Isomorphism testing of read-once functions and polynomials

Abstract

In this paper, we study the isomorphism testing problem of formulas in the Boolean and arithmetic settings. We show that isomorphism testing of read-once formulas is complete for log-space. In contrast, we observe that the problem becomes polynomial-time equivalent to the graph isomorphism problem when the input formulas can be represented as OR of two or more read-once formulas.We address the polynomial isomorphism problem, a special case of polynomial equivalence problem which in turn is important from a cryptographic perspective [Patarin EUROCRYPT '96, and Kayal SODA '11]. As our main result, we propose a deterministic polynomial time canonization scheme for polynomials computed by read-once arithmetic formulas. In contrast, we show that when the arithmetic formula is allowed to read a variable twice, this problem is at least as hard as the graph isomorphism problem.