  Rabi and Sherman 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 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. Rabiand 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 $\mathcal{O}(n)$-to-one. This puts a reasonable upper bound on the
ambiguity.
