Distinguishing properties and applications of higher order derivatives of Boolean functions

Higher order differential cryptanalysis is based on a property of higher order derivatives of Boolean functions such that derivative of a Boolean function reduces its degree at least 1 and continuously taking derivatives eventually yields a zero function. A quicker degree reduction means a lower data complexity in cryptanalysis, which can be determined by fast point at which the derivative reduces the degree at least 2.In this paper, we show that the set of the fast points of a Boolean function constitutes a linear subspace and its dimension plus the degree of the function is at most the size of the function. We also show that non-zero fast point exists in every n  -variable Boolean function of degree n-1n-1, every symmetric Boolean function of degree d   where n≢d(mod2) or every quadratic Boolean function of odd number variables, which help us distinguish a few block ciphers and propose a new design principle about degree for block cipher.