Division is in uniform TC°

Abstract

Integer division has been known since 1986 [4,13,12] to be in slightly non-uniform TC°, i.e., computable by polynomial-size, constant depth threshold circuits. This has been perhaps the outstanding natural problem known to be in a standard circuit complexity class, but not known to be in its uniform version. We show that indeed division is in uniform TC°. A key step of our proof is the discovery of a first-order formula expressing exponentiation modulo any number of polynomial size. © 2011 Springer-Verlag Berlin Heidelberg.