CRYPTANALYSIS OF AN IMPLEMENTATION SCHEME OF THE TAMED TRANSFORMATION METHOD CRYPTOSYSTEM

2004 ◽  
Vol 03 (03) ◽  
pp. 273-282 ◽  
Author(s):  
JINTAI DING ◽  
TIMOTHY HODGES
Keyword(s):  
Linear Equations ◽  
Transformation Method ◽  
Total Degree ◽  
Quadratic Polynomials ◽  
Polynomial Maps ◽  
Linearly Independent ◽  
Linear Maps ◽  
Implementation Scheme ◽  
Multivariable Polynomial ◽  
Composition Factors

A Tamed Transformation Method (TTM) cryptosystem was proposed by T. T. Moh in 1999. We describe how the first implementation scheme of the TTM system can be defeated. The computational complexity of our attack is 233 computations on the finite field with 28 elements. The cipher of the TTM systems are degree 2 polynomial maps derived from composition of invertible maps of either total degree 2 or linear maps which can be easily calculated and can be easily inverted. To ensure the system to be of degree two, the key construction of the implementation schemes of the TTM systems is a multivariable polynomial Q8(x1,…,xn) and a set of linearly independent quadratic polynomials qi(x1,…,xm), i=1,…,n such that Q8(q1,…,qn) is again a degree 2 polynomials of x1,…,xm. In this paper, we study the first implementation scheme of the TTM systems [6]. We discovered that in this implementation scheme the specific polynomial Q8 can be decomposed further into a factorization in terms of composition. By taking powers of the equality satisfied by the new composition factors, we can actually derive a set of equations, that can produce linear equations satisfied by the plaintext. These linear equations lead us to find a way to defeat this implementation scheme.

CONSTRUCTING CHAOTIC POLYNOMIAL MAPS

2009 ◽  
Vol 19 (02) ◽  
pp. 531-543 ◽  
Author(s):  
XU ZHANG ◽  
YUMING SHI ◽  
GUANRONG CHEN
Keyword(s):  
Logistic Map ◽  
Special Kind ◽  
Quadratic Polynomials ◽  
One Dimensional ◽  
Polynomial Maps ◽  
Polynomial Map ◽  
Expansion Theory ◽  
Chaotic Parameter ◽  
The Given ◽  
Nonzero Polynomial

This paper studies the construction of one-dimensional real chaotic polynomial maps. Given an arbitrary nonzero polynomial of degree m (≥ 0), two methods are derived for constructing chaotic polynomial maps of degree m + 2 by simply multiplying the given polynomial with suitably designed quadratic polynomials. Moreover, for m + 2 arbitrarily given different positive constants, a method is given to construct a chaotic polynomial map of degree 2m based on the coupled-expansion theory. Furthermore, by multiplying a real parameter to a special kind of polynomial, which has at least two different non-negative or nonpositive zeros, the chaotic parameter region of the polynomial is analyzed based on the snap-back repeller theory. As a consequence, for any given integer n ≥ 2, at least one polynomial of degree n can be constructed so that it is chaotic in the sense of both Li–Yorke and Devaney. In addition, two natural ways of generalizing the logistic map to higher-degree chaotic logistic-like maps are given. Finally, an illustrative example is provided with computer simulations for illustration.

scholarly journals The fourth dimension subgroups and polynomial maps, II

1978 ◽  
Vol 69 ◽  
pp. 1-7 ◽  
Author(s):  
Ken-Ichi Tahara
Keyword(s):  
Linear Equations ◽  
Integral Solution ◽  
Fourth Dimension ◽  
Direct Products ◽  
Kronecker Symbol ◽  
Cyclic Groups ◽  
Polynomial Maps ◽  
Polynomial Map

In our previous paper [3] we proved the following ([3, Theorem 16]) :THEOREM A. Let G be a 2-group of class 3. Let G2 and G/G2 be direct products of cyclic groups 〈yq〉 of order αq (1 ≦ q ≦ m), and of cyclic groups 〈hi〉 of order βi (1 ≦ i ≦ n) with β1 ≧ β2 ≧ · · · βn, respectively. Let xi be representatives of hi (1 ≦ i ≦ n), and put Then a homomorphism ψ:G3→T can be extended to a polynomial map from G to T of degree ≦ 4 if and only if there exists an integral solution in the following linear equations of Xiq (1 ≦ i ≦ n, 1 ≦ q ≦ m) with coefficients in T: (I)where δij is the Kronecker symbol for βi: i.e. δij = 1 or 0 according to βi = βj or βi > βj, respectively.

