大整数分解中二次筛算法实现的优化
Optimization of Quadratic Sieve Algorithm Implementation for Large Integer Factorization
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中文总结 AI 辅助
本文提出二次筛算法的两种优化：根值累加改为mdi以降低复杂度至O(n/m)，参数查找改用二分查找降至O(logn)，从而提升RSA模数分解效率。
中文摘要 AI 辅助
二次筛法是数论中的核心工具。本文针对二次筛算法提出了两种优化方法。在筛选过程中，将原始多项式根值累加步骤从原始的di改为mdi（m为小整数），可将复杂度从O(n)降至O(n/m)。另一种优化方法是，对于所有参数查找步骤，可将原始遍历查找改为高效的二分查找，从而将循环复杂度从O(n)降至O(logn)。这一改进在实际场景中降低了RSA模数分解的计算复杂度。
英文摘要
The quadratic sieve method is a core tool in number theory. In this paper, we present two optimization methods for the Quadratic Sieve algorithm. In the sieving process, the original polynomial root value accumulation step is changed from the original di to the mdi (m is a small integer), which can change the complexity from O(n) to O(n/m). Another optimization method is that for all parameter lookup steps, the original traversal lookup can be changed to an efficient binary search, which can change the complexity of the loop from O(n) to O(logn). This enhancement reduces the computational complexity of RSA modulus factorization in practical settings.
发表机构
- Electronics Information College, Qingdao University（青岛大学电子信息学院）
- Technical University of Sofia（索菲亚技术大学）
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