Related papers: Fast Fraction-Integer Method for Computing Multipl…
In this paper, we present a new numerical method to solve fractional differential equations. Given a fractional derivative of arbitrary real order, we present an approximation formula for the fractional operator that involves integer-order…
We present new algorithms for computing the low $n$ bits or the high $n$ bits of the product of two $n$-bit integers. We show that these problems may be solved in asymptotically 75% of the time required to compute the full $2n$-bit product,…
This paper introduces a new algorithm for the fundamental problem of generating a random integer from a discrete probability distribution using a source of independent and unbiased random coin flips. We prove that this algorithm, which we…
We consider the problem of private distributed matrix multiplication under limited resources. Coded computation has been shown to be an effective solution in distributed matrix multiplication, both providing privacy against the workers and…
The efficient evaluation of high-dimensional integrals is of importance in both theoretical and practical fields of science, such as data science, statistical physics, and machine learning. However, exact computation methods suffer from the…
To factor an integer N, given that it is equal to the product of two primes, it suffices to find an integer d satisfying a certain simple numerical test. In this approach, the factorization problem equates to the problem of designing an…
We develop an algorithm to invert the $j$--function in quasilinear time, and apply this to testing whether an elliptic curve has complex multiplication.
Two prominent methods for integer factorization are those based on general integer sieve and elliptic curve. The general integer sieve method can be specialized to quadratic integer sieve method. In this paper, a probability analysis for…
Encrypted computation opens up promising avenues across a plethora of application domains, including machine learning, health-care, finance, and control. Arithmetic homomorphic encryption, in particular, is a natural fit for cloud-based…
The division operation is important for many areas of data processing. Especially considering today's demand for hardware accelerators for machine learning algorithms, there is a high demand for an efficient calculation of the division…
We will derive a function that eliminates any sequence of equidistant numbers from the integer numbers, then we will derive its inverse. Then we will use the Sequence elimination function to eliminate the multiples of the prime numbers from…
Various applications such as MRI, solution of PDEs, etc. need to perform an inverse nonequispaced fast Fourier transform (NFFT), i. e., compute $M$ Fourier coefficients from given $N$ nonequispaced data. In the present paper we consider…
An efficient hybrid numerical method for multiple scattering calculations is proposed. We use the well established doubling--adding method to find the reflection function of the lowermost homogeneous slab comprising the atmosphere of our…
Multidimensional imaging, capturing image data in more than two dimensions, has been an emerging field with diverse applications. Due to the limitation of two-dimensional detectors in obtaining the high-dimensional image data, computational…
The aim of this note is to prove the inversion formula, which can be used to compute the Levi measure of an infinitely divisible distribution from its characteristic function. Obtained formula is similar to the well-known inversion formula…
In poly alphabetic substitution the plain texts letters are enciphered differently according to their position. The name poly alphabetic suggests that there are more than one key so we have used two keys combination instead of just one, in…
Bayesian probabilistic numerical methods are a set of tools providing posterior distributions on the output of numerical methods. The use of these methods is usually motivated by the fact that they can represent our uncertainty due to…
We give an $O(N\cdot \log N\cdot 2^{O(\log^*N)})$ algorithm for multiplying two $N$-bit integers that improves the $O(N\cdot \log N\cdot \log\log N)$ algorithm by Sch\"{o}nhage-Strassen. Both these algorithms use modular arithmetic.…
We present new algorithms to detect and correct errors in the product of two matrices, or the inverse of a matrix, over an arbitrary field. Our algorithms do not require any additional information or encoding other than the original inputs…
Over the last decade, it has been demonstrated that many systems in science and engineering can be modeled more accurately by fractional-order than integer-order derivatives, and many methods are developed to solve the problem of fractional…