On APN Functions with Boomerang Uniformity One over $\mathbb F_{3^n}$: Differential and Boomerang Spectra and CCZ-Inequivalence
Cryptographic construction yields infinite APN function families with boomerang uniformity one over odd-characteristic fields, proving CCZ-inequivalence to power functions.
For q=3^n with odd n>1, every sign-switch of a perfect nonlinear Dembowski-Ostrom polynomial is proven APN with boomerang uniformity one or two, attaining uniformity one for (q-3)/2 parameters. This gives the first general construction of infinite APN families achieving boomerang uniformity one over odd-characteristic finite fields. The authors determine common differential and complete boomerang spectra, ruling out CCZ equivalence with power and Ness-Helleseth binomials, and exhibit three pairwise CCZ-inequivalent PN functions for infinitely many n, with the smallest degree n=45.
- First general construction of APN families with boomerang uniformity one in odd characteristic
- Common differential spectrum rules out CCZ equivalence with power functions
- Sign-switch CCZ equivalence forces EA equivalence of original PN functions
- Three pairwise CCZ-inequivalent PN functions shown for infinitely many odd n
From the original article
1 min read at arxiv.orgLet $q=3^n$, 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 $τ=g(1)$, let $ε$ be the indicator of $\Fthree^*$, and, for $c\in\Fq$, define $\widetilde G_c(x):=g(x+c)+τε(x)$. We prove that every $\widetilde G_c$ is APN and has boomerang uniformity either one or two.
By Namhun Koo · arxiv.org
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