Additive patterns in a multiplicative group in a finite field

2003 ◽  
Vol 111 (2) ◽  
pp. 187-194 ◽  
Author(s):  
Geumlan Choi ◽  
Alexandru Zaharescu
Keyword(s):  
Finite Field ◽  
Multiplicative Group

Commutators of Matrices with Prescribed Determinant

1968 ◽  
Vol 20 ◽  
pp. 203-221 ◽  
Author(s):  
R. C. Thompson
Keyword(s):  
Finite Field ◽  
Multiplicative Group ◽  
Companion Matrix ◽  
Identity Matrix ◽  
Commutative Field

Let K be a commutative field, let GL(n, K) be the multiplicative group of all non-singular n × n matrices with elements from K, and let SL(n, K) be the subgroup of GL(n, K) consisting of all matrices in GL(n, K) with determinant one. We denote the determinant of matrix A by |A|, the identity matrix by In, the companion matrix of polynomial p(λ) by C(p(λ)), and the transpose of A by AT. The multiplicative group of nonzero elements in K is denoted by K*. We let GF(pn) denote the finite field having pn elements.

Representations by Hermitian Forms in a Finite Field of Characteristic Two

1977 ◽  
Vol 29 (1) ◽  
pp. 169-179 ◽  
Author(s):  
John D. Fulton
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Multiplicative Group ◽  
Group Homomorphism ◽  
Hermitian Forms ◽  
Characteristic Two ◽  
Field Automorphism

Throughout this paper, we let q = 2W,﹜ w a positive integer, and for u = 1 or 2, we let GF(qu) denote the finite field of cardinality qu. Let - denote the involutory field automorphism of GF(q2) with GF(q) as fixed subfield, where ā = aQ for all a in GF﹛q2). Moreover, let | | denote the norm (multiplicative group homomorphism) mapping of GF(q2) onto GF(q), where |a| — a • ā = aQ+1.

scholarly journals Improved lower bound for Diffie–Hellman problem using multiplicative group of a finite field as auxiliary group

2018 ◽  
Vol 12 (2) ◽  
pp. 101-118 ◽  
Author(s):  
Prabhat Kushwaha
Keyword(s):  
Lower Bound ◽  
Finite Field ◽  
Elliptic Curves ◽  
Multiplicative Group ◽  
Discrete Logarithm ◽  
Discrete Logarithm Problem ◽  
Reduction Algorithm ◽  
Practical Applications ◽  
Diffie Hellman ◽  
Estimate Lower

Abstract In 2004, Muzereau, Smart and Vercauteren [A. Muzereau, N. P. Smart and F. Vercauteren, The equivalence between the DHP and DLP for elliptic curves used in practical applications, LMS J. Comput. Math. 7 2004, 50–72] showed how to use a reduction algorithm of the discrete logarithm problem to Diffie–Hellman problem in order to estimate lower bound for the Diffie–Hellman problem on elliptic curves. They presented their estimates on various elliptic curves that are used in practical applications. In this paper, we show that a much tighter lower bound for the Diffie–Hellman problem on those curves can be achieved if one uses the multiplicative group of a finite field as auxiliary group. The improved lower bound estimates of the Diffie–Hellman problem on those recommended curves are also presented. Moreover, we have also extended our idea by presenting similar estimates of DHP on some more recommended curves which were not covered before. These estimates of DHP on these curves are currently the tightest which lead us towards the equivalence of the Diffie–Hellman problem and the discrete logarithm problem on these recommended elliptic curves.

scholarly journals Double Character Sums over Subgroups and Intervals

2014 ◽  
Vol 90 (3) ◽  
pp. 376-390 ◽  
Author(s):  
MEI-CHU CHANG ◽  
IGOR E. SHPARLINSKI
Keyword(s):  
Finite Field ◽  
Upper Bound ◽  
Multiplicative Group ◽  
Character Sums ◽  
Multiplicative Character ◽  
Nontrivial Estimate ◽  
Image Position ◽  
Primitive Roots ◽  
Double Character

AbstractWe estimate double sums $$\begin{eqnarray}S_{{\it\chi}}(a,{\mathcal{I}},{\mathcal{G}})=\mathop{\sum }\limits_{x\in {\mathcal{I}}}\mathop{\sum }\limits_{{\it\lambda}\in {\mathcal{G}}}{\it\chi}(x+a{\it\lambda}),\quad 1\leq a<p-1,\end{eqnarray}$$ with a multiplicative character ${\it\chi}$ modulo $p$ where ${\mathcal{I}}=\{1,\dots ,H\}$ and ${\mathcal{G}}$ is a subgroup of order $T$ of the multiplicative group of the finite field of $p$ elements. A nontrivial upper bound on $S_{{\it\chi}}(a,{\mathcal{I}},{\mathcal{G}})$ can be derived from the Burgess bound if $H\geq p^{1/4+{\it\varepsilon}}$ and from some standard elementary arguments if $T\geq p^{1/2+{\it\varepsilon}}$, where ${\it\varepsilon}>0$ is arbitrary. We obtain a nontrivial estimate in a wider range of parameters $H$ and $T$. We also estimate double sums $$\begin{eqnarray}T_{{\it\chi}}(a,{\mathcal{G}})=\mathop{\sum }\limits_{{\it\lambda},{\it\mu}\in {\mathcal{G}}}{\it\chi}(a+{\it\lambda}+{\it\mu}),\quad 1\leq a<p-1,\end{eqnarray}$$ and give an application to primitive roots modulo $p$ with three nonzero binary digits.

