Propagation characteristics of \(x\mapsto x^{-1}\) and Kloosterman sums
The inverse permutation \(\sigma: x \mapsto x^{-1}\) appears in the design of the AES cryptosystem. This permutation is considered as a very good mapping. In this paper, the authors study the permutation \(\sigma: x \mapsto x^{-1}\) on the field of order \(2^n\) by means of their component functions \(f_\lambda\). It is proved that the weights of derivatives of \(f_\lambda\) can be expressed in terms of Kloosterman sums. The authors compute some indicators of the propagation characteristics of \(\sigma\) \((\mathcal{M}(\sigma)\) and \(\mathcal{V}(\sigma))\) and they prove that the mapping \(\sigma: x \mapsto x^{-1}\) has good propagation characteristics with respect to these indicators. Moreover, they deduce several new formulas on Kloosterman sums, by using classical formulas which link any Boolean function with its derivatives.
scientific article; zbMATH DE number 5347274