LLL: A Tool for Effective Diophantine Approximation

DOI10.1007/978-3-642-02295-1_6zbMath1230.11154OpenAlexW105343697MaRDI QIDQ5188542

Guillaume Hanrot

Publication date: 5 March 2010

Published in: The LLL Algorithm (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/978-3-642-02295-1_6

zbMATH Keywords

algorithmsLLL algorithmDiophantine equationsMertens conjecture

Mathematics Subject Classification ID

Computer solution of Diophantine equations (11Y50) Number-theoretic algorithms; complexity (11Y16) Diophantine inequalities (11J25)

Related Items (4)

Non-linear polynomial selection for the number field sieve ⋮ Gradual sub-lattice reduction and a new complexity for factoring polynomials ⋮ Selected Applications of LLL in Number Theory ⋮ Explicit realization of elements of the Tate-Shafarevich group constructed from Kolyvagin classes

Cites Work

• Solving exponential diophantine equations using lattice basis reduction algorithms
• Factoring polynomials and the knapsack problem
• On Lovász' lattice reduction and the nearest lattice point problem
• A modification of the LLL reduction algorithm
• A hierarchy of polynomial time lattice basis reduction algorithms
• On the practical solution of the Thue equation
• Solving Thue equations of high degree
• Diophantine approximation
• Factoring polynomials with rational coefficients
• Small solutions to polynomial equations, and low exponent RSA vulnerabilities
• \(S\)-integral solutions to a Weierstrass equation
• On the computation of Mordell-Weil and 2-Selmer groups of elliptic curves.
• \(S\)-integral points on elliptic curves -- notes on a paper of B. M. M. de Weger.
• LLL \(\and\) ABC
• On the \(abc\) conjecture. II.
• An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers. II
• Disproof of the Mertens conjecture.
• Searching worst cases of a one-variable function using lattice reduction
• Improved Analysis of Kannan’s Shortest Lattice Vector Algorithm
• Analysis of PSLQ, an integer relation finding algorithm
• Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms
• Computing integral points on elliptic curves
• Computing all integer solutions of a genus 1 equation
• The densest lattice in twenty-four dimensions
• A sieve algorithm for the shortest lattice vector problem
• Floating-Point LLL Revisited
• Contributions to the theory of diophantine equations I. On the representation of integers by binary forms
• Linear forms in the logarithms of algebraic numbers
• THE EQUATIONS 3x²−2 = y² AND 8x²−7 = z²
• Algorithmic Number Theory
• Algorithmic Number Theory
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