# What are the security implications of knowing the private polynomial $\mathcal{F}$

First, affine transformations $$S,T$$ are defined by $$S=A_1+v_s, T=A_2+v_t$$. Let the private polynomial function $$\mathcal{F}$$ be known. The short description of the public key map is $$P(X) = T \circ \mathcal{F} \circ S(X)$$. Recall that common multivariate schemes are based on the security assumption of IP2, this is, the Isomorphism Of Polynomials with two secrets.

It results that only applying transformations $$T$$ and $$S=I_n$$ (the identity) makes the scheme vulnerable by linear algebra attacks. We can recover the transformation matrix $$T$$ since we already know that it transformed $$\mathcal{F}$$ to the public key matrix.

In addition, using $$T=I_n$$ and $$S$$ an arbitrary invertible affine function, the problem now is equivalent to IP1, which can be stated from the so called Polynomial Affine Equivalence problem (PAE) (see [1] section $$2.3$$ and $$3$$). Given $$\mathcal{F}(x)$$ and $$\mathcal{F}(S(x))$$ find $$S$$. Schemes based on IP1 can be vulnerable to attacks if not designed with care, see [2].

Then we are restricted to IP2 if we want to mantain security from this point. Recall that including abritrary invertible affine transformations $$S,T$$ would make the input-output of $$\mathcal{F}$$ unknown, even knowing $$\mathcal{F}$$. Then as $$T$$ transforms an unknown input, we learn nothing from here nor from the view of PAE or IP1, as we only know one set of quadratic equations $$B$$ but the set $$A$$ remains hidden by the transformation $$T$$.

In the previous description I tried to demonstrate that the knowledge of the private polynomial $$\mathcal{F}$$ has no effect when the scheme is based on the assumptions of IP2.

However, is there any complication if the private polynomial $$\mathcal{F}$$ is publicly known from other perspective?

My intiution says that recovering $$T$$ is fatal as $$\mathcal{F}(X), \mathcal{F^{-1}}(X)$$ are known the attacker obtains $$F^{-1}(Y)=S(X)$$. Then to solve IP1 he must establish an affine equivalence function $$S$$ between $$\mathcal{F}(X)$$ and $$\mathcal{F}(S(X))$$

If the attacker doesn't possess $$\mathcal{F}$$ knowing $$T$$ will yield $$F \circ S(X)$$ but yet he doesn't know $$\mathcal{F}(X)$$, thus IP1 cannot be followed from here.

Let's answer my question after revisiting the research. Start by quoting the following text from [2]. Note that I've changed the variables names to fit my description. Here $$P(X) = T \circ F \circ S(X)$$.
An open question is to know whether or not these schemes remain secure if the set of polynomials $$\mathcal{F}$$ is public. In this situation, the security of these schemes relies not only on the difficulty of finding a common zero of a system of non linear equations but also on the difficulty of $$\mathcal{IP}$$ (Isomorphism of Polynomials) problem (when $$\mathcal{F}$$ and $$\mathcal{P}$$ are given, the problem is to recover the pairs $$(A_1,v_s)$$ and $$(A_2,v_t)$$).
In this post the private polynomial $$\mathcal{F}$$ is public, moreover described methods reduce IP2 to IP1 if $$T$$ is known, and IP1 is totally insecure [2]. In the other hand, if $$S$$ is known $$T$$ is recovered by basic linear algebra.
Then I conclude that a cryptosystem where $$\mathcal{F}$$ is known depends on the IP2 assumption (finding $$(S,T))$$ and the $$\mathcal{MQ}$$ problem implicitly. You can follow up from [2] section $$6.2$$
EDIT: Moreover, it results that IP1 is totally broken when the private polynomial is known. However if the polynomial $$\mathcal{F}$$ is unknown, having access to an oracle then querying for canonical vectors as a plaintext will reveal $$S$$ too. A secure cryptosystem based on MPKC must satisfy the IP2 assumption from design, between others.