Computing the Determinant via the Generalized Euclidean Algorithm

Abstract

We present an algorithm with a natural geometric interpretation for computing the determinant of a matrix $B\in\mathbb{Z}^{d\times d}$. It improves upon the current fastest deterministic algorithms by a factor of $d^{\omega(1)+1-\omega(2)}\approx d^{0.1213}$, where $\omega(k)$ denotes the exponent required for multiplying a $d\times d$ matrix with a $d\times d^k$ matrix. Our approach builds on a recent result of Klein and Reuter (STOC 2025), who introduced a novel algorithmic idea for lattice basis computation that can be viewed as extending the Euclidean algorithm from $\mathbb{Z}$ to $\mathbb{Z}^d$. By adapting their techniques, we compute the determinant with the same bit complexity as applying the generalized Euclidean algorithm to an input matrix $A\in\mathbb{Z}^{d\times 2d}$ with $\|A\| = \|B\|$, namely $\tilde{O}(d^{\omega(2)}\log\|B\|)$. Prior to this work, the fastest deterministic algorithm for computing the determinant required $\tilde{O}(d^{\omega(1)+1}\log\|B\|)$ bit operations.