Higher-order differential attacks, introduced by Knudsen in 1994, are the first family of attacks against block ciphers which exploit some specific property of the polynomial representation of the cipher. Indeed, these attacks rely on the fact that, for all keys, the involved multivariate polynomial does not have maximal degree. This idea has then been generalized by several authors and has led to the notion of cube distinguishers, and more recently to the so-called division property. Both generalizations actually exploit the fact that some given monomials do not appear in the polynomials. In this talk, I will present some unified view of these attacks, and I will show how such algebraic properties propagate through the successive layers of iterated primitives. Joint work with Christina Boura (Université de Versailles St Quentin)