Further study of 2-to-1 mappings over F2n
Further study of 2-to-1 mappings over F2n
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DOI：
10.1109/iwsda46143.2019.8966103
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发表时间：
2019-10
期刊：
影响因子：
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通讯作者：
Kangquan Li;Sihem Mesnager;Longjiang Qu
中科院分区：
文献类型：
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作者：
Kangquan Li;Sihem Mesnager;Longjiang Qu
有限域上的2-to-1映射在对称密码学中起着重要的作用，如APN函数、Bent函数、半Bent函数等。最近，Mesnager和Qu[9]对有限域上的2到1映射进行了系统的研究。特别地，他们确定了任一有限域上≤4次的所有2到1映射。另外，另一个研究方向是考虑少项的2对1多项式。在他们工作的启发下，本文继续研究2-1映射，特别是特征为2的有限域上的2-1映射。首先，我们完全由Hasse-Weil界确定了环上的5次2-1多项式。此外，利用多元方法和两个多项式的结式，我们给出了两类2到1三项式和四类2到1四项式。
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 $\mathbb{F}_{2^n}$ 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 $\mathbb{F}_{2^n}$.