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. For more details on Intel HEXL, see our whitepaper

Introduction

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 prime, q, up to q~62 bits. More precisely, the polynomials live in the ring Z_q[X]/(X^N + 1). That is, when adding or multiplying two polynomials, each coefficient of the result is reduced by the prime modulus q. 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_q[X]/(X^N + 1). For efficient implementation, Intel HEXL implements the negacyclic number-theoretic transform (NTT). To multiply two polynomials, q_1(x), q_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 q 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

Building Intel HEXL

Intel HEXL can be built in several ways. Intel HEXL has been uploaded to the Microsoft vcpkg C++ package manager, which supports Linux, macOS, and Windows builds. See the vcpkg repository for instructions to build Intel HEXL with vcpkg, e.g. run vcpkg install hexl. There may be some delay in uploading latest release ports to vcpkg. Intel HEXL provides port files to build the latest version with vcpkg. For a static build, run vcpkg install hexl --overlay-ports=/path/to/hexl/port/hexl --head. For dynamic build, use the custom triplet file and run vcpkg install hexl:hexl-dynamic-build --overlay-ports=/path/to/hexl/port/hexl --head --overlay-triplets=/path/to/hexl/port/hexl. For detailed explanation, see instruction for building vcpkg port using overlays and use of custom triplet provided by vcpkg.

Intel HEXL also supports a build using the CMake build system. See below for the instructions to build Intel HEXL from source using CMake.

Dependencies

We have tested Intel HEXL on the following operating systems:

Ubuntu 20.04
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 compiling with clang++-12. We also recommend using a processor with Intel AVX512DQ support, with best performance on processors supporting Intel AVX512-IFMA52. To determine if your processor supports AVX512-IFMA52, simply look for HEXL_HAS_AVX512IFMA during the configure step (see Compiling Intel HEXL).

Compile-time options

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_DOCS	ON / OFF (default OFF)	Set to ON to enable building of documentation
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

Compiling Intel HEXL

To compile Intel HEXL from source code, first clone the repository and change directories to the where the source has been cloned.

Linux and Mac

The instructions to build Intel HEXL are common to Linux and MacOS.

Then, to configure the build, call

cmake -S . -B build

adding the desired compile-time options with a -D flag. For instance, to build Intel HEXL as a shared library, call

cmake -S . -B build -DHEXL_SHARED_LIB=ON

Then, to build Intel HEXL, call

cmake --build build

This will build the Intel HEXL library in the build/hexl/lib/ directory.

To install Intel HEXL to the installation directory, run

cmake --install build

To use a non-standard installation directory, configure the build with

cmake -S . -B build -DCMAKE_INSTALL_PREFIX=/path/to/install

before proceeding with the build and installation directions above.

Windows

To compile Intel HEXL on Windows using Visual Studio in Release mode, configure the build via

cmake -S . -B build -G "Visual Studio 16 2019" -DCMAKE_BUILD_TYPE=Release

adding the desired compile-time options with a -D flag (see Compile-time options).

To specify the desired build configuration, pass either --config Debug or --config Release to the build step and install steps. For instance, to build Intel HEXL in Release mode, call

cmake --build build --config Release

This will build the Intel HEXL library in the build/hexl/lib/ or build/hexl/Release/lib directory.

To install Intel HEXL to the installation directory, run

cmake --build build --target install --config Release

To use a non-standard installation directory, configure the build with

cmake -S . -B build -G "Visual Studio 16 2019" -DCMAKE_BUILD_TYPE=Release -DCMAKE_INSTALL_PREFIX=/path/to/install

before proceeding with the build and installation directions above.

Testing Intel HEXL

To 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 unittest

The unit-test executable itself is located at build/test/unit-test on Linux and Mac, and at build\test\Release\unit-test.exe or build\test\Debug\unit-test.exe on Windows.

Benchmarking Intel HEXL

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 bench

On Windows, run

cmake --build build --target bench --config Release

The benchmark executable itself is located at build/benchmark/bench_hexl on Linux and Mac, and at build\benchmark\Debug\bench_hexl.exe or build\benchmark\Release\bench_hexl.exe on Windows.

Using Intel HEXL

The example folder has an example of using Intel HEXL in a third-party project.

Debugging

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 -DCMAKE_BUILD_TYPE=Debug (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. In Debug mode, Intel HEXL will also link against Address Sanitizer.

Note, enabling CMAKE_BUILD_TYPE=Debug will result in a significant runtime overhead.

To enable verbose logging for the benchmarks or unit-tests in a Debug build, add the log level as a command-line argument, e.g. build/benchmark/bench_hexl --v=9. See easyloggingpp's documentation for more details.

Threading

Intel HEXL is single-threaded and thread-safe.

Community Adoption

Intel HEXL has been integrated to the following homomorphic encryption libraries:

See also the Intel Homomorphic Encryption Toolkit for example uses cases using HEXL.

Please let us know if you are aware of any other uses of Intel HEXL.

Documentation

Intel HEXL supports documentation via Doxygen. See https://intel.github.io/hexl for the latest Doxygen documentation.

To build documentation, first install doxygen and graphviz, e.g.

sudo apt-get install doxygen graphviz

Then, 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 docs

To view the generated Doxygen documentation, open the generated docs/doxygen/html/index.html file in a web browser.

Contributing

This project welcomes external contributions. To contribute to Intel HEXL, see CONTRIBUTING.md. We encourage feedback and suggestions via Github Issues as well as discussion via Github Discussions.

Repository layout

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.

Citing Intel HEXL

To cite Intel HEXL, please use the following BibTeX entry.

### Version 1.2

@misc{IntelHEXL,
    author={Boemer, Fabian and Kim, Sejun and Seifu, Gelila and de Souza, Fillipe DM and Gopal, Vinodh and others},
    title = {{I}ntel {HEXL} (release 1.2)},
    howpublished = {\url{https://github.com/intel/hexl}},
    month = september,
    year = 2021,
    key = {Intel HEXL}
}

### Version 1.1

@misc{IntelHEXL,
    author={Boemer, Fabian and Kim, Sejun and Seifu, Gelila and de Souza, Fillipe DM and Gopal, Vinodh and others},
    title = {{I}ntel {HEXL} (release 1.1)},
    howpublished = {\url{https://github.com/intel/hexl}},
    month = may,
    year = 2021,
    key = {Intel HEXL}
}

### Version 1.0

@misc{IntelHEXL,
    author={Boemer, Fabian and Kim, Sejun and Seifu, Gelila and de Souza, Fillipe DM and Gopal, Vinodh and others},
    title = {{I}ntel {HEXL} (release 1.0)},
    howpublished = {\url{https://github.com/intel/hexl}},
    month = april,
    year = 2021,
    key = {Intel HEXL}
}

Contributors

The Intel contributors to this project, sorted by last name, are

In addition to the Intel contributors listed, we are also grateful to contributions to this project that are not reflected in the Git history:

Antonis Papadimitriou

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