Homomorphic Encryption

The defining property: operations performed on ciphertexts produce an encrypted result that, when decrypted, matches the result of the same operations applied to the plaintext. For an encryption scheme E, there exists an operation ⊕ such that E(a) ⊕ E(b) = E(a + b). This allows a third party to process data without ever seeing it in the clear.

Schemes supporting only one type of operation (addition or multiplication) an unlimited number of times.

• Additive (Paillier): E(a) ⊕ E(b) = E(a + b). Used in electronic voting and private aggregation.
• Multiplicative (RSA, ElGamal): E(a) ⊗ E(b) = E(a × b). Used in blind signatures and key exchange.

PHE is computationally lightweight and practical for production systems requiring a single operation type.

Schemes supporting both addition and multiplication but only for a limited number of operations. Each operation increases ciphertext noise, and once noise exceeds a threshold, decryption fails. SHE is the building block for fully homomorphic encryption and is useful for evaluating shallow circuits like statistical averages or low-degree polynomials.

Schemes supporting arbitrary computation (unlimited additions and multiplications) on encrypted data. Achieved through bootstrapping—a technique that evaluates the decryption circuit homomorphically to refresh the ciphertext noise budget.

• BGV/BFV: Integer arithmetic, efficient for vector operations.
• CKKS: Approximate arithmetic, ideal for machine learning with floating-point numbers.
• TFHE: Fast bootstrapping for boolean circuits and lookup tables.

FHE enables private inference where a server evaluates a neural network on encrypted user data.

A ciphertext reveals no information about the underlying plaintext to a computationally bounded adversary. Formally, an encryption scheme is semantically secure if no polynomial-time adversary can distinguish between encryptions of two chosen plaintexts with probability better than random guessing.

• Achieved through randomized encryption: the same plaintext produces different ciphertexts each time.
• Homomorphic encryption schemes achieve this via noise-based hardness assumptions like Ring Learning With Errors (RLWE).
• Critical for ensuring that an untrusted cloud cannot infer anything from stored ciphertexts.

A stronger property ensuring the evaluated ciphertext reveals nothing about the function that was computed, not just the input. Without circuit privacy, the output ciphertext's noise distribution may leak which operations were performed.

• Achieved through noise flooding or sanitization techniques that add extra noise after computation.
• Essential when the function itself is proprietary, such as a bank's fraud detection model running on encrypted customer data.
• Standard FHE schemes are not circuit-private by default; additional steps are required.

A mathematical framework that provides a provable guarantee of privacy by injecting calibrated statistical noise into data or model updates. The core principle is that the presence or absence of any single training record should be indistinguishable in the output. In federated RF learning, differential privacy is often applied via the DP-SGD algorithm, which clips per-sample gradients and adds Gaussian noise before aggregation. The privacy budget is quantified by the parameter epsilon (ε)—lower values indicate stronger privacy but may degrade model accuracy.

A privacy attack where an adversary exploits access to a trained machine learning model to reconstruct sensitive features or representative samples of the private training data. In the context of RF fingerprinting, a successful inversion attack could reveal the unique hardware impairments of a specific emitter from the global model's weights. Homomorphic encryption and differential privacy are primary defenses: the former prevents the server from inspecting individual updates, while the latter mathematically bounds what can be inferred about any single training sample.