Symmetries in Random Permutations and the P vs NP Problem

In this paper, an analysis of encryption using permutations that are selected at random is presented. A construction using Feistel rounds is built upon this methodology is used to build two hashing algorithms: algorithm H and algorithm Ħ. It is shown that Ħ is a one-way function. The existence of a one-way function implies that P≠NP for an alphabet size greater than 1. Furthermore, it is shown that encryption using a random permutation can be used to build a Turing Machine that is able to correctly accept the words when, and only when, the words belong to any language with an alphabet size of 1 that is selected, in polynomial time. This construction is used to show that P=NP for an alphabet size of 1.