Our main result is that 1-BPI cannot be NP-hard unless the polynomial hierarchy collapses. The result is extended to the isomorphism problem for arithmetic circuits over large enough fields.
We use the known arithmetization of read-once branching programs and arithmetic circuits into multivariate polynomials over the rationals. Hence, another way of stating our result is: the isomorphism problem for multivariate polynomials over large enough fields is not NP-hard unless the polynomial hierarchy collapses.
We derive this result by providing a two-round interactive proof for the nonisomorphism problem for multivariate polynomials. The protocol is based on the Schwartz-Zippel Theorem to probabilistically check polynomial identities.
Finally, we show that there is a
perfect zero-knowledge interactive proof
for the isomorphism problem for multivariate polynomials.