On commuting circle mappings and simultaneous Diophantine approximations

1990 ◽  
Vol 205 (1) ◽  
pp. 105-121 ◽  
Author(s):  
Jürgen Moser
Keyword(s):  
Diophantine Approximations ◽  
Simultaneous Diophantine Approximations ◽  
Circle Mappings

scholarly journals Lyapunov exponent of random dynamical systems on the circle

2021 ◽  
pp. 1-28
Author(s):  
DOMINIQUE MALICET
Keyword(s):  
Lyapunov Exponent ◽  
Dynamical Systems ◽  
Taylor Expansion ◽  
Diophantine Condition ◽  
Quantitative Version ◽  
Rotation Numbers ◽  
Diophantine Approximations ◽  
Simultaneous Diophantine Approximations ◽  
Random Products ◽  
Circle Mappings

Abstract We consider products of an independent and identically distributed sequence in a set $\{f_1,\ldots ,f_m\}$ of orientation-preserving diffeomorphisms of the circle. We can naturally associate a Lyapunov exponent $\lambda $ . Under few assumptions, it is known that $\lambda \leq 0$ and that the equality holds if and only if $f_1,\ldots ,f_m$ are simultaneously conjugated to rotations. In this paper, we state a quantitative version of this fact in the case where $f_1,\ldots ,f_m$ are $C^k$ perturbations of rotations with rotation numbers $\rho (f_1),\ldots ,\rho (f_m)$ satisfying a simultaneous diophantine condition in the sense of Moser [On commuting circle mappings and simultaneous diophantine approximations. Math. Z.205(1) (1990), 105–121]: we give a precise estimate of $\lambda $ (Taylor expansion) and we prove that there exist a diffeomorphism g and rotations $r_i$ such that $\mbox {dist}(gf_ig^{-1},r_i)\ll |\lambda |^{{1}/{2}}$ for $i=1,\ldots , m$ . We also state analogous results for random products of $2\times 2$ matrices, without any diophantine condition.

scholarly journals Strategic Way of Factoring Modulus ``\(N = p^r q\)

2021 ◽  
pp. 74-86
Author(s):  
Sadiq Shehu ◽  
Abdullahi Hussaini ◽  
Zahriya Lawal
Keyword(s):  
Information Security ◽  
Polynomial Time ◽  
Continued Fractions ◽  
Prime Power ◽  
Electronic Information ◽  
Lll Algorithm ◽  
Key Equation ◽  
Public Keys ◽  
Diophantine Approximations ◽  
Simultaneous Diophantine Approximations

Cryptography is fundamental to the provision of a wider notion of information security. Electronic information can easily be transmitted and stored in relatively insecure environments. This research was present to factor the prime power modulus \(N = p^r q\) for \(r \geq 2\) using the RSA key equation, if \(\frac{y}{x}\) is a convergents of the continued fractions expansions of \(\frac{e}{N - \left(2^{\frac{2r+1}{r+1}} N^{\frac{r}{r+1}} - 2^{\frac{r-1}{r+1}} N^{\frac{r-1}{r+1}}\right)}\). We furthered our analysis on \(n\) prime power moduli \(N_i = p_i^r q_i\) by transforming the generalized key equations into Simultaneous Diophantine approximations and using the LLL algorithm on \(n\) prime power public keys \((N_i,e_i)\) we were able to factorize the \(n\) prime power moduli \(N_i = p_i^r q_i\), for \(i = 1,....,n\) simultaneously in polynomial time.

Simultaneous diophantine approximations and Hermite's method

1980 ◽  
Vol 21 (3) ◽  
pp. 463-470 ◽  
Author(s):  
Alain Durand
Keyword(s):  
Exponential Function ◽  
Integer Point ◽  
Rational Numbers ◽  
Rational Points ◽  
Rational Approximations ◽  
Diophantine Approximations ◽  
Simultaneous Diophantine Approximations ◽  
Image Position

In this paper we generalize a result of Mahler on rational approximations of the exponential function at rational points by proving the following theorem: let n ε N* and αl, …, αn be distinct non-zero rational numbers; there exists a constant c = c(n, αl, …, αn) ≥ 0 such thatfor every non-zero integer point (qo, ql, …, qn)and q = max {|ql|, … |qn|, 3}.

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