 2^12]</td><td>4096              </td><td>[30, 20, 30]                    </td><td>2**40    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^11 - 2^12]</td><td>4096              </td><td>[20, 20, 20]                    </td><td>2**38    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^11 - 2^12]</td><td>4096              </td><td>[19, 19, 19]                    </td><td>2**35    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^11 - 2^12]</td><td>4096              </td><td>[18, 18, 18]                    </td><td>2**33    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^11 - 2^12]</td><td>4096              </td><td>[30, 30]                        </td><td>2**25    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^11 - 2^12]</td><td>4096              </td><td>[25, 25]                        </td><td>2**20    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^11 - 2^12]</td><td>4096              </td><td>[18, 18]                        </td><td>2**16    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^11 - 2^12]</td><td>4096              </td><td>[17, 17]                        </td><td>2**15    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^11 - 2^12]</td><td>2048              </td><td>[20, 20]                        </td><td>2**18    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^11 - 2^12]</td><td>2048              </td><td>[18, 18]                        </td><td>2**16    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^11 - 2^12]</td><td>2048              </td><td>[16, 16]                        </td><td>2**14    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^21 - 2^22]</td><td>8192              </td><td>[60, 40, 60]                    </td><td>2**40    </td><td>encrypt  </td><td>decryption prec 2 ** -39</td></tr>\n",
              "<tr><td>[2^21 - 2^22]</td><td>8192              </td><td>[60, 40, 60]                    </td><td>2**40    </td><td>sum      </td><td>decryption prec 2 ** -39</td></tr>\n",
              "<tr><td>[2^21 - 2^22]</td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>2**40    </td><td>encrypt  </td><td>decryption prec 2 ** -31</td></tr>\n",
              "<tr><td>[2^21 - 2^22]</td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>2**40    </td><td>sum      </td><td>decryption prec 2 ** -30</td></tr>\n",
              "<tr><td>[2^21 - 2^22]</td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>2**21    </td><td>encrypt  </td><td>decryption prec 2 ** -15</td></tr>\n",
              "<tr><td>[2^21 - 2^22]</td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>2**21    </td><td>sum      </td><td>decryption prec 2 ** -14</td></tr>\n",
              "<tr><td>[2^21 - 2^22]</td><td>8192              </td><td>[40, 20, 40]                    </td><td>2**40    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^21 - 2^22]</td><td>8192              </td><td>[20, 20, 20]                    </td><td>2**38    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^21 - 2^22]</td><td>8192              </td><td>[60, 60]                        </td><td>2**38    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^21 - 2^22]</td><td>8192              </td><td>[40, 40]                        </td><td>2**38    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^21 - 2^22]</td><td>8192              </td><td>[17, 17]                        </td><td>2**15    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^21 - 2^22]</td><td>4096              </td><td>[40, 20, 40]                    </td><td>2**40    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^21 - 2^22]</td><td>4096              </td><td>[30, 20, 30]                    </td><td>2**40    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^21 - 2^22]</td><td>4096              </td><td>[20, 20, 20]                    </td><td>2**38    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^21 - 2^22]</td><td>4096              </td><td>[19, 19, 19]                    </td><td>2**35    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^21 - 2^22]</td><td>4096              </td><td>[18, 18, 18]                    </td><td>2**33    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^21 - 2^22]</td><td>4096              </td><td>[30, 30]                        </td><td>2**25    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^21 - 2^22]</td><td>4096              </td><td>[25, 25]                        </td><td>2**20    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^21 - 2^22]</td><td>4096              </td><td>[18, 18]                        </td><td>2**16    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^21 - 2^22]</td><td>4096              </td><td>[17, 17]                        </td><td>2**15    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^21 - 2^22]</td><td>2048              </td><td>[20, 20]                        </td><td>2**18    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^21 - 2^22]</td><td>2048              </td><td>[18, 18]                        </td><td>2**16    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^21 - 2^22]</td><td>2048              </td><td>[16, 16]                        </td><td>2**14    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^41 - 2^42]</td><td>8192              </td><td>[60, 40, 60]                    </td><td>2**40    </td><td>encrypt  </td><td>decryption prec 2 ** -39</td></tr>\n",
