Antiderivative Functions over F 2 n

Abstract : In this paper, we use a linear algebra point of view to describe the derivatives and higher order derivatives over F2n. On one hand, this new approach enables us to prove several properties of these functions, as well as the functions that have these derivatives. On the other hand, we provide a method to construct all of the higher order derivatives in given directions. We also demonstrate some properties of the higher order derivatives and their decomposition as a sum of functions with 0-linear structure. Moreover, we introduce a criterion and an algorithm to realize discrete antidifferentiation of vectorial Boolean functions. This leads us to define a new equivalence of functions, that we call differential equivalence , which links functions that share the same derivatives in directions given by some subspace. Finally, we discuss the importance of finding 2-to-1 functions.
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Pascale Charpin; Nicolas Sendrier; Jean-Pierre Tillich. WCC2015 - 9th International Workshop on Coding and Cryptography 2015 , Apr 2015, Paris, France. 2016, Proceedings of the 9th International Workshop on Coding and Cryptography 2015 WCC2015
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Valentin Suder. Antiderivative Functions over F 2 n. Pascale Charpin; Nicolas Sendrier; Jean-Pierre Tillich. WCC2015 - 9th International Workshop on Coding and Cryptography 2015 , Apr 2015, Paris, France. 2016, Proceedings of the 9th International Workshop on Coding and Cryptography 2015 WCC2015. 〈hal-01275708〉

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