The isomorphism of polynomials problem applied to multivariate quadratic cryptography

The threat quantum computing poses to traditional cryptosystems (such as RSA, elliptic-curve cryptosystems) has brought about the appearance of new systems resistant to it: among them, multivariate quadratic public-key ones. The security of the latter kind of cryptosystems is related to the isomorphism of polynomials (IP) problem. In this work, we study some aspects of the equivalence relation the IP problem induces over the set of quadratic polynomial maps and the determination of its equivalence classes. We contribute two results. First, we prove that when determining these classes, it suffices to consider the affine transformation on the left of the central vector of polynomials to be linear. Second, for a particular case, we determine an explicit system of invariants from which systems of equations whose solutions are the elements of an equivalence class can be derived.