New results on n-to-1 mappings over finite fields

n-to-1 mappings have many applications in cryptography, finite geometry, coding theory and combinatorial design. In this paper, we first use cyclotomic cosets to construct several kinds of n-to-1 mappings over Fq⁎. Then we characterize a new form of AGW-like criterion, and use it to present many classes of n-to-1 polynomials with the form xrh(xq−1) over Fq2⁎. Finally, by using monomials on the cosets of a subgroup of μq+1 and another form of AGW-like criterion, we show some n-to-1 trinomials over Fq2⁎.