Two-to-one mappings and involutions without fixed points over~$\bF_{2^n}$

In this paper, two-to-one mappings and involutions without any fixed point on finite fields of even characteristic are investigated. First, we characterize a closed relationship between them by implicit functions and develop an AGW-like criterion for \2 mappings. Using this criterion, some new constructions of \2 mappings are proposed and eight classes of \2 mappings of the form $(x^{2^k}+x+\delta)^{s}+cx$ are obtained. Finally, a number of classes of involutions without any fixed point are derived from the known \2 mappings by the relation between them.