A Lower Bound for Primality

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. In this note, 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. ∗ A preliminary version of this work appeared in Proc. 14th Annual IEEE Conference on Computational Complexity, 1999, pp. 10–14. † Supported in part by NSF grant CCR-9734918. ‡ Supported in part by NSF grant CCR-9700239. § Supported in part by ARC grant A69700294. 1 

• Type: Conference
• Year: 1999
• Where: COCO
• Authors: Eric Allender, Michael E. Saks, Igor Shparlinski
• Keywords: Algorithms, COCO 1999, Lower Bounds, Prime Numbers, Square-free Numbers