New constructions of resilient functions with strictly almost optimal nonlinearity via non-overlap spectra functions.

The design of n -variable t -resilient functions with strictly almost optimal (SAO) nonlinearity ( > 2 n − 1 − 2 n 2 , n even) appears to be a rather difficult task. The known construction methods commonly use a rather large number (exactly ∑ i = t + 1 n / 2 ( n / 2 i ) ) of affine subfunctions in n 2 variables which can induce some algebraic weaknesses, making these functions susceptible to certain types of guess and determine cryptanalysis and dynamic cube attacks. In this paper, the concept of non-overlap spectra functions is introduced, which essentially generalizes the idea of disjoint spectra functions on different variable spaces. Two general methods to obtain a large set of non-overlap spectra functions are given and a new framework for designing infinite classes of resilient functions with SAO nonlinearity is developed based on these. Unlike previous construction methods, our approach employs only a few n /2-variable affine subfunctions in the design, resulting in a more favourable algebraic structure. It is shown that these new resilient SAO functions properly include all the existing classes of resilient SAO functions as a subclass. Moreover, it is shown that the new class provides a better resistance against (fast) algebraic attacks than the known functions with SAO nonlinearity, and in addition these functions are more robust to guess and determine cryptanalysis and dynamic cube attacks. [ABSTRACT FROM AUTHOR]