scholarly journals First Integrals and Hamiltonians of Some Classes of ODEs of Maximal Symmetry

2017 ◽  
Vol 2017 ◽  
pp. 1-9
Author(s):  
J. C. Ndogmo
Keyword(s):  
Hamiltonian Systems ◽  
Linear Equations ◽  
Symmetry Algebra ◽  
First Integrals ◽  
Second Order ◽  
Maximal Symmetry ◽  
General Solutions ◽  
Linearly Independent ◽  
Complete Sets

Complete sets of linearly independent first integrals are found for the most general form of linear equations of maximal symmetry algebra of order ranging from two to eight. The corresponding Hamiltonian systems are constructed and it is shown that their general solutions can also be found by a simple superposition formula from the solutions of a scalar second-order source equation.

scholarly journals Relation between saturated and normal operators

BIBECHANA ◽  
2013 ◽  
Vol 10 ◽  
pp. 115-117
Author(s):  
Nagendra Pd Sah
Keyword(s):  
Vector Space ◽  
Finite Dimensional ◽  
Linearly Independent ◽  
Linear Maps ◽  
Normal Operators ◽  
The Ideal

A vector space X with algebra of all linear maps ∑(X) from X into itself and the ideal of all finite dimensional linear maps with dual (Conjugate) transformation T* to T from X' to itself form a relation in terms of relatively regular and linearly independent which is sufficient for mentioned title. DOI: http://dx.doi.org/10.3126/bibechana.v10i0.9342   BIBECHANA 10 (2014) 115-117

Fibonacci fixed point of renormalization

2000 ◽  
Vol 20 (5) ◽  
pp. 1287-1317 ◽  
Author(s):  
XAVIER BUFF
Keyword(s):  
Fixed Point ◽  
Lebesgue Measure ◽  
Julia Set ◽  
Positive Lebesgue Measure ◽  
Quadratic Polynomials ◽  
Polynomial Maps ◽  
Mathematical Sciences

To study the geometry of a Fibonacci map $f$ of even degree $\ell\geq 4$, Lyubich (Dynamics of quadratic polynomials, I–II. Acta Mathematica178 (1997), 185–297) defined a notion of generalized renormalization, so that $f$ is renormalizable infinitely many times. van Strien and Nowicki (Polynomial maps with a Julia set of positive Lebesgue measure: Fibonacci maps. Preprint, Institute for Mathematical Sciences, SUNY at Stony Brook, 1994) proved that the generalized renormalizations ${\cal R}^{\circ n}(f)$ converge to a cycle $\{f_1,f_2\}$ of order two depending only on $\ell$. We will explicitly relate $f_1$ and $f_2$ and show the convergence in shape of Fibonacci puzzle pieces to the Julia set of an appropriate polynomial-like map.

scholarly journals Generalized squared remainder minimization method for solving multi-term fractional differential equations

2021 ◽  
Vol 26 (1) ◽  
pp. 57-71
Author(s):  
Mir Sajjad Hashemi ◽  
Mustafa Inc ◽  
Somayeh Hajikhah
Keyword(s):  
Differential Equations ◽  
Lagrange Multiplier ◽  
Fractional Differential Equations ◽  
Minimization Problem ◽  
Linear Equations ◽  
Approximate Solutions ◽  
Minimization Method ◽  
Linearly Independent ◽  
Independent Functions ◽  
Fractional Differential

In this paper, we introduce a generalization of the squared remainder minimization method for solving multi-term fractional differential equations. We restrict our attention to linear equations. Approximate solutions of these equations are considered in terms of linearly independent functions. We change our problem into a minimization problem. Finally, the Lagrange-multiplier method is used to minimize the resultant problem. The convergence of this approach is discussed and theoretically investigated. Some relevant examples are investigated to illustrate the accuracy of the method, and obtained results are compared with other methods to show the power of applied method.

scholarly journals Study of Properties of Differential Transform Method for Solving the Linear Differential Equation

