Halt Properties and Complexity Evaluations for Optimal DeepLLL Algorithm Families

DeepLLL algorithm (Schnorr, 1994) is a famous variant of LLL lattice basis reduction algorithm, and PotLLL algorithm (Fontein et al., 2014) and $S^2$LLL algorithm (Yasuda and Yamaguchi, 2019) are recent polynomial-time variants of DeepLLL algorithm developed from cryptographic applications. However, the known polynomial bounds for computational complexity are shown only for parameter $\delta < 1$; for "optimal" parameter $\delta = 1$ which ensures the best output quality, no polynomial bounds are known, and except for LLL algorithm, it is even not formally proved that the algorithm always halts within finitely many steps. In this paper, we prove that these four algorithms always halt also with optimal parameter $\delta = 1$, and furthermore give explicit upper bounds for the numbers of loops executed during the algorithms. Unlike the known bound (Akhavi, 2003) applicable to LLL algorithm only, our upper bounds are deduced in a unified way for all of the four algorithms.