A constructive approach to a conjecture by Voskresenskii

TL;DR

This paper re-proves a conjecture by Voskresenskii that stably rational tori are rational, using explicit birational isomorphisms, and explores connections to torus-based cryptography.

Contribution

It offers a simplified proof of Klyachko's result on rationality of certain tori and discusses implications for cryptography.

Findings

Re-proved Klyachko's theorem with explicit birational isomorphisms

Established links between rational tori and cryptography

Simplified the understanding of Voskresenskii's conjecture

Abstract

Voskresenskii conjectured that stably rational tori are rational. Klyachko proved this assertion for a wide class of tori by general principles. We re-prove Klyachko's result by providing simple explicit birational isomorphisms, and elaborate on some links to torus-based cryptography.