Differential Properties of $xmapsto x^2^t-1$

We provide an extensive study of the differential properties of the functions x→ x2t-1 over \BBF 2n, for 1 <; t <; n. We notably show that the differential spectra of these functions are determined by the number of roots of the linear polynomials x2t+bx2+(b+1)x where b varies in \BBF 2n. We prove a strong relationship between the differential spectra of x→ x2t-1 and x→ x2s-1 for s = n-t+1. As a direct consequence, this result enlightens a connection between the differential properties of the cube function and of the inverse function. We also determine the complete differential spectra of x → x7 by means of the value of some Kloosterman sums, and of x → x2t-1 for t ∈ {[n/2], [n /2]+1, n-2}.