Investigate Intel HEXL to determine the core difficulty of implementing FHE with AVX extensions

https://arxiv.org/abs/2103.16400

> Modern implementations of homomorphic encryption (HE) rely heavily on polynomial arithmetic over a finite field. This is particularly true of the CKKS, BFV, and BGV HE schemes. Two of the biggest performance bottlenecks in HE primitives and applications are polynomial modular multiplication and the forward and inverse number-theoretic transform (NTT). Here, we introduce Intel Homomorphic Encryption Acceleration Library (Intel HEXL), a C++ library which provides optimized implementations of polynomial arithmetic for Intel processors. Intel HEXL takes advantage of the recent Intel Advanced Vector Extensions 512 (Intel AVX512) instruction set to provide state-of-the-art implementations of the NTT and modular multiplication. On the forward and inverse NTT, Intel HEXL provides up to 7.2x and 6.7x speedup, respectively, over a native C++ implementation. Intel HEXL also provides up to 6.0x speedup on the element-wise vector-vector modular multiplication, and 1.7x speedup on the element-wise vector-scalar modular multiplication. Intel HEXL is available open-source at this https URL under the Apache 2.0 license and has been adopted by the Microsoft SEAL and PALISADE homomorphic encryption libraries.

In particular, what is the discrepancy between a typical NTT implementation compiled with LLVM's `opt -loop-vectorize` (cf. [this SO post](https://stackoverflow.com/questions/22548397/can-i-generate-avx-vectorized-code-using-llvm-jit)) and the hand-written AVX implementation?