Further study of 2-to-1 mappings over F2 n

Abstract

2-to-1 mappings over finite fields play important roles in symmetric cryptography, such as APN functions, bent functions, semi-bent functions and so on. Very recently, Mesnager and Qu [9] provided a systematic study of 2-to-1 mappings over finite fields. Particularly, they determined all 2-to-1 mappings of degree ≤ 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 can be found in [9].Motivated by their work, in this present paper, we continue studying 2-to-1 mappings, particularly, over finite fields with characteristic 2. Firstly, we determine 2-to-1 polynomials with degree 5 over F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> completely by the Hasse-Weil bound. Besides, using the multivariate method and the resultant of two polynomials, we present two classes of 2-to-1 trinomials and four classes of 2-to-1 quadrinomials over F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> .