72 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
PN functions, complete mappings and quasigroup difference sets
We investigate pairs of permutations of such that
is a permutation for every . We show that
necessarily for some complete mapping of
, and call the permutation a perfect nonlinear
(PN) function. If , then is a PcN function, which have
been considered in the literature, lately. With a binary operation on
involving , we obtain a
quasigroup, and show that the graph of a PN function is a difference
set in the respective quasigroup. We further point to variants of symmetric
designs obtained from such quasigroup difference sets. Finally, we analyze an
equivalence (naturally defined via the automorphism group of the respective
quasigroup) for PN functions, respectively, the difference sets in the
corresponding quasigroup
On the characters of Sylow -subgroups of finite Chevalley groups for arbitrary primes
We develop in this work a method to parametrize the set of
irreducible characters of a Sylow -subgroup of a finite Chevalley group
which is valid for arbitrary primes , in particular when is a
very bad prime for . As an application, we parametrize
when .Comment: 22 page
On construction and (non)existence of c-(almost) perfect nonlinear functions
Functions with low differential uniformity have relevant applications in cryptography. Recently, functions with low c-differential uniformity attracted lots of attention. In particular, so-called APcN and PcN functions (generalization of APN and PN functions) have been investigated. Here, we provide a characterization of such functions via quadratic polynomials as well as non-existence results.publishedVersio
Analysis, classification and construction of optimal cryptographic Boolean functions
Modern cryptography is deeply founded on mathematical theory and vectorial Boolean functions play an important role in it. In this context, some cryptographic properties of Boolean functions are defined. In simple terms, these properties evaluate the quality of the cryptographic algorithm in which the functions are implemented.
One cryptographic property is the differential uniformity, introduced by Nyberg in 1993. This property is related to the differential attack, introduced by Biham and Shamir in 1990. The corresponding optimal functions are called Almost Perfect Nonlinear functions, shortly APN. APN functions have been constructed, studied and classified up to equivalence relations. Very important is their classification in infinite families, i.e. constructing APN functions that are defined for infinitely many dimensions. In spite of an intensive study of these maps, many fundamental problems related to APN functions are still open and relatively few infinite families are known so far.
In this thesis we present some constructions of APN functions and study some of their properties. Specifically, we consider a known construction, L1(x^3)+L2(x^9) with L1 and L2 linear maps, and we introduce two new constructions, the isotopic shift and the generalised isotopic shift. In particular, using the two isotopic shift constructing techniques, in dimensions 8 and 9 we obtain new APN functions and we cover many unclassified cases of APN maps. Here new stands for inequivalent (in respect to the so-called CCZ-equivalence) to already known ones.
Afterwards, we study two infinite families of APN functions and their generalisations. We show that all these families are equivalent to each other and they are included in another known family. For many years it was not known whether all the constructed infinite families of APN maps were pairwise inequivalent. With our work, we reduce the list to those inequivalent to each other.
Furthermore, we consider optimal functions with respect to the differential uniformity in fields of odd characteristic. These functions, called planar, have been valuable for the construction of new commutative semifields. Planar functions present often a close connection with APN maps. Indeed, the idea behind the isotopic shift construction comes from the study of isotopic equivalence, which is defined for quadratic planar functions. We completely characterise the mentioned equivalence by means of the isotopic shift and the extended affine equivalence. We show that the isotopic shift construction leads also to inequivalent planar functions and we analyse some particular cases of this construction.
Finally, we study another cryptographic property, the boomerang uniformity, introduced by Cid et al. in 2018. This property is related to the boomerang attack, presented by Wagner in 1999. Here, we study the boomerang uniformity for some known classes of permutation polynomials.Doktorgradsavhandlin
Computational search for isotopic semifields and planar functions in characteristic 3
In this thesis, we investigate the possibility of finding new planar functions and corresponding semifields in characteristic 3 by the construction of isotopic semifields from the known families and sporadic instances of planar functions. Using the conditions laid out by Coulter and Henderson, we are able to deduce that a number of the known infinite families can never produce CCZ-inequivalent functions via isotopism. For the remaining families, we computationally investigate the isotopism classes of their instances over finite fields of order 3^n for n ≤ 8. We find previously unknown isotopisms between the semifields corresponding to some of the known planar functions for n = 6 and n = 8. This allows us to refine the known classification of planar functions up to isotopism, and to provide an updated, partial classification up to isotopism over finite fields of order 3^n for n ≤ 8.Masteroppgave i informatikkINF399MAMN-INFMAMN-PRO
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