  The scaled inverse of a nonzero element $a(x)\in \mathbb{Z}[x]/f(x)$, where
$f(x)$ is an irreducible polynomial over $\mathbb{Z}$, is the element $b(x)\in
\mathbb{Z}[x]/f(x)$ such that $a(x)b(x)=c \pmod{f(x)}$ for the smallest
possible positive integer scale $c$. In this paper, we investigate the scaled
inverse of $(x^i-x^j)$ modulo cyclotomic polynomial of the form $\Phi_{p^s}(x)$
or $\Phi_{p^s q^t}(x)$, where $p, q$ are primes with $p<q$ and $s, t$ are
positive integers. Our main results are that the coefficient size of the scaled
inverse of $(x^i-x^j)$ is bounded by $p-1$ with the scale $p$ modulo
$\Phi_{p^s}(x)$, and is bounded by $q-1$ with the scale not greater than $q$
modulo $\Phi_{p^s q^t}(x)$. Previously, the analogous result on cyclotomic
polynomials of the form $\Phi_{2^n}(x)$ gave rise to many lattice-based
cryptosystems, especially, zero-knowledge proofs. Our result provides more
flexible choice of cyclotomic polynomials in such cryptosystems. Along the way
of proving the theorems, we also prove several properties of
$\{x^k\}_{k\in\mathbb{Z}}$ in $\mathbb{Z}[x]/\Phi_{pq}(x)$ which might be of
independent interest.
