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 f 1 ……f n Z n Output : A reduced basis (b 1 ……b n ) of the lattice L=Σ 1 ≦ i ≦ n Zf i Z n

LLL Algorithm 1. for i =1,…,n do b i  f i compute the GSO G*,M Q n*n, i  2 2.while i ≦ n do 3. for j= i-1,i-2,…,1 do 4. b i  b i - 「 μ ij 」 b j update the GSO {replacement step}

LLL Algorithm 5. if i>1 and then exchange b i-1 and b i and update the GSO, i  i-1 else i  i+1 6. return b 1,…,b n

Compare LLL algorithm (i) 滿足,1 ≦ j ＜ i ≦ n (ii) 滿足 ，每一個基底元 素不會太小於前一個基底元素，一般 δ 為 3/4 ，則 所以會

Compare New basis reduction algorithm (i) 滿足,1 ≦ j ＜ i ≦ n (ii) 滿足 ，則 δ 為 100/105 大於原來數 3/4 ， ( 實際是 )

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