              "<tr><td>[2^41 - 2^42]</td><td>8192              </td><td>[60, 40, 60]                    </td><td>2**40    </td><td>sum      </td><td>decryption prec 2 ** -39</td></tr>\n",
              "<tr><td>[2^41 - 2^42]</td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>2**40    </td><td>encrypt  </td><td>decryption prec 2 ** -39</td></tr>\n",
              "<tr><td>[2^41 - 2^42]</td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>2**40    </td><td>sum      </td><td>decryption prec 2 ** -39</td></tr>\n",
              "<tr><td>[2^41 - 2^42]</td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>2**21    </td><td>encrypt  </td><td>decryption prec 2 ** -20</td></tr>\n",
              "<tr><td>[2^41 - 2^42]</td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>2**21    </td><td>sum      </td><td>decryption prec 2 ** -20</td></tr>\n",
              "<tr><td>[2^41 - 2^42]</td><td>8192              </td><td>[40, 20, 40]                    </td><td>2**40    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^41 - 2^42]</td><td>8192              </td><td>[20, 20, 20]                    </td><td>2**38    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^41 - 2^42]</td><td>8192              </td><td>[60, 60]                        </td><td>2**38    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^41 - 2^42]</td><td>8192              </td><td>[40, 40]                        </td><td>2**38    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^41 - 2^42]</td><td>8192              </td><td>[17, 17]                        </td><td>2**15    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^41 - 2^42]</td><td>4096              </td><td>[40, 20, 40]                    </td><td>2**40    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^41 - 2^42]</td><td>4096              </td><td>[30, 20, 30]                    </td><td>2**40    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^41 - 2^42]</td><td>4096              </td><td>[20, 20, 20]                    </td><td>2**38    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^41 - 2^42]</td><td>4096              </td><td>[19, 19, 19]                    </td><td>2**35    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^41 - 2^42]</td><td>4096              </td><td>[18, 18, 18]                    </td><td>2**33    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^41 - 2^42]</td><td>4096              </td><td>[30, 30]                        </td><td>2**25    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^41 - 2^42]</td><td>4096              </td><td>[25, 25]                        </td><td>2**20    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^41 - 2^42]</td><td>4096              </td><td>[18, 18]                        </td><td>2**16    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^41 - 2^42]</td><td>4096              </td><td>[17, 17]                        </td><td>2**15    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^41 - 2^42]</td><td>2048              </td><td>[20, 20]                        </td><td>2**18    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^41 - 2^42]</td><td>2048              </td><td>[18, 18]                        </td><td>2**16    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^41 - 2^42]</td><td>2048              </td><td>[16, 16]                        </td><td>2**14    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^51 - 2^52]</td><td>8192              </td><td>[60, 40, 60]                    </td><td>2**40    </td><td>encrypt  </td><td>decryption prec 2 ** -39</td></tr>\n",
              "<tr><td>[2^51 - 2^52]</td><td>8192              </td><td>[60, 40, 60]                    </td><td>2**40    </td><td>sum      </td><td>decryption prec 2 ** -39</td></tr>\n",
              "<tr><td>[2^51 - 2^52]</td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>2**40    </td><td>encrypt  </td><td>decryption prec 2 ** -39</td></tr>\n",
              "<tr><td>[2^51 - 2^52]</td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>2**40    </td><td>sum      </td><td>decryption prec 2 ** -39</td></tr>\n",
              "<tr><td>[2^51 - 2^52]</td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>2**21    </td><td>encrypt  </td><td>decryption prec 2 ** -20</td></tr>\n",
              "<tr><td>[2^51 - 2^52]</td><td>8192              </td><td>[40, 21, 21, 21, 21, 21, 21, 40]</td><td>2**21    </td><td>sum      </td><td>decryption prec 2 ** -20</td></tr>\n",
