The SPQlios library provides fast arithmetic for Fully Homomorphic Encryption, and other lattice constructions that arise in post quantum cryptography.

Namely, it is divided into 4 sections:

• The low-level DFT section support FFT over 64-bit floats, as well as NTT modulo one fixed 120-bit modulus. It is an upgrade of the original spqlios-fft module embedded in the TFHE library since 2016. The DFT section exposes the traditional DFT, inverse-DFT, and coefficient-wise multiplications in DFT space.
• The VEC_ZNX section exposes fast algebra over vectors of small integer polynomial modulo $X^N+1$. It proposed in particular efficient (prepared) vector-matrix products, scalar-vector products, convolution products, and element-wise products, operations that naturally occurs on gadget-decomposed Ring-LWE coordinates.
• The RNX section is a simpler variant of VEC_ZNX, to represent single polynomials modulo $X^N+1$ (over the reals or over the torus) when the coefficient precision fits on 64-bit doubles. The small vector-matrix API of the RNX section is particularly adapted to reproducing the fastest CGGI-based bootstrappings.
• The ZN section focuses over vector and matrix algebra over scalars (used by scalar LWE, or scalar key-switches, but also on non-ring schemes like Frodo, FrodoPIR, and SimplePIR).

SPQlios is more than a library, it is also a good target for hardware developers. On one hand, the arithmetic operations that are defined in the library have a clear standalone mathematical definition. And at the same time, the amount of work in each operations is sufficiently large so that meaningful functions only require a few of these.

This makes the SPQlios API a high value target for hardware acceleration, that targets FHE.

SPQlios itself is not an FHE library: there is no ciphertext, plaintext or key. It is a mathematical library that exposes efficient algebra over polynomials. Using the functions exposed, it is possible to quickly build efficient FHE libraries, with support for the main schemes based on Ring-LWE: BFV, BGV, CGGI, DM, CKKS.

The SPQLIOS-FFT library is a C library that can be compiled with a standard C compiler, and depends only on libc and libm. The API interface can be used in a regular C code, and any other language via classical foreign APIs.

The unittests and integration tests are in an optional part of the code, and are written in C++. These tests rely on benchmark, and gtest libraries, and therefore require a C++17 compiler.

Currently, the project has been tested with the gcc,g++ >= 11.3.0 compiler under Linux (x86_64). In the future, we plan to extend the compatibility to other compilers, platforms and operating systems.

The library uses a classical cmake build mechanism: use cmake to create a build folder in the top level directory and run make from inside it. This assumes that the standard tool cmake is already installed on the system, and an up-to-date c++ compiler (i.e. g++ >=11.3.0) as well.

It will compile the shared library in optimized mode, and make install install it to the desired prefix folder (by default /usr/local/lib).

If you want to choose additional compile options (i.e. other installation folder, debug mode, tests), you need to run cmake manually and pass the desired options:

mkdir build
cd build
cmake ../src -CMAKE_INSTALL_PREFIX=/usr/
make

The available options are the following: