In 2003, Leonid A. Levin presented the idea of a combinatorial complete one-way function and a sketch of the proof that Tiling represents such a function. In this paper, we present two new one-way functions based on semi-Thue string rewriting systems and a version of the Post Correspondence Problem and prove their completeness. Besides, we present an alternative proof of Levin's result. We also discuss the properties a combinatorial problem should have in order to hold a complete one-way function. @InProceedings{kojevnikov_et_al:LIPIcs:2008:1365, author = {Arist Kojevnikov and Sergey I. Nikolenko}, title = {40. New Combinatorial Complete One-Way Functions}, booktitle = {25th International Symposium on Theoretical Aspects of Computer Science (STACS 2008)}, pages = {457--466}, series = {Leibniz International Proceedings in Informatics}, year = {2008}, volume = {1}, editor = {Susanne Albers and Pascal Weil}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany}, address = {Dagstuhl, Germany}, URL = {http://drops.dagstuhl.de/opus/volltexte/2008/1365}, URN = {urn:nbn:de:0030-drops-13652}, annote = {Keywords: } }