New results of $0$-APN power functions over $\mathbb{F}_{2^n}$
(2210.02207)Abstract
Partially APN functions attract researchers' particular interest recently. It plays an important role in studying APN functions. In this paper, based on the multivariate method and resultant elimination, we propose several new infinite classes of $0$-APN power functions over $\mathbb{F}{2n}$. Furthermore, two infinite classes of $0$-APN power functions $xd$ over $\mathbb{F}{2n}$ are characterized completely where $(2k-1)d\equiv 2m-1~({\rm mod}\ 2n-1)$ or $(2k+1)d\equiv 2m+1~({\rm mod}\ 2n-1)$ for some positive integers $n, m, k$. These infinite classes of $0$-APN power functions can explain some examples of exponents of Table $1$ in \cite{BKRS2020}.
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