On the algebraic degree stability of vectorial Boolean functions when restricted to affine subspaces

We study the behaviour of the algebraic degree of vectorial Boolean functions when their inputs are restricted to an affine subspace of their domain. Functions which maintain their degree on all subspaces of as high a codimension as possible are particularly interesting for cryptographic applications. For functions which are power functions $x^d$ in their univariate representation, we fully characterize the exponents $d$ for which the algebraic degree of the function stays unchanged when the input is restricted to spaces of codimension 1 or 2. For codimensions $k\ge 3$, we give a sufficient condition for the algebraic degree to stay unchanged. We apply these results to the multiplicative inverse function, as well as to the Kasami functions. We define an optimality notion regarding the stability of the degree on subspaces, and determine a number of optimal functions, including the multiplicative inverse function and the quadratic APN functions. We also give an explicit formula for counting the functions that keep their algebraic degree unchanged when restricted to hyperplanes.