Rabi and Sherman [RS97] present a cryptographic paradigm based on associative, one-way functions that are strong (i.e., hard to invert even if one of their arguments is given) and total. Hemaspaandra and Rothe [HR99] proved that such powerful one-way functions exist exactly if (standard) one-way functions exist, thus showing that the associative one-way function approach is as plausible as previous approaches. In the present paper, we study the degree of ambiguity of one-way functions. Rabi and Sherman showed that no associative one-way function (over a universe having at least two elements) can be unambiguous (i.e., one-to-one). Nonetheless, we prove that if standard, unambiguous, one-way functions exist, then there exist strong, total, associative, one-way functions that are O(n)-to-one. This puts a reasonable upper bound on the ambiguity. Our other main results are: 1. P 6= FewP if and only if there exists an (nO(1))-to-one, strong, total AOWF. 2. No O(1)-to-one total, associative functions exist in [see article for equations]. 3. For every nondecreasing, unbounded, total, recursive function g : N → N, there is a g(n)-to-one, total, commutative, associative, recursive function in [see article for equations]

Publication Date



Low Ambiguity in Strong, Total, Associative, One-Way Functions, C. Homan, Uni- versity of Rochester Department of Computer Science Technical Report 734, August 2000, revised December 2002. Note: imported from RIT’s Digital Media Library running on DSpace to RIT Scholar Works in February 2014.

Document Type

Technical Report

Department, Program, or Center

Computer Science (GCCIS)


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