Related papers: A Note on Integer Factorization Using Lattices
We present a lattice algorithm specifically designed for some classical applications of lattice reduction. The applications are for lattice bases with a generalized knapsack-type structure, where the target vectors are boundably short. For…
This note presents the basic mathematical structure of a new integer factorization method based on systems of linear Diophantine equations.
By applying Schmidt's lattice method, we prove results on simultaneous Diophantine approximation modulo 1 for systems of polynomials in a single prime variable provided that certain local conditions are met.
Recently in joint work with E. Sert, we proved sharp boundedness results on discrete fractional integral operators along binary quadratic forms. Present work vastly enhances the scope of those results by extending boundedness to bivariate…
The main objective of this thesis is a classification project for integral lattices. Using Kneser's neighbour method we have developed the computer program tn to classify complete genera of integral lattices. Main results are detailed…
We describe an algorithm for arbitrary-precision computation of the elementary functions (exp, log, sin, atan, etc.) which, after a cheap precomputation, gives roughly a factor-two speedup over previous state-of-the-art algorithms at…
The availability of working quantum computers has led to several proposals and claims of quantum advantage. In 2023, this has included claims that quantum computers can successfully factor large integers, by optimizing the search for nearby…
In this paper we describe a deep learning--based probabilistic algorithm for integer factorisation. We use Lawrence's extension of Fermat's factorisation algorithm to reduce the integer factorisation problem to a binary classification…
A lattice of integers is the collection of all linear combinations of a set of vectors for which all entries of the vectors are integers and all coefficients in the linear combinations are also integers. Lattice reduction refers to the…
Computing a basis for the exponent lattice of algebraic numbers is a basic problem in the field of computational number theory with applications to many other areas. The main cost of a well-known algorithm…
We propose a deterministic algorithm based on Coppersmith's method that employs a rank-3 lattice to address factoring-related problems. An interesting aspect of our approach is that we utilize the second vector in the LLL-reduced basis to…
We investigate the integer solutions of Diophantine equations related to Lehmer's totient conjecture. We give an algorithm that computes all nontrivial spoof Lehmer factorizations with a fixed number of factors, and enumerate all nontrivial…
Shor's algorithm contains a classical post-processing part for which we aim to create an efficient, understandable method aside from continued fractions. Let r be an unknown positive integer. Assume that with some constant probability we…
We approximate $d$-variate periodic functions in weighted Korobov spaces with general weight parameters using $n$ function values at lattice points. We do not limit $n$ to be a prime number, as in currently available literature, but allow…
We revisit the problem of integer factorization with number-theoretic oracles, including a well-known problem: can we factor an integer $N$ unconditionally, in deterministic polynomial time, given the value of the Euler totient $$\Phi$(N)$?…
The optimization of submodular functions on the integer lattice has received much attention recently, but the objective functions of many applications are non-submodular. We provide two approximation algorithms for maximizing a…
We obtain conditions for a trigonometric polynomial t of one variable to equal or be approximated by |p|^2 where p has frequencies in a Bohr set of integers obtained by projecting lattice points in the open planar region bounded by the…
We investigate the problem of computing tensor product multiplicities for complex semisimple Lie algebras. Even though computing these numbers is #P-hard in general, we show that if the rank of the Lie algebra is assumed fixed, then there…
We give an algorithm that finds a sequence of approximations with Dirichlet coefficients bounded by a constant only depending on the dimension. The algorithm uses the LLL-algorithm for lattice basis reduction. We present a version of the…
Since the invention of the famous LLL algorithm, lattice reduction has been an extremely useful tool in computational number theory. By construction, the LLL algorithm deals with lattices living in a vector space endowed with a positive…