Floating-Point LLL Revisited

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https://hal.inria.fr/inria-00070616

Submitted on 19 May 2006

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Floating-Point LLL Revisited

Phong Q. Nguyen, Damien Stehlé

To cite this version:

Phong Q. Nguyen, Damien Stehlé. Floating-Point LLL Revisited. [Research Report] RR-5387, INRIA.

2004, pp.28. �inria-00070616�

ISRN INRIA/RR--5387--FR+ENG

a p p o r t

d e r e c h e r c h e

THÈME 2

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Floating-Point LLL Revisited

Phong Q. Nguyen, Damien Stehlé

N° 5387

Novembre 2004

Unité de recherche INRIA Lorraine

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dim(L) ≥ 2^¥

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vol(L)

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[b^∗

1, . . . ,b^∗

d]VXe acU¡[]lg`gJL[]\z[]U"acYW«¢acM¡ViYif sWNž¤LULNqsAlNq_FdWlegVi}zNFYifAa]e(«¹[]YiYX[bŽe»

b^∗

i

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b₁, . . . ,b_i−1^¥ !N J"ab}zN

b^∗

i =b_i−P_i−1

j=1µ_i,jb^∗

j

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ji/kb^∗

jk^2¥

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b_ieOeac`gVXeg¤"Nqe

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b_ieaclNºViUp`NF\zNFl}zNq_F`[]leL`gJLN

b^∗

i

eacULsA`gJLN

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R^_ o"aclg`([c«<`gJLN

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R = (r_i,j)^

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i Ž£N>J"ab}zN

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j < i ≤d¥2±UViULs5Vi}pVXs5d"acYŒ}zNq_F`[]l

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η = 1/2 acULs–VXe3`’f,oWVX_bacYiYif a]_JWVXNF}zNqs¬hpfegdL_q_qNqeegVi}zNFYif¾egVi”bNž‘¢lNqs5dL_FViUW\ŸViULs5Vi},VXs5d"acY2}zNq_F`[]lehpfƒViUL_FlNba]eViUW\ŸViULsWNž­

i^¥

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δ V—«2`gJLNh"a]eVXe£VXeegVi”bNž‘¢lNqs5dL_qNqsŽ(Vi`gJ«¢a]_F`[]l

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`gJLN

(d−1) 2[b}^“]eg”_q[]ULs5Vi`gVX[]ULe

(δ−µ²_i,i−1)kb^∗

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kb₁k ≤β^(d−1)/4vol(L)^1/d ,^Ž(JLNFlN β= 1/(δ−η²).

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δ ^acULs η aclNlNqego*Nq_F`gVi}zNFYif²`[

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κ egdL_J¬`gJ"ac`

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κ

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[b₁, . . . ,b_d] ^VXe3acY—‘

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g V

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