Nonlinearity of Boolean functions: an algorithmic approach based on multivariate polynomials

Abstract: We compute the nonlinearity of Boolean functions with Groebner basis techniques, providing two algorithms: one over the binary field and the other over the rationals. We also estimate their complexity. Then we show how to improve our rational algorithm, arriving at a worst-case complexity of $O(n2^n)$ operations over the integers, that is, sums and doublings. This way, with a different approach, we reach the same complexity of established algorithms, such as those based on the fast Walsh transform.