Upper bounds on the multiplicative complexity of symmetric Boolean functions

A special metric of interest about Boolean functions is multiplicative complexity (MC): the minimum number of AND gates sufficient to implement a function with a Boolean circuit over the basis {XOR, AND, NOT}. In this paper we study the MC of symmetric Boolean functions, whose output is invariant upon reordering of the input variables. Based on the Hamming weight method from Muller and Preparata (J. ACM 22(2), 195–201, 1975), we introduce new techniques that yield circuits with fewer AND gates than upper bounded by Boyar et al. (Theor. Comput. Sci. 235(1), 43–57, 2000) and by Boyar and Peralta (Theor. Comput. Sci. 396(1–3), 223–246, 2008). We generate circuits for all such functions with up to 25 variables. As a special focus, we report concrete upper bounds for the MC of elementary symmetric functions \({{\Sigma }^{n}_{k}}\) and counting functions \({E^{n}_{k}}\) with up to n = 25 input variables. In particular, this allows us to answer two questions posed in 2008: both the elementary symmetric \({{\Sigma }^{8}_{4}}\) and the counting \({E^{8}_{4}}\) functions have MC 6. Furthermore, we show upper bounds for the maximum MC in the class of n-variable symmetric Boolean functions, for each n up to 132.