A connection between the boomerang uniformity and the extended differential in odd characteristic and applications
Mohit Pal, Pantelimon Stanica
TL;DR
This work builds a bridge between boomerang connectivity, differential uniformity, and $c$-differential uniformity in odd characteristic by proving that, for odd APN functions, boomerang entries coincide with $(-1)$-differential entries via ${\mathcal{B}}_f(a,b)={_{-1}}\Delta_f(a,-b)$. It then leverages this link to derive identities for the boomerang spectrum of odd APN power functions and to compute the boomerang spectrum of the inverse function across several parameter regimes, illustrating how $(-1)$-differential data governs BCT behavior. The paper introduces two low-differential-uniformity constructions by modifying the inverse: (i) composing the inverse with a 3-cycle, yielding $\Delta_f=3$ (or $5$ when $p=13$ and $n$ even; $4$ otherwise), and (ii) adding a quadratic term $uX^2$ to obtain $\Delta_f\le 4$, with CCZ/EA considerations and switching bounds extended to odd characteristic. It also provides upper bounds for differential uniformity under switching and extends Charpin–Kyureghyan-type results to odd characteristic, enhancing the design toolkit for cryptographic S-boxes with favorable boomerang and differential properties.
Abstract
This paper makes the first bridge between the classical differential/boomerang uniformity and the newly introduced $c$-differential uniformity. We show that the boomerang uniformity of an odd APN function is given by the maximum of the entries (except for the first row/column) of the function's $(-1)$-Difference Distribution Table. In fact, the boomerang uniformity of an odd permutation APN function equals its $(-1)$-differential uniformity. We next apply this result to easily compute the boomerang uniformity of several odd APN functions. In the second part we give two classes of differentially low-uniform functions obtained by modifying the inverse function. The first class of permutations (CCZ-inequivalent to the inverse) over a finite field $\mathbb{F}_{p^n}$ ($p$, an odd prime) is obtained from the composition of the inverse function with an order-$3$ cycle permutation, with differential uniformity $3$ if $p=3$ and $n$ is odd; $5$ if $p=13$ and $n$ is even; and $4$ otherwise. The second class is a family of binomials and we show that their differential uniformity equals~$4$. We next complete the open case of $p=3$ in the investigation started by G\" olo\u glu and McGuire (2014), for $p\geq 5$, and continued by Kölsch (2021), for $p=2$, $n\geq 5$, on the characterization of $L_1(X^{p^n-2})+L_2(X)$ (with linearized $L_1,L_2$) being a permutation polynomial. Finally, we extend to odd characteristic a result of Charpin and Kyureghyan (2010) providing an upper bound for the differential uniformity of the function and its switched version via a trace function.