Halt Properties and Complexity Evaluations for Optimal DeepLLL Algorithm Families

TL;DR

This paper proves that DeepLLL, PotLLL, and $S^2$LLL algorithms always terminate with the optimal parameter setting and provides explicit bounds on their running times, extending known results beyond LLL.

Contribution

It establishes the halting of these polynomial-time variants at the optimal parameter and derives unified explicit bounds on their complexity.

Findings

All four algorithms always halt with $oldsymbol{ ext{delta} = 1}$.

Explicit upper bounds for the number of loops are provided.

The bounds are unified for all four algorithms, extending previous results.

Abstract

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.