We consider the possibility of basing one-way functions on NP-Hardness; that is, we study possible reductions from a worst-case decision problem to the task of average-case inverting a polynomial-time computable function J. Our main findings are the following two negative results: 1. If given y one can efficiently compute |f-1(y)| then the existence of a (randomized) reduction of NP to the task of inverting / implies that coN P ⊆ AM. Thus, it follows that such reductions cannot exist unless coNP ⊆ AM. 2. For any function f, the existence of a (randomized) non-adaptive reduction of NP to the task of average-case inverting f implies that coNP ⊆ AM. Our work builds upon and improves on the previous works of Feigenbaum and Fortnow (SIAM Journal on Computing, 1993) and Bogdanov and Trevisan (44th FOCS, 2003), while capitalizing on the additional "computational structure" of the search problem associated with the task of inverting polynomial-time computable functions. We believe that our results illustrate the gain of directly studying the context of one-way functions rather than inferring results for it from a the general study of worst-case to average-case reductions.