A constructive approach to a conjecture by Voskresenskii
  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. 
Mathieu Florence and Michel van Garrel
math.AG