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SLIDE REDUCTION, SUCCESSIVE MINIMA AND SEVERAL APPLICATIONS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 88 / Issue 3 / December 2013
- Published online by Cambridge University Press:
- 30 April 2013, pp. 390-406
- Print publication:
- December 2013
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- Article
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Gama and Nguyen [‘Finding short lattice vectors within Mordell’s inequality’, in: Proceedings of the 40th Annual ACM Symposium on Theory of Computing, New York, 2008, 257–278] have presented slide reduction which is currently the best SVP approximation algorithm in theory. In this paper, we prove the upper and lower bounds for the ratios
$\Vert { \mathbf{b} }_{i}^{\ast } \Vert / {\lambda }_{i} (\mathbf{L} )$ and 
$\Vert {\mathbf{b} }_{i} \Vert / {\lambda }_{i} (\mathbf{L} )$, where 
${\mathbf{b} }_{1} , \ldots , {\mathbf{b} }_{n} $ is a slide reduced basis and 
${\lambda }_{1} (\mathbf{L} ), \ldots , {\lambda }_{n} (\mathbf{L} )$ denote the successive minima of the lattice 
$\mathbf{L} $. We define generalised slide reduction and use slide reduction to approximate 
$i$-SIVP, SMP and CVP. We also present a critical slide reduced basis for blocksize 2.
