We show that addition mod 2n is CCZ-equivalent to a quadratic vectorial Boolean function. We use this to reduce the solution of systems di erential equations of addition to the solution of a system of linear equations and to derive a fully explicit formula for the correlation coe cients, which leads to new results about the Walsh transform of addition mod 2n . The results have applications in the cryptanalysis of cyptographic primitives which use addition mod 2n .