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
Keyword(s):  
Finite Field ◽  
Chebyshev Polynomials

On the Security of Public-Key Algorithms Based on Chebyshev Polynomials over the Finite Field $Z_N$

2010 ◽  
Vol 59 (10) ◽  
pp. 1392-1401 ◽  
Author(s):  
Xiaofeng Liao ◽  
Fei Chen ◽  
Kwok-wo Wong
Keyword(s):  
Finite Field ◽  
Chebyshev Polynomials ◽  
Public Key

Non-acyclic SL2-representations of Twist Knots, -3-Dehn Surgeries, and L-functions

2021 ◽  
Author(s):  
Ryoto Tange ◽  
Anh T Tran ◽  
Jun Ueki
Keyword(s):  
Finite Field ◽  
Chebyshev Polynomials ◽  
Dehn Surgery ◽  
Character Variety ◽  
Common Tangent ◽  
Characteristic P ◽  
Reidemeister Torsion ◽  
Universal Deformations ◽  
Discrete Valuation Rings ◽  
Valuation Rings

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.

Cryptanalysis of Multiplicative Coupled Cryptosystems Based on the Chebyshev Polynomials

2016 ◽  
Vol 26 (07) ◽  
pp. 1650112 ◽  
Author(s):  
Ali Shakiba ◽  
Mohammad Reza Hooshmandasl ◽  
Mohsen Alambardar Meybodi
Keyword(s):  
Finite Field ◽  
Chebyshev Polynomials ◽  
Public Key ◽  
Floating Point ◽  
Real Numbers ◽  
Public Key Cryptosystems ◽  
Floating Point Arithmetic ◽  
Security Models ◽  
Point Arithmetic

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.

Beta dynamical systems over the formal fields

2014 ◽  
Vol 51 (4) ◽  
pp. 454-465
Author(s):  
Lu-Ming Shen ◽  
Huiping Jing
Keyword(s):  
Dynamical Systems ◽  
Finite Field ◽  
Haar Measure ◽  
Laurent Series ◽  
Formal Laurent Series ◽  
Almost All

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.

Valued Field Graph and Some Related Parameters

2020 ◽  
pp. 389-395
Author(s):  
G. Suresh Singh ◽  
P. K. Prasobha
Keyword(s):  
Finite Field ◽  
Independence Number ◽  
Valued Field ◽  
Vertex Set ◽  
Adic Valuation

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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