Article

Planar Functions and Planes of Lenz-Barlotti Class II

February 1997
Designs Codes and Cryptography 10(2):167-184

Source
DBLP

Authors:

University of Delaware

Planar functions were introduced by Dembowski and Ostrom [4] to describe projective planes possessing a collineation group with particular properties. Several classes of planar functions over a finite field are described, including a class whose associated affine planes are not translation planes or dual translation planes. This resolves in the negative a question posed in [4]. These planar functions define at least one such affine plane of order 3e for every e = 4 and their projective closures are of Lenz-Barlotti type II. All previously known planes of type II are obtained by derivation or lifting. At least when e is odd, the planes described here cannot be obtained in this manner.

Independent sets in polarity graphs

Preprint

Jan 2016

Given a projective plane

\Sigma

and a polarity

\theta

\Sigma

, the corresponding polarity graph is the graph whose vertices are the points of

\Sigma

, and two distinct points

p_1

and

p_2

are adjacent if

p_1

is incident to

p_2^{ \theta}

\Sigma

. A well-known example of a polarity graph is the Erd\H{o}s-R\'{e}nyi orthogonal polarity graph

ER_q

, which appears frequently in a variety of extremal problems. Eigenvalue methods provide an upper bound on the independence number of any polarity graph. Mubayi and Williford showed that in the case of

ER_q

, the eigenvalue method gives the correct upper bound in order of magnitude. We prove that this is also true for other families of polarity graphs. This includes a family of polarity graphs for which the polarity is neither orthogonal nor unitary. We conjecture that any polarity graph of a projective plane of order q has an independent set of size

\Omega (q^{3/2})

. Some related results are also obtained.

New Classes of Ternary Bent Functions from the Coulter-Matthews Bent Functions

Preprint

Jul 2017

It has been an active research issue for many years to construct new bent functions. For k odd with

\gcd(n, k)=1

, and

a\in\mathbb{F}_{3^n}^{*}

, the function

f(x)=Tr(ax^{\frac{3^k+1}{2}})

is weakly regular bent over

\mathbb{F}_{3^n}

, where

Tr(\cdot):\mathbb{F}_{3^n}\rightarrow\mathbb{F}_3

is the trace function. This is the well-known Coulter-Matthews bent function. In this paper, we determine the dual function of f(x) completely. As a consequence, we find many classes of ternary bent functions not reported in the literature previously. Such bent functions are not quadratic if

k>1

, and have

\left(\left(\frac{1+\sqrt{5}}{2}\right)^{w+1}-\right.

\left.\left(\frac{1-\sqrt{5}}{2}\right)^{w+1}\right)/\sqrt{5}

\left(\left(\frac{1+\sqrt{5}}{2}\right)^{n-w+1}-\right.

\left.\left(\frac{1-\sqrt{5}}{2}\right)^{n-w+1}\right)/\sqrt{5}

trace terms, where

0<w<n

and

wk\equiv 1\ (\bmod\;n)

. Among them, five special cases are especially interesting: for the case of k=(n+1)/2, the number of trace terms is

\left(\left(\frac{1+\sqrt{5}}{2}\right)^{n-1}-\right.

\left.\left(\frac{1-\sqrt{5}}{2}\right)^{n-1}\right)/\sqrt{5}

; for the case of

k=n-1

, the number of trace terms is

\left(\left(\frac{1+\sqrt{5}}{2}\right)^n-\right.

\left.\left(\frac{1-\sqrt{5}}{2}\right)^n\right)/\sqrt{5}

; for the case of

k=(n-1)/2

, the number of trace terms is

\left(\left(\frac{1+\sqrt{5}}{2}\right)^{n-1}-\right.

\left.\left(\frac{1-\sqrt{5}}{2}\right)^{n-1}\right)/\sqrt{5}

; for the case of

(n, k)=(5t+4, 4t+3)

(5t+1, 4t+1)

with

t\geq 1

, the number of trace terms is 8; and for the case of

(n, k)=(7t+6, 6t+5)

(7t+1, 6t+1)

with

t\geq 1

, the number of trace terms is 21. As a byproduct, we find new classes of ternary bent functions with only 8 or 21 trace terms.

Unitals in shift planes of odd order

Preprint

Aug 2015

A finite shift plane can be equivalently defined via abelian relative difference sets as well as planar functions. In this paper, we present a generic way to construct unitals in finite shift planes of odd orders

q^2

. We investigate various geometric and combinatorial properties of them, such as the self-duality, the existences of O'Nan configurations, the Wilbrink's conditions, the designs formed by circles and so on. We also show that our unitals are inequivalent to the unitals derived from unitary polarities in the same shift planes. As designs, our unitals are also not isomorphic to the classical unitals (the Hermitian curves).

A Sequence Construction of Cyclic Codes over Finite Fields

Preprint

Nov 2016

Cunsheng Ding

Cyclic codes over finite fields are widely implemented in data storage systems, communication systems, and consumer electronics, as they have very efficient encoding and decoding algorithms. They are also important in theory, as they are closely connected to several areas in mathematics. There are a few fundamental ways of constructing all cyclic codes over finite fields, including the generator matrix approach, the generator polynomial approach, the generating idempotent approach, and the q-polynomial approach. Another one is a sequence approach, which has been intensively investigated in the past decade. The objective of this paper is to survey the progress in the past decade in this direction.

