A More Efficient Algorithm
for Lattice Basis Reduction
C.P.SCHNORR
Journal of algorithm 9,47-62(1988)
報告者
張圻毓
Outline



LLL Algorithm
Compare
Time
LLL Algorithm




Input :
Linearly independent column vector
f1……fn  Zn
Output :
A reduced basis (b1……bn) of the lattice
L=Σ1≦i ≦n Zfi 
Zn
LLL Algorithm






1. for i =1,…,n do bi  fi
compute the GSO G*,M Qn*n , 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}
LLL Algorithm





* 2
i 1
* 2
i
5. if i>1 and
b
2b
then exchange bi-1 and bi and update
the GSO , i  i-1
else i i+1
6. return b1,…,bn
Compare





LLL algorithm
1


(i)滿足 ij
2 ,1≦j＜i≦n
2
 2
 2
(ii)滿足 bi     i ,i 1  bi 1 ，每一個基底元
素不會太小於前一個基底元素，一般δ為
2
2
1
*
*
2
3/4，則    i ,i 1  所以會 bi 1  2 bi
2
Compare





New basis reduction algorithm
(i)滿足  ij  0.55 ,1≦j＜i≦n
2
 2
 2
(ii)滿足 bi     i ,i 1  bi 1 ，則δ為
100/105大於原來數3/4，   i2,i 1  0.64988
* 2
* 2
(實際是1.538745)
bi 1  1.55 bi
Time







LLL algorithm
O(n 4 log B) arithmetic operation on
O ( n log B ) -bit
New algorithm
O(n 4 log B) arithmetic operation on
O(n  log B)-bit
(B bounds the euclidean length of the input
2
2
vectors,ie b1 ,..., bn ) B