Computing the $p$-adic Canonical Quadratic Form in Polynomial Time

An $n$-ary integral quadratic form is a formal expression $Q(x_1,\mathrm{.}\mathrm{.},x_n)={\sum }_{1\le i,j\le n}{a}_{ij}{x}_i{x}_j$ in $n$-variables $x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n$, where $a_{ij}=a_{ji}\in \mathbb{Z}$. We present a randomized polynomial time algorithm that given a quadratic form $Q(x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n)$, a prime $p$, and a positive integer $k$ outputs a $$\mathtt{U}$ \in $\text{GL}$_n($\mathbb{Z}$/p^k$\mathbb{Z}$)$suchthat$$\mathtt{U}$$transforms$Q$ to its $p$-adic canonical form.