The Complexity of Factors of Multivariate Polynomials
arXiv:1812.06828 · doi:10.1007/s10208-002-0059-5
Abstract
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 polynomially bounded degree over fields of characteristic zero. The proof relies on a polynomial upper bound on the approximative complexity of a factor g of a polynomial f in terms of the (approximative) complexity of f and the degree of the factor g. 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 (two-sided error) decision complexity.
This is an updated version of a paper published in J. FoCM in 2004. Here we have corrected an error in the statement and proof of Theorem 5.7