Journal ArticleDOI
Complexity questions in number theory
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In this article, an analysis of the complexity of known solution algorithms for four problems of number theory is given, including the solving of Diophantine equations and inequalities and the seeking of diophantine approximations and solutions of quadratic DDEs.Abstract:
An analysis of the complexity of the known solution algorithms is given for four problems of number theory — the solving of Diophantine equations and inequalities and the seeking of Diophantine approximations and solutions of quadratic Diophantine equations. A comparison is made of the various algorithms on the basis of their time complexity. The relation of time complexity to the sizes of the intermediate numbers is particularly stressed. A machine independent description of complexity classes is given and some open problems are formulated.read more
Citations
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Book ChapterDOI
Open problems in number theoretic complexity, II
TL;DR: This chapter presents a collection of 36 open problems in number theoretic complexity, showing how questions about the integers have natural generalizations to rings of integers in an algebraic number field, and questions about elliptic curves may generalize to arbitrary abelian varieties.
Proceedings ArticleDOI
A linear space algorithm for computing the hermite normal form
TL;DR: The presented algorithm has the same time complexity of the asymptotically fastest (but space inefficient) algorithms and a heuristic algorithm for HNF that achieves a substantial speedup when run on randomly generated input matrices.
Journal Article
A Linear Space Algorithm for Computing the Hermite Normal Form
TL;DR: In this article, the Hermite Normal Form of an n × n integer matrix is computed in O(n 2 log M) space, where M is a bound on the entries of the input matrix.
Proceedings ArticleDOI
On the worst-case complexity of integer Gaussian elimination
Xin Gui Fang,George Havas +1 more
TL;DR: It is shown that there is an exponential length lower bound on the operands for a well-deflned variant of Gaussian elimination when applied to Smith and Hermite normal form calculation, and the analysis provides guidance as to how integer matrix algorithms based onGaussian elimination may be further developed for better performance.
Proceedings ArticleDOI
Algorithmic number theory-the complexity contribution
TL;DR: A brief history of the symbiotic relationship between number theory and complexity theory will be presented and some of the technical aspects underlying 'modern' methods of primality testing and factoring will be described.
References
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Proceedings ArticleDOI
The complexity of theorem-proving procedures
TL;DR: It is shown that any recognition problem solved by a polynomial time-bounded nondeterministic Turing machine can be “reduced” to the problem of determining whether a given propositional formula is a tautology.
Book
An Introduction to the Geometry of Numbers
TL;DR: In this article, the authors introduce the concept of the quotient space and the notion of automorphs for diophantine approximations of diophantas in the Euclidean space.
Journal ArticleDOI
Integer Programming with a Fixed Number of Variables
TL;DR: It is shown that the integer linear programming problem with a fixed number of variables is polynomially solvable.