Gauss

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Gauss Algorithm RevisitedBrigitte
ValleeJournal of Algorithm 12, 1991

• ???
• ???95/01/25
• ????95/2/27

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• Paper????
• 1.An historical survey
• 2.Different formulations of Gauss algorithm
• 3.Comparison between these algorithms
• 4.The possible antecedents of a configuration
• 5.The worst-case configuration of
  Gaussalgorithm
• 6.Comparing Gauss algorithm and Euclid s
  algorithm
• 7.The complexity of the LLL algorithm
• ??????
• 1.??????(??)
• 2.Gauss algorithm????(1)
• 3.Different formulations of Gauss
  algorithm(2)
• 4.?Euclids Algorithm?Continued Fraction??(6)

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1.??????

• ??????????
• ??Gaussalgorithm???Euclids algorithm???case,
• ??Gaussalgorithm?worst-case ???Euclids
  algorithm?worst-case ?
• ????????LLL algorithm???, LLL algorithm??Gauss
  algorithm????,????gauss(t)?
• LLL algorithm?t??1????polynomial time algorithm
• ???t??1?,LLL algorithm??polynomial time
  algorithm?
• ?t??1?, LLL algorithm????????????Gauss algorithm

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2.Gauss algorithm????

• Lagrange11??????reduction theory of binary
  quadratic forms,????????????????????
• ?????bases of lattice?quadratic
  forms??????Gauss3(1953)
• Dirchlet1(1850)?????????????
• Gauss?????,???reduction of binary quadratic
  form???Gausss algorithm?
• 1983,Lovasz basis reduction algorithm(L3-algorith
  m)

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3.Different formulations of Gauss algorithm

• Gauss algorithm
• Gauss-acute algorithm

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Gauss algorithm(1/2)

• ????Z2?lattice L???basis(u,v),?u?basis(u,v)?????,?
  ?????v??????x(v,u)
• K(v,u) ww e(v-mu), m??Z, e1
• x(v,u)??ru.v/u2??,?m????r??,?e????r-m
• Gauss algorithm
• Repeat
• 1. If u2gtv2, ?u and v??
• 2. vx(v,u)
• Until u2?v2

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Gauss algorithm(2/2)

• ??Gauss???,????????Gram matrix G(u,v), ??????
• ?lattice L?????basis(u,v),
• Gauss algorithm??L????basis???????

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Gauss-acute algorithm

• ??Gauss-acute?????
• ?????A.B.C????
• u AB , v AC
• ???(u?v ? v2 )?C????,?????A.B????????,??u?v???????
  A.B.C?????????,Gauss-acute???

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??Gauss,Gauss(t),Gauss-acute
Name Gauss Gauss(t) (LLL Algorithm) Gauss-acute
?????? Repeat 1.If u2gt v2 swap(u,v) 2.uv-? ?u (??) (??)
???? until u2?v2 until u2?t2v2 until u?v ltv2
???? v2? u2 0? u?v ?½ u2 v2? (1/t2)u2 0? u?v ?½ u2 u?v lt v2 0? u?v ?½ u2
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4.?Euclids Algorithm?Continued Fraction??

• Gauss algorithm??Euclids algorithm????,Euclids
  algorithm????????????a?b(a?b)????
• a bq r with
• ???(a,b)???(b,r)
• ?Gauss algorithm???????????
• ?Z2 ,Gauss algorithm????????

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• (w,w)?(u,v)????lattice L???bases,??????????
• (1)????u?v?????
• (2)????w?w????basis(u,v)?
• w puqv ? wpuqv
• p,p,q,q????,??q lt q
• ?Gauss algorithm??,????w?
• w0p0uq0v
• ???p0/q0,?p/q?????convergent
• ?pq - pq 1 ??

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Example

• ?a(78,37) b(34,16)
• ??? A7453 B1412
• ??? n3244 r?2 T125
• ??? t(10,5) a(34,16) b(10,5) A1412 B125
• ??? n420 r?3 T17
• ??? t(4,1) a(10,5) b(4,1) A125 B17
• ??? n45 r?3 T8
• ??? t(-2,2) a(4,1) b(-2,2) A17 B8
• ??? n-6 r?-1 T13
• ??? 13?8 output (-2,2)

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• u(78,37) v(34,16)
• ????23,3 ,convergent?
• (w,w)?(u,v)????lattice L???bases
• w p(78,37) q (34,16)
• wp (78,37) q (34,16)
• ?Gauss algorithm??,????w
• (10,5)1 (78,37) -2 (34,16)
• (-4,-1)3 (78,37) -7 (34,16)
• (-2,2)10 (78,37) -23 (34,16)