MATHEMATICS OF DATA PRIVACY

The mathematics of data privacy involves applying mathematical techniques to ensure that sensitive information remains secure and private. This field intersects with cryptography, statistics, and information theory to develop methods for protecting data while still allowing useful analysis and operations. Here are some key concepts and techniques:

1. Differential Privacy:

• Concept: Provides a framework for quantifying and ensuring the privacy of individuals in a dataset. The idea is to add noise to the data in such a way that the presence or absence of any single individual's data does not significantly affect the outcome of any analysis.
• Mathematical Formalism:
  • Differential privacy is usually expressed as follows: A mechanism $M$ is $(\epsilon, \delta)$-differentially private if for all datasets $D$ and $D'$ that differ by one entry, and for all possible outputs $O$, $$\Pr[M(D) \in O] \leq e^\epsilon \Pr[M(D') \in O] + \delta$$
  • Here, $\epsilon$ is the privacy parameter (lower values indicate stronger privacy), and $\delta$ is a small probability of failure.

2. k-Anonymity:

• Concept: Ensures that each record in a dataset is indistinguishable from at least $k-1$ other records with respect to certain identifying attributes. This helps protect against re-identification.
• Mathematical Formalism:
  • For a dataset to be $k$-anonymous, each combination of quasi-identifiers (attributes that can be used to identify individuals) must appear at least $k$ times in the dataset.

3. l-Diversity:

• Concept: An extension of k-anonymity that ensures that each equivalence class (set of records sharing the same quasi-identifiers) contains at least $l$ distinct values for sensitive attributes.
• Mathematical Formalism:
  • A dataset is $l$-diverse if every equivalence class in the anonymized dataset contains at least $l$ different values for sensitive attributes.

4. t-Closeness:

• Concept: Further extends $l$-diversity by ensuring that the distribution of sensitive attributes in each equivalence class is close to the distribution in the entire dataset.
• Mathematical Formalism:
  • For each equivalence class $C$ and sensitive attribute $A$, the distance between the distribution of $A$ in $C$ and in the entire dataset should be within a threshold $t$, typically measured using metrics like the Earth Mover’s Distance (EMD).

5. Secure Multi-Party Computation (MPC):

• Concept: Allows multiple parties to jointly compute a function over their inputs while keeping those inputs private.
• Mathematical Formalism:
  • The protocol ensures that each party's input is hidden and only the final result is revealed. Techniques involve secret sharing, cryptographic commitments, and homomorphic encryption.

6. Homomorphic Encryption:

• Concept: Allows computations to be performed on encrypted data without decrypting it, thus preserving privacy.
• Mathematical Formalism:
  • A cryptographic scheme is homomorphic if it supports operations such that given ciphertexts $C_1$ and $C_2$ corresponding to plaintexts $m_1$ and $m_2$, the operation on ciphertexts $C_1$ and $C_2$ corresponds to the same operation on plaintexts $m_1$ and $m_2$.

7. Information Theory and Privacy:

• Concept: Uses concepts from information theory to quantify privacy and measure the amount of information leaked by a system.
• Mathematical Formalism:
  • Shannon Entropy: Measures the uncertainty in a dataset.
  • Mutual Information: Measures the amount of information obtained about one variable by observing another.

8. Data Masking and Obfuscation:

• Concept: Techniques to hide or obscure sensitive data to protect privacy.
• Mathematical Formalism:
  • Randomization: Adding noise to data, often using statistical distributions, to prevent exact recovery of original values.

9. Zero-Knowledge Proofs (ZKPs):

• Concept: Allows one party to prove to another that they know a value without revealing the value itself.
• Mathematical Formalism:
  • ZKPs are based on complex mathematical constructs like interactive proofs and commitment schemes.

These techniques and concepts are central to ensuring data privacy and security, particularly as data collection and analysis become increasingly sophisticated. They are used in various applications, including secure data sharing, privacy-preserving data mining, and protection against unauthorized access and misuse of sensitive information.