Constructing Boolean Functions With Potential Optimal Algebraic Immunity   Based on Additive Decompositions of Finite Fields
  We propose a general approach to construct cryptographic significant Boolean functions of $(r+1)m$ variables based on the additive decomposition $\mathbb{F}_{2^{rm}}\times\mathbb{F}_{2^m}$ of the finite field $\mathbb{F}_{2^{(r+1)m}}$, where $r$ is odd and $m\geq3$. A class of unbalanced functions are constructed first via this approach, which coincides with a variant of the unbalanced class of generalized Tu-Deng functions in the case $r=1$. This class of functions have high algebraic degree, but their algebraic immunity does not exceeds $m$, which is impossible to be optimal when $r>1$. By modifying these unbalanced functions, we obtain a class of balanced functions which have optimal algebraic degree and high nonlinearity (shown by a lower bound we prove). These functions have optimal algebraic immunity provided a combinatorial conjecture on binary strings which generalizes the Tu-Deng conjecture is true. Computer investigations show that, at least for small values of number of variables, functions from this class also behave well against fast algebraic attacks. 
Baofeng Wu, Qingfang Jin, Zhuojun Liu, Dongdai Lin
cs.CR