Is NTRU still hard if $G$ is set to 1? I'm looking at the description of NTRUEncrypt given on page 21 of http://archive.dimacs.rutgers.edu/Workshops/Post-Quantum/Slides/Silverman.pdf and using its notation. So in NTRU there are always two private parameters $F$ and $G$. However, decryption only requires knowledge of $F$. I'm confused about what the purpose of $G$ is because it doesn't come into the decryption and I don't see why removing $G$ hurts the security of NTRU - more specifically, is NTRU still hard if $G$ is always set to 1 (i.e., $(1,0,0,...,0)\in \{-1,0,1\}^N$)?
If $G$ is set to $1$, then the adversary can easily decrypt the ciphertext because in this case $h = pf^{-1} \mod q,  p$ is coprime with $q$, then inverting  $p \mod q$ is possible and  after that he calculates $f \mod q$ from $h$, then he calculates $f^{-1} \mod p$ from $f$