Tabulating Pseudoprimes and Tabulating Liars

Abstract

This article explores the asymptotic complexity of two problems related to the Miller-Rabin-Selfridge primality test. The first problem is to tabulate strong pseudoprimes to a single fixed base a . It is now proven that tabulating up to x requires O ( x ) arithmetic operations and O ( x log x ) bits of space. The second problem is to find all strong liars and witnesses, given a fixed odd composite n . This appears to be unstudied, and a randomized algorithm is presented that requires an expected O ((log n ) 2 + | S ( n )|) operations (here S ( n ) is the set of strong liars). Although interesting in their own right, a notable application is the search for sets of composites with no reliable witnesses.