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We provide an extensive study of the differential properties of the functions$x\mapsto x^{2^t-1}$ over $\F$, for $2 \leq t \leq n-1$. We notably show thatthe differential spectra of these functions are determined by the number ofroots of the linear polynomials $x^{2^t}+bx^2+(b+1)x$ where $b$ varies in$\F$.We prove a strong relationship between the differential spectra of$x\mapsto x^{2^t-1}$ and $x\mapsto x^{2^{s}-1}$ for $s= n-t+1$. As a directconsequence, this result enlightens a connection between the differentialproperties of the cube function and of the inverse function. We also determinethe complete differential spectra of $x \mapsto x^7$ by means of the value ofsome Kloosterman sums, and of $x \mapsto x^{2^t-1}$ for $t \in \{\lfloorn/2\rfloor, \lceil n/2\rceil+1, n-2\}$.

Differential properties of functions x -> x^{2^t-1} -- extended version