# Further study of 2-to-1 mappings over F_2^n

2-to-1 mappings over finite fields play an important role in symmetric cryptography, in particular in the constructions of APN functions, bent functions, semi-bent functions and so on. Very recently, Mesnager and Qu <cit.> provided a systematic study of 2-to-1 mappings over finite fields. In particular, they determined all 2-to-1 mappings of degree at most 4 over any finite fields. In addition, another research direction is to consider 2-to-1 polynomials with few terms. Some results about 2-to-1 monomials and binomials have been obtained in <cit.>. Motivated by their work, in this present paper, we push further the study of 2-to-1 mappings, particularly, over finite fields with characteristic 2 (binary case being the most interesting for applications). Firstly, we completely determine 2-to-1 polynomials with degree 5 over F_2^n using the well known Hasse-Weil bound. Besides, we consider 2-to-1 mappings with few terms, mainly trinomials and quadrinomials. Using the multivariate method and the resultant of two polynomials, we present two classes of 2-to-1 trinomials, which explain all the examples of 2-to-1 trinomials of the form x^k+β x^ℓ + α x∈F_2^n[x] over F_2^n with n< 7, and derive twelve classes of 2-to-1 quadrinomials with trivial coefficients over F_2^n.

READ FULL TEXT