Improving the BKZ Reduction Algorithm by Quick Reordering Technique

Abstract

In this paper, we propose a simple method to improve the BKZ algorithm with small blocksize. At first, we observe that reordering the LLL-reduced basis vectors by increasing norm will change the distribution of search nodes in the enumeration tree, which gives a chance to reduce the enumeration search nodes with non-negligible probability. Thus the runtime of enumeration algorithm is accelerated approximately by a factor of two. We explain this phenomenon from a theoretical point of view, which follows the Gama-Nguyen-Regev’s analysis [6]. Then we apply this reordering technique on the BKZ algorithm and implement it in the open source library NTL. Our experimental results in dimensions 100–120 with blocksize 15–30 show that on LLL-reduced bases, our modified NTL-BKZ outputs a vector shorter than the original NTL-BKZ with probability 40%–46% with LLL Lovász constant $\delta _{LLL}=0.99$. Furthermore, in the instances where the improved BKZ found a same or shorter vector, the runtime is up to 2.02 times faster when setting the blocksize $\beta =25$ with $\delta _{LLL}=0.99$.