Counting Boolean functions with specified values in their Walsh spectrum

The problem of counting Boolean functions with specified number s of Walsh coefficients ω in their Walsh spectrum is discussed in this paper. Strategies to solve this problem shall help solving many more problems related to desired cryptographic features of Boolean functions such as nonlinearity, resiliency, algebraic immunity, etc. In an attempt to study this problem, we present a new framework of solutions. We give results for |ω|≥2n−1 and for all s, in line with a previous work of Wu (1998) [12]. We also provide various results such as existence and construction for some s when ω=0, multiplicities for all ω and naive bounds on s for ω>2n/2.