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

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 some close 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 $x^{r}h\left(x^s\right)$ from the cyclotomic perspective, and several classes of n-to-1 mappings of the form $g(x^{q^k}-x+\delta )+cx$ are provided.     

On the differential spectrum of a differentially 3-uniform power function

In this paper, we investigate the cardinality of the intersection of four particular cyclotomic classes of order two over the finite field with characteristic 5 by making most use of the cyclotomic numbers of orders two and four. As a consequence, a class of power functions over ${\mathbb{F}}_{5^n}$ is shown to be differentially 3-uniform and its differential spectrum is also completely determined.     

Generalized almost perfect nonlinear binomials and trinomials over fields of prime-square order

Let $p>3$ be a prime. We show that, for each integer d with $p\le d\le 2(p-1)$, there exists a generalized almost perfect nonlinear (GAPN) binomial or trinomial over ${\mathbb{F}}_{p^2}$ of algebraic degree d. We start by deriving sufficient conditions for the function $G:{\mathbb{F}}_{p^2}\to {\mathbb{F}}_{p^2},X\mapsto X^{d_1}+u{X}^{d_2}$ to be GAPN in the case where one of the terms of G is GAPN. We then give explicit constructions of GAPN binomials over ${\mathbb{F}}_{p^2}$ of any odd algebraic degree between p and $2(p-1)$ and, in the case where p is not a Mersenne prime, also of any even algebraic degree in this range. To obtain GAPN functions of even algebraic degree also in the general case, we finally show how to construct GAPN trinomials over ${\mathbb{F}}_{p^2}$ of any even algebraic degree between p and $2(p-1)$ by applying a characterization of a special form of GAPN binomials by Özbudak and Sălăgean. Our constructed functions are the first GAPN functions of even algebraic degree over extension fields of odd characteristic reported so far.