Related papers: A Summation-Based Algorithm For Integer Factorizat…

Consider a sparse polynomial in several variables given explicitly as a sum of non-zero terms with coefficients in an effective field. In this paper, we present several algorithms for factoring such polynomials and related tasks (such as…

Converting binary integers to variable-length decimal strings is a fundamental operation in computing. Conventional fast approaches rely on recursive division and small lookup tables. We propose a SIMD-based algorithm that leverages integer…

The factorization of a large digit integer in polynomial time is a challenging computational task to decipher. The exponential growth of computation can be alleviated if the factorization problem is changed to an optimization problem with…

Shor's algorithm has seriously challenged information security based on public key cryptosystems. However, to break the widely used RSA-2048 scheme, one needs millions of physical qubits, which is far beyond current technical capabilities.…

A compression algorithm is presented that uses the set of prime numbers. Sequences of numbers are correlated with the prime numbers, and labeled with the integers. The algorithm can be iterated on data sets, generating factors of doubles on…

We show that $n$-bit integers can be factorized by independently running a quantum circuit with $\tilde{O}(n^{3/2})$ gates for $\sqrt{n}+4$ times, and then using polynomial-time classical post-processing. The correctness of the algorithm…

We have developed a framework to convert an arbitrary integer factorization problem to an executable Ising model by first writing it as an optimization function and then transforming the k-bit coupling ($k\geq 3$) terms to quadratic terms…

In this paper, we will describe a new factorization algorithm based on the continuous representation of Gauss sums, generalizable to orders j>2. Such an algorithm allows one, for the first time, to find all the factors of a number N in a…

Factorization of polynomials arises in numerous areas in symbolic computation. It is an important capability in many symbolic and algebraic computation. There are two type of factorization of polynomials. One is convention polynomial…

Decimal multiplication is the task of multiplying two numbers in base $10^N.$ Specifically, we focus on the number-theoretic transform (NTT) family of algorithms. Using only portable techniques, we achieve a 3x-5x speedup over the mpdecimal…

In many important applications -- such as search engines and relational database systems -- data is stored in the form of arrays of integers. Encoding and, most importantly, decoding of these arrays consumes considerable CPU time.…

We give a geometric approach to integer factorization. This approach is based on special approximations of segments of the curve that is represented by $y=n/x$, where $n$ is the integer whose factorization we need.

We explore an algorithm for approximating roots of integers, discuss its motivation and derivation, and analyze its convergence rates with varying parameters and inputs. We also perform comparisons with established methods for approximating…

In this article we develop an algorithm which computes a divisor of an integer $N$, which is assumed to be neither prime nor the power of a prime. The algorithm uses discrete time heat diffusion on a finite graph. If $N$ has $m$ distinct…

Fast algorithms for integer and polynomial multiplication play an important role in scientific computing as well as in other disciplines. In 1971, Sch{\"o}nhage and Strassen designed an algorithm that improved the multiplication time for…

We report on two new records: the factorization of RSA-240, a 795-bit number, and a discrete logarithm computation over a 795-bit prime field. Previous records were the factorization of RSA-768 in 2009 and a 768-bit discrete logarithm…

The Fast Reciprocal Square Root Algorithm is a well-established approximation technique consisting of two stages: first, a coarse approximation is obtained by manipulating the bit pattern of the floating point argument using integer…

Matrix factorization methods are important tools in data mining and analysis. They can be used for many tasks, ranging from dimensionality reduction to visualization. In this paper we concentrate on the use of matrix factorizations for…

A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time of at most a polynomial factor. This may not…

Simplification of fractional powers of positive rational numbers and of sums, products and powers of such numbers is taught in beginning algebra. Such numbers can often be expressed in many ways, as this article discusses in some detail.…