On APN Functions with Boomerang Uniformity One over $\mathbb F_{3^n}$: Differential and Boomerang Spectra and CCZ-Inequivalence

RESUMO

Let $q=3^n$, where $n>1$ is odd, and let $g:\Fq\to\Fq$ be a perfect nonlinear (PN) function represented by a Dembowski--Ostrom (DO) polynomial. Put $\tau=g(1)$, let $\epsilon$ be the indicator of $\Fthree^*$, and, for $c\in\Fq$, define $\widetilde G_c(x):=g(x+c)+\tau\epsilon(x)$. We prove that every $\widetilde G_c$ is APN and has boomerang uniformity either one or two. More precisely, \[ \beta_{\widetilde G_c}=1 \quad\Longleftrightarrow\quad c\in\mathcal C_g :=\{c\in\Fq\setminus\Fthree:g(c)+\tau\notin g(\Fq)\}, \qquad |\mathcal C_g|=\frac{q-3}{2}, \] whereas $\beta_{\widetilde G_c}=2$ for the remaining $(q+3)/2$ parameters. We determine the common differential spectrum and complete boomerang spectra of all the functions $\widetilde G_c$. Since boomerang uniformity one is the least possible for an APN function over a finite field of odd characteristic, this gives, to the best of our knowledge, the first general construction yielding infinite families of APN functions attaining this optimum. This common differential spectrum rules out CCZ equivalence with every power function and every Ness--Helleseth-type binomial. We also prove that CCZ equivalence between sign-switches of DO PN functions forces EA equivalence between the original PN functions. Using the orders of the nuclei of the associated presemifields, we exhibit, for infinitely many odd $n$, three pairwise CCZ-inequivalent PN functions over $\F_{3^n}$, one from each of the Gold $f_1$, Ding--Yuan $f_3$, and Bierbrauer $f_5$ families. Consequently, over each such field, our construction produces three pairwise CCZ-inequivalent APN functions with boomerang uniformity one. The smallest extension degree obtained in this way is $n=45$.