Equivalence of F - algebras and cubic forms

Abstract

We study the isomorphism problem of two "natural" algebraic structures - \mathbbFF -algebras and cubic forms. We prove that the F -algebra isomorphism problem reduces in polynomial time to the cubic forms equivalence problem. This answers a question asked in [AS05]. For finite fields of the form 3Λ(#F - 1), this result implies that the two problems are infact equivalent. This result also has the following interesting consequence: Graph Isomorphism ≤ ^P_m F -algebra Isomorphism ≤ ^P_mCubic Form Equivalence.