Rejecting Claimed Speedup of 2^𝛽/2 in Extreme Pruning and Revising BKZ 2.0 for Better Speedup

BKZ 2.0 algorithm is one of the claimant lattice reduction algorithms which incorporates extreme pruning as its main phase. The nonextreme pruning and extreme pruning in the original paper by GammaNguyenRegev (in 2010) respectively are claimed to reach the maximum speedup of 2^(β/4) and 2^(β/2) over fullenumeration. For 37≤β, this paper verifies the claimed speedup of 2^(β/4) by an optimal nonextreme pruned enumeration, while for a practical block size of 100≤β≤250, the upperbound for speedup of extremepruned enumeration when blocks are preprocessed by BKZ with enumeration (as SVPsolver) is estimated from 2^(β/6.6) to 2^(β/4.4), and when blocks are preprocessed by BKZ with sieving is estimated from 2^(β/9.8) to 2^(β/3.4) ! By using these upperbounds for speedup by extremepruning, all former security analyses which use the claimed speedup of 2^(β/2), should be revised  (or rejected). Also, this paper proposes a revised version of BKZ 2.0, so that for a practical block size of 100≤β≤250 and an infinite number of rounds N≈∞, the lowerbound of its speedup over BKZ 2.0 is estimated by a factor of ρ_0 in range of 2^12≤ρ_0≤2^15.5 when blocks are preprocessed by BKZ with enumeration, and also lowerbound of its speedup is estimated in the range of 0≤ρ_0≤2^20.5 when blocks are preprocessed by BKZ with sieving. Moreover, for a finite number of rounds N, speedup of our revised BKZ 2.0 over original BKZ 2.0 can be estimated by O(min⁡(N,ρ_0 ) ).