Optimizing polynomial convolution for NTRUEncrypt.

$\sf{ NTRUEncrypt}$  is one of the most promising candidates for quantum-safe cryptography. In this paper, we focus on the $\sf{ NTRU743}$  parameter set. We give a report on all known attacks against this parameter set and show that it delivers 256 bits of security against classical attackers and 128 bits of security against quantum attackers. We then present a parameter-dependent optimization using a tailored hierarchy of multiplication algorithms as well as the Intel AVX2 instructions, and show that this optimization is constant-time. Our implementation is two to three times faster than the reference implementation of $\sf{ NTRUEncrypt}$ .