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Tabulating Pseudoprimes and Tabulating Liars

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(xlog 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.