For testing Efficient, Verifiable and Large enough $\large q$ over $\mathbb{Z}_{q}$ NTRU / NTT Multiplication in Academia Sinica IIS.

1. For Pqc_ntruhps2048677_primeANDroot, We use Decode.fr/Prime-Numbers-Search to find 1000 Closest Primes larger than $\large q'$ ($\large{q' > 2 * (\pm} \Large{\frac{q}{2}} \large{)^2 * n = 2 * (} \Large{\frac{2048}{2}} \large{)^2 * 677}$ ) and put in Prime_list because of the Specification of ntruhps2048677 (Details see in Multi-Parameter Support with NTTs for NTRU...).

The prime $\large q'$ must be $\large q_{i}' = 1536k + 1$

2. We tested and used the Prime_list to find three Roots $\large{x = yz;\ (y^3)\ mod\ q' =(z^{512})\ mod\ q'=1}$ over $\large \mathbb{Z}_{2048}$ for NTT.