The stability of the algebraic degree of Boolean functions when restricted to affine spaces

Abstract

We study the $n$-variable Boolean functions which keep their algebraic degree unchanged when they are restricted to any (affine) hyperplane, or more generally to any affine space of a given co-dimension $k$. For cryptographic applications it is of interest to determine functions $f$ which have a relatively high degree and also maintain this degree when restricted to affine spaces of co-dimension $k$ for $k$ ranging from 1 to as high a value as possible. This highest value will be called the restriction degree stability of $f$, denoted by $deg\_stab(\mathrm{f})$. We give several necessary and/or sufficient conditions for $f$ to maintain its degree on spaces of co-dimension $k$; we show that this property is related to the property of having ``fast points'' as well as to other properties and parameters. The value of $deg\_stab(\mathrm{f})$ is determined for functions of degrees $r\in \{1,2,n-2,n-1,n\}$ and for functions which are direct sums of monomials; we also determine the symmetric functions which maintain their degree on any hyperplane. Furthermore, we give an explicit formula for the number of functions which maintain their degree on all hyperplanes. Finally, using our previous results and some computer assistance, we determine the behaviour of all the functions in 8 variables, therefore determining the optimal ones (i.e. with highest value of $deg\_stab(\mathrm{f})$) for each degree.