{"id":3297,"date":"2023-12-17T18:14:35","date_gmt":"2023-12-17T09:14:35","guid":{"rendered":"https:\/\/saraheee.com\/?p=3297"},"modified":"2023-12-17T18:14:38","modified_gmt":"2023-12-17T09:14:38","slug":"dp2-database-anonymization-differential-privacy","status":"publish","type":"post","link":"https:\/\/saraheee.com\/ko\/2023\/12\/dp2-database-anonymization-differential-privacy\/","title":{"rendered":"[DP#2] Database anonymization – differential privacy"},"content":{"rendered":"

The Mathematics Behind Differential Privacy<\/h2>\n\n\n\n

\\(Pr[\\mathcal{M}(D) \\in \\mathcal{S}] \\leq e^{\\varepsilon} \\cdot Pr[\\mathcal{M}(D’) \\in \\mathcal{S}]\\)<\/p>\n\n\n\n

Sensitivity and Query Function<\/h4>\n\n\n\n

Sensitivity and Query Function: \\(\\Delta f = max_{D, D^{‘}} ||f(D) – f(D^{‘})||\\)<\/p>\n\n\n\n

D and D’ are neighboring datasets
D and D’ differ in exactly one element<\/p>\n\n\n\n

Name<\/td>Alice<\/td>Bob<\/td>Charly<\/td>Dave<\/td>Eve<\/td>Ferris<\/td>George<\/td>Harvey<\/td>Iris<\/td><\/tr>
Age<\/td>29<\/td>22<\/td>27<\/td>43<\/td>52<\/td>47<\/td>30<\/td>36<\/td>32<\/td><\/tr><\/tbody><\/table>
D: mean_age(D) = 35.3, max change = 52<\/figcaption><\/figure>\n\n\n\n
Name<\/td>Alice<\/td>Bob<\/td>Charly<\/td>Dave<\/td><\/td>Ferris<\/td>George<\/td>Harvey<\/td>Iris<\/td><\/tr>
Age<\/td>29<\/td>22<\/td>27<\/td>43<\/td><\/td>47<\/td>30<\/td>36<\/td>32<\/td><\/tr><\/tbody><\/table>
D’<\/figcaption><\/figure>\n\n\n\n

(e.g., count(D) = 9, count(D’) = 8 – difference will always be one – \u0394f = 1
mean_age(D) = 35.3, mean_age(D’) = 33.3 – \u0394f = 2<\/p>\n\n\n\n

Formal Definition of Differential Privacy<\/h4>\n\n\n\n

Differential privacy provides a mathematical framework to quantify and control the privacy guarantees of a data analysis system.
It is defined in terms of a parameter called \u03b5, which represents the privacy budget or the level of privacy protection provided.<\/p>\n\n\n\n

Detailed Explanation of the Differential-Privacy Equation<\/h4>\n\n\n\n

\\(Pr[\\mathcal{M}(D) \\in \\mathcal{S}] \\leq e^{\\varepsilon} \\cdot Pr[\\mathcal{M}(D’) \\in \\mathcal{S}]\\)<\/p>\n\n\n\n

The probability that a mechanism M, applied on a database D,
produces a certain output S is approximately the same
if applied to a neighboring database, with the level of similarity controlled by the privacy parameter \\(\\varepsilon\\).
it is with the level of similarity controlled by the privacy parameter Epsilon
and this is now well this is e to the power of Epsilon,
this way it’s a bit easier to calculate the Epsilon because it’s a we cat just use logarithm<\/mark><\/p>\n\n\n\n

The probability distribution of the output remains nearly unchanged whether an individual’s data is included or excluded.<\/p>\n\n\n\n

How to Add Noise to Achieve Differential Privacy<\/h2>\n\n\n\n

Sensitivity and Noise<\/h4>\n\n\n\n
\"\"<\/td>\"\"<\/td><\/tr><\/tbody><\/table>
Laplacian Noise, Gaussian Noise<\/figcaption><\/figure>\n\n\n\n

it’s a bit more easy and I think it’s more straightforward to understand how it works and it’s basically the same thing with gaussian noises the math behind it is just a little bit more complicated
how to implement it in Python<\/mark><\/p>\n\n\n\n

laplacian function: Probability Distribution function
Two exponential functions<\/p>\n\n\n\n

Lap(\u00b5, b), \u00b5 = 0, b = \u0394f \/\u03b5<\/mark><\/p>\n\n\n\n

\u00b5: the value which luckily for us is always going to be zero
laplacian function these two exponentials are always mirrored at the y-axis so at point zero<\/mark><\/p>\n\n\n\n

b: the parameter where the sensitivity comes into place so b equals the sensitivity Delta F divided by Epsilon
obviously I should remember that Epsilon is the Privacy parameter of the differential privacy equation<\/mark><\/p>\n\n\n\n

count-query: \u0394f = 1
b = 1\/\u03b5<\/p>\n\n\n

\n
\"\"<\/figure><\/div>\n\n\n

different values of Epsilon I have different values or different outcomes of the probability distribution function of the laplacian function<\/mark><\/p>\n\n\n\n

Epsilon(\u03b5) is the privacy parameter of differential privacy
\u03b5 = 1.0 – biggest Peak or the largest Peak
\u03b5 = 0.2 – very low (the total probability is higher however the peak is lower)<\/mark><\/p>\n\n\n\n

Python<\/h4>\n\n\n\n
import numpy as np\n\nsensitivity = 1.00\nepsilon = 0.5\nb = sensitivity \/ epsilon\n\nn = 200 # number of individuals\nnoise = np.random.laplace(0, b, n)\nprint(noise)<\/code><\/pre>\n\n\n\n
Step-by-Step implementation of k-anonymity with the Mondrian Algorithm in Python<\/h2>\n\n\n\n
pip install pandas
python mondrian.py<\/p>\n\n\n\n
import pandas as pd\n\nk = 3\ndata = pd.read_csv('data.csv')\nprint(data.head())\n\ndef mondrian(data, k):\n    partition[]\n\n    if(len(data)) <= (2*k-1):\n        partition.append(data)\n        return [data]\n\n    # define QIs\n    qis = ['Age']\n\n    # sort by QIs\n    data = data.sort_values(by=qis)\n\n    # number of total values\n    si = data['Age'].count()\n\n    mid = si\/2\n\n    lhs = data[:mid]\n    rhs = data[mid:]\n\n    partition.extend(mondrian(lhs, k))\n    partition.extend(mondrian(rhs, k))\n\n    return partition\n\nresult_partitions = mondrian(data, k)\nresults_final = []\n\nfor i, partition in enumerate(result_partitions, start=1):\n    part_min = partition['Age'].min()\n    part_max = partition['Age'].max()\n\n    gen = f\"|{part_min} - {part_max}|\"\n\n    partitions['Age'] = gen\n    results_final.append(partition)\n\n    print(f\"Partition, length={len(partition)}:\")\n\n    if(len(partition)<k):\n        print(\"Error: Partition too small\")\n    else:\n        print(partition)\n\nanonymized_data = pd.concat(results_final, ignore_index=True)\nanonymized_data.to_csv('anon.csv', index=False)\n\n#    k = 3\n#    len - partition = 6 <- 2*k\n#    partition1 = 3\n#    partition2 = 2<\/code><\/pre>\n\n\n\n
<\/p>\n\n\n\n
References<\/h3>\n\n\n\n