Intel:registered: HEXL is an open-source library which provides efficient implementations of integer arithmetic on Galois fields. Such arithmetic is prevalent in cryptography, particularly in homomorphic encryption (HE) schemes. Intel HEXL targets integer arithmetic with word-sized primes, typically 40-60 bits. Intel HEXL provides an API for 64-bit unsigned integers and targets Intel CPUs.
Many cryptographic applications, particularly homomorphic encryption (HE), rely on integer polynomial arithmetic in a finite field. HE, which enables computation on encrypted data, typically uses polynomials with degree N a power of two roughly in the range N=[2^{10}, 2^{17}]. The coefficients of these polynomials are in a finite field with a word-sized primes, p, up to p~62 bits. More precisely, the polynomials live in the ring Z_p[X]/(X^N + 1). That is, when adding or multiplying two polynomials, each coefficient of the result is reduced by the prime modulus p. When multiplying two polynomials, the resulting polynomials of degree 2N is additionally reduced by taking the remainder when dividing by X^N+1.
The primary bottleneck in many HE applications is polynomial-polynomial multiplication in Z_p[X]/(X^N + 1). For efficient implementation, Intel HEXL implements the negacyclic number-theoretic transform (NTT). To multiply two polynomials, p_1(x), p_2(x) using the NTT, we perform the FwdNTT on the two input polynomials, then perform an element-wise modular multiplication, and perform the InvNTT on the result.
Intel HEXL implements the following functions:
- The forward and inverse negacyclic number-theoretic transform (NTT)
- Element-wise vector-vector modular multiplication
- Element-wise vector-scalar modular multiplication with optional addition
- Element-wise modular multiplication
For each function, the library implements one or several Intel(R) AVX-512 implementations, as well as a less performant, more readable native C++ implementation. Intel HEXL will automatically choose the best implementation for the given CPU Intel(R) AVX-512 feature set. In particular, when the modulus p is less than 2^{50}, the AVX512IFMA instruction set available on Intel IceLake server and IceLake client will provide a more efficient implementation.
For additional functionality, see the public headers, located in include/hexl
We have tested Intel HEXL on the following operating systems:
- Ubuntu 18.04
- macOS 10.15
- Microsoft Windows 10
Intel HEXL requires the following dependencies:
| Dependency | Version |
|---|---|
| CMake | >= 3.5.1 |
| Compiler | gcc >= 7.0, clang++ >= 5.0, MSVC >= 2019 |
For best performance, we recommend using a processor with AVX512-IFMA52 support, and a recent compiler (gcc >= 8.0, clang++ >= 6.0). To determine if your process supports AVX512-IFMA52, simply look for HEXL_HAS_AVX512IFMA during the configure step (see Compiling Intel HEXL).
In addition to the standard CMake build options, Intel HEXL supports several compile-time flags to configure the build. For convenience, they are listed below:
| CMake option | Values | |
|---|---|---|
| HEXL_BENCHMARK | ON / OFF (default ON) | Set to ON to enable benchmark suite via Google benchmark |
| HEXL_COVERAGE | ON / OFF (default OFF) | Set to ON to enable coverage report of unit-tests |
| HEXL_DEBUG | ON / OFF (default OFF) | Set to ON to enable debugging at large runtime penalty |
| HEXL_DOCS | ON / OFF (default OFF) | Set to ON to enable building of documentation |
| HEXL_ENABLE_ADDRESS_SANITIZER | ON / OFF (default OFF) | Set to ON to enable building with address sanitizer (ASan) |
| HEXL_ENABLE_THREAD_SANITIZER | ON / OFF (default OFF) | Set to ON to enable building with thread sanitizer (TSan) |
| HEXL_ENABLE_UB_SANITIZER | ON / OFF (default OFF) | Set to ON to enable building with undefined behavior sanitizer (UBSan) |
| HEXL_EXPORT | ON / OFF (default OFF) | Set to ON to enable export of Intel HEXL for use in 3rd-party project |
| HEXL_SHARED_LIB | ON / OFF (default OFF) | Set to ON to enable building shared library |
| HEXL_TESTING | ON / OFF (default ON) | Set to ON to enable building of unit-tests |
| HEXL_TREAT_WARNING_AS_ERROR | ON / OFF (default OFF) | Set to ON to treat all warnings as error |
The instructions to build Intel HEXL are common between Linux, MacOS, and Windows.
To compile Intel HEXL from source code, first clone the repository into your current directory. Then, to configure the build, call
cmake -S . -B buildadding the desired compile-time options with a -D flag. For instance, to build Intel HEXL with debugging capabilities, call
cmake -S . -B build -DHEXL_DEBUG=ONThen, to build Intel HEXL, call
cmake --build buildThis will build the Intel HEXL library in the build/hexl/lib/ directory.
To install Intel HEXL to the installation directory, run
cmake --install buildTo use a non-standard installation directory, configure the build with
cmake -S . -B build -DCMAKE_INSTALL_PREFIX=/path/to/installTo run a set of unit tests via Googletest, configure and build Intel HEXL with -DHEXL_TESTING=ON (see Compile-time options).
Then, run
cmake --build build --target unittestThe unit-test executable itself is located at build/test/unit-test
To run a set of benchmarks via Google benchmark, configure and build Intel HEXL with -DHEXL_BENCHMARK=ON (see Compile-time options).
Then, run
cmake --build build --target benchThe benchmark executable itself is located at build/benchmark/bench_hexl
The example folder has an example of using Intel HEXL in a third-party project.
For optimal performance, Intel HEXL does not perform input validation. In many cases the time required for the validation would be longer than the execution of the function itself. To debug Intel HEXL, configure and build Intel HEXL with -DHEXL_DEBUG=ON (see Compile-time options). This will generate a debug version of the library, e.g. libhexl_debug.a, that can be used to debug the execution.
Note, enabling HEXL_DEBUG=ON will result in a significant runtime overhead.
Intel HEXL is single-threaded and thread-safe.
See https://intel.github.io/hexl for Doxygen documentation.
Intel HEXL supports documentation via Doxygen and sphinx.
To build documentation, first install doxygen and graphviz, e.g.
sudo apt-get install doxygen graphvizThen, configure Intel HEXL with -DHEXL_DOCS=ON (see Compile-time options).
To build Doxygen documentation, after configuring Intel HEXL with -DHEXL_DOCS=ON, run
cmake --build build --target doxygen
To view the generated Doxygen documentation, open the generated build/docs/doxygen/html/index.html file in a web browser.
To build the sphinx documentation, install sphinx and required dependencies breathe, m2r2, e.g.
sudo apt-get install python3-sphinx
pip3 install breathe m2r2Then, after configuring Intel HEXL with -DHEXL_DOCS=ON, run
cmake --build build --target docsTo view the generated Sphinx documentation, open the generated build/docs/sphinx/html/index.html file in a web browser.
At this time, Intel HEXL does not accept external contributions. We encourage feedback and suggestions via issues.
For Intel developers, use pre-commit to validate the formatting of the code.
Before contributing, please run
cmake --build build --target check unittestand make sure pre-commit checks and all unit tests pass.
Public headers reside in the hexl/include folder.
Private headers, e.g. those containing Intel(R) AVX-512 code should not be put in this folder.
Find published paper at https://arxiv.org/abs/2103.16400.
To cite Intel HEXL, please use the following BibTeX entry.
@misc{IntelHEXL,
title = {{I}ntel {HEXL} (release 1.0)},
howpublished = {\url{https://arxiv.org/abs/2103.16400}},
month = mar,
year = 2021,
key = {Intel HEXL}
}