Chicago Journal of Theoretical Computer Science

Volume 1998

Article 1

Published by MIT Press. Copyright 1997 Massachusetts Institute of Technology.

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The Isomorphism Problem for Read-Once Branching Programs and Arithmetic Circuits

Thomas Thierauf (Universität Ulm)
10 March 1998
Special Issue on Computational Complexity from the 1996 Dagstuhl-Seminar, Eric Allender editor

We investigate the computational complexity of the isomorphism problem for read-once branching programs (1-BPI): upon input of two read-once branching programs B_0 and B_1, decide whether there exists a permutation of the variables of B_1 such that it becomes equivalent to B_0.

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 for probabilistically checking polynomial identities.

Finally, we show that there is a perfect zero-knowledge interactive proof for the isomorphism problem for multivariate polynomials.

DOI: 10.4086/cjtcs.1998.001
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