Further investigation on differential properties of the generalized Ness–Helleseth function

Article

Full-text available

Nov 2024
DESIGN CODE CRYPTOGR

Let n be an odd positive integer, p be an odd prime with

p\equiv 3\pmod 4

d_{1} = {{p^{n}-1}\over {2}} -1

and

d_{2} =p^{n}-2

. The function defined by

f_u(x)=ux^{d_{1}}+x^{d_{2}}

is called the generalized Ness–Helleseth function over

\mathbb {F}_{p^n}

, where

u\in \mathbb {F}_{p^n}

. It was initially studied by Ness and Helleseth in the ternary case. In this paper, for

p^n \equiv 3 \pmod 4

and

p^n \ge 7

, we provide the necessary and sufficient condition for

f_u(x)

to be an APN function. In addition, for each u satisfying

\chi (u+1) = \chi (u-1)

, the differential spectrum of

f_u(x)

is investigated, and it is expressed in terms of some quadratic character sums of cubic polynomials, where

\chi (\cdot )

denotes the quadratic character of

{\mathbb {F}}_{p^n}

Further Investigation on Differential Properties of the Generalized Ness-Helleseth Function

Preprint

Aug 2024

Let n be an odd positive integer, p be a prime with

p\equiv3\pmod4

d_{1} = {{p^{n}-1}\over {2}} -1

and

d_{2} =p^{n}-2

. The function defined by

f_u(x)=ux^{d_{1}}+x^{d_{2}}

is called the generalized Ness-Helleseth function over

\mathbb{F}_{p^n}

, where

u\in\mathbb{F}_{p^n}

. It was initially studied by Ness and Helleseth in the ternary case. In this paper, for

p^n \equiv 3 \pmod 4

and

p^n \ge7

, we provide the necessary and sufficient condition for

f_u(x)

to be an APN function. In addition, for each u satisfying

\chi(u+1) = \chi(u-1)

, the differential spectrum of

f_u(x)

is investigated, and it is expressed in terms of some quadratic character sums of cubic polynomials, where

\chi(\cdot)

denotes the quadratic character of

\mathbb{F}_{p^n}

A class of functions and their application in constructing semisymmetric designs

Article

Full-text available

Jul 2024
DESIGN CODE CRYPTOGR

We introduce the notion of a semiplanar function of index

\lambda

, generalising several previous concepts. We show how semiplanar functions can be used to construct semisymmetric designs using an incidence structure determined by the function. Issues regarding the connectivity of the structure are then considered. The question of existence is addressed by establishing monomial examples over finite fields, and we examine how composition with linearized polynomials can lead to further classes of examples. We end by returning to the incidence structure and considering maximal intersection sets when the incidence structure is constructed using a particular class of functions.

A Degree Bound for Planar Functions

Preprint

Full-text available

Jul 2024

Using Stickelberger's theorem on Gauss sums, we show that if F is a planar function on a finite field

\mathbb{F}_q

, then for all non-zero functions

G : \mathbb{F}_q \to \mathbb{F}_q

, we have \begin{equation*} \mathrm{deg } \ G \circ F - \mathrm{deg } \ G \le \frac{n(p-1)}{2}\,, \end{equation*} where

q = p^n

with p a prime and n a positive integer, and

\mathrm{deg } \ F

is the algebraic degree of F, i.e., the degree of the corresponding multivariate polynomial over

\mathbb{F}_p

. This bound leads to a simpler proof of the classification of planar polynomials over

\mathbb{F}_p

and planar monomials over

\mathbb{F}_{p^2}

. As a new result, using the same degree bound, we complete the classification of planar monomials for all

n = 2^k

with

p>5

and k a non-negative integer. Finally, we state a conjecture on the sum of the base-p digits of integers modulo

q-1

that implies the complete classification of planar monomials over finite fields of characteristic

p>5

Few-weight linear codes over

\mathbb{F}_p

from t-to-one mappings

Preprint

Full-text available

Nov 2023

René Rodríguez-Aldama

x^{p^{\alpha}+1}

, for a suitable choice of the exponent

\alpha

, so that, when

p>2

and

n\not\equiv 0 \pmod{4}

, these monomials are planar. We study the properties of such monomials in detail for each integer

n>1

and any prime number p. In particular, we show that they are t-to-one, where the parameter t depends on the field

\mathbb{F}_{p^n}

and it takes the values 1, 2 or p+1. Moreover, we give a simple proof of the fact that the functions are

\delta

-uniform with

\delta\in \{\ 1, 4, p\ \}

. This result describes the differential behaviour of these monomials for any p and n. For the second class of functions, we consider an affine equivalent trinomial to

x^{p^{\alpha}+1}

, namely,

x^{p^{\alpha}+1} +\lambda x^{p^{\alpha}}+\lambda^{p^{\alpha}}x

for

\lambda\in \mathbb{F}_{p^n}^*

. We prove that these trinomials satisfy certain regularity properties, which are useful for the specification of linear codes with three or four weights that are different than the monomial construction. These families of codes contain projective codes and optimal codes (with respect to the Griesmer bound). Remarkably, they contain infinite families of self-orthogonal and minimal p-ary linear codes for every prime number p. Our findings highlight the utility of studying affine equivalent functions, which is often overlooked in this context.

On the Planarity of Certain Dembowski-Ostrom PolynomialsBazı Dembowski-Ostrom Polinomlarının Planaritesi Üzerine

Article

Full-text available

Nov 2022

Planar mappings, defined by Dembowski and Ostrom, are identified as a means to construct projective planes. Then, many important applications of planar mappings appear in different fields such as cryptography and coding theory. In this paper, we provide sufficient and necessary conditions for the planarity of certain Dembowski-Ostrom polynomials over the finite field extension of degree three with odd characteristic. In particular, we completely determine the coefficients of the given Dembowski-Ostrom polynomials to be planar.

Refining the search space of Alltop monomials

Article

Full-text available

Nov 2022
J ALGEBR COMB

Alltop functions have applications to code-division multiple access (CDMA) systems and mutually unbiased bases (MUBs). Alltop functions construct MUBs and CDMA signal sets, and this motivates the continued search for further Alltop functions. The discovery of further Alltop functions is hindered by the computational complexity of verification. This paper narrows the search space through the introduction of a characteristic cubic polynomial and provides a verification process for Alltop monomials with a reduced computational complexity. Computational results lead to the conjecture that there are no further Alltop monomials.

