Microsoft SEAL

Provides a complete implementation of the Brakerski/Fan-Vercauteren (BFV) scheme for exact integer arithmetic. The BFV scheme operates on modular arithmetic, making it ideal for applications requiring precise results like encrypted database queries, private set intersection, and financial computation. SEAL's BFV implementation supports ciphertext packing for SIMD parallelism, relinearization to control ciphertext growth after multiplication, and modulus switching for noise management without decryption.

Implements the Cheon-Kim-Kim-Song (CKKS) scheme optimized for approximate fixed-point arithmetic on real numbers. CKKS is the preferred scheme for encrypted machine learning inference because it natively handles floating-point operations like neural network weights and activations. Key features include:

• Rescaling to maintain stable precision after multiplication
• Ciphertext rotation via Galois keys for matrix operations
• Support for evaluating polynomial approximations of non-linear functions like sigmoid and ReLU

SEAL provides transparent noise budget tracking throughout the computation lifecycle. Each ciphertext carries metadata indicating its remaining noise budget in bits. The library automatically checks whether sufficient budget remains before performing operations, preventing silent decryption failures. Developers can query the noise budget at any point to debug circuit depth issues or optimize computation order. This is critical for leveled HE where the multiplicative depth must be known before encryption.

Leverages Single Instruction Multiple Data (SIMD) through ciphertext packing, encoding up to 16,384 plaintext values into a single ciphertext depending on the polynomial modulus degree. A single homomorphic addition or multiplication simultaneously operates on all packed slots, providing massive amortized throughput. SEAL supports homomorphic rotation of packed slots using Galois keys, enabling efficient implementation of convolution, matrix multiplication, and other structured linear algebra operations in the encrypted domain.

Written in modern C++17 with no external dependencies beyond a standard library, SEAL compiles on Windows, Linux, macOS, and Android. The library provides official wrappers for .NET (C#) and Python through community-maintained bindings like PySEAL. This broad platform support enables integration into diverse environments from cloud servers to edge devices. The build system uses CMake with optional Intel HEXL integration for hardware-accelerated number theoretic transforms on Intel processors.

SEAL ships with extensive C++ examples demonstrating real-world usage patterns including encrypted database lookup, private set intersection, logistic regression training, and CKKS-based neural network inference. The documentation covers:

• Parameter selection guidelines for 128-bit and 192-bit security levels
• Serialization of keys and ciphertexts for storage and network transmission
• Memory management best practices for large polynomial operands
• Detailed API reference for all public classes and methods