TY - JOUR

T1 - Statistical mechanical models of integer factorization problem

AU - Nakajima, Chihiro H.

AU - Ohzeki, Masayuki

N1 - Publisher Copyright:
© 2017 The Physical Society of Japan.
Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 2017/1/15

Y1 - 2017/1/15

N2 - 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 hardness. The 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 (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 that are each related to two marked 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.

AB - 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 hardness. The 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 (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 that are each related to two marked 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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U2 - 10.7566/JPSJ.86.014001

DO - 10.7566/JPSJ.86.014001

M3 - Article

AN - SCOPUS:85007478254

VL - 86

JO - Journal of the Physical Society of Japan

JF - Journal of the Physical Society of Japan

SN - 0031-9015

IS - 1

M1 - 014001

ER -