Bounds on the OBDD-size of integer multiplication via universal hashing

Abstract

Bryant [On the complexity of VLSI implementations and graph representations of boolean functions with applications to integer multiplication, IEEE Trans. Comput. 40 (1991) 205–213] has shown that any OBDD for the function MUL n - 1 , n , i.e. the middle bit of the n-bit multiplication, requires at least 2 n / 8 nodes. In this paper a stronger lower bound of essentially 2 n / 2 / 61 is proven by a new technique, using a universal family of hash functions. As a consequence, one cannot hope anymore to verify e.g. 128-bit multiplication circuits using OBDD -techniques because the representation of the middle bit of such a multiplier requires more than 3 · 10 17 OBDD -nodes. Further, a first non-trivial upper bound of 7 / 3 · 2 4 n / 3 for the OBDD -size of MUL n - 1 , n is provided.