On a classification of planar functions in characteristic three

Article

Full-text available

Feb 2025

Planar functions are functions over a finite field that have optimal combinatorial properties and they have applications in several branches of mathematics, including algebra, projective geometry and cryptography. There are two relevant equivalence relations for planar functions, that are isotopic equivalence and CCZ-equivalence. Classification of planar functions is performed via CCZ-equivalence which arises from cryptographic applications. In the case of quadratic planar functions, isotopic equivalence, coming from connections to commutative semifields, is more general than CCZ-equivalence and isotopic transformations can be considered as a construction method providing up to two CCZ-inequivalent mappings. In this paper, we first survey known infinite classes and sporadic cases of planar functions up to CCZ-equivalence, aiming to exclude equivalent cases and to identify those with the potential to provide additional functions via isotopic equivalence. In particular, for fields of order

\boldsymbol{3^n}

with

\boldsymbol{n\le 11}

, we completely resolve if and when isotopic equivalence provides different CCZ-classes for all currently known planar functions. Further, we perform an extensive computational investigation on some of these fields and find seven new sporadic planar functions over

\boldsymbol{\mathbb {F}_{3^6}}

and two over

\boldsymbol{\mathbb {F}_{3^9}}

. Finally, we give new simple quadrinomial representatives for the Dickson family of planar functions.

Giesbrecht's algorithm, the HFE cryptosystem and Ore's

p^s

-polynomials

Preprint

Nov 2016

We report on a recent implementation of Giesbrecht's algorithm for factoring polynomials in a skew-polynomial ring. We also discuss the equivalence between factoring polynomials in a skew-polynomial ring and decomposing

p^s

-polynomials over a finite field, and how Giesbrecht's algorithm is outlined in some detail by Ore in the 1930's. We end with some observations on the security of the Hidden Field Equation (HFE) cryptosystem, where p-polynomials play a central role.

Planar polynomials and an extremal problem of Fischer and Matousek

Preprint

Feb 2017

Let G be a 3-partite graph with k vertices in each part and suppose that between any two parts, there is no cycle of length four. Fischer and Matou\u{s}ek asked for the maximum number of triangles in such a graph. A simple construction involving arbitrary projective planes shows that there is such a graph with

(1 - o(1)) k^{3/2}

triangles, and a double counting argument shows that one cannot have more than

(1+o(1)) k^{7/4}

triangles. Using affine planes defined by specific planar polynomials over finite fields, we improve the lower bound to

(1 - o(1)) k^{5/3}

The Weight Distributions of Two Classes of Linear Codes From Perfect Nonlinear Functions

Preprint

Jun 2023

In this paper, all the possibilities for the value distribution of a perfect nonlinear function from

\mathbb{F}_{p^m}

\mathbb{F}_p

are determined, where p is an odd prime number and

m\in\mathbb{N}_+

. As an application, we determine the weight distributions of two classes of linear codes over

\mathbb{F}_p

constructed from perfect nonlinear functions.

Characterizations of a Class of Planar Functions over Finite Fields

Preprint

Mar 2023

Planar functions, introduced by Dembowski and Ostrom, have attracted much attention in the last decade. As shown in this paper, we present a new class of planar functions of the form

\operatorname{Tr}(ax^{q+1})+\ell(x^2)

on an extension of the finite field

\mathbb F_{q^n}/\mathbb F_q

. Specifically, we investigate those functions on

\mathbb F_{q^2}/\mathbb F_q

and construct several typical kinds of planar functions. We also completely characterize them on

\mathbb F_{q^3}/\mathbb F_q

. When the degree of extension is higher, it will be proved that such planar functions do not exist given certain conditions.

Several classes of new weakly regular bent functions outside RF

\mathcal{R}\mathcal{F}

, their duals and some related (minimal) codes with few weights

Article

Full-text available

Mar 2023
DESIGN CODE CRYPTOGR

Boosted by cryptography and coding theory applications and rich connections to objects from geometry and combinatorics, bent functions and related functions developed into a lively research area. In the mid-seventies, Rothaus initially introduced bent functions in the Boolean case, but later, they extended in the p-ary case where p is any prime. Such an extension brought more rich connections to bent functions. In addition to the theory of Fourier transform, handling p-ary bent functions (where p is an odd prime), analyzing their properties and designing them require using the theory of cyclotomic fields. Such a family is classified into weakly regular bent and regular bent. Compared to the Boolean case, the class of p-ary bent functions inherits a much larger variety of properties. For example, it contains the class of dual-bent functions as a proper subclass, which again includes weakly regular bent functions as a proper subclass. Weakly regular bent functions can also be employed in many domains. In particular, they have been widely used in designing good linear codes for several applications (such as secret sharing and two-party computation), association schemes, and strongly regular graphs. More exploration is still needed despite a lot of interest and attention in this topic to increase our knowledge of bent functions and design them. In particular, only a few constructions of weakly regular bent functions have been presented in the literature. Nice construction methods of these functions have been given by Tang et al (IEEE Trans Inf Theory 62(3):1166–1176, 2016). All the known infinite families of weakly regular bent functions belong to a class (denoted by Tang et al. in the previous reference) RF

\mathcal{R}\mathcal{F}

. This paper is devoted to weakly regular bent functions, and its objective is twofold. First, it aims to generate new infinite families of weakly regular bent functions living outside RF

\mathcal{R}\mathcal{F}

and secondly, to exploit the constructed functions to design new families of p-ary linear codes and investigate their use for some standard application after studying this minimality based on their weight distributions. More specifically, we present several classes of weakly regular bent functions obtained from monomial bent functions by modifying the values of some known weakly regular bent functions on some subsets of the finite field Fpn

\mathbb {F}_{p^n}

(n is a positive integer). Our weakly regular bent functions are of degree either p or p+12

\frac{p+1}{2}

. We also explicitly determine their corresponding dual functions. Finally, we exploit our constructions to derive four new classes of linear codes with three to seven weights. We also show that two of them lead to minimal codes, which are more appropriate for several concrete applications.

