  Cubical cochains are equipped with an associative product, dual to the Serre
diagonal, lifting the graded commutative structure in cohomology. In this work
we introduce through explicit combinatorial methods an extension of this
product to a full ${E_\infty}$-structure. As an application we prove that the
Cartan-Serre map, which relates the cubical and simplicial singular cochains of
spaces, is a quasi-isomorphism of ${E_\infty}$-algebras.
