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We propose a general approach to construct cryptographic significant Booleanfunctions 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 unbalancedfunctions are constructed first via this approach, which coincides with avariant of the unbalanced class of generalized Tu-Deng functions in the case$r=1$. This class of functions have high algebraic degree, but their algebraicimmunity does not exceeds $m$, which is impossible to be optimal when $r>1$. Bymodifying these unbalanced functions, we obtain a class of balanced functionswhich have optimal algebraic degree and high nonlinearity (shown by a lowerbound we prove). These functions have optimal algebraic immunity provided acombinatorial conjecture on binary strings which generalizes the Tu-Dengconjecture is true. Computer investigations show that, at least for smallvalues of number of variables, functions from this class also behave wellagainst fast algebraic attacks.

Constructing Boolean Functions With Potential Optimal Algebraic Immunity Based on Additive Decompositions of Finite Fields