Statistical mechanical models of integer factorization problem

TL;DR

This paper models the integer factorization problem using statistical mechanics, revealing a first-order phase transition and the exponential difficulty of finding correct divisors due to isolated ground states.

Contribution

It introduces a novel statistical mechanical formulation of integer factorization, analyzing the energy landscape and phase transitions to understand computational hardness.

Findings

Ground state is isolated from low energy states, proportional to system size.

Microcanonical entropy shows features indicating phase transitions.

First-order phase transition observed in the model.

Abstract

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.