Paper 2024/576 Original: eprint.iacr.org/2024/576 ↗
On the complexity of the problem of solving systems of tropical polynomial equations of degree two
Abstract
In this paper, we investigate the computational complexity of the problem of solving a one-sided system of equations of degree two of a special form over the max-plus algebra. Also, we consider the asymptotic density of solvable systems of this form. Such systems have appeared during the analysis of some tropical cryptography protocols that were recently suggested. We show how this problem is related to the integer linear programming problem and prove that this problem is NP-complete. We show that the asymptotic density of solvable systems of this form with some restrictions on the coefficients, the number of variables, and the number of equations is 0. As a corollary, we prove that this problem (with some restrictions on the coefficients, the number of variables, and the number of equations) is decidable generically in polynomial time.
Metadata
- Available format(s)
PDF- Category
- Foundations
- Publication info
- Preprint.
- History
- 2024-12-23: last updated
- 2024-04-15: published
- Short URL
- https://ia.cr/2024/576
- License
- https://creativecommons.org/licenses/by/4.0/
BibTeX
@misc{cryptoeprint:2024/576,
author = {Ivan Buchinskiy and Matvei Kotov and Alexander Treier},
title = {On the complexity of the problem of solving systems of tropical polynomial equations of degree two},
howpublished = {Cryptology {ePrint} Archive, Paper 2024/576},
year = {2024},
url = {https://eprint.iacr.org/2024/576}
}