Korkin-Zolotarev bases and successive minima of a lattice and its reciprocal lattice |
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Authors: | J C Lagarias H W Lenstra Jr C P Schnorr |
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Institution: | (1) AT&T Bell Laboratories, Murray Hill, New Jersey, USA;(2) Department of Mathematics, University of California, Berkeley, California, USA;(3) UniversitÄt Frankfurt, Frankfurt, F. R. Germany |
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Abstract: | Let
i(L), i(L*) denote the successive minima of a latticeL and its reciprocal latticeL
*, and let b1,..., b
n
] be a basis ofL that is reduced in the sense of Korkin and Zolotarev. We prove that and, where and
j denotes Hermite's constant. As a consequence the inequalities are obtained forn7. Given a basisB of a latticeL in
m
of rankn andx
m
, we define polynomial time computable quantities(B) and(x,B) that are lower bounds for 1(L) and(x,L), where(x,L) is the Euclidean distance fromx to the closest vector inL. If in additionB is reciprocal to a Korkin-Zolotarev basis ofL
*, then 1(L)
n
*
(B) and.The research of the second author was supported by NSF contract DMS 87-06176. The research of the third author was performed at the University of California, Berkeley, with support from NSF grant 21823, and at AT&T Bell Laboratories. |
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Keywords: | 11 H 06 11 H 50 |
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