Abstract
In this paper we investigate various differential properties of generalized Boolean functions defined on \({\mathbb F}_2^n\) with values in \({\mathbb Z}_{2^k}\), \(k\ge 2\). We characterize linear structures for the generalized Boolean functions in terms of their binary expansion components, and find all symmetric generalized bent functions. Next, we show that there are no symmetric balanced functions defined on \({\mathbb F}_2^n\) with values in a group of order \(2^k, k\ge 2\), a contrast to the classical case for \(k=1\), commonly known as the bisection of binomial coefficients. Further, we characterize the avalanche features of a generalized Boolean function in terms of differentials. Lastly, we show that a partially gbent function is plateaued.
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Acknowledgement
This paper was started while the second and fourth named authors visited the third named author at the Institute of Algebra and Geometry, of Otto von Guericke University Magdeburg. They thank the host and the institute for hospitality and excellent working conditions.
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Martinsen, T., Meidl, W., Pott, A., Stănică, P. (2018). On Symmetry and Differential Properties of Generalized Boolean Functions. In: Budaghyan, L., Rodríguez-Henríquez, F. (eds) Arithmetic of Finite Fields. WAIFI 2018. Lecture Notes in Computer Science(), vol 11321. Springer, Cham. https://doi.org/10.1007/978-3-030-05153-2_11
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