linear change of variables (a similar reduction holds between ${{\mathbb}{F}}$-algebra Isomorphism and cubic equivalence of polynomials). This strongly suggests that solving equivalence problems efficiently is a very challenging algorithmic task.

In cryptography, the hardness of deciding equivalence between two sets of $m$ polynomials with respect to an invertible linear change of variables is the security core of several cryptographic schemes: the seminal zero-knowledge <span style="font-variant:small-caps;">ID</span> scheme of [@DBLP:conf/eurocrypt/Patarin96], and more recently group/proxy signature schemes ([@DBLP:conf/nss/TangX12; @Tang2014; @DBLP:conf/ispec/YangTY11]). Note that there is a subtle difference between the equivalence problem considered in [@DBLP:conf/stacs/AgrawalS06; @Kayal2011; @Saxena2006] and the one considered in cryptographic applications.

Whilst [@DBLP:conf/stacs/AgrawalS06; @Kayal2011; @Saxena2006] restrict their attention to $m=1$, arbitrary $m \geq 1$ is usually considered in cryptographic applications. In the former case, the problem is called *Polynomial Equivalence* ([`PolyEquiv`]{}), whereas it is called *Isomorphism of Polynomials with One Secret* ([`IP1S`]{}) problem in the latter case. We emphasize that the hardness of equivalence can drastically vary in function of $m$. An interesting example is the case of quadratic forms. The problem is completely solved when $m=1$, but no polynomial-time algorithm exists for deciding simultaneous equivalence of quadratic forms. In this paper, we make a step ahead to close this gap 