A More Efficient Algorithm for Lattice Basis Reduction

1
A More Efficient Algorithm for Lattice Basis
Reduction

• C.P.SCHNORR
• Journal of algorithm 9,47-62(1988)
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2
Outline

• LLL Algorithm
• Compare
• Time

3
LLL Algorithm

• Input
• Linearly independent column vector f1fn Zn
• Output
• A reduced basis (b1bn) of the lattice LS1?i ?n
  Zfi Zn

4
LLL Algorithm

• 1. for i 1,,n do bi ? fi
• compute the GSO G,M Qnn , i ? 2
• 2.while i ?n do
• 3. for j i-1,i-2,,1 do
• 4. bi ? bi - ?µij?bj
• update the GSO replacement step

5
LLL Algorithm

• 5. if igt1 and
• then exchange bi-1 and bi and update
• the GSO , i ? i-1
• else i? i1
• 6. return b1,,bn

6
Compare

• LLL algorithm
• (i)?? ,1?jlti?n
• (ii)?? ,??????
• ?????????????,??d?
• 3/4,? ???

7
Compare

• New basis reduction algorithm
• (i)?? ,1?jlti?n
• (ii)?? ,?d?
• 100/105?????3/4,
• (???1.538745)

8
Time

• LLL algorithm
• arithmetic operation on
  
• -bit
• New algorithm
• arithmetic operation on
  
• -bit
• (B bounds the euclidean length of the input
  vectors,ie )