# Differential properties of functions $x \mapsto x^{2^t-1}$ -- extended version

Abstract : 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 that the differential spectra of these functions are determined by the number of roots 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 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 $x \mapsto x^7$ by means of the value of some Kloosterman sums, and of $x \mapsto x^{2^t-1}$ for $t \in \{\lfloor n/2\rfloor, \lceil n/2\rceil+1, n-2\}$.
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Cited literature [21 references]

https://hal.inria.fr/inria-00616674
Contributor : Anne Canteaut Connect in order to contact the contributor
Submitted on : Thursday, August 25, 2011 - 9:17:07 AM
Last modification on : Wednesday, February 17, 2021 - 10:26:08 AM
Long-term archiving on: : Saturday, November 26, 2011 - 2:20:36 AM

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### Identifiers

• HAL Id : inria-00616674, version 2
• ARXIV : 1108.4753

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### Citation

Céline Blondeau, Anne Canteaut, Pascale Charpin. Differential properties of functions $x \mapsto x^{2^t-1}$ -- extended version. [Research Report] INRIA. 2011, pp.32. ⟨inria-00616674v2⟩