Distinguishing Properties of Higher Order Derivatives of Boolean Functions

Higher order differential cryptanalysis is based on the property of higher order derivatives of Boolean functions that the degree of a Boolean function can be reduced by at least 1 by taking a derivative on the function at any point. We define fast point as the point at which the degree can be reduced by at least 2. In this paper, we show that the fast points of a n-variable Boolean function form a linear subspace and its dimension plus the algebraic degree of the function is at most n. We also show that non-trivial fast point exists in every n-variable Boolean function of degree n− 1, every symmetric Boolean function of degree d where n 6≡ d (mod 2) and every quadratic Boolean function of odd number variables. Moreover we show the property of fast points for n-variable Boolean functions of degree n− 2.