## 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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### References:

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