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13.
New constructions of resilient functions with strictly almost optimal nonlinearity via non-overlap spectra functions
Wei, Yongzhuang ...
The design of ▫$n$▫-variable ▫$t$▫-resilient functions with strictly almost optimal (SAO) nonlinearity ▫$(>2^{n-1}-2^\frac{n}{2}$▫, ▫$n$▫ even) appears to be a rather difficult task. The known ...construction methods commonly use a rather large number (exactly ▫$\sum_{i=t+1}^{n/2}\binom{n/2}{i}$▫) of affine subfunctions in ▫$\frac{n}{2}$▫ 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.
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