A lower bound for primality

Abstract

Recent work by Bernasconi, Damm, and Shparlinski showed that the set of square-free numbers is not in AC0 and raised as an open question whether similar (or stronger) lower bounds could be proved for the set of prime numbers. We show that the Boolean majority function is AC0-Turing reducible to the set of prime numbers (represented in binary). From known lower bounds on Maj (due to Razborov and Smolensky) we conclude that primality cannot be tested in AC0[p] for any prime p. Similar results are obtained for the set of square-free numbers and for the problem of computing the greatest common divisor of two numbers.