Algebraic Immunity Hierarchy of Boolean Functions

Algebraic immunity of Boolean functions is a very important concept in recently introduced algebraic attacks of stream cipher. For a n-variable Boolean function f , the algebraic immunity AIn(f) takes values in {0, 1, . . . , d2 e}. For every k in this range, denote Bn,k the set of all n-variable Boolean functions with algebraic immunity k, and we know that Bn,k is always non-empty. According to the algebraic immunity, we can form a hierarchy of Boolean functions. Trivially, |Bn,0| = 2. In general, about this integer sequence |Bn,k|, k = 0, 1, . . . , d2 e, very few results are known. In this paper, we show an explicit formula for |Bn,1|. That is, we obtain an exact formula for the number of Boolean functions with algebraic immunity one. This is the first exact formula for the terms in the above integer sequence. We also give a tight upper bound for nonlinearity of Boolean functions with algebraic immunity one.