On Matrix Algebras Isomorphic to Finite Fields and Planar Dembowski-Ostrom Monomials

Preprint

Full-text available

Nov 2022

Let p be a prime and n a positive integer. We present a deterministic algorithm for deciding whether the matrix algebra

\mathbb{F}_p[A_1,\dots,A_t]

with

A_1,\dots,A_t \in \mathrm{GL}(n,\mathbb{F}_p)

is a finite field, performing at most

\mathcal{O}(tn^6\log(p))

elementary operations in

\mathbb{F}_p

. We then show how this new algorithm can be used to decide the isotopy of a commutative presemifield of odd order to a finite field in polynomial time. More precisely, for a Dembowski-Ostrom (DO) polynomial

g \in \mathbb{F}_{p^n}[x]

, we associate to g a set of

n \times n

matrices with coefficients in

\mathbb{F}_p

, denoted

\mathrm{Quot}(\mathcal{D}_g)

, that stays invariant up to matrix similarity when applying extended-affine equivalence transformations to g. In the case where g is a planar DO polynomial,

\mathrm{Quot}(\mathcal{D}_g)

is the set of quotients

XY^{-1}

with

Y \neq 0,X

being elements from the spread set of the corresponding commutative presemifield. We then show that

\mathrm{Quot}(\mathcal{D}_g)

forms a field of order

p^n

if and only if g is equivalent to the planar monomial

x^2

, i.e., if and only if the commutative presemifield associated to g is isotopic to a finite field. Finally, we analyze the structure of

\mathrm{Quot}(\mathcal{D}_g)

for all planar DO monomials, i.e., for commutative presemifields of odd order being isotopic to a finite field or a commutative twisted field. More precisely, for g being equivalent to a planar DO monomial, we show that every non-zero element

X \in \mathrm{Quot}(\mathcal{D}_g)

generates a field

\mathbb{F}_p[X] \subseteq \mathrm{Quot}(\mathcal{D}_g)

. In particular,

\mathrm{Quot}(\mathcal{D}_g)

contains the field

\mathbb{F}_{p^n}

Dual and Hull code in the first two generic constructions and relationship with the Walsh transform of cryptographic functions

Article

Full-text available

Apr 2025
APPL ALGEBR ENG COMM

Virginio Fratianni

We contribute to the knowledge of linear codes from special polynomials and functions, which have been studied intensively in the past few years. Such codes have several applications in secret sharing, authentication codes, association schemes and strongly regular graphs. To the best of our knowledge, this is the first work in which the dual and hull codes are studied in the framework of the two generic constructions; in particular, we propose a Gram–Schmidt process to compute them explicitly. We also determine a necessary condition expressed by employing the Walsh transform for a codeword of

\mathcal {C}

to belong in the dual. This achievement was generally obtained when the functions were weakly regularly bent. We shall give a novel description of the Hull code in the framework of the two generic constructions. Our primary interest is constructing linear codes of fixed Hull dimension and determining the (Hamming) weight of the codewords in their duals.

On the Dual of the Coulter-Matthews Bent Functions

Preprint

Apr 2016

For any bent function, it is very interesting to determine its dual function because the dual function is also bent in certain cases. For k odd and

\gcd(n, k)=1

, it is known that the Coulter-Matthews bent function

f(x)=Tr(ax^{\frac{3^k+1}{2}})

is weakly regular bent over

\mathbb{F}_{3^n}

, where

a\in\mathbb{F}_{3^n}^{*}

, and

Tr(\cdot):\mathbb{F}_{3^n}\rightarrow\mathbb{F}_3

is the trace function. In this paper, we investigate the dual function of f(x), and dig out an universal formula. In particular, for two cases, we determine the formula explicitly: for the case of n=3t+1 and k=2t+1 with

t\geq 2

, the dual function is given by

Tr\left(-\frac{x^{3^{2t+1}+3^{t+1}+2}}{a^{3^{2t+1}+3^{t+1}+1}}-\frac{x^{3^{2t}+1}}{a^{-3^{2t}+3^{t}+1}}+\frac{x^{2}}{a^{-3^{2t+1}+3^{t+1}+1}}\right);

and for the case of n=3t+2 and k=2t+1 with

t\geq 2

, the dual function is given by

Tr\left(-\frac{x^{3^{2t+2}+1}}{a^{3^{2t+2}-3^{t+1}+3}}-\frac{x^{2\cdot3^{2t+1}+3^{t+1}+1}}{a^{3^{2t+2}+3^{t+1}+1}}+\frac{x^2}{a^{-3^{2t+2}+3^{t+1}+3}}\right).

As a byproduct, we find two new classes of ternary bent functions with only three terms. Moreover, we also prove that in certain cases f(x) is regular bent.

On the 2-ranks of a class of unitals

Preprint

Oct 2015

Let

U_\theta

be a unital defined in a shift plane of odd order

q^2

, which are constructed recently by the authors. In particular, when the shift plane is desarguesian,

U_\theta

is a special Buekenhout-Metz unital formed by a union of ovals. We investigate the dimensions of the binary codes derived from

U_\theta

. By using Kloosterman sums, we obtain a new lower bound on the aforementioned dimensions which improves the result obtained by Leung and Xiang in 2009. In particular, for

q=3^m

, this new lower bound equals

\frac{2}{3}(q^3+q^2-2q)-1

for even m and

\frac{2}{3}(q^3+q^2+q)-1

for odd m.

Revisiting products of the form X times a linearized polynomial L(X)

Article

Full-text available

Oct 2024
DESIGN CODE CRYPTOGR

Christof Beierle

For a q -polynomial L over a finite field

\mathbb {F}_{q^n}

F q n , we characterize the differential spectrum of the function

f_L:\mathbb {F}_{q^n} \rightarrow \mathbb {F}_{q^n}, x \mapsto x \cdot L(x)

f L : F q n → F q n , x ↦ x · L ( x ) and show that, for

n \le 5

n ≤ 5 , it is completely determined by the image of the rational function

r_L :\mathbb {F}_{q^n}^* \rightarrow \mathbb {F}_{q^n}, x \mapsto L(x)/x

r L : F q n ∗ → F q n , x ↦ L ( x ) / x . This result follows from the classification of the pairs ( L , M ) of q -polynomials in

\mathbb {F}_{q^n}[X]

F q n [ X ] ,

n \le 5

n ≤ 5 , for which

r_L

r L and

r_M

r M have the same image, obtained in Csajbók et al. (Ars Math Contemp 16(2):585–608, 2019). For the case of

n>5

n > 5 , we pose an open question on the dimensions of the kernels of

x \mapsto L(x) - ax

x ↦ L ( x ) - a x for

a \in \mathbb {F}_{q^n}

a ∈ F q n . We further present a link between functions

f_L

f L of differential uniformity bounded above by q and scattered q -polynomials and show that, for odd values of q , we can construct CCZ-inequivalent functions

f_M

f M with bounded differential uniformity from a given function

f_L

f L fulfilling certain properties.

