We suggest a construction of the minimal polynomial
m
β
k
of
β
k
∈
F
q
n
over
F
q
from the minimal polynomial
f
=
m
β
for all positive integers
k
whose prime factors divide
q
-
1
. The computations ...of our construction are carried out in
F
q
. The key observation leading to our construction is that for
k
∣
q
-
1
holds
m
β
k
(
X
k
)
=
∏
j
=
1
k
t
ζ
k
-
j
n
f
(
ζ
k
j
X
)
,
where
t
=
max
{
m
∈
N
:
m
∣
gcd
(
n
,
k
)
,
f
(
X
)
=
g
(
X
m
)
,
g
∈
F
q
X
}
and
ζ
k
is a primitive
k
-th root of unity in
F
q
. The construction allows to construct a large number of irreducible polynomials over
F
q
of the same degree. Since different applications require different properties, this large number allows the selection of the candidates with the desired properties.
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We study the cycle structure of permutations
F
(
x
)
=
x
+
γ
f
(
x
)
on
F
q
n
, where
f
:
F
q
n
→
F
q
. We show that for a 1-homogeneous function
f
the cycle structure of
F
can be determined by ...calculating the cycle structure of certain induced mappings on parallel lines of
γ
F
q
. Using this observation we describe explicitly the cycle structure of two families of permutations over
F
q
2
:
x
+
γ
Tr
(
x
2
q
-
1
)
, where
q
≡
-
1
(
mod
3
)
and
γ
∈
F
q
2
, with
γ
3
=
-
1
27
and
x
+
γ
Tr
x
2
2
s
-
1
+
3
·
2
s
-
1
+
1
3
, where
q
=
2
s
,
s
odd and
γ
∈
F
q
2
, with
γ
(
q
+
1
)
/
3
=
1
.
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In this paper we consider in detail the composition of an irreducible polynomial with
X
2
and suggest a recurrent construction of irreducible polynomials of fixed degree over finite fields of odd ...characteristics. More precisely, given an irreducible polynomial of degree
n
and order
2
r
t
with
t
odd, the construction produces
o
r
d
t
(
2
)
irreducible polynomials of degree
n
and order
t
. The construction can be used for example to search irreducible polynomials with specific requirements on its coefficients.
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Image sets of perfectly nonlinear maps Kölsch, Lukas; Kriepke, Björn; Kyureghyan, Gohar M.
Designs, codes, and cryptography,
2023/1, Volume:
91, Issue:
1
Journal Article
Peer reviewed
Open access
We consider image sets of differentially
d
-uniform maps of finite fields. We present a lower bound on the image size of such maps and study their preimage distribution. Further, we focus on a ...particularly interesting case of APN maps on binary fields
F
2
n
. We show that APN maps with the minimal image size are very close to being 3-to-1. We prove that for
n
even the image sets of several important families of APN maps are minimal, and as a consequence they have the classical Walsh spectrum. Finally, we present upper bounds on the image size of APN maps. For a non-bijective almost bent map
f
, these results imply
2
n
+
1
3
+
1
≤
|
Im
(
f
)
|
≤
2
n
-
2
(
n
-
1
)
/
2
.
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5.
Editorial: Coding and Cryptography 2019 Canteaut, Anne; Kyureghyan, Gohar; Pott, Alexander ...
Designs, codes, and cryptography,
09/2020, Volume:
88, Issue:
9
Journal Article
Peer reviewed
Open access
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A new class of monomial bent functions Canteaut, Anne; Charpin, Pascale; Kyureghyan, Gohar M.
Finite fields and their applications,
01/2008, Volume:
14, Issue:
1
Journal Article
Peer reviewed
Open access
We study the Boolean functions
f
λ
:
F
2
n
→
F
2
,
n
=
6
r
, of the form
f
(
x
)
=
Tr
(
λ
x
d
)
with
d
=
2
2
r
+
2
r
+
1
and
λ
∈
F
2
n
. Our main result is the characterization of those
λ for which
...f
λ
are bent. We show also that the set of these cubic bent functions contains a subset, which with the constantly zero function forms a vector space of dimension 2
r over
F
2
. Further we determine the Walsh spectra of some related quadratic functions, the derivatives of the functions
f
λ
.
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On inversion in $Z_{2^n-1} Kyureghyan, Gohar; Suder, Valentin
Finite fields and their applications,
01/2014, Volume:
25
Journal Article
Peer reviewed
Open access
In this paper we determined explicitly the multiplicative inverses of the Dobbertin and Welch APN exponents in Z2n−1, and we described the binary weights of the inverses of the Gold and Kasami ...exponents. We studied the function Invd(n), which for a fixed positive integer d maps integers n⩾1 to the least positive residue of the inverse of d modulo 2n−1, if it exists. In particular, we showed that the function Invd is completely determined by its values for 1⩽n⩽θd, where θd is the order of 2 modulo the largest odd divisor of d.
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This volume contains the proceedings of the 11th International Conference on Finite Fields and their Applications (Fq11), held July 22-26, 2013, in Magdeburg, Germany.Finite Fields are fundamental ...structures in mathematics. They lead to interesting deep problems in number theory, play a major role in combinatorics and finite geometry, and have a vast amount of applications in computer science.Papers in this volume cover these aspects of finite fields as well as applications in coding theory and cryptography.
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