On the Unimodular Isomorphism Problem of Convex Lattice Polytopes

TL;DR

This paper investigates the computational complexity of the unimodular isomorphism problem for convex lattice polytopes, establishes its hardness, and introduces a zero-knowledge proof system and an algorithm for solving it.

Contribution

It proves UIP is graph isomorphism hard, develops a zero-knowledge proof protocol, and presents an algorithm to determine unimodular affine transformations between polytopes.

Findings

UIP is graph isomorphism hard.

A statistical zero-knowledge proof system for UIP is developed.

An algorithm to compute all unimodular affine transformations between polytopes is proposed.

Abstract

This paper studies the \emph{unimodular isomorphism problem} (UIP) of convex lattice polytopes: given two convex lattice polytopes $P$ and $P'$, decide whether there exists a unimodular affine transformation mapping $P$ to $P'$. We show that UIP is graph isomorphism hard, while the polytope congruence problem and the combinatorial polytope isomorphism problem (Akutsu, 1998; Kaibel, Schwartz, 2003) were shown to be graph isomorphism complete, and both the lattice isomorphism problem ( $\mathrm{Sikiri\acute{c}}$, $\mathrm{Sch\ddot{u}rmann}$, Vallentin, 2009) and the projective/affine polytope isomorphism problem (Kaibel, Schwartz, 2003) were shown to be graph isomorphism hard. Furthermore, inspired by protocols for lattice (non-) isomorphism (Ducas, van Woerden, 2022; Haviv, Regev, 2014), we present a statistical zero-knowledge proof system for unimodular isomorphism of lattice polytopes. Finally, we propose an algorithm that given two lattice polytopes computes all unimodular affine transformations mapping one polytope to another and, in particular, decides UIP.