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Non-linéarité des fonctions booléennes : applications de la théorie des fonctions booléennes et des codes en cryptographie

Abstract : This thesis mainly focuses on coding theory and boolean functions which are connected with cryptography. Two research's trends are followed: the first part is dedicated to the nonlinearity of boolean functions whereas the second one shows cryptographic applications of objects coming from these theory. Interested in Patterson and Wiedemann's conjecture, we propose to generalize their construction based on union of orbits under the action of a group from which the determination of the minimum spectral magnitude reduces to two sub-problems that we intensely study: the evaluation of Gauss sums and the estimation of some partial exponential sums. Several conditions and ideas which may help to confirm the conjecture are thus detailed. This work allows us to construct functions with a nonlinearity greater than the asymptotic mean. Moreover, thanks to this technique, we obtain an example of a quadratic spread which is highly nonlinear and close to the Patterson and Wiedemann bound in IS variables. In the second part, we first turn our attention to the field of lightweight cryptographic protocols. Boolean functions with specific resistance to differential cryptanalysis are introduced in the HB+ protocol to strengthen it against man-in-the-middle attacks. Starting from a second protocol using the idea of background noise, we exploit the link with the wiretap channel theory of Wyncr to show how to increase the security. At last we deal with authentication of noisy data and secure sketches in which a modification of the McEliece's cryptosystem is explained in order to restrict the access to checking functions.
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Submitted on : Thursday, February 21, 2008 - 4:10:14 PM
Last modification on : Friday, July 13, 2018 - 2:36:01 PM
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  • HAL Id : tel-00258334, version 1


Julien Bringer. Non-linéarité des fonctions booléennes : applications de la théorie des fonctions booléennes et des codes en cryptographie. Mathématiques [math]. Université du Sud Toulon Var, 2007. Français. ⟨tel-00258334⟩



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