Some Constructions of Perfect c-Nonlinear and Pseudo-Perfect c-Nonlinear Functions

Article

Full-text available

Oct 2024
INT J FOUND COMPUT S

Xiangdong Cheng

The perfect nonlinear functions play an important role in block ciphers and have been widely investigated in the literature. Subsequently, the perfect c-nonlinear functions were generalized by Ellingsen et al. in 2020. In this paper, we study the c-differential uniformity of some functions. We first construct some perfect c-nonlinear functions from known ones. Additionally, pseudo-perfect c-nonlinear functions are also investigated and a necessary and sufficient condition for a function to be pseudo-perfect c-nonlinear is presented. Meanwhile, some pseudo-perfect c-nonlinear functions are constructed. Remarkably, we completely give the difference distribution table of a function with respect to the pseudo c-derivative.

Determining Sidon polynomials on Sidon sets over finite fields

Article

Sep 2024

Let p be a prime, and

q=p^n

be a prime power. In his works on Sidon sets over

\mathbb{F}_q \times \mathbb{F}_q

, Cilleruelo conjectured about polynomials that could generate q-element Sidon sets over

\mathbb{F}_q\times \mathbb{F}_q

. Here, we derive some criteria for determining polynomials that could generate q-element Sidon set over

\mathbb{F}_q\times \mathbb{F}_q

. Using these criteria, we prove that certain classes of monomials and cubic polynomials over

\mathbb{F}_p

cannot be used to generate p-element Sidon set over

\mathbb{F}_p\times \mathbb{F}_p

. We also discover a connection between the needed polynomials and planar polynomials.

On a classification of planar functions in characteristic three

Preprint

Full-text available

Jul 2024

3^n

with

n\le 11

\mathbb{F}_{3^6}

and two over

\mathbb{F}_{3^9}

. Finally, we give new simple quadrinomial representatives for the Dickson family of planar functions.

More on the number of distinct values of a class of functions

Preprint

Full-text available

Jun 2024

In a previous article, the authors determined the first (and at the time of writing, the only) non-trivial upper bound for the cardinality of the image set for several classes of functions, including planar functions. Here, we show that the upper bound cannot be tight for planar functions over

\mathbb F_q

, with the possible exception of q=343. We further show that if such an exceptional planar function exists, then it implies the existence of a projective plane of order 18. This follows from more general results, which apply to wider classes of functions.

Value Distributions of Perfect Nonlinear Functions

Article

Full-text available

Sep 2023
COMBINATORICA

In this paper, we study the value distributions of perfect nonlinear functions, i.e., we investigate the sizes of image and preimage sets. Using purely combinatorial tools, we develop a framework that deals with perfect nonlinear functions in the most general setting, generalizing several results that were achieved under specific constraints. For the particularly interesting elementary abelian case, we derive several new strong conditions and classification results on the value distributions. Moreover, we show that most of the classical constructions of perfect nonlinear functions have very specific value distributions, in the sense that they are almost balanced. Consequently, we completely determine the possible value distributions of vectorial Boolean bent functions with output dimension at most 4. Finally, using the discrete Fourier transform, we show that in some cases value distributions can be used to determine whether a given function is perfect nonlinear, or to decide whether given perfect nonlinear functions are equivalent.

Binomials and Trinomials as Planar Functions on Cubic Extensions of Finite Fields

Preprint

Mar 2023

Planar functions, introduced by Dembowski and Ostrom, are functions from a finite field to itself that give rise to finite projective planes. They exist, however, only for finite fields of odd characteristic. They have attracted much attention in the last decade thanks to their interest in theory and those deep and various applications in many fields. This paper focuses on planar functions on a cubic extension

\mathbb F_{q^3}/\mathbb F_q

. Specifically, we investigate planar binomials and trinomials polynomials of the form

\sum_{0\le i\le j<3}a_{ij}x^{q^i+q^j}

\mathbb F_{q^3}

, completely characterizing them and determine the equivalence class of those planar polynomials toward their classification. Our achievements are obtained using connections with algebraic projective curves and classical algebraic tools over finite fields.

The Apparent Structure of Dense Sidon Sets

Article

Feb 2023
ELECTRON J COMB

The correspondence between perfect difference sets and transitive projective planes is well-known. We observe that all known dense (i.e., close to square-root size) Sidon subsets of abelian groups come from projective planes through a similar construction. We classify the Sidon sets arising in this manner from desarguesian planes and find essentially no new examples. There are many further examples arising from nondesarguesian planes. We conjecture that all dense Sidon sets arise from finite projective planes in this way. If true, this implies that all abelian groups of most orders do not have dense Sidon subsets. In particular if

\sigma_n

denotes the size of the largest Sidon subset of

\mathbb{Z}/n\mathbb{Z}

, this implies

\liminf_{n \to \infty} \sigma_n / n^{1/2} < 1

. We also give a brief bestiary of somewhat smaller Sidon sets with a variety of algebraic origins, and for some of them provide an overarching pattern.

Value Distributions of Perfect Nonlinear Functions

Preprint

Full-text available

Feb 2023

Linear Codes over R= R1R2R3 and Their Applications in Secret Sharing Schemes

Conference Paper

Full-text available

Apr 2025

Karima Chatouh

On matrix algebras isomorphic to finite fields and planar Dembowski-Ostrom monomials

Article

Mar 2025

Optimal quinary cyclic codes with three zeros

Article

Jan 2025

Linear codes from planar functions and related covering codes

Article

Jan 2025

On the c-differential uniformity of certain maps over finite fields

Preprint

Full-text available

Apr 2020

We give some classes of power maps with low c-differential uniformity over finite fields of odd characteristic, {for

c=-1

}. Moreover, we give a necessary and sufficient condition for a linearized polynomial to be a perfect c-nonlinear function and investigate conditions when perturbations of perfect c-nonlinear (or not) function via an arbitrary Boolean or p-ary function is perfect c-nonlinear. In the process, we obtain a class of polynomials that are perfect c-nonlinear for all

c\neq 1

, in every characteristic. The affine, extended affine and CCZ-equivalence is also looked at, as it relates to c-differential uniformity.

