solve [`IP`]{}by reducing it to a system of nonlinear equations whose variables are the unknown coefficients of the matrices. It was conjectured in [@FaugerePerret06], but never proved, that the corresponding system of nonlinear equations can be solved in polynomial time as soon as the [`IP`]{}instances considered are not homogeneous. Indeed, by slicing of the polynomials degree by degree, one can find equations in the coefficients of the transformation allowing one to recover the transformation. More recently, [@DBLP:conf/eurocrypt/BouillaguetFV13] presented exponential (in the number of variables $n$) algorithms for solving quadratic homogeneous instances of [`IP`]{}over finite fields. This situation is clearly unsatisfying, and suggests that an important open problem for [`IP`]{}is to identify large class of instances which can be solved in (randomized) polynomial time.

An important special case of [`IP`]{}is the *[`IP`]{}problem with one secret* ([`IP1S`]{}for short), where $B$ is the identity matrix. From a cryptographic point of view, the most natural case encountered for equivalence problems is inhomogeneous polynomials with affine transformations. For [`IP1S`]{}, we show that such a case can be handled in the same way as homogeneous instances with linear transformations (see Proposition \[prop:homo\]). As a side remark, we mention that there exist more efficient methods to handle 