Characterizations and constructions of n-to-1 mappings over finite fields
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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 𝔽_p^m by means of the Walsh transform. Then, we completely determine 3-to-1 polynomials with degree no more than 4 over 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^rh( x^s ) from the cyclotomic perspective, and several classes of n-to-1 mappings of the form g( x^q^k -x +δ) +cx are provided.
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