Locally-APN Binomials with Low Boomerang Uniformity in Odd Characteristic
Namhun Koo, Soonhak Kwon, Minwoo Ko, Byunguk Kim
TL;DR
This work addresses the differential and boomerang properties of locally-APN binomials over odd-characteristic finite fields, focusing on functions $F_r(x)=x^r+x^{r+(q-1)/2}$ with $q\equiv3\pmod4$. It gives a concrete criterion: if $(x+1)^r-x^r=b$ has at most one solution in $S_{00}$ for every $b\neq0$ and $\gcd(r,q-1)\mid 2$, then $F_r$ is locally-APN with differential uniformity bounded by $2$ and boomerang uniformity also bounded by $2$, with the exponents in Table 1 satisfying the hypothesis. The paper further analyzes the differential spectra for the cases $r=3$ and $r=(2q-1)/3$, establishing locally-APN behavior and explicit spectra, and determines the boomerang spectrum of $F_2$ when $p=3$, showing a tight bound $\beta_{F_2}=1$ in many cases. These results extend the known set of locally-APN binomials in odd characteristic and provide exact spectra expressed in terms of character sums, offering guidance for constructing robust S-boxes and motivating search for additional exponents.
Abstract
Recently, several studies have shown that when $q\equiv3\pmod{4}$, the function $F_r(x)=x^r+x^{r+\frac{q-1}{2}}$ defined over $\mathbb{F}_q$ is locally-APN and has boomerang uniformity at most~$2$. In this paper, we extend these results by showing that if there is at most one $x\in \mathbb{F}_q$ with $χ(x)=χ(x+1)=1$ satisfying $(x+1)^r - x^r = b$ for all $b\in \mathbb{F}_q^*$ and $\gcd(r,q-1)\mid 2$, then $F_r$ is locally-APN with boomerang uniformity at most $2$. Moreover, we study the differential spectra of $F_3$ and $F_{\frac{2q-1}{3}}$, and the boomerang spectrum of $F_2$ when $p=3$.