Few-weight linear codes over F_p from t-to-one mappings

Article

Oct 2024
FINITE FIELDS TH APP

René Rodríguez-Aldama

For any prime number p, we provide two classes of linear codes with few weights over a p-ary alphabet. These codes are based on a well-known generic construction (the defining-set method), stemming on a class of monomials and a class of trinomials over finite fields. The considered monomials are Dembowski-Ostrom monomials x^{p^\alpha+1}, for a suitable choice of the exponent \alpha, so that, when p>2 and n\not\equiv 0 \pmod{4}, these monomials are planar. We study the properties of such monomials in detail for each integer n>1 and any prime number p. In particular, we show that they are t-to-one, where the parameter t depends on the field \mathbb{F}_{p^n} and it takes the values or 1,2, or p+1. Moreover, we give a simple proof of the fact that the functions are \delta-uniform with \delta\in\{1,4, p\}. This result describes the differential behavior of these monomials for any p and n. For the second class of functions, we consider an affine equivalent trinomial to x^{p^\alpha+1}, namely, for, x^{p^\alpha+1}+\lambda x^{p^\alpha}+\lambda^{p^\alpha}x for \lambda\in\mathbb{F}_{p^n}^*. We prove that these trinomials satisfy certain regularity properties, which are useful for the specification of linear codes with three or four weights that are different than the monomial construction. These families of codes contain projective codes and optimal codes (with respect to the Griesmer bound). Remarkably, they contain infinite families of self-orthogonal and minimal p-ary linear codes for every prime number p. Our findings highlight the utility of studying affine equivalent functions, which is often overlooked in this context.

More Differential Properties of the Ness-Helleseth Function

Article

Aug 2024

Let n ≥ 3 be an odd integer, d ₁ =3 ⁿ -1/2-1, d ₂ =3 ⁿ -2 and u be an element of the finite field F _{3n} . This paper shows that f_u(x)=ux^d1+x^d2 is an almost perfect nonlinear (APN) function on F ₃ n if and only if χ( u +1)= χ( u -1)=χ( u ), where χ(∙) denotes the quadratic character of F3n. This settles the open problem raised by Ness and Helleseth in IEEE Trans. Inf. Theory 53(7): 2581-2586, 2007, where only the sufficiency part of the result was proved. Furthermore, we investigate the differential spectra of f_u(x) for elements u satisfying χ( u +1)=χ( u -1) and express them in terms of several quadratic character sums of cubic polynomials.

On a class of APN power functions over odd characteristic finite fields: Their differential spectrum and c-differential properties

Article

Apr 2024
DISCRETE MATH

Characterizations of a class of planar functions over finite fields

Article

Mar 2024
FINITE FIELDS TH APP

Exceptionality of (X+1) + (X−1)

Article

Mar 2024
FINITE FIELDS TH APP

The Weight Distributions of Two Classes of Linear Codes From Perfect Nonlinear Functions

Article

Jan 2023

In this paper, we employ general results on the value distributions of perfect nonlinear functions from F _{pm} to F _p to give a unified approach to determining the weight distributions of two classes of linear codes over F _p constructed from perfect nonlinear functions, where p is an odd prime and m is an odd number. When m is even, we give some mild additional conditions for similar conclusions to hold.

Commutative quasigroups and semifields from planar functions

Article

Nov 2023

Ales Drapal

https://www.tandfonline.com/eprint/AEH87DVDDGU4DHIPP2RQ/full?target=10.1080/00927872.2023.2275392 https://www.tandfonline.com/action/showCitFormats?doi=10.1080%2F00927872.2023.2275392

Subfield Codes of Several Few-Weight Linear Codes Parameterized by Functions and Their Consequences

Article

Jan 2023

Subfield codes of linear codes over finite fields have recently received much attention since they can produce optimal codes, which may have applications in secret sharing, authentication codes and association schemes. In this paper, we first present a construction framework of 3-dimensional linear codes

{\mathcal{ C}}_{f,g}

over

\mathbb {F}_{q^{m}}

parameterized by any two functions

f, g

over

\mathbb {F}_{q^{m}}

, and then study the properties of six types of

{\mathcal{ C}}_{f,g}

, its punctured code

{\mathcal{ C}}^{\ast}_{f,g}

and their corresponding subfield codes over

\mathbb {F}_{q}

. The classification of

{\mathcal{ C}}_{f,g}

is based on special choices of f,g as trace function, norm function, almost bent function, Boolean bent function or a combination of these functions. For the first two types of

{\mathcal{ C}}_{f,g}

, we explicitly determine the weight distributions and dualities of

{\mathcal{ C}}_{f,g}, {\mathcal{ C}}_{f,g}^{\ast}

and their subfield codes over

\mathbb {F}_{q}

. The remaining four types of

{\mathcal{ C}}_{f,g}

are restricted to q=2 , and the weight distributions and dualities of the subfields code

{\mathcal{ C}}_{f,g}^{(q)}

and

{\cal C_{f,g}^{\ast}}^{(q)}

are completely determined. Most of the resultant linear codes (over

\mathbb {F}_{q^{m}}

or over

\mathbb {F}_{q}

) have few weights. Some of them are optimal and some have the best-known parameters according to the tables maintained at https://www.codetables.de . In fact, 16 infinite families of optimal linear codes are produced in this paper. As a byproduct, a family of

[2^{4m-2},2m+1,2^{4m-3}]

quaternary Hermitian self-orthogonal codes are obtained with

m \geq 2

. As an application, we present several infinite families of 2-designs or 3-designs with some of the codes presented in this paper.

Investigating rational perfect nonlinear functions

Article

Full-text available

May 2023

Perfect nonlinear (PN) functions over a finite field, whose study is also motivated by practical applications to Cryptography, have been the subject of several recent papers where the main problems, such as effective constructions and non-existence results, are considered. So far, all contributions have focused on PN functions represented by polynomials, and their constructions. Unfortunately, for polynomial PN functions, the approach based on Hasse–Weil type bounds applied to function fields can only provide non-existence results in a small degree regime. In this paper, we investigate the non-existence problem of rational perfect nonlinear functions over a finite field. Our approach makes it possible to use deep results about the number of points of algebraic varieties over finite fields. Our main result is that no PN rational function f / g with

f,g\in \mathbb {F}_q[X]

f , g ∈ F q [ X ] exists when certain mild arithmetical conditions involving the degree of f ( X ) and g ( X ) are satisfied.

