Fermat's well-known factorization algorithm is based on finding a representation of natural numbers $N$ as the difference of squares. In 1895, Lawrence generalized this idea and applied it to multiples $kN$ of the original number. A systematic approach to choose suitable values for $k$ has been introduced by Lehman in 1974, which resulted in the first deterministic factorization algorithm considerably faster than trial division. In this paper, we construct a time-space tradeoff for Lawrence's generalization and apply it together with Lehman's result to obtain a deterministic integer factorization algorithm with runtime complexity $O(N^{2/9+o(1)})$. This is the first exponential improvement since the establishment of the $O(N^{1/4+o(1)})$ bound in 1977.

中文翻译：

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雷曼确定性整数分解方法的时空权衡

费马著名的因式分解算法基于找到自然数 $N$ 作为平方差的表示。1895 年，劳伦斯推广了这个想法，并将其应用于原始数的倍数 $kN$。Lehman 于 1974 年引入了一种为 $k$ 选择合适值的系统方法，这导致第一个确定性分解算法比试除法快得多。在本文中，我们为 Lawrence 的泛化构建了一个时空权衡，并将其与 Lehman 的结果一起应用，以获得运行时复杂度为 $O(N^{2/9+o(1)})$ 的确定性整数分解算法。这是自 1977 年建立 $O(N^{1/4+o(1)})$ 界限以来的第一次指数级改进。