The complete extended differential spectrum of a power map in finite fields of odd characteristics

The concept of the (scaled) [Formula: see text]-differential uniformity found its first application in a paper by Anbar et al. [P[Formula: see text]N functions, complete mappings and quasigroup difference sets, J. Combin. Des. 31(12) (2023) 667–690], where it was shown that the graph of a P[Formula: see text]N function corresponds to a difference set in a quasigroup. Further, in a very recent paper by Pal and Stănică [A connection between the boomerang uniformity and the extended differential in odd characteristic and applications, Adv. Math. Commun. (2025), doi:10.3934/amc.2024059], a connection was discovered showing that in fact, the boomerang uniformity for an odd APN function in odd characteristic equals its [Formula: see text]-differential uniformity when [Formula: see text], if the function is a permutation, otherwise it is the maximum of the [Formula: see text]-DDT entries, disregarding the first row/column. While we take our inspiration from the extended differentials, our work’s goal is to investigate the preimages (these problems tend to be quite difficult, in general) of various polynomials that are obtained via those extended differentials, in odd characteristic. Precisely, in the paper, we concentrate on the map [Formula: see text] on [Formula: see text] and using combinatorial and number theory techniques, we compute its detailed [Formula: see text]-differential spectrum for all values of [Formula: see text] (the spectrum for [Formula: see text] is known — see Choi et al. [Differential spectrum of some power functions in odd prime characteristic, Finite Fields Appl. 21 (2013) 11–29]). As a consequence, we find that there are many values of [Formula: see text], where the function has low [Formula: see text]-differential uniformity. For example, for [Formula: see text], via elliptic curves techniques, we find explicit expressions for the spectrum, and as a consequence, we show that the function is APcN, when [Formula: see text] is even and [Formula: see text], or [Formula: see text] ([Formula: see text] arbitrary); when [Formula: see text] and [Formula: see text] is odd, the [Formula: see text]-differential uniformity is [Formula: see text].