Homomorphic Encryption

Supports unlimited operations of a single type—either addition or multiplication, but not both. RSA encryption is additively homomorphic, while ElGamal is multiplicatively homomorphic. PHE schemes are computationally lightweight and practical for specific use cases like encrypted vote tallying or privacy-preserving aggregate statistics. In fraud detection, PHE can compute encrypted sums of transaction amounts across institutions without revealing individual values.

Permits both addition and multiplication but only for circuits of limited depth. Each operation introduces noise that accumulates; once the noise exceeds a threshold, decryption fails. SHE schemes like BGV and BFV use modulus switching and noise management to extend computational depth. They are well-suited for evaluating low-degree polynomials, such as computing encrypted risk scores from transaction features without exposing raw data.

Supports arbitrary computation on encrypted data with no theoretical limit on circuit depth. Gentry's 2009 breakthrough introduced bootstrapping—recursively refreshing ciphertext noise by homomorphically evaluating the decryption circuit itself. Modern FHE schemes (CKKS, TFHE) enable practical deep neural network inference on encrypted inputs. For fraud analytics, FHE allows a third party to run a proprietary ML model over encrypted transaction data and return an encrypted fraud score without ever seeing the plaintext.

All modern HE schemes derive security from hard problems on mathematical lattices, primarily the Ring Learning With Errors (RLWE) problem. RLWE is believed to be resistant to attacks by both classical and quantum computers, making HE post-quantum secure. The hardness assumption is that distinguishing noisy linear equations from random is computationally infeasible. This cryptographic foundation ensures that encrypted fraud data remains protected even against future quantum adversaries.

The Cheon-Kim-Kim-Song (CKKS) scheme operates on approximate real and complex numbers, making it the preferred choice for machine learning workloads. Unlike exact integer schemes, CKKS treats least significant bits as noise and performs fixed-point arithmetic natively. It supports efficient SIMD-style packing, encoding thousands of values into a single ciphertext. In fraud detection, CKKS enables encrypted logistic regression and neural network inference with minimal precision loss.

Extends FHE to multi-party scenarios where data encrypted under different keys can be jointly computed upon. Threshold FHE requires a quorum of parties to collaboratively decrypt results, preventing any single entity from accessing plaintext. Multi-Key FHE allows computation across ciphertexts encrypted under distinct keys without prior coordination. This enables consortium-based fraud detection where multiple banks contribute encrypted data and jointly decrypt only aggregate anomaly scores.

The strongest form of homomorphic encryption, supporting arbitrary computations (unlimited additions and multiplications) on ciphertexts. FHE enables a third party to perform complex fraud scoring algorithms on encrypted transaction data without ever seeing the plaintext.

• Bootstrapping: A noise-reduction technique that enables unlimited depth circuits
• Lattice-based: Relies on the hardness of the Learning With Errors (LWE) problem
• Performance: Historically 1,000,000x slower than plaintext, but hardware acceleration is closing the gap
• Use case: A bank encrypts transaction logs and sends them to a cloud-based fraud model that returns encrypted risk scores without decrypting the data

A restricted form of HE that supports only one operation type—either addition or multiplication, but not both. PHE is computationally lightweight and practical for specific fraud analytics tasks.

• Additive (Paillier): Enables encrypted sums for aggregating transaction values across institutions
• Multiplicative (RSA/ElGamal): Enables encrypted products for computing risk ratios
• Performance: Near-plaintext speed, making it viable for real-time fraud scoring pipelines
• Limitation: Cannot perform complex ML inference requiring both operations

An intermediate scheme that supports a limited number of both additions and multiplications before noise overwhelms the ciphertext. SHE bridges the gap between PHE's efficiency and FHE's expressiveness.

• Bounded depth: Can evaluate shallow circuits like logistic regression or decision trees
• No bootstrapping: Avoids the heavy computational cost of FHE's noise management
• Ideal for: Running a pre-trained fraud classifier on encrypted data where the model architecture is known to fit within the circuit depth budget
• Trade-off: More expressive than PHE, faster than FHE, but requires careful circuit design

Combining HE with federated learning creates a double-blind privacy guarantee: the server aggregates encrypted model updates without seeing individual contributions, and clients never expose raw transaction data.

• Secure aggregation: Clients encrypt gradient updates with HE before sending to the central server
• Defense against gradient leakage: Prevents the server from reconstructing training samples from model updates
• Use case: A consortium of banks jointly trains a fraud detection model where each bank's transaction patterns remain cryptographically hidden from competitors
• Integration: Often paired with Secure Multi-Party Computation (SMPC) for key management

The mathematical foundation underlying most modern HE schemes. Lattice problems like Learning With Errors (LWE) and Ring-LWE provide security based on the hardness of finding short vectors in high-dimensional lattices.

• Post-quantum secure: Believed to resist attacks from both classical and quantum computers
• Worst-case to average-case reduction: Security proofs that are stronger than traditional factoring-based cryptography
• Noise growth: Each homomorphic operation adds noise to the ciphertext; managing this noise is the central engineering challenge
• Schemes: BGV, BFV, CKKS, and TFHE are all lattice-based HE constructions

A leveled homomorphic encryption scheme optimized for approximate fixed-point arithmetic on real numbers, making it the preferred choice for machine learning inference on encrypted data.

• Packed ciphertexts: Encodes multiple plaintext values into a single ciphertext using SIMD-style batching, enabling vectorized operations
• Rescaling: Manages noise by truncating least significant bits after multiplications, analogous to floating-point rounding
• Ideal for: Running neural network inference where exact precision is not required, such as fraud probability scoring
• Limitation: Not suitable for exact integer arithmetic required by financial accounting