On a generalization of the Euler totient function

2021 ◽  
Author(s):  
Wenguang Zhai
Keyword(s):  
Euler Totient Function

scholarly journals Omega theorems related to the general Euler totient function

2014 ◽  
Vol 412 (1) ◽  
pp. 401-415 ◽  
Author(s):  
Jerzy Kaczorowski ◽  
Kazimierz Wiertelak
Keyword(s):  
Euler Totient Function

scholarly journals Matrix-Based RSA Encryption of Streaming Data

2021 ◽  
Author(s):  
Michael Prendergast
Keyword(s):  
Streaming Data ◽  
New Method ◽  
Encryption Algorithm ◽  
Public Key ◽  
Invertible Matrix ◽  
Integer Coefficients ◽  
Private Key ◽  
Matrix Generalization ◽  
Euler Totient Function ◽  
The Matrix

This paper describes a new method for performing secure encryption of blocks of streaming data. This algorithm is an extension of the RSA encryption algorithm. Instead of using a public key (e,n) where n is the product of two large primes and e is relatively prime to the Euler Totient function, φ(n), one uses a public key (n,m,E), where m is the rank of the matrix E and E is an invertible matrix in GL(m,φ(n)). When m is 1, this last condition is equivalent to saying that E is relatively prime to φ(n), which is a requirement for standard RSA encryption. Rather than a secret private key (d,φ(n)) where d is the inverse of e (mod φ(n)), the private key is (D,φ(n)), where D is the inverse of E (mod (φ(n)). The key to making this generalization work is a matrix generalization of the scalar exponentiation operator that maps the set of m-dimensional vectors with integer coefficients modulo n, onto itself.

scholarly journals A note on primes in certain residue classes

2018 ◽  
Vol 14 (08) ◽  
pp. 2219-2223
Author(s):  
Paolo Leonetti ◽  
Carlo Sanna
Keyword(s):  
Positive Integer ◽  
Asymptotic Density ◽  
Residue Classes ◽  
Euler Totient Function ◽  
The One ◽  
Positive Integers

Given positive integers [Formula: see text], we prove that the set of primes [Formula: see text] such that [Formula: see text] for [Formula: see text] admits asymptotic density relative to the set of all primes which is at least [Formula: see text], where [Formula: see text] is the Euler totient function. This result is similar to the one of Heilbronn and Rohrbach, which says that the set of positive integer [Formula: see text] such that [Formula: see text] for [Formula: see text] admits asymptotic density which is at least [Formula: see text].

Asymptotic evaluations for some double sums in number theory

2015 ◽  
Vol 11 (07) ◽  
pp. 2073-2085 ◽  
Author(s):  
Magdalena Bănescu ◽  
Dumitru Popa
Keyword(s):  
Number Theory ◽  
Euler Totient Function

In this paper, we indicate a method to obtain asymptotic evaluations for double sums from the asymptotic evaluations of a single sum. As applications, we show that all results in the recent paper [New extensions of some classical theorems in number theory, J. Number Theory133(13) (2013) 3771–3795] can be used to obtain some asymptotic evaluations for double sums. Further, we give asymptotic evaluations for double sums which involve the divisor and the Euler totient function.

scholarly journals On sparsely totient numbers

1991 ◽  
Vol 33 (3) ◽  
pp. 350-358
Author(s):  
Glyn Harman
Keyword(s):  
Positive Integer ◽  
Prime Factor ◽  
Multiplicative Structure ◽  
Euler Totient Function ◽  
Image Position

Following Masser and Shiu [6] we say that a positive integer n is sparsely totient ifHere φ is the familiar Euler totient function. We write ℱ for the set of sparsely totient numbers. In [6] several results are proved about the multiplicative structure of ℱ. If we write P(n) for the largest prime factor of n then it was shown (Theorem 2 of [6]) thatand infinitely often

scholarly journals UNSOLVED PROBLEMS

2013 ◽  
Vol 56 (1) ◽  
pp. 109-229 ◽  
Author(s):  
Oto Strauch
Keyword(s):  
Uniform Distribution ◽  
Distribution Theory ◽  
Benford’S Law ◽  
Moment Problems ◽  
Positive Density ◽  
Sum Of Digits ◽  
Research Activities ◽  
Euler Totient Function ◽  
New Research ◽  
Van Der Corput Sequence

ABSTRACT In this paper there are given problems from the Unsolved Problems Section on the homepage of the journal Uniform Distribution Theory <http://www.boku.ac.at/MATH/udt/unsolvedproblems.pdf> It contains 38 items and 5 overviews collected by the author and by Editors of UDT. They are focused at uniform distribution theory, more accurate, distri- bution functions of sequences, logarithm of primes, Euler totient function, van der Corput sequence, ratio sequences, set of integers of positive density, exponen- tial sequences, moment problems, Benford’s law, Gauss-Kuzmin theorem, Duffin- Schaeffer conjecture, extremes fQ fQ F(x,y)dg(x,y) over copulas g(x,y), sum- -of-digits sequence, etc. Some of them inspired new research activities. The aim of this printed version is publicity.

scholarly journals On a generalization of the Euler totient function

2012 ◽  
Vol 170 (1) ◽  
pp. 27-48 ◽  
Author(s):  
Jerzy Kaczorowski
Keyword(s):  
Euler Totient Function

scholarly journals Oscillations of the remainder term related to the Euler totient function

2010 ◽  
Vol 130 (12) ◽  
pp. 2683-2700 ◽  
Author(s):  
Jerzy Kaczorowski ◽  
Kazimierz Wiertelak
Keyword(s):  
Remainder Term ◽  
Euler Totient Function
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