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I'm trying to understand the GGH graded encoding scheme, but something there leaves me very confused and I can not figure out how to explain it:

Let $R := \mathbb{Z}[X]/(X^n+1)$, where $n$ is a power of $2$ (hence $X^n+1$ is irreducible in $\mathbb{Z}[X]$); $q := 2^a$ for a large enough $a$; $R_q := R/qR$, i.e. $R_q = \mathbb{Z}_q[X]/(X^n+1)$.

In the scheme, we choose $z$ uniformly at random in $R_q$. Then the scheme uses this $z$ to divide other elements in $R_q$, like $[c/z]_q$. But is it correct? I believe $R_q$ is not field (it is not even integral domain) and so not every element there has it's multiplicative inverse.

Or maybe I misunderstood the notation?

Thank you for any advice.

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I believe $R_q$ is not field

Yes, it is not a field in general. $R_q$ is a field if, and only if, $q$ is a prime and $X^n+1$ is irreducible over $\mathbb{Z}_q[X]$.

However, you are not forced to pick $q$ as a power of two. As section 4.2 of the original paper explains, the only restriction over $q$ is

$$q \ge 2^{8\kappa \lambda}n^{o(\kappa)}$$

In particular, you can let $q$ to be prime (as it is supposed to be in Lemma 5).

And even if you don't select $q$ as a prime, you just have to check that $z$ has an inverse in $R_q$ and that the order of $z^{-1}$ is bigger than $\kappa$. If this is the case, then you are fine, if not, then sample another $z$.

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