Differential properties of functions $x \mapsto x^{2^t-1}$ -- extended version - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Reports (Research Report) Year : 2011

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

Céline Blondeau
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  • PersonId : 905946
Anne Canteaut
Pascale Charpin
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  • PersonId : 905949

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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Dates and versions

inria-00616674 , version 1 (23-08-2011)
inria-00616674 , version 2 (25-08-2011)

Identifiers

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

Cite

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⟩
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