              "<tr><td>[2^51 - 2^52]</td><td>8192              </td><td>[40, 20, 40]                    </td><td>2**40    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^51 - 2^52]</td><td>8192              </td><td>[20, 20, 20]                    </td><td>2**38    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^51 - 2^52]</td><td>8192              </td><td>[60, 60]                        </td><td>2**38    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^51 - 2^52]</td><td>8192              </td><td>[40, 40]                        </td><td>2**38    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^51 - 2^52]</td><td>8192              </td><td>[17, 17]                        </td><td>2**15    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^51 - 2^52]</td><td>4096              </td><td>[40, 20, 40]                    </td><td>2**40    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^51 - 2^52]</td><td>4096              </td><td>[30, 20, 30]                    </td><td>2**40    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^51 - 2^52]</td><td>4096              </td><td>[20, 20, 20]                    </td><td>2**38    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^51 - 2^52]</td><td>4096              </td><td>[19, 19, 19]                    </td><td>2**35    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^51 - 2^52]</td><td>4096              </td><td>[18, 18, 18]                    </td><td>2**33    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^51 - 2^52]</td><td>4096              </td><td>[30, 30]                        </td><td>2**25    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^51 - 2^52]</td><td>4096              </td><td>[25, 25]                        </td><td>2**20    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^51 - 2^52]</td><td>4096              </td><td>[18, 18]                        </td><td>2**16    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^51 - 2^52]</td><td>4096              </td><td>[17, 17]                        </td><td>2**15    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^51 - 2^52]</td><td>2048              </td><td>[20, 20]                        </td><td>2**18    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^51 - 2^52]</td><td>2048              </td><td>[18, 18]                        </td><td>2**16    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^51 - 2^52]</td><td>2048              </td><td>[16, 16]                        </td><td>2**14    </td><td>encrypt  </td><td>encryption failed       </td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>8192              </td><td>[60, 40, 40, 60]                </td><td>2**40    </td><td>mul      </td><td>decryption prec 2 ** -23</td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>8192              </td><td>[40, 21, 21, 40]                </td><td>2**21    </td><td>mul      </td><td>decryption prec 2 ** -2 </td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>8192              </td><td>[40, 20, 20, 40]                </td><td>2**40    </td><td>mul      </td><td>failed                  </td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>8192              </td><td>[20, 20, 20]                    </td><td>2**38    </td><td>mul      </td><td>failed                  </td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>4096              </td><td>[40, 20, 40]                    </td><td>2**40    </td><td>mul      </td><td>failed                  </td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>4096              </td><td>[30, 20, 30]                    </td><td>2**40    </td><td>mul      </td><td>failed                  </td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>4096              </td><td>[20, 20, 20]                    </td><td>2**38    </td><td>mul      </td><td>failed                  </td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>4096              </td><td>[19, 19, 19]                    </td><td>2**35    </td><td>mul      </td><td>failed                  </td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>4096              </td><td>[18, 18, 18]                    </td><td>2**33    </td><td>mul      </td><td>failed                  </td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>4096              </td><td>[30, 30, 30]                    </td><td>2**25    </td><td>mul      </td><td>decryption prec 2 ** 0  </td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>4096              </td><td>[25, 25, 25]                    </td><td>2**20    </td><td>mul      </td><td>decryption prec 2 ** 0  </td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>4096              </td><td>[18, 18, 18]                    </td><td>2**16    </td><td>mul      </td><td>decryption prec 2 ** 0  </td></tr>\n",
