Möbius transformations and characterizations of hyper-bent functions from Dillon-like exponents with coefficients in extension fields.

In this paper, we are interested in the characterization of the hyper-bentness property of all functions defined over the Galois field $ \mathbb{F}_{2^{2m}} $ in polynomial forms as sums of multiple trace terms obtained via Dillon-like exponents with coefficients in the extension Galois field $ \mathbb{F}_{2^{2m}} $. Characterizations of hyper-bentness were provided firstly by Charpin and Gong fourteen years ago for functions defined over $ \mathbb{F}_{2^{2m}} $, with coefficients in $ \mathbb{F}_{2^{m}} $, in terms of some exponential sums over $ \mathbb{F}_{2^m}{} $ involving Dickson polynomials. Mesnager extended their results in 2011 by considering adding a trace monomial function defined over specific fields of small cardinalities.By employing the Möbius transformation, we succeed in characterizing the hyper-bentness property of functions with Dillon-like exponents with coefficients in the whole $ \mathbb{F}_{2^{2m}} $. We emphasize that this ascension offers more choices on the coefficients (and therefore in exhibiting possible new hyper-bent functions) by keeping the evaluations of the related exponential sums in the $ \mathbb{F}_{2^m}{} $ as well as the computation of the number of rational points on the associate hyperelliptic curves on the same field in the spirit of previous achievements due to Lisoněk (2011) and Flori and Mesnager (2012-2013) in this context. This strategy gives a generalization of the result due to Charpin and Gong, and its effectiveness can be seen from a new infinite family of binomial hyper-bent functions. Their coefficients are in an extension field. Nevertheless, as all the previously found functions, they still lie in the well-known class denoted by $ \mathcal{PS}_{ap}^{\#} $, initially studied in 2006 by Carlet and Gaborit. [ABSTRACT FROM AUTHOR]