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abstract: |
    Let $\mathbf{f}=(f_1,\ldots,f_m)$ and $\mathbf{g}=(g_1,\ldots,g_m)$ be two sets of $m\geq 1$ nonlinear polynomials in $\mathbb{K}[x_1,\ldots,x_n]$ ($\mathbb{K}$ being a field). We consider the computational problem of finding – if any – an invertible transformation on the variables mapping $\mathbf{f}$ to $\mathbf{g}$. The corresponding equivalence problem is known as *Isomorphism of Polynomials with one Secret* (`IP1S`) and is a fundamental problem in multivariate cryptography. Amongst its applications, we can cite Graph Isomorphism (`GI`) which reduces to equivalence of cubic polynomials with respect to an invertible linear change of variables, according to Agrawal and Saxena. The main result is a randomized polynomial-time algorithm for solving `IP1S` for quadratic instances – a particular case of importance in cryptography.

    To this end, we show that `IP1S` for quadratic polynomials can be reduced to a variant of the classical module isomorphism problem in representation theory. We show that we can essentially *linearize* the problem by reducing quadratic-`IP1S` to test the orthogonal simultaneous similarity of symmetric matrices; this latter problem was shown by Chistov, Ivanyos and Karpinski (<span style="font-variant:small-caps;">ISSAC</span> 1997) to be equivalent to finding an invertible matrix in the linear space $\mathbb{K}^{n \times n}$ of $n \times n$ matrices over $\mathbb{K}$ and to compute 