              "<tr><td>[2^-1 - 2^0] </td><td>2048              </td><td>[18, 18, 18]                    </td><td>2**16    </td><td>mul      </td><td>decryption prec 2 ** -1 </td></tr>\n",
              "<tr><td>[2^0 - 2^1]  </td><td>8192              </td><td>[60, 40, 40, 60]                </td><td>2**40    </td><td>mul      </td><td>decryption prec 2 ** -22</td></tr>\n",
              "<tr><td>[2^0 - 2^1]  </td><td>8192              </td><td>[40, 21, 21, 40]                </td><td>2**21    </td><td>mul      </td><td>decryption prec 2 ** -2 </td></tr>\n",
              "<tr><td>[2^0 - 2^1]  </td><td>8192              </td><td>[40, 20, 20, 40]                </td><td>2**40    </td><td>mul      </td><td>failed                  </td></tr>\n",
              "<tr><td>[2^0 - 2^1]  </td><td>4096              </td><td>[40, 20, 40]                    </td><td>2**40    </td><td>mul      </td><td>failed                  </td></tr>\n",
              "<tr><td>[2^0 - 2^1]  </td><td>4096              </td><td>[30, 20, 30]                    </td><td>2**40    </td><td>mul      </td><td>failed                  </td></tr>\n",
              "<tr><td>[2^0 - 2^1]  </td><td>4096              </td><td>[18, 18, 18]                    </td><td>2**16    </td><td>mul      </td><td>decryption prec 2 ** 0  </td></tr>\n",
              "<tr><td>[2^0 - 2^1]  </td><td>2048              </td><td>[18, 18, 18]                    </td><td>2**16    </td><td>mul      </td><td>decryption prec 2 ** 0  </td></tr>\n",
              "<tr><td>[2^1 - 2^2]  </td><td>8192              </td><td>[60, 40, 40, 60]                </td><td>2**40    </td><td>mul      </td><td>decryption prec 2 ** -19</td></tr>\n",
              "<tr><td>[2^1 - 2^2]  </td><td>8192              </td><td>[40, 20, 20, 40]                </td><td>2**40    </td><td>mul      </td><td>failed                  </td></tr>\n",
              "<tr><td>[2^1 - 2^2]  </td><td>4096              </td><td>[40, 20, 40]                    </td><td>2**40    </td><td>mul      </td><td>failed                  </td></tr>\n",
              "<tr><td>[2^1 - 2^2]  </td><td>4096              </td><td>[30, 20, 30]                    </td><td>2**40    </td><td>mul      </td><td>failed                  </td></tr>\n",
              "<tr><td>[2^5 - 2^6]  </td><td>8192              </td><td>[60, 40, 40, 60]                </td><td>2**40    </td><td>mul      </td><td>decryption prec 2 ** -11</td></tr>\n",
              "<tr><td>[2^5 - 2^6]  </td><td>8192              </td><td>[40, 20, 20, 40]                </td><td>2**40    </td><td>mul      </td><td>failed                  </td></tr>\n",
              "<tr><td>[2^5 - 2^6]  </td><td>4096              </td><td>[40, 20, 40]                    </td><td>2**40    </td><td>mul      </td><td>failed                  </td></tr>\n",
              "<tr><td>[2^5 - 2^6]  </td><td>4096              </td><td>[30, 20, 30]                    </td><td>2**40    </td><td>mul      </td><td>failed                  </td></tr>\n",
              "<tr><td>[2^11 - 2^12]</td><td>8192              </td><td>[40, 20, 20, 40]                </td><td>2**40    </td><td>mul      </td><td>failed                  </td></tr>\n",
              "<tr><td>[2^11 - 2^12]</td><td>4096              </td><td>[40, 20, 40]                    </td><td>2**40    </td><td>mul      </td><td>failed                  </td></tr>\n",
              "<tr><td>[2^21 - 2^22]</td><td>8192              </td><td>[40, 20, 20, 40]                </td><td>2**40    </td><td>mul      </td><td>failed                  </td></tr>\n",
              "</tbody>\n",
              "</table>"
            ],
            "text/plain": [
              "<IPython.core.display.HTML object>"
            ]
          },
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          "output_type": "display_data"
        }
      ],
      "source": [
        "#@title Understanding the ciphertext precision impact { display-mode: \"form\" }\n",
        "\n",
        "\n",
        "data = [ random.random()]\n",
        "\n",
        "enc_type = ts.ENCRYPTION_TYPE.ASYMMETRIC\n",
        "ct_size_benchmarks = [[\"Value range\", \"Polynomial modulus\", \"Coefficient modulus sizes\", \"Precision\", \"Operation\", \"Status\"]]\n",
        "\n",
        "\n",
        "for data_pow in [-1, 0, 1, 5, 11, 21, 41, 51]:\n",
        "    data = [ random.uniform(2 ** data_pow, 2 ** (data_pow + 1))]\n",
        "    for (poly_mod, coeff_mod_bit_sizes, prec) in [\n",
        "        (8192, [60, 40, 60], 40),\n",
        "        (8192, [40, 21, 21, 21, 21, 21, 21, 40], 40),\n",
        "        (8192, [40, 21, 21, 21, 21, 21, 21, 40], 21),\n",
        "        (8192, [40, 20, 40], 40),\n",
        "        (8192, [20, 20, 20], 38),\n",
        "        (8192, [60, 60], 38),\n",
        "        (8192, [40, 40], 38),\n",
        "        (8192, [17, 17], 15),\n",
        "        (4096, [40, 20, 40], 40),\n",
        "        (4096, [30, 20, 30], 40),\n",
