Theory of linear and integer programming
Theory of linear and integer programming
Fast reduction and composition of binary quadratic forms
ISSAC '91 Proceedings of the 1991 international symposium on Symbolic and algebraic computation
Short vectors of planar lattices via continued fractions
Information Processing Letters
The Design and Analysis of Computer Algorithms
The Design and Analysis of Computer Algorithms
An affine point of view on minima finding in integer lattices of lower dimensions
EUROCAL '87 Proceedings of the European Conference on Computer Algebra
A 3-Dimensional Lattice Reduction Algorithm
CaLC '01 Revised Papers from the International Conference on Cryptography and Lattices
Fast unimodular reduction: planar integer lattices
SFCS '92 Proceedings of the 33rd Annual Symposium on Foundations of Computer Science
An LLL-reduction algorithm with quasi-linear time complexity: extended abstract
Proceedings of the forty-third annual ACM symposium on Theory of computing
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We show that a positive definite integral ternary form can be reduced with O(M(s) log2 s) bit operations, where s is the binary encoding length of the form and M(s) is the bit-complexity of s-bit integer multiplication. This result is achieved in two steps. First we prove that the the classical Gaussian algorithm for ternary form reduction, in the variant of Lagarias, has this worst case running time. Then we show that, given a ternary form which is reduced in the Gaussian sense, it takes only a constant number of arithmetic operations and a constant number of binary-form reductions to fully reduce the form. Finally we describe how this algorithm can be generalized to higher dimensions. Lattice basis reduction and shortest vector computation in fifixed dimension d can be done with O(M(s) logd-1 s) bit-operations.