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We report on improved practical algorithms for lattice basis reduction. We propose a practical oating point v e r s i o n o f t h e L 3 {algorithm of Lenstra, Lenstra, Lovv asz (1982). We present a v ariant o f t h e L 3 { algorithm with \deep insertions" and a practical algorithm for block Korkin{Zolotarev reduction, a concept introduced by S c hnorr(More)
The general subset sum problem is NP-complete. However, there are two algorithms, one due to Brickell and the other to Lagarias and Odlyzko, which in polynomial time solve almost all subset sum problems of suuciently low density. Both methods rely on basis reduction algorithms to nd short non-zero vectors in special lattices. The Lagarias-Odlyzko algorithm(More)
Let Ai(L), Ai(L*) denote the successive minima of a lattice L and its reciprocal lattice L*, and let [bl,..., bn] be a basis of L that is reduced in the sense of Korkin and Zolotarev. We prove that [4/(/+ 3)]),i(L) 2 _< [bi[ 2 < [(i + 3)/4])~i(L) 2 and Ibil2An_i+l(L*) 2 <_ [(i + 3)/4][(n-i + 4)/417~ 2, where "y~ =-min(Tj : 1 < j _< n} and 7j denotes(More)
We introduce algorithms for lattice basis reduction that are improvements of the famous L 3-algorithm. If a random L 3 {reduced lattice basis b1; : : : ; bn is given such that the vector of reduced Gram{ Schmidt coeecients (fi;jg 1 j < i n) is uniformly distributed in 0; 1) (n 2) , then the pruned enumeration nds with positive probability a shortest lattice(More)