        "        (4096, [20, 20, 20], 38),\n",
        "        (4096, [19, 19, 19], 35),\n",
        "        (4096, [18, 18, 18], 33),\n",
        "        (4096, [30, 30], 25),\n",
        "        (4096, [25, 25], 20),\n",
        "        (4096, [18, 18], 16),\n",
        "        (4096, [17, 17], 15),\n",
        "        (2048, [20, 20], 18),\n",
        "        (2048, [18, 18], 16),\n",
        "        (2048, [16, 16], 14),\n",
        "    ]:\n",
        "        val_str = \"[2^{} - 2^{}]\".format(data_pow, data_pow + 1)\n",
        "        context = ts.context(\n",
        "            scheme=ts.SCHEME_TYPE.CKKS,\n",
        "            poly_modulus_degree=poly_mod,\n",
        "            coeff_mod_bit_sizes=coeff_mod_bit_sizes,\n",
        "            encryption_type=enc_type,\n",
        "        )\n",
        "        scale = 2 ** prec\n",
        "        try:\n",
        "            ckks_vec = ts.ckks_vector(context, data, scale)\n",
        "        except BaseException as e:\n",
        "            ct_size_benchmarks.append([val_str, poly_mod, coeff_mod_bit_sizes, \"2**{}\".format(prec), \"encrypt\", \"encryption failed\"])\n",
        "            continue\n",
        "    \n",
        "        decrypted = decrypt(ckks_vec)\n",
        "        for dec_prec in reversed(range(prec)):\n",
        "            if pytest.approx(decrypted, abs=2 ** -dec_prec) == data:\n",
        "                ct_size_benchmarks.append([val_str, poly_mod, coeff_mod_bit_sizes, \"2**{}\".format(prec), \"encrypt\", \"decryption prec 2 ** {}\".format(-dec_prec)])\n",
        "                break\n",
        "        ckks_sum = ckks_vec + ckks_vec\n",
        "        decrypted = decrypt(ckks_sum)\n",
        "        for dec_prec in reversed(range(prec)):\n",
        "            if pytest.approx(decrypted, abs=2 ** -dec_prec) == [data[0] + data[0]]:\n",
        "                ct_size_benchmarks.append([val_str, poly_mod, coeff_mod_bit_sizes, \"2**{}\".format(prec), \"sum\", \"decryption prec 2 ** {}\".format(-dec_prec)])\n",
        "                break\n",
        "                \n",
        "\n",
        "# We add more depth for the multiplication scenario\n",
        "for data_pow in [-1, 0, 1, 5, 11, 21, 41, 51]:\n",
        "    data = [ random.uniform(2 ** data_pow, 2 ** (data_pow + 1))]\n",
        "    for (poly_mod, coeff_mod_bit_sizes, prec) in [\n",
        "        (8192, [60, 40, 40, 60], 40),\n",
        "        (8192, [40, 21, 21, 40], 40),\n",
        "        (8192, [40, 21, 21, 40], 21),\n",
        "        (8192, [40, 20, 20, 40], 40),\n",
        "        (8192, [20, 20, 20], 38),\n",
        "        (4096, [40, 20, 40], 40),\n",
        "        (4096, [30, 20, 30], 40),\n",
        "        (4096, [20, 20, 20], 38),\n",
        "        (4096, [19, 19, 19], 35),\n",
        "        (4096, [18, 18, 18], 33),\n",
        "        (4096, [30, 30, 30], 25),\n",
        "        (4096, [25, 25, 25], 20),\n",
        "        (4096, [18, 18, 18], 16),\n",
        "        (2048, [18, 18, 18], 16),\n",
        "    ]:\n",
        "        val_str = \"[2^{} - 2^{}]\".format(data_pow, data_pow + 1)\n",
        "        context = ts.context(\n",
        "            scheme=ts.SCHEME_TYPE.CKKS,\n",
        "            poly_modulus_degree=poly_mod,\n",
        "            coeff_mod_bit_sizes=coeff_mod_bit_sizes,\n",
        "            encryption_type=enc_type,\n",
        "        )\n",
        "        scale = 2 ** prec\n",
        "        try:\n",
        "            ckks_vec = ts.ckks_vector(context, data, scale)\n",
        "        except BaseException as e:\n",
        "            continue\n",
        "                \n",
        "        try:\n",
        "            ckks_mul = ckks_vec * ckks_vec\n",
        "        except:\n",
        "            ct_size_benchmarks.append([val_str, poly_mod, coeff_mod_bit_sizes, \"2**{}\".format(prec), \"mul\", \"failed\"])\n",
        "            continue\n",
        "        decrypted = decrypt(ckks_mul)\n",
        "        for dec_prec in reversed(range(prec)):\n",
        "            if pytest.approx(decrypted, abs=2 ** -dec_prec) == [data[0] * data[0]]:\n",
        "                ct_size_benchmarks.append([val_str, poly_mod, coeff_mod_bit_sizes, \"2**{}\".format(prec), \"mul\", \"decryption prec 2 ** {}\".format(-dec_prec)])\n",
        "                break\n",
        "                \n",
        "display(HTML(tabulate.tabulate(ct_size_benchmarks, tablefmt='html')))"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "3v1juerA_sYR"
      },
      "source": [
        "## A WARNING ABOUT HOMOMORPHIC ENCRYPTION\n",
        "\n",
        "Your encrypted system is only as secure as your keys.\n",
        "\n",
        "![](https://imgs.xkcd.com/comics/security.png)"
      ]
    }
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