We introduce the definition ofm-to-1 mappings between two finite sets, which unifies and generalizes the definitions of 2-to-1 andn-to-1 mappings in recent literature. We also characterize thesem-to-1 mappings in terms of the generalized local criterion and thus provide three generic constructions ofm-to-1 mappings, which unify and generalize the previous known constructions. Using these constructions, the problem whetherxrh(xs) ism-to-1 on the multiplicative groupF∗qis converted into that whether an associated polynomialxr1h(x)s1ism2-to-1 on the order ℓ subgroupUℓ ofF∗q, wherem2=m/(r,s) and ℓ = (q− 1)/s. Furthermore, them2-to-1 property ofxr1h(x)s1onUℓ is studied in detail in five different cases. In addition, a recursive construction ofm-to-1 mappings fromm-to-1 mappings is proposed.

Permutation polynomials have been a subject of study for a long time and have applications in many areas of science and engineering. However, only a small number of specific classes of permutation polynomials are described in the literature so far. In this paper we present a number of permutation trinomials over finite fields, which are of different forms.