Differential properties of functions x 7 → x 2 t − 1 – extended version ∗ –

We provide an extensive study of the differential properties of the functions x 7→ x t −1 over F2n , 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 x t + bx + (b + 1)x where b varies in F2n .We prove a strong relationship between the differential spectra of x 7→ x t −1 and x 7→ x s −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 7→ x by means of the value of some Kloosterman sums, and of x 7→ x t −1 for t ∈ {⌊n/2⌋, ⌈n/2⌉+1, n− 2}.