Characterizations and constructions of n-to-1 mappings over finite fields

DOI10.1016/J.FFA.2022.102126arXiv2201.10290MaRDI QIDQ6389128

Chao Li, Tailin Niu, Kangquan Li, Longjiang Qu

Publication date: 25 January 2022

Abstract: $n$-to-$1$ mappings have wide applications in many areas, especially in cryptography, finite geometry, coding theory and combinatorial design. In this paper, many classes of $n$-to-$1$ mappings over finite fields are studied. First, we provide a characterization of general $n$-to-$1$ mappings over $mathbb{F}_{p^m}$ by means of the Walsh transform. Then, we completely determine $3$-to-$1$ polynomials with degree no more than $4$ over $mathbb{F}_{p^m}$. Furthermore, we obtain an AGW-like criterion for characterizing an equivalent relationship between the $n$-to-$1$ property of a mapping over finite set $A$ and that of another mapping over a subset of $A$. Finally, we apply the AGW-like criterion into several forms of polynomials and obtain some explicit $n$-to-$1$ mappings. Especially, three explicit constructions of the form $x^{r}hleft(x^{s}ight)$ from the cyclotomic perspective, and several classes of $n$-to-$1$ mappings of the form $gleft(x^{q^k}-x+deltaight)+cx$ are provided.

Mathematics Subject Classification ID

Polynomials over finite fields (11T06)

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