Low c -differential uniformity of the swapped inverse function in odd characteristic

Article

Sep 2023
DISCRETE APPL MATH

Differential spectrum of a class of APN power functions

Article

Full-text available

May 2023
DESIGN CODE CRYPTOGR

APN power functions are fundamental objects as they are used in various theoretical and practical applications in cryptography, coding theory, combinatorial design, and related topics. Let p be an odd prime and n be a positive integer. Let

F(x)=x^d

be a power function over

{\mathbb {F}}_{p^n}

, where

d=\frac{3p^n-1}{4}

when

p^n\equiv 3\pmod 8

and

d=\frac{p^n+1}{4}

when

p^n\equiv 7\pmod 8

. When

p^n>7

, F is an APN function, which is proved by Helleseth et al. (IEEE Trans Inform Theory 45(2):475–485, 1999). In this paper, we study the differential spectrum of F. By investigating some system of equations, the number of solutions of certain system of equations and consequently the differential spectrum of F can be expressed by quadratic character sums over

{\mathbb {F}}_{p^n}

. By the theory of elliptic curves over finite fields, the differential spectrum of F can be investigated by a given p. It is the fourth infinite family of APN power functions with nontrivial differential spectrum.

Application of Secret Sharing Scheme in Many Linear codes over ℤ𝑝𝑅1𝑅2

Conference Paper

Jan 2023

Karima Chatouh

The class of Simplex and Macdonald codes is a very important class of linear codes from both theoretical and practical points of view see [1], [2], and [3], being easier to implement due to their rich algebraic structure, such that these codes related to the concept of secret sharing schemes, which have important applications in many cryptographic applications, secure multiparty computations, and threshold cryptography, as the secret can be recovered once a subset of the participants shares their information see [4]. In this paper, we present simplex and MacDonald codes over ℤ𝑝𝑅1𝑅2 . The properties of these codes are studied, particularly the weight enumerators and Gray images of the simplex and MacDonald codes over the ring ℤ𝑝𝑅1𝑅2 . We use the dual of the Gray images of simplex and MacDonald codes over the ring ℤ𝑝𝑅1𝑅2 to obtain secret sharing schemes. We draw on many properties to understand the access structure of these secret sharing schemes.

On the Inverses and Their Hamming Weights of Known APN, 4-differentially Uniform and CPP Exponents over F 2 n

Article

Jan 2022

This article was initially motivated by an interesting work of Kyureghyan and Suder, in 2014, whereas an open problem it was stated to express the Hamming weights of the inverses of Kasami exponent 2 ^{2 k} - 2 ^k + 1 modulo 2 ⁿ - 1 in terms of k and n when it exists. In 2020, Kölsch solved the open problem by developing the modular add-with-carry approach first formally introduced in 2001. By these two nice works, the Hamming weights of inverses of all known APN and 4-differentially uniform exponents over F _{2 n } have been determined. However, their common difficulty is guessing the inverse related structures by performing some numerical experiments. As also mentioned by Fu in [Discrete Math. 345, 112658, 2022], where inversions of some exponents are revisited, it is unclear how their approaches can be generalized to others except that they studied. In this paper, we first present a new way to find the Hamming weights of inverses of Kasami exponents, which might be generalized to other exponents which are independent of the field size. Although this provides an alternative simpler solution to the aforementioned open problem and the found representations for inverses are fundamentally different from ones found by Kölsch, the representations still do not express the least positive residues of inverses in a closed form. Then, we present a method to find explicitly the least positive residue of the modular inverse without a prior guess and experiment when the exponent depends on the field size. With this method, we determine the inverses and their Hamming weights of all known APN, 4-differentially uniform and complete permutation polynomial (CPP) exponents over F _{2 n } , which solves a research problem raised in [Des. Codes, Cryptogr. 88, 2597-2621, 2020].

Bent functions and their connections to combinatorics

Chapter

Jun 2013

This volume contains nine survey articles based on the invited lectures given at the 24th British Combinatorial Conference, held at Royal Holloway, University of London in July 2013. This biennial conference is a well-established international event, with speakers from around the world. The volume provides an up-to-date overview of current research in several areas of combinatorics, including graph theory, matroid theory and automatic counting, as well as connections to coding theory and Bent functions. Each article is clearly written and assumes little prior knowledge on the part of the reader. The authors are some of the world's foremost researchers in their fields, and here they summarise existing results and give a unique preview of cutting-edge developments. The book provides a valuable survey of the present state of knowledge in combinatorics, and will be useful to researchers and advanced graduate students, primarily in mathematics but also in computer science and statistics.

Two low differentially uniform power permutations over odd characteristic finite fields: APN and differentially 4-uniform functions

Preprint

Oct 2022

Permutation polynomials over finite fields are fundamental objects as they are used in various theoretical and practical applications in cryptography, coding theory, combinatorial design, and related topics. This family of polynomials constitutes an active research area in which advances are being made constantly. In particular, constructing infinite classes of permutation polynomials over finite fields with good differential properties (namely, low) remains an exciting problem despite much research in this direction for many years. This article exhibits low differentially uniform power permutations over finite fields of odd characteristic. Specifically, its objective is twofold concerning the power functions

F(x)=x^{\frac{p^n+3}{2}}

defined over the finite field

F_{p^n}

of order

p^n

, where p is an odd prime, and n is a positive integer. The first is to complement some former results initiated by Helleseth and Sandberg in \cite{HS} by solving the open problem left open for more than twenty years concerning the determination of the differential spectrum of F when

p^n\equiv3\pmod 4

and

p\neq 3

. The second is to determine the exact value of its differential uniformity. Our achievements are obtained firstly by evaluating some exponential sums over

F_{p^n}

(which amounts to evaluating the number of

F_{p^n}

-rational points on some related curves and secondly by computing the number of solutions in

(F_{p^n})^4

of a system of equations presented by Helleseth, Rong, and Sandberg in ["New families of almost perfect nonlinear power mappings," IEEE Trans. Inform. Theory, vol. 45. no. 2, 1999], naturally appears while determining the differential spectrum of F. We show that in the considered case (

p^n\equiv3\pmod 4

and

p\neq 3

), F is an APN power permutation when

p^n=11

, and a differentially 4-uniform power permutation otherwise.