scholarly journals Cliques and colorings in generalized Paley graphs and an approach to synchronization

2015 ◽  
Vol 14 (06) ◽  
pp. 1550088 ◽  
Author(s):  
Csaba Schneider ◽  
Ana C. Silva
Keyword(s):  
Finite Field ◽  
Directed Graph ◽  
Undirected Graph ◽  
Multiplicative Group ◽  
Permutation Groups ◽  
Chromatic Numbers ◽  
Paley Graph ◽  
Paley Graphs

Given a finite field, one can form a directed graph using the field elements as vertices and connecting two vertices if their difference lies in a fixed subgroup of the multiplicative group. If -1 is contained in this fixed subgroup, then we obtain an undirected graph that is referred to as a generalized Paley graph. In this paper, we study generalized Paley graphs whose clique and chromatic numbers coincide and link this theory to the study of the synchronization property in 1-dimensional primitive affine permutation groups.

scholarly journals Translates of subgroups of the multiplicative group of a finite field

1975 ◽  
Vol 78 (4) ◽  
pp. 285-289 ◽  
Author(s):  
P.J Cameron ◽  
J.I Hall ◽  
J.H van Lint ◽  
T.A Springer ◽  
H.C.A van Tilborg
Keyword(s):  
Finite Field ◽  
Multiplicative Group

scholarly journals Construction of metrics on the set of elliptic curves over a finite field

2021 ◽  
Vol 109 (123) ◽  
pp. 125-141
Author(s):  
Keisuke Hakuta
Keyword(s):  
Finite Field ◽  
Elliptic Curves ◽  
Multiplicative Group ◽  
Supersingular Elliptic Curves ◽  
Characteristic Two

We consider metrics on the set of elliptic curves in short Weierstrass form over a finite field of characteristic greater than three. The metrics have been first found by Mishra and Gupta (2008). Vetro (2011) constructs other metrics which are independent on the choice of a generator of the multiplicative group of the underlying finite field, whereas the metrics found by Mishra and Gupta, are dependent on the choice of a generator of the multiplicative group of the underlying finite field. Hakuta (2015, 2018) constructs metrics on the set of non-supersingular elliptic curves in shortWeierstrass form over a finite field of characteristic two and three, respectively. The aim of this paper is to point out that the metric found by Mishra and Gupta is in fact not a metric. We also construct new metrics which are slightly modified versions of the metric found by Mishra and Gupta.

scholarly journals Repeated-root constacyclic codes of length 6lmpn

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Tingting Wu ◽  
Shixin Zhu ◽  
Li Liu ◽  
Lanqiang Li
Keyword(s):  
Finite Field ◽  
Disjoint Union ◽  
Multiplicative Group ◽  
Primitive Element ◽  
Constacyclic Codes ◽  
Negacyclic Codes ◽  
Positive Integers

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{F}_{q} $\end{document}</tex-math></inline-formula> be a finite field with character <inline-formula><tex-math id="M2">\begin{document}$ p $\end{document}</tex-math></inline-formula>. In this paper, the multiplicative group <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{F}_{q}^{*} = \mathbb{F}_{q}\setminus\{0\} $\end{document}</tex-math></inline-formula> is decomposed into a mutually disjoint union of <inline-formula><tex-math id="M4">\begin{document}$ \gcd(6l^mp^n,q-1) $\end{document}</tex-math></inline-formula> cosets over subgroup <inline-formula><tex-math id="M5">\begin{document}$ &lt;\xi^{6l^mp^n}&gt; $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M6">\begin{document}$ \xi $\end{document}</tex-math></inline-formula> is a primitive element of <inline-formula><tex-math id="M7">\begin{document}$ \mathbb{F}_{q} $\end{document}</tex-math></inline-formula>. Based on the decomposition, the structure of constacyclic codes of length <inline-formula><tex-math id="M8">\begin{document}$ 6l^mp^n $\end{document}</tex-math></inline-formula> over finite field <inline-formula><tex-math id="M9">\begin{document}$ \mathbb{F}_{q} $\end{document}</tex-math></inline-formula> and their duals is established in terms of their generator polynomials, where <inline-formula><tex-math id="M10">\begin{document}$ p\neq{3} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M11">\begin{document}$ l\neq{3} $\end{document}</tex-math></inline-formula> are distinct odd primes, <inline-formula><tex-math id="M12">\begin{document}$ m $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M13">\begin{document}$ n $\end{document}</tex-math></inline-formula> are positive integers. In addition, we determine the characterization and enumeration of all linear complementary dual(LCD) negacyclic codes and self-dual constacyclic codes of length <inline-formula><tex-math id="M14">\begin{document}$ 6l^mp^n $\end{document}</tex-math></inline-formula> over <inline-formula><tex-math id="M15">\begin{document}$ \mathbb{F}_{q} $\end{document}</tex-math></inline-formula>.</p>

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