On the Higher Order Nonlinearities of Algebraic Immune Functions

One of the most basic requirements concerning Boolean functions used in cryptosystems is that they must have high algebraic degrees. This simple criterion is not always well adapted to the concrete situation in which Boolean functions are used in symmetric cryptography, since changing one or several output bits of a Boolean function considerably changes its algebraic degree while it may not change its robustness. The proper characteristic is the r-th order nonlinearity profile (which includes the first-order nonlinearity). However, studying it is difficult and almost no paper, in the literature, has ever been able to give general effective results on it. The values of the nonlinearity profile are known for very few functions and these functions have little cryptographic interest. A recent paper has given a lower bound on the nonlinearity profile of functions, given their algebraic immunity. We improve upon it, and we deduce that it is enough, for a Boolean function, to have high algebraic immunity, for having non-weak low order nonlinearity profile (even when it cannot be evaluated), except maybe for the first order.