ON THE SCALED INVERSE OF (xi − xj) MODULO CYCLOTOMIC POLYNOMIAL OF THE FORM Φps (x) OR Φpsqt (x)

• Published in: Journal of the Korean Mathematical Society pp. 621 - 634
• Main Authors: 이기우, 천정희, 김동우, 김두형
• Format: Journal Article
• Language: English
• Published: 대한수학회 01.05.2022
• Subjects: 수학
• ISSN: 0304-9914; 2234-3008
• DOI: 10.4134/JKMS.j210446

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. KCI Citation Count: 0