A More Efficient Algorithm for Lattice Basis Reduction

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……fnZn • Output : • A reduced basis (b1……bn) of the lattice L=Σ1≦i ≦nZfiZn

LLL Algorithm • 1. for i =1,…,n dobi 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 • 5. if i>1 and • then exchange bi-1 and bi and update • the GSO , i  i-1 • else i i+1 • 6. return b1,…,bn

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1.538745)

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

A More Efficient Algorithm for Lattice Basis Reduction

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