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Lattice reduction algorithms have numerous applications in number theory,algebra, as well as in cryptanalysis. The most famous algorithm for latticereduction is the LLL algorithm. In polynomial time it computes a reduced basiswith provable output quality. One early improvement of the LLL algorithm wasLLL with deep insertions (DeepLLL). The output of this version of LLL hashigher quality in practice but the running time seems to explode. Weakervariants of DeepLLL, where the insertions are restricted to blocks, behavenicely in practice concerning the running time. However no proof of polynomialrunning time is known. In this paper PotLLL, a new variant of DeepLLL withprovably polynomial running time, is presented. We compare the practicalbehavior of the new algorithm to classical LLL, BKZ as well as blockwisevariants of DeepLLL regarding both the output quality and running time.

PotLLL: A Polynomial Time Version of LLL With Deep Insertions