Related papers: A Summation-Based Algorithm For Integer Factorizat…
This paper presents the concept of digit polynomials, which leads to a deterministic and unconditional integer factorization algorithm with the runtime complexity $\mathcal{O}(N^{1/4+\epsilon})$. Strassen's well known factoring approach is…
We propose a new formulation of the problem of prime factorization of integers. With replica exchange Monte Carlo simulation, the behavior which is seemed to indicate exponential computational hardness is observed. But this formulation is…
Most popular word embedding techniques involve implicit or explicit factorization of a word co-occurrence based matrix into low rank factors. In this paper, we aim to generalize this trend by using numerical methods to factor higher-order…
The number of steps any classical computer requires in order to find the prime factors of an $l$-digit integer $N$ increases exponentially with $l$, at least using algorithms known at present. Factoring large integers is therefore…
We demonstrate a method for general linear optical networks that allows one to factorize any SU($n$) matrix in terms of two SU($n-1)$ blocks coupled by an SU(2) entangling beam splitter. The process can be recursively continued in an…
We present an algorithm to reduce the computational effort for the multiplication of a given matrix with an unknown column vector. The algorithm decomposes the given matrix into a product of matrices whose entries are either zero or integer…
Boolean matrix factorization (BMF) approximates a given binary input matrix as the product of two smaller binary factors. As opposed to binary matrix factorization which uses standard arithmetic, BMF uses the Boolean OR and Boolean AND…
We use the periodicity properties of generalized Gauss sums to factor numbers. Moreover, we derive rules for finding the factors and illustrate this factorization scheme for various examples. This algorithm relies solely on interference and…
By establishing an interesting connection between ordinary Bell polynomials and rational convolution powers, some composition and inverse relations of Bell polynomials as well as explicit expressions for convolution roots of sequences are…
Factoring large integers using a quantum computer is an outstanding research problem that can illustrate true quantum advantage over classical computers. Exponential time order is required in order to find the prime factors of an integer by…
A scheme to encode arbitrarily long integer pairs on degenerate optical parametric oscillations multiplexed in time is proposed. The classical entanglement between the polarization directions and the phases of the oscillating pulses,…
In addition to recent developments in computing speed and memory, methodological advances have contributed to significant gains in the performance of stochastic simulation. In this paper, we focus on variance reduction for matrix…
Numerical data processing is a key task across different fields of computer technology use. However, even simple summation of values is not precise due to the floating point representation use. This paper presents a practical algorithm for…
Integer division instruction is generally expensive in most architectures. If the divisor is constant, the division can be transformed into combinations of several inexpensive integer instructions. This article discusses the classic…
Let $a,b\in \mathbb{N}$ be fixed and coprime such that $a>b$, and let $N$ be any number of the form $a^n\pm b^n$, $n\in\mathbb{N}$. We will generalize a result of Bostan, Gaudry and Schost and prove that we may compute the prime…
Today's PCs can directly manipulate numbers not longer than 64 bits because the size of the CPU registers and the data-path are limited. Consequently, arithmetic operations such as addition, can only be performed on numbers of that length.…
The road to computing on quantum devices has been accelerated by the promises that come from using Shor's algorithm to reduce the complexity of prime factorization. However, this promise hast not yet been realized due to noisy qubits and…
The key-generation algorithm for the RSA cryptosystem is specified in several standards, such as PKCS#1, IEEE 1363-2000, FIPS 186-3, ANSI X9.44, or ISO/IEC 18033-2. All of them substantially differ in their requirements. This indicates that…
I present two new methods for exactly summing a set of floating-point numbers, and then correctly rounding to the nearest floating-point number. Higher accuracy than simple summation (rounding after each addition) is important in many…
Quantum integer factorization is a potential quantum computing solution that may revolutionize cryptography. Nevertheless, a scalable and efficient quantum algorithm for noisy intermediate-scale quantum computers looks far-fetched. We…