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 \lq\lq optimal\rq\rq{} 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.
翻译:DeepLLL算法(Schnorr,1994年)是LLL Lat基削减算法和PotLLL算法(Fontein等人,2014年)和$S ⁇ 2$LLL 算法(Yasuda和Yamaguchi,2019年)的著名变体,是最近从加密应用中开发的DeepLLLLL算法的多元时变体。然而,已知的计算复杂性的多元多边界限仅显示于 $delta < 1$;对于确保最佳输出质量最佳质量的 $\delta 和 PotLLLLLL 参数 =1$1$的参数 和 PotLLLLLL 运算法(Fontein等人,2014年) 和 $S $2$2$LLLLLL 算法(Yudo,2019年) 和$2$2$LLLLL 运算法(Y) 运算法的著名变体。但除了LLLLLLL 运算法之外,我们甚至没有正式证明,这些算法总是在有限的许多步骤中停止。在本文中总是用最优参数 $1美元= 1美元,我们的所有算法中总是停止使用最优参数。我们这四种算法中,我们四个法中,我们用最优参数也总是用最优数法计算法计算法计算法计算法中最优值,我们所有法中最优的上限是明确的四法。我们的所有算法。