Related papers: Counting RSA-integers

Let $p$ be a prime number. We say that a positive integer $n$ is a Sylow $p$-number if there exists a finite group having exactly $n$ Sylow $p$-subgroups. When $p=2$, every odd integer is a Sylow $2$-number. In contrast, when $p$ is odd,…

We obtain the expected asymptotic formula for the number of primes $p<N=2^n$ with $r$ prescribed (arbitrarly placed) binary digits, provided $r<cn$ for a suitable constant $c>0$. This result improves on our earlier result where $r$ was…

This paper proposes an alternative approach to formally establishing the correctness of the RSA public key cryptosystem. The methodology presented herein deviates slightly from conventional proofs found in existing literature. Specifically,…

In \cite{Tm}, the authors give two public key encryptions based on third order linear sequences modulo $n^2$, where $n=pq$ is an RSA integer. In their scheme (3), there are two mistakes in the decryption procedure

The RSA cryptosystem could be easily broken with large scale general purpose quantum computers running Shor's factorization algorithm. Being such devices still in their infancy, a quantum annealing approach to integer factorization has…

In this paper we present a new efficient algorithm for factoring the RSA and the Rabin moduli in the particular case when the difference between their two prime factors is bounded. As an extension, we also give some theoretical results on…

We consider Shor's quantum factoring algorithm in the setting of noisy quantum gates. Under a generic model of random noise for (controlled) rotation gates, we prove that the algorithm does not factor integers of the form $pq$ when the…

There are several methods to measure computing power. On the other hand, Bit Length (BL) can be considered a metric to measure the strength of an asymmetric encryption method. We review here ways to determine the security, given an span of…

We study whether sufficiently large integers can be written in the form cp+T_x, where p is either zero or a prime congruent to r mod d, and T_x=x(x+1)/2 is a triangular number. We also investigate whether there are infinitely many positive…

It is well known that Shor's quantum algorithm for integer factorization can break down the RSA public-key cryptosystem, which is widely used in many cryptographic applications. Thus, public-key cryptosystems in the quantum computational…

Many modern asymmetric encryption methods rely on prime numbers, as they have distinctive properties. For instance, the security of RSA cryptosystem relies on the computational difficulty of factoring a large composite number in its prime…

An efficient integer factorization algorithm would reduce the security of all variants of the RSA cryptographic scheme to zero. Despite the passage of years, no method for efficiently factoring large semiprime numbers in a classical…

We present four combinatorial proofs of Morgado's formula for the number $\varrho(n)$ of non-congruent regular integers modulo $n$, corresponding to sequence A055653 in the On-Line Encyclopedia of Integer Sequences (OEIS), where an integer…

In this paper we give a polynomial time algorithm to compute $\varphi(N)$ for an RSA module $N$ using as input the order modulo $N$ of a randomly chosen integer. This provides a new insight in the very important problem of factoring an RSA…

Quantum algorithms are at the heart of the ongoing efforts to use quantum mechanics to solve computational problems unsolvable on ordinary classical computers. Their common feature is the use of genuine quantum properties such as…

In this paper, we prove the lower bound for the number of balancing non-Wieferich primes in arithmetic progressions. More precisely, for any given integer $r\geq2$ there are $\gg\log x$ balancing non-Wieferich primes $p\leq x$ such that…

The theoretical aspects of four integer factorization algorithms are discussed in details in this note. The focus is on the performances of these algorithms on the subset of hard to factor balanced integers N = pq, p < q < 2p. The running…

Given the escalating importance of cybersecurity, it becomes increasingly beneficial for a diverse community to comprehend fundamental security mechanisms. Among these, the RSA algorithm stands out as a crucial component in public-key…

The security of RSA algorithm depends upon the positive integer N, which is the multiple of two precise large prime numbers. Factorization of such great numbers is a problematic process. There are many algorithms has been implemented in the…

Let $p,q>1$ be two relatively prime integers and $\mathbb{N}$ the set of nonnegative integers. Let $f_{p,q}(n)$ be the number of different expressions of $n$ written as a sum of distinct terms taken from…