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Some theorems on planar mappings. (English) Zbl 1180.94056

von zur Gathen, Joachim (ed.) et al., Arithmetic of finite fields. 2nd international workshop, WAIFI 2008, Siena, Italy, July 6–9, 2008. Proceedings. Berlin: Springer (ISBN 978-3-540-69498-4/pbk). Lecture Notes in Computer Science 5130, 117-122 (2008).
Summary: A mapping \(f\colon {\mathbb{F}}_p^n\to {\mathbb{F}}_p^n\) is called planar if for every nonzero \(a \in {\mathbb{F}}_p^n\) the difference mapping \(D_{f,a}\colon x \rightarrowtail f(x + a) - f(x)\) is a permutation of \({\mathbb{F}}_p^n\). In this note we prove that two planar functions are Carlet-Charpin-Zinoviev equivalent (CCZ-equivalent) exactly when they are extended affine equivalent (EA-equivalent). We give a sharp lower bound on the size of the image set of a planar function. Further we observe that all currently known main examples of planar functions have image sets of that minimal size.
Here two mappings \(f, g\colon {\mathbb{F}}_p^n\to {\mathbb{F}}_p^n\) are called extended affine equivalent (EA-equivalent), if \(g = A_1 \circ f \circ A_2 + A\) for some affine permutations \(A_1, A_2\) and an affine mapping \(A\).
For the entire collection see [Zbl 1141.11003].

MSC:

94A60 Cryptography
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
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