On the Scaled Inverse of $(x^i-x^j)$ modulo Cyclotomic Polynomial of the form $Phi_{p^s}(x)$ or $Phi_{p^s q^t}(x)$

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 ${x^k}_{kinmathbb{Z}}$ in $mathbb{Z}[x]/Phi_{pq}(x)$ which might be of independent interest.