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- W4221143145 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^rhleft( x^s right) $ from the cyclotomic perspective, and several classes of $n$-to-$1$ mappings of the form $ gleft( x^{q^k} -x +delta right) +cx$ are provided." @default.
- W4221143145 created "2022-04-03" @default.
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- W4221143145 date "2022-01-25" @default.
- W4221143145 modified "2023-10-16" @default.
- W4221143145 title "Characterizations and constructions of n-to-1 mappings over finite fields" @default.
- W4221143145 doi "https://doi.org/10.48550/arxiv.2201.10290" @default.
- W4221143145 hasPublicationYear "2022" @default.
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