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

A Class of n-to-1 Binomials over Finite Fields

“…More recently, Gao et al [7] generalized the definition of 2-to-1 mappings to n-to-1 mappings. Niu et al [8] showed some approaches to construct n-to-1 mappings of finite fields. Therefore, to find more classes of n-to-1 mappings of finite fields still is an open problem.…”

“…Definition 1 [8] Let f be a mapping from one finite set A to another finite set B. Then f is called an n-to-1 mapping if one of the following two cases holds:…”

“…Lemma 1 [8] Let f (x) = ax d be a monomial over F q , where a ¹ 0, and A be a subset in F q . Then f is an n-to-1 mapping over A if and only if gcd(d#A) = n.…”

“…Lemma 2 [8] Let AS and S ˉbe finite sets with #S = #S ˉ, and #A º #S(mod n). Let f: A ® A, g: S ® S ˉ, λ: A ® S, and λ ˉ: A ® S ˉbe maps such that λ ˉ f = g  λ, where both λ and λ ˉare surjective.…”

“…Lemma 3 [8] Let q be a prime power, r be a positive integer such that gcd(rq -1) = 1 and n|(q + 1). Let f (x) = x r h(x q -1 ), where h(x)Î F q 2 [x] such that h(x) ¹ 0 if x ¹ 0.…”