scholarly journals Permutation Properties of Chebyshev Polynomials of the Second Kind over a Finite Field

1995 ◽  
Vol 1 (1) ◽  
pp. 115-125 ◽  
Author(s):  
M. Henderson ◽  
R. Matthews
2010 ◽  
Vol 59 (10) ◽  
pp. 1392-1401 ◽  
Author(s):  
Xiaofeng Liao ◽  
Fei Chen ◽  
Kwok-wo Wong

Author(s):  
Ryoto Tange ◽  
Anh T Tran ◽  
Jun Ueki

Abstract We study irreducible $\mathop{\textrm{SL}}\nolimits _2$-representations of twist knots. We first determine all non-acyclic $\mathop{\textrm{SL}}\nolimits _2({\mathbb{C}})$-representations, which turn out to lie on a line denoted as $x=y$ in ${\mathbb{R}}^2$. Our main tools are character variety, Reidemeister torsion, and Chebyshev polynomials. We also verify a certain common tangent property, which yields a result on $L$-functions, that is, the orders of the knot modules associated to the universal deformations. Secondly, we prove that a representation is on the line $x=y$ if and only if it factors through the $-3$-Dehn surgery, and is non-acyclic if and only if the image of a certain element is of order 3. Finally, we study absolutely irreducible non-acyclic representations $\overline{\rho }$ over a finite field with characteristic $p>2$, to concretely determine all non-trivial $L$-functions $L_{{\boldsymbol{\rho }}}$ of the universal deformations over complete discrete valuation rings. We show among other things that $L_{{\boldsymbol{\rho }}}$  $\dot{=}$  $k_n(x)^2$ holds for a certain series $k_n(x)$ of polynomials.


2016 ◽  
Vol 26 (07) ◽  
pp. 1650112 ◽  
Author(s):  
Ali Shakiba ◽  
Mohammad Reza Hooshmandasl ◽  
Mohsen Alambardar Meybodi

In this work, we propose a class of public-key cryptosystems called multiplicative coupled cryptosystem, or MCC for short, as well as discuss its security within three different models. Moreover, we discuss a chaotic instance of MCC based on the first and the second types of Chebyshev polynomials over real numbers for these three security models. To avoid round-off errors in floating point arithmetic as well as to enhance the security of the chaotic instance discussed, the Chebyshev polynomials of the first and the second types over a finite field are employed. We also consider the efficiency of the proposed MCCs. The discussions throughout the paper are supported by practical examples.


2014 ◽  
Vol 51 (4) ◽  
pp. 454-465
Author(s):  
Lu-Ming Shen ◽  
Huiping Jing

Let \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q ((X^{ - 1} ))$$ \end{document} denote the formal field of all formal Laurent series x = Σ n=ν∞anX−n in an indeterminate X, with coefficients an lying in a given finite field \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q$$ \end{document}. For any \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\beta \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document} with deg β > 1, it is known that for almost all \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$x \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document} (with respect to the Haar measure), x is β-normal. In this paper, we show the inverse direction, i.e., for any x, for almost all \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\beta \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document}, x is β-normal.


Author(s):  
G. Suresh Singh ◽  
P. K. Prasobha

Let $K$ be any finite field. For any prime $p$, the $p$-adic valuation map is given by $\psi_{p}:K/\{0\} \to \R^+\bigcup\{0\}$ is given by $\psi_{p}(r) = n$ where $r = p^n \frac{a}{b}$, where $p,a,b$ are relatively prime. The field $K$ together with a valuation is called valued field. Also, any field $K$ has the trivial valuation determined by $\psi{(K)} = \{0,1\}$. Through out the paper K represents $\Z_q$. In this paper, we construct the graph corresponding to the valuation map called the valued field graph, denoted by $VFG_{p}(\Z_{q})$ whose vertex set is $\{v_0,v_1,v_2,\ldots, v_{q-1}\}$ where two vertices $v_i$ and $v_j$ are adjacent if $\psi_{p}(i) = j$ or $\psi_{p}(j) = i$. Here, we tried to characterize the valued field graph in $\Z_q$. Also we analyse various graph theoretical parameters such as diameter, independence number etc.


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