2019 ◽  
Vol 8 (2) ◽  
pp. 50-56
Author(s):  
Nandita Das
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Equations ◽  
Transformation Method ◽  
Differential Transformation Method ◽  
Transform Method ◽  
Differential Transformation ◽  
Differential Transform ◽  
Linear Differential ◽  
Non Linear Equations

The differential transformation method (DTM) is an alternative procedure for obtaining an analytic Taylor series solution of differential linear and non-linear equations. However, the proofs of the properties of equation have been long ignored in the DTM literature. In this paper, we present an analytical solution for linear properties of differential equations by using the differential transformation method. This method has been discussed showing the proof of the equation which are presented to show the ability of the method for linear systems of differential equations. Most authors assume the knowledge of these properties, so they do not bother to prove the properties. The properties are therefore proved to serve as a reference for any work that would want to use the properties without proofs. This work argues that we can obtain the solution of differential equation through these proofs by using the DTM. The result also show that the technique introduced here is accurate and easy to apply.

MHD nanofluid flow around a permeable stretching sheet with thermal radiation and viscous dissipation

2021 ◽  
pp. 095440622110230
Author(s):  
Rana MA Muntazir ◽  
Muhammad Mushtaq ◽  
Shamaila Shahzadi ◽  
Kanwali Jabeen
Keyword(s):  
Thermal Radiation ◽  
Viscous Dissipation ◽  
Stretching Sheet ◽  
Volume Fraction ◽  
Linear Equations ◽  
Transformation Method ◽  
Nano Particles ◽  
Physical Parameters ◽  
Analytical Technique ◽  
Field Environment

In this research article, we have investigated the unsteady MHD nanofluids flow problem around a permeable linearly stretching sheet under the influence of thermal radiation and viscous dissipation. Transfer of mass and heat analysis is considered for various kinds of nano-particles such as [Formula: see text] , Cu, Ag and TiO2. Many research studies had been concluded that thermal conductivity of traditional fluid accelerates 15–40% as nano-particles are mixed in to a base fluid, this theory however depends upon the adding mechanism of the nano-particles. Although it depends upon volume fraction, agglomeration or size of nano-particles etc. But it can be concluded from this study, in a magnetic field environment not only the fluid flow is more consistent than regular fluid but also the rate of heat transfer increases. We have tabulated the results of the four different types of nanofluid and graphically presented the behavior of [Formula: see text] and Cu nanofluids. These distinct MHD nanofluids are used to explore the parametric features of heat and mass transfer phenomena along a permeable stretching sheet. The impacts of various physical parameters and physical quantities are analyzed. It is observed that from this study that the heat transfers rate of Cu nanofluid is higher than [Formula: see text] nanofluid. The coupled non-linear equations are solved by semi-analytical technique i.e Differential Transformation Method (DTM) along with Pade-approximation and found to be in well agreement with already reported work in literature. The graphical illustration and tabular results represent the physical importance of the work. It was observed and concluded that the temperature profiles in case of Cu nanofluid presents significantly high as compared with [Formula: see text] nanofluid. Also, the thicknesses of velocity, thermal and concentration profile decreases by increasing suction/injection and unsteady parameter. These parameters used to control the flow rate.

scholarly journals Analytical and numerical results of fractional differential-difference equations

2015 ◽  
Vol 7 (2) ◽  
pp. 186-199 ◽  
Author(s):  
Najeeb Alam Khan ◽  
Fatima Riaz
Keyword(s):  
Difference Equation ◽  
Nonlinear Dynamical Systems ◽  
Linear Equations ◽  
Transformation Method ◽  
Test Problems ◽  
Nonlinear Dynamical ◽  
Fractional Differential ◽  
Differential Difference Equation ◽  
Sensitivity Approach ◽  
Infinite Linear

Abstract In this paper, we examine the fractional differential-difference equation (FDDE) by employing the proposed sensitivity approach (SA) and Adomian transformation method (ADTM). In SA the nonlinear differential-difference equation is converted to infinite linear equations which have a wide criterion to solve for the analytical solution. By ADTM, the FDDE is converted into ordinary differential-difference equation that can be solved. We test both the techniques through some test problems which are arising in nonlinear dynamical systems and found that ADTM is equivalently appropriate and simpler method to handle than SA.

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