整数分解问题的统计力学模型

C. Nakajima, Masayuki Ohzeki
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引用次数: 0

摘要

我们通过对统计力学哈密顿量基态的搜索问题的表述来表述整数分解问题。找到合数的正确除数所需要的第一次通过时间表明了指数级的计算难度。分析两个宏观量的状态密度,即能量和正确解的汉明距离,得出基态(正确解)与其他低能态完全隔离的结论,距离与系统大小成正比。此外,模型的微正则熵分布具有两个特殊的特征,它们分别与蒙特卡罗模拟或模拟退火采样的能量区域的两个剧烈变化有关。因此,我们在模型中发现了一种特殊的一阶相变。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Statistical mechanical models of integer factorization problem
We formulate the integer factorization problem via a formulation of the searching problem for the ground state of a statistical mechanical Hamiltonian. The first passage time required to find a correct divisor of a composite number signifies the exponential computational hard- ness. Analysis of the density of states of two macroscopic quantities, i.e. the energy and the Hamming distance from the correct solutions, leads to the conclusion that the ground state (the correct solution) is completely isolated from the other low energy states, with the distance being proportional to the system size. In addition, the profile of the microcanonical entropy of the model has two peculiar features which are each related to two dramatic changes in the energy region sampled via Monte Carlo simulation or simulated annealing. Hence, we find a peculiar first-order phase transition in our model.
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