Propagation characteristics of x x−1 and Kloosterman sums

We study the inverse permutation on the field of order 2n by means of their component functions fλ. We prove that the weights of derivatives of fλ can be expressed in terms of Kloosterman sums. We are then able to compute some indicators of the propagation characteristics of σ. We can claim that σ, which is considered as a good cryptographic mapping regarding several criteria, is moreover such that the functions fλ have good propagation properties with respect to these indicators. We further deduce several new formulas on Kloosterman sums, by using classical formulas which link any Boolean function with its derivatives.