Abstract
Let \(g\) be a function over \(\mathbb {F}_q\). If there exist a function \(f\) and \(a\in \mathbb {F}_q^*\) such that \(g(x)=f(x+a)-f(x)\), then we call \(g\) a differential function and call \(f\) a differential-inverse of \(g\). We present two criteria to decide whether a given \(g\) is a differential function. The set of the degrees of all differential functions over \({\mathbb {F}}_q\) is determined. Then we give a lower bound and an upper bound on the number of differential functions over \({\mathbb {F}}_q\). Besides, we show how to construct differential inverses of a given differential function. At last, some applications of our results are introduced.
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Acknowledgments
The authors would like to thank the anonymous referees for the helpful comments, which improve much both on the technical quality and the editorial quality of the paper. The work was supported by the the Research Project of National University of Defense Technology (No. CJ 13-02-01), the open research fund of Science and Technology on Information Assurance Laboratory (Grant No. KJ-12-02), the Hunan Provincial Innovation Foundation for Postgraduate (No. CX2013B007) and the Innovation Foundation of NUDT (No. B130201).
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Xiong, H., Qu, L., Li, C. et al. Some results on the differential functions over finite fields. AAECC 25, 189–195 (2014). https://doi.org/10.1007/s00200-014-0220-9
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DOI: https://doi.org/10.1007/s00200-014-0220-9