The NTRU crytposystem makes use of the ring $(\mathbb{Z}/3\mathbb{Z})[X]/(X^N-1)$, which I thought it was the ring of truncated polynomials with degree at most $N-1$, and coefficients reduced modulo $3$, i.e., in the set $\{0,1,2\}$. However, many algorithms for working with these polynomials within the cryptosystem specify that the coefficients are in the set $\{-1,0,1\}$.
Where do these coefficients come from? Is there any relation between the two sets? Maybe I'm not understanding properly the meaning of $(\mathbb{Z}/3\mathbb{Z})[X]/(X^N-1)$.
Also, let $f(X)$ and $g(x)$ be such polynomials with coefficients in the set $\{-1,0,1\}$. What is the correct way of performing the operation $$ h(X) = f(X) + g(X) \bmod 3 $$ Summing each coefficient and making a modulo operation seems not to be the right way, becuase a computer will always return values in the set $\{0,1,2\}$. I would appreciate any help on this.
The set $\Bbb Z{/}3\Bbb Z$ is where the coefficients come from and we can use the representatives $\{0,1,2\}$ for those or $\{-1,0,1\}$ which is the same as $2 \equiv -1 \pmod{3}$. It's more "symmetric" to use the latter one. Modding out by $X^N-1$ only makes some cyclic shifts equivalent (repesenting polynomials as vectors or tuples) and has no bearing on the coefficients of the resulting polynomial classes. Addition is easy for such polynomials as $-1$ and $1$ cancel, $1 + 1$ becomes $-1$ and $-1 + -1$ becomes $1$, and $0$ is the neutral element (power by power of course, or coordinatewise, in tuples).