3 research outputs found
The differential properties of certain permutation polynomials over finite fields
Finding functions, particularly permutations, with good differential
properties has received a lot of attention due to their possible applications.
For instance, in combinatorial design theory, a correspondence of perfect
-nonlinear functions and difference sets in some quasigroups was recently
shown [1]. Additionally, in a recent manuscript by Pal and Stanica [20], a very
interesting connection between the -differential uniformity and boomerang
uniformity when was pointed out, showing that that they are the same for
an odd APN permutations. This makes the construction of functions with low
-differential uniformity an intriguing problem. We investigate the
-differential uniformity of some classes of permutation polynomials. As a
result, we add four more classes of permutation polynomials to the family of
functions that only contains a few (non-trivial) perfect -nonlinear
functions over finite fields of even characteristic. Moreover, we include a
class of permutation polynomials with low -differential uniformity over the
field of characteristic~. As a byproduct, our proofs shows the permutation
property of these classes. To solve the involved equations over finite fields,
we use various techniques, in particular, we find explicitly many Walsh
transform coefficients and Weil sums that may be of an independent interest
The second-order zero differential spectra of some functions over finite fields
It was shown by Boukerrou et al.~\cite{Bouk} [IACR Trans. Symmetric Cryptol.
1 2020, 331--362] that the -boomerang uniformity (which is the same as the
second-order zero differential uniformity in even characteristic) of perfect
nonlinear functions is~ on \F_{p^n} ( prime) and the one of almost
perfect nonlinear functions on \F_{2^n} is~. It is natural to inquire what
happens with APN or other low differential uniform functions in even and odd
characteristics. Here, we explicitly determine the second-order zero
differential spectra of several maps with low differential uniformity. In
particular, we compute the second-order zero differential spectra for some
almost perfect nonlinear (APN) functions, pushing further the study started in
Boukerrou et al.~\cite{Bouk} and continued in Li et al. \cite{LYT} [Cryptogr.
Commun. 14.3 (2022), 653--662], and it turns out that our considered functions
also have low second-order zero differential uniformity
The second-order zero differential spectra of some APN and other maps over finite fields
The Feistel Boomerang Connectivity Table and the related notion of
-Boomerang uniformity (also known as the second-order zero differential
uniformity) has been recently introduced by Boukerrou et al.~\cite{Bouk}. These
tools shall provide a major impetus in the analysis of the security of the
Feistel network-based ciphers. In the same paper, a characterization of almost
perfect nonlinear functions (APN) over fields of even characteristic in terms
of second-order zero differential uniformity was also given. Here, we find a
sufficient condition for an odd or even function over fields of odd
characteristic to be an APN function, in terms of second-order zero
differential uniformity. Moreover, we compute the second-order zero
differential spectra of several APN or other low differential uniform
functions, and show that our considered functions also have low second-order
zero differential uniformity, though it may vary widely, unlike the case for
even characteristic when it is always zero