Multivariate cryptography is one of the candidates for post-quantum cryptography. Multivariate schemes are usually constructed by applying two secret affine invertible transformations \({\mathcal {S}},{\mathcal {T}}\) to a set of multivariate polynomials \(\mathcal {F}\) (often quadratic). The polynomials \(\mathcal {F}\) possess a trapdoor that allows the legitimate user to find a solution of the corresponding system, while the public polynomials \({\mathcal {G}}={\mathcal {S}}\circ {\mathcal {F}}\circ {\mathcal {T}}\) look like random polynomials. The polynomials \({\mathcal {G}}\) and \({\mathcal {F}}\) are said to be affine equivalent. In this article, we present a more general way of constructing a multivariate scheme by considering the CCZ equivalence, which has been introduced and studied in the context of vectorial Boolean functions.
