A Division Algorithm for the Gaussian Integers' Minimal Euclidean Function
Hester Graves
TL;DR
This work delivers the first division algorithm for the Gaussian integers ${\mathbb{Z}}[i]$ with respect to the minimal Euclidean function ${\phi}_{{\mathbb{Z}}[i]}$, the pointwise minimum over all Euclidean functions. Building on Graves' explicit formula for ${\phi}_{{\mathbb{Z}}[i]}$ via valuations and the ${w_m}$ sequence, and leveraging $(1+i)$-ary expansions, the paper shows how to perform Euclidean division in ${\mathbb{Z}}[i]$ using ${\phi}_{{\mathbb{Z}}[i]}$. When a Gauss remainder already yields a descent in ${\phi}$, standard adjustments suffice; otherwise, the authors construct alternate remainders within the coset $r+b\mathbb{Z}[i]$ to guarantee a reduced ${\phi}$ in the next step, even in challenging sign-alignment cases. The resulting algorithm runs in ${O}(\log N)$ time for ${N}$-bit inputs, enabling efficient computation of the Euclidean algorithm in ${\mathbb{Z}}[i]$ under the minimal Euclidean function and advancing practical computations in this classical number-theoretic setting.
Abstract
The usual division algorithms on $\mathbb{Z}$ and $\mathbb{Z}[i]$ measure the size of remainders using the norm function. These rings are Euclidean with respect to several functions. The pointwise minimum of all Euclidean functions $f: R \setminus 0 \rightarrow \mathbb{N}$ on a Euclidean domain $R$ is itself a Euclidean function, called the minimal Euclidean function and denoted by $φ_R$. The integers, $\mathbb{Z}$, and the Gaussians, $\mathbb{Z}[i]$, are the only rings of integers of number fields for which we have a formula to compute their minimal Euclidean functions, $φ_{\mathbb{Z}}$ and $φ_{\mathbb{Z}[i]}$. This paper presents the first division algorithm for $\mathbb{Z}[i]$ relative to $φ_{\mathbb{Z}[i]}$, empowering readers to perform the Euclidean algorithm on $\mathbb{Z}[i]$ using its minimal Euclidean function.