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APN trinomials and hexanomials

In this paper we give a new family of APN trinomials of the form $X^{2^k+1} + (\mathsf{tr}^{n}_{m}(X))^{2^k+1}$ on $\mathbb{F}_{2^n}$ where $\mathsf{gcd}(k,n)=1$ and $n = 2m = 4t$, and prove its important properties. The family satisfies for all $n = 4t$ an interesting property of the Kim function which is, up to equivalence, the only known APN function equivalent to a permutation on $\mathbb{F}_{2^{2m}}$. As another contribution of the paper, we consider a family of hexanomials $g_{C,k}$ which was shown to be differentially $2^{\mathsf{gcd}(m,k)}$-uniform by Budaghyan and Carlet (2008) when a quadrinomial $P_{C,k}$ has no roots in a specific subgroup. In this paper, for all $(m,k)$ pairs, we characterize, construct and count all $C \in \mathbb{F}_{2^n}$ satisfying the condition. Bracken, Tan and Tan (2014) and Qu, Tan and Li (2014) constructed some elements $C$ satisfying the condition when $m \equiv 2 \textrm{or} 4 \pmod{6}$ and $m \equiv 0 \pmod{6}$ respectively, both requiring $\mathsf{gcd}(m,k) = 1$. Bluher (2013) proved that such $C$ exists if and only if $k \ne m$ without characterizing, constructing or counting those $C$. To prove the results, we effectively use a Trace-$0$/Trace-$1$ (relative to the subfield $\mathbb{F}_{2^m}$) decomposition of $\mathbb{F}_{2^n}$.