Among other contributions, Schroeppel was the first to recognize the sub-exponential running time of certain integer factoring algorithms. While not entirely rigorous, his proof that Morrison and Brillhart's continued fraction factoring algorithm ran in roughly steps was an important milestone in factoring and laid a foundation for much later work, including the current "champion" factoring algorithm, the number field sieve.
Schroeppel analyzed Morrison and Brillhart's algorithm,[4] and saw how to cut the run time to roughly by modifications that allowed sieving. This improvement doubled the size of numbers that could be factored in a given amount of time. Coming around the time of the RSA algorithm, which depends on the difficulty of factoring for its security, this was a critically important result.
Due to Schroeppel's apparent prejudice against publishing (though he freely circulated his ideas within the research community), and in spite of Pomerance noting that his quadratic sieve factoring algorithm owed a debt to Schroeppel's earlier work, the latter's contribution is often overlooked. (See the section on "Smooth Numbers" on pages 1476–1477 of Pomerance's "A Tale of Two Sieves," Notices of the AMS, Vol. 43, No. 12, December 1996.)
