Abstract
We give some classes of power maps with low c-differential uniformity over finite fields of odd characteristic, for \(c=-1\). Moreover, we give a necessary and sufficient condition for a linearized polynomial to be a perfect c-nonlinear function and investigate conditions when perturbations of perfect c-nonlinear (or not) function via an arbitrary Boolean or p-ary function is perfect c-nonlinear. In the process, we obtain a class of polynomials that are perfect c-nonlinear for all \(c\ne 1\), in every characteristic. The affine, extended affine and CCZ-equivalence is also looked at, as it relates to c-differential uniformity.
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Acknowledgements
The authors would like to express their sincere appreciation for the reviewers’ careful reading, beneficial comments and suggestions, and to the editors for the prompt handling of our paper. The research of Sartaj Ul Hasan is partially supported by Start-up Research Grant SRG/2019/000295 from the Science and Engineering Research Board, Government of India.
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Hasan, S.U., Pal, M., Riera, C. et al. On the c-differential uniformity of certain maps over finite fields. Des. Codes Cryptogr. 89, 221–239 (2021). https://doi.org/10.1007/s10623-020-00812-0
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DOI: https://doi.org/10.1007/s10623-020-00812-0
Keywords
- Boolean and p-ary functions
- c-Differentials
- Walsh transform
- Differential uniformity
- Perfect and almost perfect c-nonlinearity
- Dickson polynomial