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Published in:
2006 | OriginalPaper | Chapter
A Note of Perfect Nonlinear Functions
Authors : Xiyong Zhang, Hua Guo, Jinjiang Yuan
Published in: Cryptology and Network Security
Publisher: Springer Berlin Heidelberg
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Perfect nonlinear functions are of importance in cryptography. By using Galois rings and investigating the character values of corresponding relative difference sets, we construct a perfect nonlinear function from
$\mathbb{Z}^{n}_{p_{2}}$
to
$\mathbb{Z}^{m}_{p_{2}}$
where 2
m
is possibly larger than the largest divisor of
n
. Meanwhile we prove that there exists a perfect nonlinear function from
$\mathbb{Z}^{2}_{2_{p}}$
to
$\mathbb{Z}_{2_{p}}$
if and only if
p
=2, and that there doesn’t exist a perfect nonlinear function from
$\mathbb{Z}^{2n}_{2k_{l}}$
to
$\mathbb{Z}^{m}_{2k_{l}}$
if
m
>
n
and
l
(
l
is odd) is
self-conjugate
modulo 2
k
(
k
≥1) .