we consider the counting problem \#[`IP1S`]{}associated to [`IP1S`]{}for quadratic (homogeneous) polynomials in its canonical form (as defined in Theorem \[th:normal\]). Note that such a counting problem is also related to cryptographic concerns. It corresponds to evaluating the number of equivalent secret-keys in [`MPKC`]{}, see [@DFPW11; @WP11]. Given homogeneous quadratic polynomials $\big( {{\mathbf}{f}}=(f_1,\ldots,f_m),{{\mathbf}{g}}=(g_1,\ldots,g_m)\big) \in
{{\mathbb}{K}}[x_1,\ldots,x_n]^m \times {{\mathbb}{K}}[x_1,\ldots,x_n]^m$, we want to count the number of invertible matrices $A \in \operatorname{GL}_n({{\mathbb}{K}})$ such that $
{{\mathbf}{g}}({{\mathbf}{x}})={{\mathbf}{f}}(A \cdot {{\mathbf}{x}}).$ To do so, we define:

Let ${{\mathbf}{f}}=(f_1,\ldots,f_m) \in {{\mathbb}{K}}[x_1,\ldots,x_n]^m$, we shall call *automorphism group of ${{\mathbf}{f}}$* the set: $$\mathcal{G}_{{{\mathbf}{f}}}=\{ A \in \operatorname{GL}_n({{\mathbb}{K}}) \mid {{\mathbf}{f}}(A\cdot {{\mathbf}{x}}) ={{\mathbf}{f}}({{\mathbf}{x}})
  \}.$$

If ${{\mathbf}{f}}\sim {{\mathbf}{g}}$, the automorphism groups of ${{\mathbf}{f}}$ and ${{\mathbf}{g}}$ are similar. Thus, the size of the automorphism group of ${{\mathbf}{f}}$ allows us to count the number of invertible matrices mapping ${{\mathbf}{f}}$ to ${{\mathbf}{g}}$. For quadratic homogeneous polynomials, the automorphism group coincides with the subset of regular matrices in the centralizer $\operatorname{{\mathcal}{C}}({{\mathcal}{H}})$ of the Hessian matrices ${\mathcal}{H}$ associated to ${{\mathbf}{f}}$. Taking $\alpha$ an algebraic element of degree $m$ over ${{\mathbb}{K}}={{\mathbb}{F}}_q$, let us assume the Jordan normal form of $H=\sum_{i=i}^m\alpha^{i-1}\,H_i$ has Jordan blocks of sizes $s_{i,1}\leq\cdots\leq s_{i,d_i}$ associated to eigenvalue $\zeta_i$, for $i,1\leq i\leq r$. Then, as a consequence of 