Some classes of translation planes

Article

Full-text available

Jun 1984

Derived Fisher translation planes

Article

Jan 1990

Norman L. Johnson

Planes of order n with collineation groups of order n 2

Article

Jun 1968

The Analytic Representation of Substitutions on a Power of a Prime Number of Letters with a Discussion of the Linear Group.

Article

Jan 1897
ANN MATH

Leonard Eugene Dickson

Finite Fields Encyclopedia of Mathematics

Article

Über eine Klasse von Permutationspolynomen und die dadurch dargestellten Gruppen

Article

Jul 1968
J REINE ANGEW MATH

Wilfried Nöbauer

Semi-Translation Planes

Article

Apr 1964

T. G. Ostrom

On Planes of Lenz-Barlotti Class II-1

Article

Jul 1974

On planar functions

Article

Aug 1990

Yutaka Hiramine

Projective planes of prime order p that admit collineation groups of order p 2

Article

Oct 1987

Norman L. Johnson

Without Abstract

The dual L�neburg planes

Article

Jun 1966

T. G. Ostrom

On perspectivities of finite projective planes

Article

Jan 1959

A. Wagner

A note on affine planes with transitive collineation groups

Article

Feb 1977

Michael J. Kallaher

On point-transitive affine planes

Article

Sep 1982

William M. Kantor

Finite affine planes are constructed admitting nonabelian sharply point-transitive collineation groups. These planes are of two sorts: dual translation planes, and planes of type II.1 derived from them.

Ovoids and translation planes revisited

Article

Apr 1991

Norman L. Johnson

Kantor has previously described the translation planes which may be obtained by projecting sections of ovoids in +(8, q)-spaces to ovoids in corresponding +(6, q)-spaces. Since the Klein correspondence associates spreads in 4-dimensional vector spaces with ovoids in +(6, q)-spaces, there are corresponding translation planes of order q 2 and kernel containing GF(q). In this article, we revisit some of these translation planes and give some presentations of the spreads. Motivated by various properties of the planes, we study, in general, translation planes which admit certain homology groups and/or elation groups. In particular, we develop new constructions of projective planes of Lenz-Barlotti class II-1.Finally, we show how certain projective planes of order q 2 of Lenz-Barlotti class II-1 may be considered equivalent to flocks of quadratic cones in PG(3, q).

Some unusual translation planes of order 64

Article

Dec 1984

Planar functions over finite fields

Article

Sep 1989

Letp>2 be a prime. A functionf: GF(p)GF(p) is planar if for everyaGF(p) *, the functionf(x+a–f(x) is a permutation ofGF(p). Our main result is that every planar function is a quadratic polynomial. As a consequence we derive the following characterization of desarguesian planes of prime order. IfP is a protective plane of prime orderp admitting a collineation group of orderp 2, thenP is the Galois planePG(2,p). The study of such collineation groups and planar functions was initiated by Dembowski and Ostrom [3] and our results are generalizations of some results of Johnson [8].We have recently learned that results equivalent to ours have simultaneously been obtained by Y. Hiramine and D. Gluck.

A structure theory for two-dimensional translation planes of order q2 that admit collineation groups of order q2

Article

Apr 1989

This paper is devoted to the study of translation planes of order q 2 and kernel GF(q) that admit a collineation group of order q 2 in the linear translation complement. We give a representation of this group by a suitable set of matrices depending on some functions over GF(q). Using this representation we obtain several results concerning the existence and the collineation group of the plane.

A note on permutation polynomials and finite geometries

Article

Feb 1990
DISCRETE MATH

David Gluck

A polynomial ƒ over a finite field F is called a difference permutation polynomial if the mapping x → ƒ(x + a) − ƒ(x) is a permutation of F for each nonzero element a of F. Difference permutation polynomials give rise to affine planes. We show that when F = GF(p), where p is a prime, the only difference permutation polynomials over F are quadratic.

A conjecture on affine planes of prime order

Article

Sep 1989

Yutaka Hiramine

In this article we show that any affine plane of prime order with a collineation group transitive on the affine points is Desarguesian.

Pitman Monographs and Surveys in Pure and Appl

Jan 1993

R Lidl
G L Mullen
G Turnwald
Dickson

R. Lidl, G.L. Mullen, and G. Turnwald, Dickson Polynomials, Pitman Monographs and Surveys in Pure and Appl. Math., vol. 65, Longman Scientific and Technical, Essex, England, 1993.

Dickson polynomials, Pitman Monographs and Surveys in Pure and Appl

Jan 1993
65

R Lidl
G L Mullen
G Turnwald

R. Lidl, G. L. Mullen, and G. Turnwald, Dickson polynomials, Pitman Monographs and Surveys in Pure and Appl. Math., Longman Scientific and Technical, Essex, England, 65 (1993).

Affine planes and permutation polynomials, Coding Theory and Design Theory, part II (Design Theory), The IMA

Jan 1990
99-100

D Gluck

D. Gluck, Affine planes and permutation polynomials, Coding Theory and Design Theory, part II (Design Theory), The IMA Volumes in Mathematics and its Applications, Vol. 21, Springer-Verlag (1990) pp. 99–100.

Jan 1968
215-219

W Nöbauer
J Reine Uber Eine Klasse Von Permutationspolynomen Und Die Dadurch Dargestellten Gruppen
Angew

W. Nöbauer, ¨ Uber eine Klasse von Permutationspolynomen und die dadurch dargestellten Gruppen, J. Reine Angew. Math. 231 (1968), 215–219.

Projective planes of order p that admit collineation groups of order p2, <i&gt

. N L Johnson

A structure theory for two-dimensional translation planes of order q2 that admit collineation groups of order q2, Geom

. M Billiotti
V Jha
N L Johnson
G Menichetti

Planes of order n with collineation groups of order n2, <i&gt

. P Dembowski
T G Ostrom

The dual Lüneburg planes, <i&gt

. T G Ostrom

Planar Functions and Planes of Lenz-Barlotti Class II

Abstract

No full-text available

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