Constructing Higher Nonlinear Odd-Variable RSBFs With Optimal AI and Almost Optimal FAI

Rotation symmetric Boolean functions (RSBFs) are nowadays studied a lot because of its easy operations and good performance in cryptosystem. This paper constructs a new class of odd-variable RSBFs with optimal algebraic immunity (AI). The nonlinearity of the new function,
$2^{n-1}-\binom {n-1}{k}+2^{k-4}(k-3)(k-2)$
, is the highest among all existing RSBFs with optimal AI and known nonlinearity, and its algebraic degree is also almost highest. Besides, the class of functions have almost optimal fast algebraic immunity (FAI) at least for
$n < 17$
, which is actually the highest possible value for the designated number of variables.