Paper 2021/098
Image sets of perfectly nonlinear maps
Lukas Kölsch, Björn Kriepke, and Gohar Kyureghyan
Abstract
We consider image sets of $d$uniform maps of finite fields. We present a lower bound on the image size of such maps and study their preimage distribution, by extending methods used for planar maps. We apply the results to study $d$uniform DembowsiOstrom polynomials. Further, we focus on a particularly interesting case of APN maps on binary fields $\F_{2^n}$. For these maps our lower bound coincides with previous bounds. We show that APN maps fulfilling the lower bound have a very special preimage distribution. We observe that for an even $n$ the image sets of several wellstudied families of APN maps are minimal. In particular, for $n$ even, a DembowskiOstrom polynomial of form $f(x) =f'(x^3)$ is APN if and only if $f$ is almost3to1, that is when its image set is minimal. Also, any almost3to1 componentwise plateaued map is necessarily APN, if $n$ is even. For $n$ odd, we believe that the lower bound is not sharp. For $n$ odd, we present APN DembowskiOstrom polynomials of form $f'(x^3)$ on $\F_{2^n}$ with image sizes $ 2^{n1}$ and $5\cdot 2^{n3}$. We present results connecting the image sets of special APN maps with their Walsh spectrum. Especially, we show that a large class of APN maps has the classical Walsh spectrum. Finally, we present upper bounds on the image size of APN maps. In particular, we show that the image set of a nonbijective almost bent map contains at most $2^n2^{(n1)/2}$ elements.
Metadata
 Available format(s)
 Category
 Secretkey cryptography
 Publication info
 Preprint. MINOR revision.
 Keywords
 Image setAPN mapdifferential uniformityWalsh spectrumquadratic mapDembowskiOstrom polynomialplateaued function
 Contact author(s)

lukas koelsch @ unirostock de
bjoern kriepke @ unirostock de
gohar kyureghyan @ unirostock de  History
 20210127: received
 Short URL
 https://ia.cr/2021/098
 License

CC BY
BibTeX
@misc{cryptoeprint:2021/098, author = {Lukas Kölsch and Björn Kriepke and Gohar Kyureghyan}, title = {Image sets of perfectly nonlinear maps}, howpublished = {Cryptology ePrint Archive, Paper 2021/098}, year = {2021}, note = {\url{https://eprint.iacr.org/2021/098}}, url = {https://eprint.iacr.org/2021/098} }