# Differential Properties of {$x\mapsto x^{2^{t}-1}$}

Abstract : 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}.
Document type :
Journal articles
Domain :

https://hal.archives-ouvertes.fr/hal-00610099
Contributor : Céline Blondeau <>
Submitted on : Thursday, July 21, 2011 - 9:57:57 AM
Last modification on : Friday, May 25, 2018 - 12:02:05 PM

### Citation

Céline Blondeau, Anne Canteaut, Pascale Charpin. Differential Properties of {$x\mapsto x^{2^{t}-1}$}. IEEE Transactions on Information Theory, Institute of Electrical and Electronics Engineers, 2011, 57 (12), pp.8127 - 8137. ⟨10.1109/TIT.2011.2169129⟩. ⟨hal-00610099⟩

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