2019/10/15 by Li, Kangquan, Mesnager, Sihem, Qu, Longjiang
#FOS: Computer and information sciences #Information Theory (cs.IT)
paper · doi:10.48550/arxiv.1910.06654
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 \citeMQ2019 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 \citeMQ2019. 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 \mathbbF2n 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 xk+βxℓ + αx∈\mathbbF2n[x] over \mathbbF2n with n≤ 7, and derive twelve classes of 2-to-1 quadrinomials with trivial coefficients over \mathbbF2n.