A Randomised Algorithm for Checking the Normality of Cryptographic Boolean Functions

A Boolean function is called normal if it is constant on flats of certain dimensions. This property is relevant for the construction and analysis of cryptosystems. This paper presents an asymmetric Monte Carlo algorithm to determine whether a given Boolean function is normal. Our algorithm is far faster than the best known (deterministic) algorithm of Daum et al. In a first phase, it checks for flats of low dimension whether the given Boolean function is constant on them and combines such flats to flats of higher dimension in a second phase. This way, the algorithm is much faster than exhaustive search. Moreover, the algorithm benefits from randomising the first phase. In addition, by evaluating several flats implicitly in parallel, the time-complexity of the algorithm decreases further.