Generating a Quadratic Forms from a Given Genus

TL;DR

This paper presents a randomized algorithm that efficiently generates a quadratic form within a specified genus, with complexity depending polynomially on dimension and determinant size, under GRH assumptions.

Contribution

It introduces a novel randomized algorithm for generating quadratic forms from a given genus with polynomial complexity under GRH.

Findings

Algorithm operates in poly$(n, ext{log } d)$ time

Works for non-empty genus in $n$ dimensions

Assumes Generalized Riemann Hypothesis for complexity bounds

Abstract

Given a non-empty genus in $n$ dimensions with determinant $d$, we give a randomized algorithm that outputs a quadratic form from this genus. The time complexity of the algorithm is poly$(n,\log d)$; assuming Generalized Riemann Hypothesis (GRH).