On pseudoprime RSA moduli

In 2010, Dieulefait and Urroz considered the notion of malleability of an RSA modulus. They proved that, given some information on the factors of numbers coprime to n, where n is an RSA modulus, there exists an algorithm that finds a proper factor of n in time \(O(\log n)\). As a particular case of their algorithm, just some knowledge of the factors of \(2^n\pm 1\) is enough to factor n except possibly when n is a base-2 pseudoprime. They went on to prove that the set of these exceptional RSA moduli with prime factors between z and 2z has size at most \(O(z^2/(\log z)^3)\). In the present paper, we improve this bound significantly and show that the counting function for these RSA moduli is bounded above by \(O(z^{8/5}/(\log z)^2)\). In addition, as a related problem, we prove an upper bound of \(O(z^{4/5}/(\log z)^{2/5})\) for the number of base-2 pseudoprimes up to z that are products of two primes.