Boolean functions with maximum algebraic immunity: further extensions of the Carlet–Feng construction

The algebraic immunity of Boolean functions is studied in this paper. More precisely, having the prominent Carlet–Feng construction as a starting point, we propose a new method to construct a large number of functions with maximum algebraic immunity. The new method is based on deriving new properties of minimal codewords of the punctured Reed–Muller code RM ⋆ ( ⌊ n - 1 2 ⌋ , n ) for any n , allowing—if n is odd—for efficiently generating large classes of new functions that cannot be obtained by other known constructions. It is shown that high nonlinearity, as well as good behavior against fast algebraic attacks, is also attainable.