On the Higher Order Nonlinearities of Algebraic Immune Functions

Abstract

Abstract. 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.

Further properties of several classes of Boolean functions with optimum algebraic immunity

“…Correspondingly, the elements y 1 , y 2 , · · · , y |U | of U are listed as: 1, 2). From the definition above, it can be verified T and U satisfy the conditions in Equality (5). Furthermore, S = T and V = U also satisfy the conditions in Equality (5).…”

“…This shows f is included in the class of functions described as Proposition 3.1 if T , S, U and V satisfy the conditions in (5).…”

“…In fact, it must keep high degree even if a few output bits are modified. In other words, it must have high nonlinearity profile [5].…”

“…Let f ∈ B n be defined by (4). If the sets T , S, U and V satisfy the conditions in (5), then AI(f ) = n/2.…”

A construction of Boolean functions with good cryptographic properties

“…If we can find g of low degree and h of algebraic degree not much larger than n/2 such that f g = h, then f is considered to be weak against fast algebraic attacks [11,17]. The higher order nonlinearities of functions with high (fast) algebraic immunity is also not very low [6,26,37].…”