Disaggregation of Diophantine Equation with Boolean Variables

“…The disaggregation problem was first formulated in 1972 but it is not solved till now, though, in the case of Boolean variables, it was solved by the authors in [9][10]. In this article, a method of disaggregation of a linear equation with nonnegative integer variables is proposed.…”

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confidence: 99%

Reducing an integer linear equation to an equivalent system

Mamedov

Mardanov

2006

Cybern Syst Anal

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“…However, after decades, research studies on the techniques dealing with the disaggregation problem are limited. Mardanov and Mamedov studied the disaggregation problem with unknowns being binary in their paper [33] and [34], and later extended their results with unknowns taking values over a more general but still bounded set in [32].…”

Section: Introduction 1backgroundmentioning

confidence: 99%

Tackling A Class of Hard Subset-Sum Problems: Integration of Lattice Attacks with Disaggregation Techniques

Lu¹,

Li²,

Jiang³

2022

Preprint

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Subset-sum problems belong to the NP class and play an important role in both complexity theory and knapsack-based cryptosystems, which have been proved in the literature to become hardest when the so-called density approaches one. Lattice attacks, which are acknowledged in the literature as the most effective methods, fail occasionally even when the number of unknown variables is of medium size. In this paper we propose a modular disaggregation technique and a simplified lattice formulation based on which two lattice attack algorithms are further designed. We introduce the new concept "jump points" in our disaggregation technique, and derive inequality conditions to identify superior jump points which can more easily cut-off non-desirable short integer solutions. Empirical tests have been conducted to show that integrating the disaggregation technique with lattice attacks can effectively raise success ratios to 100% for randomly generated problems with density one and of dimensions up to 100. Finally, statistical regressions are conducted to test significant features, thus revealing reasonable factors behind the empirical success of our algorithms and techniques proposed in this paper.

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“…In the literature the discussion on this problem is rare. Mardanov and Mamedov in [6] gave a sufficient condition to split an original equation into two by a good modular multiplication and provided a heuristic to do so. In this paper we are going to tackle the disaggregation problem systematically with basis reduction.…”

Section: Introductionmentioning

confidence: 99%

A disaggregation approach for solving linear Diophantine equations

Wu¹,

Zhang²

2018

ams

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Finding integer solutions to a set of linear equations has been a challenging problem with application in many areas, such as knapsack optimization and cryptography. In this paper, we find a new method from the angle of disaggregation to solve a system of linear Diophantine equations. The disaggregation process, which employs the LLL procedure in the literature, keeps generating new valid linear equations until the system of equations becomes high-ranked and easy to solve.

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Disaggregation of Diophantine Equation with Boolean Variables

Cited by 3 publications

References 6 publications

Reducing an integer linear equation to an equivalent system

Reducing an integer linear equation to an equivalent system

Tackling A Class of Hard Subset-Sum Problems: Integration of Lattice Attacks with Disaggregation Techniques

A disaggregation approach for solving linear Diophantine equations

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