Efficient Oblivious Branching Programs for Threshold and Mod Functions

In his survey paper on branching programs, Razborov Raz91] asked the following question: Does every rectiier-switching network computing the majority of n bits have size n 1++(1) ? We answer this question in the negative by constructing a simple oblivious branching program of size O n log 3 n log log n log log log n ! for computing any threshold function. This improves the previously best known upper bound of O(n 3=2) due to Lupanov Lup65]. We also construct oblivious branching programs of size o(n log 4 n) for computing general mod functions. All previously known constructions for computing general mod functions have size (n 3=2).