# On the Optimality of Gauss's Algorithm over Euclidean Imaginary Quadratic Fields

## TL;DR

This paper investigates the effectiveness of Gauss's algorithm for reducing algebraic lattices over imaginary quadratic fields, demonstrating polynomial-time success when the ring is Euclidean.

## Contribution

It proves that the algebraic variant of Gauss's algorithm finds a basis corresponding to the successive minima over Euclidean imaginary quadratic fields in polynomial time.

## Key findings

- Gauss's algorithm is effective over Euclidean imaginary quadratic fields.
- The algorithm finds a basis aligned with the lattice's successive minima.
- Polynomial-time complexity is established for the reduction process.

## Abstract

In this paper, we continue our previous work on the reduction of algebraic lattices over imaginary quadratic fields for the special case when the lattice is spanned over a two dimensional basis. In particular, we show that the algebraicvariant of Gauss algorithm returns a basis that corresponds to the successive minima of the lattice in polynomial time if the chosen ring is Euclidean.