Matrix-Based RSA Encryption of Streaming Data

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.

On Fixed Points of Iterations Between the Order of Appearance and the Euler Totient Function

Let Fn be the nth Fibonacci number. The order of appearance z(n) of a natural number n is defined as the smallest positive integer k such that Fk≡0(modn). In this paper, we shall find all positive solutions of the Diophantine equation z(φ(n))=n, where φ is the Euler totient function.

Iterating the Sum of Möbius Divisor Function and Euler Totient Function

In this paper, according to some numerical computational evidence, we investigate and prove certain identities and properties on the absolute Möbius divisor functions and Euler totient function when they are iterated. Subsequently, the relationship between the absolute Möbius divisor function with Fermat primes has been researched and some results have been obtained.

A new upper bound for numbers with the Lehmer property and its application to repunit numbers

A composite positive integer [Formula: see text] has the Lehmer property if [Formula: see text] divides [Formula: see text] where [Formula: see text] is an Euler totient function. In this paper, we shall prove that if [Formula: see text] has the Lehmer property, then [Formula: see text], where [Formula: see text] is the number of prime divisors of [Formula: see text]. We apply this bound to repunit numbers and prove that there are at most finitely many numbers with the Lehmer property in the set [Formula: see text] where [Formula: see text] denotes the highest power of 2 that divides [Formula: see text], and [Formula: see text] is a fixed real number.