Sampling a Uniform Random Solution of a Quadratic Equation Modulo $p^k$

TL;DR

This paper introduces a randomized polynomial-time algorithm for uniformly sampling solutions to quadratic equations modulo prime powers, advancing computational methods in number theory.

Contribution

It presents the first efficient randomized algorithm for uniform sampling solutions of quadratic equations modulo p^k.

Findings

Algorithm runs in polynomial time in n, k, log p, and log t.

Successfully samples solutions uniformly at random.

Applicable to a broad class of quadratic forms.

Abstract

An $n$-ary integral quadratic form is a formal expression $Q(x_1,...,x_n)=\sum_{1\leq i,j\leq n}a_{ij}x_ix_j$ in $n$-variables $x_1,...,x_n$, where $a_{ij}=a_{ji} \in \mathbb{Z}$. We present a poly$(n,k, \log p, \log t)$ randomized algorithm that given a quadratic form $Q(x_1,...,x_n)$, a prime $p$, a positive integer $k$ and an integer $t$, samples a uniform solution of $Q(x_1,...,x_n)\equiv t \bmod{p^k}$.