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Antiderivative functions over \(\mathbb {F}_{2^n}\)

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Abstract

In this paper, we use a linear algebra point of view to describe the derivatives and higher order derivatives over \(\mathbb {F}_{2^n}\). 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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Notes

  1. Remark that \(Ker(\alpha )\cap Ker(\beta ) \subset Ker(\alpha +\beta )\Leftarrow \Delta _{\alpha +\beta }(\Delta _{\alpha ,\beta }F)(x)=0\) from Eq. (1).

  2. A function is quadratic if and only if all of its derivatives are at most affines (Berger T: Private communication, 2014).

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  1. Communications Security Lab, University of Waterloo, Waterloo, Canada

    Valentin Suder

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  1. Valentin Suder
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Correspondence to Valentin Suder.

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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding and Cryptography”.

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Suder, V. Antiderivative functions over \(\mathbb {F}_{2^n}\) . Des. Codes Cryptogr. 82, 435–447 (2017). https://doi.org/10.1007/s10623-016-0186-7

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