some nondegeneracy assumption, Theorem \[th:probmain\] can be turned into a deterministic algorithm solving [`IP1S`]{}over the base field ${{\mathbb}{K}}$ or an extension thereof. That is:

\[th:main\] Under Assumption \[as:regular\] and the assumption that one of the quadratic form is nondegenerate, there is a deterministic polynomial-time algorithm for solving quadratic-[`IP1S`]{}over an extension of ${{\mathbb}{K}}$ of polynomial degree in $n$. Furthermore, if the space of matrices satisfying equations  has dimension $1$, then the algorithm can solve quadratic-[`IP1S`]{}over ${{\mathbb}{K}}$.

Let us note that assuming that one of the Hessian matrix is invertible is not a strong assumption when the size of ${{\mathbb}{K}}$ is not too small. Indeed, the probability of picking a random invertible symmetric matrix over ${{\mathbb}{F}}_q$ is $$\frac{\prod_{i=1}^{n}(1-q^{-i})}
{\prod_{i=1}^{\lfloor n/2\rfloor}(1-q^{-2i})}
=\prod_{i=1}^{\lceil n/2\rceil}(1-q^{-2i+1}),$$ see [@Carlitz1954 Equations 4.7 and 4.8].

If $m\geq 3$, for random matrices $H_1,\ldots,H_m$, the set of solutions of equations  is a $1$-dimensional matrix space. This allows us to solve quadratic-[`IP1S`]{}in polynomial-time over ${{\mathbb}{K}}$. In Section \[ss:bench\], we present our timings for solving [`IP1S`]{}. These experiments confirm that for randomly chosen matrices, our method solves [`IP1S`]{}over ${{\mathbb}{K}}$. We remark also that our method succeeds to solve [`IP1S`]{}over ${{\mathbb}{F}}_2$ for public-keys whose sizes are much bigger than practical ones.

In Section \[s:enumeration\], 