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

Abstract

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 $a(x)b(x)=c(\mathrm{mod}f(x))$ 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.