The Complexity of Factors of Multivariate Polynomials

The existence of string functions, which are not polynomial time computable, but whose graph is checkable in polynomial time, is a basic assumption in cryptography. We prove that in the framework of algebraic complexity, there are no such families of polynomial functions of -bounded degree over fields of characteristic zero. The proof relies on a polynomial upper bound on the approximative complexity of a factor of a polynomial in terms of the (approximative) complexity of and the degree of the factor . This extends a result by Kaltofen (STOC 1986). The concept of approximative complexity allows to cope with the case that a factor has an exponential multiplicity, by using a perturbation argument. Our result extends to randomized (twosided error) decision complexity.