We know that trapdoor permutations can be used to construct all kinds of basic cryptographic primitives, including trapdoor functions, public-key encryption, private information retrieval, oblivious transfer, key agreement, and those known to be equivalent to one-way functions such as digital signature, private-key encryption, bit commitment, pseudo-random generator and pseudo-random functions. On the other hand, trapdoor functions are not as powerful as trapdoor permutations, so the structural property of permutations seem to be something special that deserves a more careful study. In this paper, we investigate the relationships between one-way permutations and all these basic cryptographic primitives. Following previous work, we focus on an important type of reductions called black-box reductions. We prove that no such reductions exist from one-way permutations to either trapdoor functions or private information retrieval. Together with previous results, all the relationships with one-way permutations have now been established, and we know that no such reductions exist from one-way permutations to any of these primitives except trapdoor permutations. This may have the following meaning, with respect to black-box reductions. We know that one-way permutations imply none of the primitives in “public cryptography”, where additional properties are required on top of “one-wayness” [12], so permutations cannot be traded for any of these additional properties. On the other hand, we now know that none of these additional properties can be traded for permutations either. Thus, permutation seems to be something orthogonal to those additional properties on top of one-wayness. Like previous non-reducibility results [12, 23, 17, 7, 9, 8, 6], our proofs follow the oracle separation paradigm of Impagliazzo and Rudich [12].